Application of 1‑D discrete wavelet transform based compressed sensing matrices for speech compressionYuvraj V. Parkale* and Sanjay L. Nalbalwar

Abstract

Background: Compressed sensing is a novel signal compression technique in which signal is compressed while sensing. The compressed signal is recovered with the only few numbers of observations compared to conventional Shannon–Nyquist sampling, and thus reduces the storage requirements. In this study, we have proposed the 1-D discrete wavelet transform (DWT) based sensing matrices for speech signal compres-sion. The present study investigates the performance analysis of the different DWT based sensing matrices such as: Daubechies, Coiflets, Symlets, Battle, Beylkin and Vaidy-anathan wavelet families.

Results: First, we have proposed the Daubechies wavelet family based sensing matrices. The experimental result indicates that the db10 wavelet based sensing matrix exhibits the better performance compared to other Daubechies wavelet based sensing matrices. Second, we have proposed the Coiflets wavelet family based sensing matri-ces. The result shows that the coif5 wavelet based sensing matrix exhibits the best performance. Third, we have proposed the sensing matrices based on Symlets wavelet family. The result indicates that the sym9 wavelet based sensing matrix demonstrates the less reconstruction time and the less relative error, and thus exhibits the good performance compared to other Symlets wavelet based sensing matrices. Next, we have proposed the DWT based sensing matrices using the Battle, Beylkin and the Vaidyanathan wavelet families. The Beylkin wavelet based sensing matrix demonstrates the less reconstruction time and relative error, and thus exhibits the good performance compared to the Battle and the Vaidyanathan wavelet based sensing matrices. Further, an attempt was made to find out the best-proposed DWT based sensing matrix, and the result reveals that sym9 wavelet based sensing matrix shows the better perfor-mance among all other proposed matrices. Subsequently, the study demonstrates the performance analysis of the sym9 wavelet based sensing matrix and state-of-the-art random and deterministic sensing matrices.

Conclusions: The result reveals that the proposed sym9 wavelet matrix exhibits the better performance compared to state-of-the-art sensing matrices. Finally, speech quality is evaluated using the MOS, PESQ and the information based measures. The test result confirms that the proposed sym9 wavelet based sensing matrix shows the better MOS and PESQ score indicating the good quality of speech.

Keywords: Speech compression, Compressed sensing (CS), Discrete wavelet transform (DWT), Mean opinion score (MOS), Perceptual evaluation of speech quality (PESQ)

Open Access

© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

RESEARCH

Parkale and Nalbalwar SpringerPlus (2016) 5:2048 DOI 10.1186/s40064‑016‑3740‑x

*Correspondence: yuvrajparkale@gmail.com Department of Electronics and Telecommunication Engineering, Dr. Babasaheb Ambedkar Technological University, Lonere, Maharashtra, India

Page 2 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

IntroductionConventional signal processing methods such as Fourier transform and a short time Fourier transform (STFT) are inadequate for the analysis of non-stationary signals which have abrupt transitions superimposed on the lower frequency backgrounds such as the speech, music and bio-electric signals. The wavelet transform (WT) (Daubechie Ingrid 1992) overcomes these drawbacks and provides both the time resolution and frequency resolution of a signal. The basic idea of the wavelet transform is to represent the signal to be analyzed as a superposition of wavelets. The wavelet transform is the most popu-lar signal analysis tool, and it is successfully used in different application areas such as speech or audio and image compression.

Given an input signal x of length N, the wavelet transform consists of log2 N decom-position levels. The input signal decomposition is accomplished through a series filtering and downsampling processes. The reconstruction of the original signal is accomplished through an upsampling, series filtering and adding all the sub-bands. Figure  1 shows the block diagram of 1-D forward wavelet transform with 2-level decomposition (Mal-lat 2009; Meyer 1993). The input signal is filtered using the low-pass filter (u) and the high-pass filter (v). A filtering is achieved by computing a linear convolution between the input signal and the filter coefficients. The two filters are chosen such that, they are orthogonal to each other and provides a perfect reconstruction of the original signal x. Therefore, the quadrature mirror filter (QMF) is commonly used for the perfect recon-struction of a two-channel filter bank.

Wavelet analysis provides approximation coefficients and detail coefficients. The low frequency information about the signal is given by the approximation, while the high frequency information is given by the detail coefficients. Since the low frequency signal is of more importance than the high frequency signal, the output of the low-pass filter is used as an input for the next decomposition stages; whereas the output of high-pass filter is used at the time of signal reconstruction. The wavelet coefficients are computed by using a series filtering and downsampling processes. The wavelet coefficients (f) are given by:

where W is the N × N wavelet matrix and defined as: W = WI, where I is N × N identity matrix.

(1)f = Wx

(i 1)vf+

(i 2)vf+

(i)uf

1-level Detail Coefficient (CD1)

Detail Coefficient (CD2)

Approximation Coefficient (CA2)

u

v

2

2

u

v 2

2

Approx. Coefficient (CA1)

(i 1)uf+

(i 2)uf+

Fig. 1 Block diagram of 1-D fast forward wavelet transform with 2-level decomposition

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Thus, the classical approach of data compression is to employ the discrete wavelet transform (DWT) based methods (Skodras and Ebrahimi 2001) prior to the transmis-sion. However, these methods includes the complicated multiplications, exhaustive coefficient search and sorting procedure along with the arithmetic encoding of the sig-nificant coefficients with their locations, which consequently results in a huge storage requirement and power consumption. Furthermore, the smooth oscillatory signals such as the speech or music signals will be compressed more efficiently in the wavelet packet basis compared to the wavelet representation. Coifman and Wickerhauser (1992) pro-posed the algorithm for an efficient data compression based on the Shannon entropy for the best basis selection. The orthogonal wavelet packets and localized trigonometric functions are exploited as a basis. This allows an efficient compression of a voice and image signals; however, at the cost of an additional computation in searching the best wavelet packet basis.

The research work presented on CS by Donoho (2006), Baraniuk (2007), Candes and Wakin (2008), and Donoho and Tsaig (2006) have energized the research in many appli-cation areas like medical image processing (Lustig et al. 2008), wireless sensor networks (Guan et al. 2011), analog-to-information converters (AIC) (Laska et al. 2007), commu-nications and networks (Berger et al. 2010), radar (Qu and Yang 2012), etc.

In the paper Liu et al. (2014) successfully implemented the CS based compression and the wavelet based compression procedure on the field programmable gate array (FPGA). The result shows that the CS based procedure achieves the better performance com-pared to the wavelet compression in terms of power consumption and the number of computing resources required. Furthermore, the sparse binary sensing matrix achieves the desired signal compression, but at the price of the higher signal reconstruction time and the higher sensing matrix construction time.

Candes et al. (2006a, b) proposed an i.i.d. (independent identical distribution) Gauss-ian or Bernoulli random sensing matrices for the compressed sensing. However, the practical implementation of these sensing matrices requires the huge computational cost and memory storage requirements, and therefore considered as inappropriate for large scale applications.

Rauhut (2009), Haupt et al. (2010), Xu et al. (2014), Yin et al. (2010), and Sebert et al. (2008) exploited the Toeplitz and Circulant sensing matrices which effectively recover the original signal with the reduction in the computational cost and the memory requirement.

As an alternative to the random sensing matrices, the authors in Arash and Farokh (2011) proposed the deterministic construction of sensing matrices such as binary, bipo-lar and the ternary matrices. Several authors have proposed the deterministic construc-tion of sensing matrices using the codes such as the sparse binary matrices based on the low density parity check (LDPC) code (Lu and Kpalma 2012), chirp sensing codes (Applebauma et al. 2009), scrambled block Hadamard matrices (Gan et al. 2008), Reed–Muller sensing codes (Howard et al. 2008) and the Vandermond matrices (DeVore 2007).

The restricted isometry property (RIP) is just a sufficient condition for an exact sig-nal recovery. Even though, the deterministic sensing matrices are an incapable to satisfy RIP condition, they are very useful in practice because of the deterministic nature of the

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sampler and might be able to advance some features like compression ratio and compu-tational complexity.

The successful implementation of the CS technique is depends on the efficient design of the sensing matrices which are used to compress the given signal. Since, the DWT shows a very good energy compaction property, it can be used for designing the sensing matrices. In this study, we have proposed the 1-D discrete wavelet transform (DWT) based sensing matrices for speech signal compression. The major contributions of the research paper are the proposed 1-D DWT sensing matrices based on different wavelet families such as the Daubechies, Coiflets, Symlets, Battle, Beylkin and the Vaidyanathan wavelet families. Furthermore, the proposed DWT based sensing matrices are compared with state-of-the-art random and the deterministic sensing matrices. Besides, the speech quality is evaluated using mean opinion sore (MOS) and the perceptual evaluation of speech quality (PESQ) measures.

The paper is organized as follows. Section two briefly introduces the compressed sensing (CS) theory with signal acquisition and reconstruction model. Section  three describes the proposed methodology for the discrete wavelet transform (DWT) matrix. Experimental results and discussion are presented in section  four. Finally, section five presents the conclusions.

Compressed sensing (CS) frameworkBackground

Compressed sensing is a novel signal compression technique in which signal is acquired and compressed simultaneously. The signal is recovered with the only few number of observations compared to the conventional Shannon–Nyquist sampling which requires observations that are twice the signal bandwidth. Compressed sensing is performed with two basic steps: signal acquisition and signal reconstruction.

CS signal acquisition model

Compressed sensing technique is illustrated as follows:

where f is the input signal of length N × 1, y is the compressed output signal of length M × 1, and Φ is M × N sensing matrix.

The input signal f is sparse in some sparsifying domain (Ψ) and given as:

where x is the non-sparse input signal. Combined form of Eqs. (2) and (3) is given as:

The two basic conditions should be satisfied for the successful implementation of the CS.

1. Sensing matrix (Φ) and sparsity transform (Ψ) should be incoherent to each other.2. The Φ should satisfy the restricted isometric property (RIP) (Candes and Tao 2006)

and defined as follow:

(2)y = �f

(3)f = �x

(4)y = �f = ΦΨ x

(5)(1− δk) �x�22 ≤ ��x�22 ≤ (1+ δk) �x�

22

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where δk ∊ (0, 1) is called as restricted isometric constant of the matrix and k is the number of non-zero coefficients.

CS signal reconstruction model

Since, the compressed sensing technique use only a few number of observations, there are large number of solutions. Therefore, the different optimization based algorithms are used to find the exact sparse solution. The basic algorithms are based on the norm minimization such as L0-norm, L1-norm and L2-norm. Out of these three, L1-norm is widely used, because of its ability to recover the exact sparse solution along with the efficient reconstruction speed. Presently, there are different recovery algorithm available such as the basis pursuit (BP) (Chen et al. 2001), orthogonal matching pursuit (OMP) (Tropp and Gilbert 2007), etc.

The proposed 1‑D discrete wavelet transform (DWT) matrix1‑D DWT matrix

For a signal x of length N = 2n and a low-pass filter (u), the ith level wavelet decomposi-tion (Vidakovic 1999; Wang and Vieira 2010) is given by an Eqs. (6) and (7). Where, v is the high-pass filter.

And

The reconstruction of f i−1u from fu

i and fvi can be obtained by

The 1-D DWT matrix forms are given as below:

and

where, f (i)u is the 2n−i dimensional low pass vector in the ith level and f (i)v the high-pass, while f (i−1)

u is the 2n−i+1 dimensional low-pass vector in the (i − 1)th level. The two 2n−i by 2n−i+1 wavelet filter matrices are given below.

(6)f (i)u (j) =2n−i+1∑

k=1

u(k − 2j) f (i−1)u (k) where, j = 1, 2, . . . , 2

n−i

(7)f (i)v (j) =2n−i+1∑

k=1

v(k − 2j) f (i−1)u (k) where, j = 1, 2, . . . , 2

n−i

(8)f (i−1)u (j) =

2n−i∑

k=1

u(j − 2k) f (i)u (k)+2n−i∑

k=1

v(j − 2k) f (i)v (k)

(9)f (i)u = U(i) f (i−1)

u

(10)f (i)v = V(i) f (i−1)

v

(11)U(i) =

u(−1) 0 0 0 · · · u(−3) u(−2)u(−3) u(−2) u(−1) 0 · · · u(−5) u(−4)

......

... .... . .

......

0 0 0 0 · · · u(−1) 0

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And

Thus, the ith scale wavelet transform can be represented as:

This gives the wavelet matrix of 1-level decomposition. The wavelet matrix for different levels of decomposition is given as below.

Above equation can be represented as,

Here, the numbers of signal decomposition levels are restricted to 2n−i+1 ≥ L. Where, L is the length of the filter.

Thus, the final wavelet transform matrix is given by an Eq. (16).

Design procedure for the proposed 1‑D DWT based sensing matrices

Following are the procedural steps to construct 1-D DWT based sensing matrices.

1. Create a desired quadrature mirror filters (QMF) such as Daubechies, Coiflets, Sym-lets, Beylkin, Vaidyanathan and Battle filters. For example db1 (Haar) filter is given as f = [1 1] and the db2 filter is formed as follows:

2. Create the N × N Identity matrix.

(12)V(i) =

v(−1) 0 0 0 · · · v(−3) v(−2)v(−3) v(−2) v(−1) 0 · · · v(−5) v(−4)

......

... .... . .

......

0 0 0 0 · · · v(−1) 0

(13)

[

f (i)u

f (i)v

]

=

[

U (i)

V (i)

]

f (i−1)u

(14)f (i−1)u = U

(i−1) f (i−2)u

(15)

f (i)u

f (i)v

f (i−1)v

.

.

.

f (2)v

f (1)v

=

U (i)U (i−1) · · ·U (1)

V (i)U (i−1) · · ·U (1)

V (i)U (i−2) · · ·U (1)

.

.

.

V (2)U (1)

V (1)

x

(16)W =

U (i)U (i−1) · · ·U (1)

V (i)U (i−1) · · ·U (1)

V (i)U (i−2) · · ·U (1)

...

V (2)U (1)

V (1)

(17)f =

[

0.482962913145 0.8365163037380.224143868042 −0.129409522551

]

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3. Perform 1-D forward wavelet transform on the N × N Identity matrix. Thus, the N × N wavelet transform matrix is generated.

4. Select the first m number of rows to form the m × N DWT sensing matrix. Where, m is the minimum number of measurements.

Experimental results and discussionMethodology

The proposed work is evaluated on the CMU/CSTR KDT US English TIMIT database for speech synthesis by Carnegie Mellon University and Edinburgh University (Edin-burgh 2002). The details of the database used are as follows: File name: Kdt_001.wav, channel: 1(Mono), bit rate: 256 kbps, audio sample rate: 16 kHz, total duration: 3 s. The number of samples (N) selected are 2048 and the total duration of analyzed speech sig-nal is 0.128 s for simulation. The experimental work is performed using MATLAB 7.8.0 (R2009a) software with Intel (R) CORE 2 Duo CPU, 3 GB RAM system specifications. The discrete cosine transform (DCT) is used as the sparsifying basis for speech signal because of its high sparsity. The speech compression is performed using the sensing matrices based on the different DWT families (Donoho et al. 2007). The basis pursuit (BP) (Chen et al. 2001) is used as signal recovery algorithm for speech signal.

The performance of the reconstructed speech signal is evaluated using the metrics like compression ratio (CR), root mean square error (RMSE), relative error, signal to noise ratio (SNR), signal reconstruction time and sensing matrix construction time.

CR is obtained using relation,

where N is the length of speech signal and M is the number of measurements taken from sensing matrix.

RMSE is given as below:

where x(n) is the original signal and x̃(n) is the reconstructed signal.Relative error is defined as:

where x(n) is the original signal and x̃(n) is the reconstructed signal.SNR is obtained as,

where x(n) is the original signal and x̃(n) is the reconstructed signal.

(18)CR =M

N

(19)RMSE =

√

∑Nn=1 (x(n)− x̃(n))2

N

(20)Rel.Error =

∥

∥x̃(n)− x(n)∥

∥

2

�x(n)�2

(21)SNR(db) = 20 log

(

�x(n)�2∥

∥x(n)− x̃(n)∥

∥

2

)

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Besides, signal reconstruction time is computed to provide the amount of time required to recover the original signal using reconstruction algorithm. The amount of time required to construct the sensing matrix is also an important parameter and should be minimum.

Performance analysis of the Daubechies wavelet family based sensing matrices

This section demonstrates the performance analysis of the different DWT sensing matri-ces based on Daubechies wavelet family such as db1, db2, db3, db4, db5, db6, db7, db8, db9, db10. The speech signal of length 2048 is taken with 50% sparsity level, preserving the only 1024 number of non-zeros. For a different number of measurements (m), cor-responding compression ratios (CR), signal reconstruction time (s), relative error, root mean square error (RMSE) and signal-to-noise ratio (SNR) are calculated (Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10).

It is noted from Fig. 2 that the db1 (Haar) wavelet based sensing matrix requires less reconstruction time compared to all other Daubechies wavelet based sensing matrices. The second best choice will be db2 or db10, closely followed by the db9 wavelet based sensing matrix. From Fig.  3, it can be observed that the db10 wavelet based sensing matrix shows the minimum relative error compared to all other matrices. From Fig. 4, it can be observed that the db10 wavelet sensing matrix exhibits the high SNR (particularly from CR = 0.3 to CR = 1) compared to other sensing matrices.

Thus, it is evident from Figs. 2, 3 and 4 that overall the db10 wavelet based sensing matrix shows the good balance between signal reconstruction error and signal recon-struction time. Moreover, the db9 also shows a close performance to the db10 and may be the second best choice.

Performance analysis of the Coiflets wavelet family based sensing matrices

This section demonstrates the performance analysis of the different DWT sensing matri-ces based on Coiflets wavelet family such as coif1, coif2, coif3, coif4 and coif5 (Tables 11, 12, 13, 14, 15).

It is noted from Fig. 5 that the coif5 and coif4 wavelet based sensing matrix shows a close performance and requires the less reconstruction time compared to all other Coi-flets wavelet based sensing matrices. From Fig. 6, it can be observed that coif5 wavelet based sensing matrix shows the minimum relative error compared to all other matrices. Also, from Fig. 7, it is seen that coif5 wavelet based sensing matrix exhibits the high SNR compared to other sensing matrices.

Thus, overall the coif5 wavelet based sensing matrix shows the good performance, since it requires the less reconstruction time, minimum relative error and the high SNR compared to other Coiflets wavelet based sensing matrices. In addition, the coif4 may be the second choice of sensing matrix.

Performance analysis of the Symlets wavelet family based sensing matrices

This section demonstrates the performance analysis of the different DWT sensing matri-ces based on Symlets wavelet family such as sym4, sym5, sym6, sym7, sym8, sym9 and sym10 (Tables 16, 17, 18, 19, 20, 21, 22).

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Tabl

e 1

Perf

orm

ance

ana

lysi

s of

 the

prop

osed

db1

(Haa

r) w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n

time

(s)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2416

1.28

1382

0.05

850.

9774

0.19

852.

5840

58

2048

410

0.2

5010

2416

2.33

1942

0.05

210.

8711

1.19

842.

5231

52

2048

512

0.25

5010

2416

3.75

2369

0.05

210.

8707

1.20

242.

5143

15

2048

614

0.3

5010

2416

4.24

2975

0.04

880.

8153

1.77

402.

5707

37

2048

849

0.4

5010

2414

5.85

3003

0.04

790.

7997

1.94

182.

5814

21

2048

1024

0.5

5010

2414

9.48

6588

0.04

780.

7993

1.94

542.

5818

56

2048

1229

0.6

5010

2414

11.0

8041

60.

0471

0.78

772.

0723

2.59

3704

2048

1434

0.7

5010

2413

14.4

4079

60.

0468

0.78

242.

1310

2.54

6346

2048

1536

0.75

5010

2412

17.7

9912

70.

0468

0.78

162.

1406

2.52

3620

2048

1638

0.8

5010

2411

16.5

6827

90.

0468

0.78

162.

1406

2.48

4885

2048

1843

0.9

5010

2411

20.6

2963

60.

0468

0.78

162.

1407

2.51

8902

2048

2048

1.0

5010

249

22.5

2503

20.

0468

0.78

152.

1409

2.61

2855

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Tabl

e 2

Perf

orm

ance

ana

lysi

s of

 the

prop

osed

db2

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m)

Com

pres

sion

ra

tio

(CR =

 m/N

)

Spar

sity

le

vel =

 (k/N

) × 

100

(%)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n

time

(s)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2415

1.33

6137

0.05

960.

9955

0.03

912.

1598

16

2048

410

0.2

5010

2415

2.44

4694

0.05

520.

9224

0.70

212.

1878

08

2048

512

0.25

5010

2415

3.62

2582

0.05

520.

9222

0.70

342.

1032

15

2048

614

0.3

5010

2414

3.91

9803

0.05

170.

8635

1.27

512.

3377

19

2048

849

0.4

5010

2413

5.74

0394

0.05

020.

8387

1.52

782.

1059

89

2048

1024

0.5

5010

2412

8.50

8652

0.05

010.

8376

1.53

912.

3195

50

2048

1229

0.6

5010

2412

10.6

2200

70.

0475

0.79

312.

0137

2.05

5366

2048

1434

0.7

5010

2412

13.7

1603

50.

0469

0.78

382.

1160

2.37

8571

2048

1536

0.75

5010

2411

15.9

9153

40.

0468

0.78

172.

1395

2.13

5815

2048

1638

0.8

5010

2411

17.0

8593

70.

0468

0.78

162.

1404

2.07

8314

2048

1843

0.9

5010

2411

27.2

1734

90.

0468

0.78

162.

1405

2.42

3662

2048

2048

1.0

5010

249

29.8

3231

10.

0468

0.78

152.

1409

2.36

5919

Page 11 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 3

Perf

orm

ance

ana

lysi

s of

 the

prop

osed

db3

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2418

1.62

6908

0.05

670.

9483

0.46

152.

2533

46

2048

410

0.2

5010

2415

2.43

4635

0.05

450.

9106

0.81

372.

1609

87

2048

512

0.25

5010

2415

3.63

6753

0.05

450.

9099

0.81

992.

2525

68

2048

614

0.3

5010

2415

4.03

3450

0.05

020.

8383

1.53

232.

2016

34

2048

849

0.4

5010

2413

5.92

9769

0.04

960.

8282

1.63

732.

2857

77

2048

1024

0.5

5010

2413

9.18

0948

0.04

960.

8282

1.63

722.

3090

93

2048

1229

0.6

5010

2413

12.4

8021

20.

0477

0.79

631.

9788

2.42

7516

2048

1434

0.7

5010

2413

16.5

5480

20.

0469

0.78

292.

1258

2.50

9123

2048

1536

0.75

5010

2412

27.2

3386

10.

0468

0.78

172.

1396

2.17

5306

2048

1638

0.8

5010

2411

16.8

2110

00.

0468

0.78

162.

1406

2.28

3585

2048

1843

0.9

5010

2411

28.6

7397

60.

0468

0.78

162.

1407

2.19

2527

2048

2048

1.0

5010

249

30.9

0555

20.

0468

0.78

152.

1409

2.26

2969

Page 12 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 4

Perf

orm

ance

ana

lysi

s of

 the

prop

osed

db4

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ra

tio (C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n

time

(s)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2415

2.30

1267

0.05

450.

9099

0.81

982.

2607

42

2048

410

0.2

5010

2415

2.33

2355

0.05

240.

8749

1.16

072.

5072

63

2048

512

0.25

5010

2415

3.69

6636

0.05

230.

8744

1.16

572.

3705

97

2048

614

0.3

5010

2415

4.30

8034

0.04

960.

8283

1.63

632.

1611

84

2048

849

0.4

5010

2413

5.85

6042

0.04

840.

8092

1.83

872.

1994

77

2048

1024

0.5

5010

2413

19.0

6277

30.

0484

0.80

931.

8376

2.21

9947

2048

1229

0.6

5010

2413

11.0

0184

40.

0472

0.78

872.

0622

2.08

0746

2048

1434

0.7

5010

2412

13.3

5980

60.

0468

0.78

232.

1325

2.13

4800

2048

1536

0.75

5010

2412

16.7

4898

30.

0468

0.78

162.

1405

2.33

7097

2048

1638

0.8

5010

2411

16.1

9774

80.

0468

0.78

162.

1407

2.02

5083

2048

1843

0.9

5010

2411

20.3

0889

00.

0468

0.78

162.

1407

2.07

4552

2048

2048

1.0

5010

249

21.9

1614

50.

0468

0.78

152.

1409

2.14

9653

Page 13 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 5

Perf

orm

ance

ana

lysi

s of

 the

prop

osed

db5

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2415

1.30

8421

0.05

500.

9192

0.73

162.

2404

30

2048

410

0.2

5010

2415

2.16

4346

0.05

080.

8496

1.41

622.

2113

80

2048

512

0.25

5010

2415

3.39

9650

0.05

090.

8498

1.41

382.

2507

45

2048

614

0.3

5010

2415

3.92

4939

0.04

910.

8201

1.72

212.

4122

86

2048

849

0.4

5010

2414

5.93

5914

0.04

810.

8030

1.90

572.

2700

19

2048

1024

0.5

5010

2414

9.35

3480

0.04

810.

8031

1.90

482.

3373

83

2048

1229

0.6

5010

2413

11.1

1847

70.

0470

0.78

482.

1053

2.21

6672

2048

1434

0.7

5010

2413

17.9

7501

40.

0468

0.78

222.

1335

2.24

5930

2048

1536

0.75

5010

2411

15.5

3842

60.

0468

0.78

162.

1405

2.44

0602

2048

1638

0.8

5010

2411

16.5

3987

90.

0468

0.78

162.

1407

2.18

2187

2048

1843

0.9

5010

2411

20.4

4285

90.

0468

0.78

162.

1407

2.23

5392

2048

2048

1.0

5010

249

22.6

4166

40.

0468

0.78

152.

1409

2.21

0355

Page 14 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 6

Perf

orm

ance

ana

lysi

s of

 the

prop

osed

db6

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n

time

(s)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2415

1.25

7052

0.05

540.

9260

0.66

772.

2911

40

2048

410

0.2

5010

2415

2.23

2353

0.05

060.

8458

1.45

442.

3561

09

2048

512

0.25

5010

2416

3.71

0359

0.05

060.

8454

1.45

842.

3572

69

2048

614

0.3

5010

2415

4.00

1119

0.04

940.

8259

1.66

172.

3066

22

2048

849

0.4

5010

2415

6.42

5953

0.04

870.

8142

1.78

552.

3678

48

2048

1024

0.5

5010

2414

9.85

939

0.04

870.

8138

1.78

932.

3894

22

2048

1229

0.6

5010

2413

15.5

4062

50.

0469

0.78

452.

1080

2.69

8509

2048

1434

0.7

5010

2413

18.1

7286

80.

0468

0.78

272.

1286

2.62

7686

2048

1536

0.75

5010

2412

21.3

2204

50.

0468

0.78

162.

1404

2.69

9761

2048

1638

0.8

5010

2412

30.7

4930

10.

0468

0.78

162.

1406

2.60

8611

2048

1843

0.9

5010

2412

39.6

6230

60.

0468

0.78

162.

1406

2.70

6883

2048

2048

1.0

5010

249

37.1

5061

80.

0468

0.78

152.

1409

2.58

1647

Page 15 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 7

Perf

orm

ance

ana

lysi

s of

 the

prop

osed

db7

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2416

1.54

4091

0.05

640.

9417

0.52

162.

5578

82

2048

410

0.2

5010

2416

2.31

0240

0.05

140.

8584

1.32

602.

5039

21

2048

512

0.25

5010

2416

3.65

5857

0.05

140.

8585

1.32

472.

3716

64

2048

614

0.3

5010

2415

4.05

0385

0.05

110.

8537

1.37

392.

3916

05

2048

849

0.4

5010

2414

5.77

2598

0.05

010.

8379

1.53

622.

5028

02

2048

1024

0.5

5010

2414

9.53

6010

0.05

010.

8375

1.53

992.

4504

85

2048

1229

0.6

5010

2412

10.2

8168

90.

0470

0.78

482.

1049

2.45

8048

2048

1434

0.7

5010

2412

13.5

5498

60.

0468

0.78

272.

1278

2.50

5254

2048

1536

0.75

5010

2412

17.1

0936

20.

0468

0.78

162.

1404

2.61

6728

2048

1638

0.8

5010

2412

18.1

3484

10.

0468

0.78

162.

1406

2.54

3920

2048

1843

0.9

5010

2411

21.1

1526

60.

0468

0.78

162.

1406

2.51

2582

2048

2048

1.0

5010

249

22.7

5380

20.

0468

0.78

152.

1409

2.49

2282

Page 16 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 8

Perf

orm

ance

ana

lysi

s of

 the

prop

osed

db8

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m)

Com

pres

sion

ra

tio (C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f  ite

ratio

ns

requ

ired

Sign

al re

cons

truc

tion

time

(s)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2416

1.37

9793

0.05

620.

9397

0.54

062.

4454

16

2048

410

0.2

5010

2416

2.31

9429

0.05

140.

8597

1.31

322.

5956

14

2048

512

0.25

5010

2416

3.67

2272

0.05

140.

8594

1.31

612.

3962

96

2048

614

0.3

5010

2416

4.35

2767

0.05

040.

8417

1.49

732.

4736

40

2048

849

0.4

5010

2414

5.84

4768

0.04

960.

8283

1.63

652.

4374

63

2048

1024

0.5

5010

2413

8.95

1393

0.04

950.

8274

1.64

582.

3593

80

2048

1229

0.6

5010

2413

11.0

7483

60.

0469

0.78

452.

1084

2.57

3343

2048

1434

0.7

5010

2413

14.5

7421

30.

0468

0.78

242.

1318

2.60

9980

2048

1536

0.75

5010

2411

15.6

4966

20.

0468

0.78

162.

1405

2.55

9536

2048

1638

0.8

5010

2411

16.8

7915

80.

0468

0.78

162.

1406

2.55

6170

2048

1843

0.9

5010

2411

20.5

3528

70.

0468

0.78

162.

1406

2.56

9219

2048

2048

1.0

5010

249

22.8

9290

10.

0468

0.78

152.

1409

2.51

3244

Page 17 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 9

Perf

orm

ance

ana

lysi

s of

 the

prop

osed

db9

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n

time

(s)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2415

1.25

2129

0.05

830.

9750

0.22

032.

6494

78

2048

410

0.2

5010

2417

2.52

7130

0.05

180.

8660

1.24

922.

5579

30

2048

512

0.25

5010

2417

3.97

2292

0.05

180.

8652

1.25

822.

5319

26

2048

614

0.3

5010

2416

4.22

1050

0.04

900.

8182

1.74

252.

5724

60

2048

849

0.4

5010

2415

6.27

0259

0.04

780.

7990

1.94

952.

6835

71

2048

1024

0.5

5010

2414

9.59

3764

0.04

780.

7983

1.95

712.

5330

85

2048

1229

0.6

5010

2413

11.0

7561

90.

0470

0.78

532.

0992

2.49

9444

2048

1434

0.7

5010

2413

14.5

1897

40.

0468

0.78

242.

1309

2.53

9711

2048

1536

0.75

5010

2412

17.1

7863

80.

0468

0.78

162.

1405

2.48

9187

2048

1638

0.8

5010

2412

18.0

4313

40.

0468

0.78

162.

1406

2.50

6067

2048

1843

0.9

5010

2412

22.3

7087

40.

0468

0.78

162.

1407

2.56

9387

2048

2048

1.0

5010

249

22.7

5173

20.

0468

0.78

152.

1409

2.49

7201

Page 18 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 10

Per

form

ance

ana

lysi

s of

 the

prop

osed

db1

0 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) 

× 1

00 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2416

1.28

1382

0.05

850.

9774

0.19

852.

5840

58

2048

410

0.2

5010

2416

2.33

1942

0.05

210.

8711

1.19

842.

5231

52

2048

512

0.25

5010

2416

3.75

2369

0.05

210.

8707

1.20

242.

5143

15

2048

614

0.3

5010

2416

4.24

2975

0.04

880.

8153

1.77

402.

5707

37

2048

849

0.4

5010

2414

5.85

3003

0.04

790.

7997

1.94

182.

5814

21

2048

1024

0.5

5010

2414

9.48

6588

0.04

780.

7993

1.94

542.

5818

56

2048

1229

0.6

5010

2414

11.0

8041

60.

0471

0.78

772.

0723

2.59

3704

2048

1434

0.7

5010

2413

14.4

4079

60.

0468

0.78

242.

1310

2.54

6346

2048

1536

0.75

5010

2412

17.7

9912

70.

0468

0.78

162.

1406

2.52

3620

2048

1638

0.8

5010

2411

16.5

6827

90.

0468

0.78

162.

1406

2.48

4885

2048

1843

0.9

5010

2411

20.6

2963

60.

0468

0.78

162.

1407

2.51

8902

2048

2048

1.0

5010

249

22.5

2503

20.

0468

0.78

152.

1409

2.61

2855

Page 19 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Ti

me

(Sec

onds

) db1(Haar) Matrixdb2 Matrixdb3 Matrixdb4 Matrixdb5 Matrixdb6 Matrixdb7 Matrixdb8 Matrixdb9 Matrixdb10 Matrix

Fig. 2 Effect of compression ratio on signal reconstruction time for different Daubechies wavelet sensing matrices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.75

0.8

0.85

0.9

0.95

1

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Er

ror (

Rel

ativ

e er

ror) db1(Haar) Matrix

db2 Matrixdb3 Matrixdb4 Matrixdb5 Matrixdb6 Matrixdb7 Matrixdb8 Matrixdb9 Matrixdb10 Matrix

Fig. 3 Effect of compression ratio on relative error for different Daubechies wavelet sensing matrices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

Compression Ratio (CR)

Sign

al-to

-Noi

se R

atio

(SN

R) (

db)

db1(Haar) Matrixdb2 Matrixdb3 Matrixdb4 Matrixdb5 Matrixdb6 Matrixdb7 Matrixdb8 Matrixdb9 Matrixdb10 Matrix

Fig. 4 Effect of compression ratio on signal-to-noise ratio for different Daubechies wavelet sensing matrices

Page 20 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 11

Per

form

ance

ana

lysi

s of

 the

prop

osed

coi

f1 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ra

tio (C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2417

1.35

4238

0.05

790.

9668

0.29

302.

0988

81

2048

410

0.2

5010

2415

2.87

4402

0.05

500.

9197

0.72

712.

0414

93

2048

512

0.25

5010

2415

10.4

6867

00.

0550

0.91

960.

7279

2.37

5475

2048

614

0.3

5010

2414

5.95

5084

0.05

190.

8675

1.23

462.

3081

89

2048

849

0.4

5010

2414

8.20

2050

0.05

130.

8579

1.33

152.

2001

96

2048

1024

0.5

5010

2413

11.5

7571

00.

0513

0.85

681.

3425

2.11

1921

2048

1229

0.6

5010

2413

18.5

7986

90.

0476

0.79

521.

9901

2.12

7905

2048

1434

0.7

5010

2412

15.6

9299

90.

0469

0.78

302.

1245

2.48

7333

2048

1536

0.75

5010

2412

17.8

1684

90.

0468

0.78

172.

1396

2.39

6088

2048

1638

0.8

5010

2412

18.9

9833

70.

0468

0.78

162.

1406

2.22

6200

2048

1843

0.9

5010

2412

23.6

1259

10.

0468

0.78

162.

1406

2.37

9104

2048

2048

1.0

5010

249

23.5

0743

50.

0468

0.78

152.

1409

2.34

3294

Page 21 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 12

Per

form

ance

ana

lysi

s of

 the

prop

osed

coi

f2 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2416

1.34

4408

0.05

890.

9847

0.13

382.

6560

92

2048

410

0.2

5010

2415

2.31

1920

0.05

420.

9050

0.86

742.

6342

35

2048

512

0.25

5010

2415

3.53

5317

0.05

420.

9052

0.86

492.

3542

71

2048

614

0.3

5010

2415

4.01

7180

0.05

310.

8869

1.04

282.

5957

62

2048

849

0.4

5010

2414

6.10

7924

0.05

270.

8812

1.09

902.

5151

27

2048

1024

0.5

5010

2413

8.92

7963

0.05

250.

8773

1.13

752.

5318

53

2048

1229

0.6

5010

2413

11.3

2333

10.

0481

0.80

351.

9008

2.50

3729

2048

1434

0.7

5010

2413

21.7

3885

60.

0469

0.78

322.

1229

2.65

7620

2048

1536

0.75

5010

2412

27.1

1614

60.

0468

0.78

172.

1392

2.59

8041

2048

1638

0.8

5010

2412

26.3

1215

60.

0468

0.78

162.

1405

2.35

8481

2048

1843

0.9

5010

2411

32.6

4676

70.

0468

0.78

162.

1406

2.56

4254

2048

2048

1.0

5010

249

34.1

6260

20.

0468

0.78

152.

1409

2.61

7750

Page 22 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 13

Per

form

ance

ana

lysi

s of

 the

prop

osed

coi

f3 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2416

1.31

3659

0.05

610.

9375

0.56

043.

0764

15

2048

410

0.2

5010

2416

2.30

0134

0.05

290.

8835

1.07

612.

8836

24

2048

512

0.25

5010

2416

3.83

6235

0.05

290.

8834

1.07

702.

9772

64

2048

614

0.3

5010

2415

4.09

5096

0.04

920.

8228

1.69

442.

9916

58

2048

849

0.4

5010

2414

6.30

7728

0.04

850.

8105

1.82

442.

7343

66

2048

1024

0.5

5010

2414

12.4

2792

70.

0485

0.81

051.

8247

3.00

0826

2048

1229

0.6

5010

2413

14.1

8072

00.

0475

0.79

312.

0136

2.73

2994

2048

1434

0.7

5010

2413

26.6

9505

40.

0468

0.78

242.

1319

2.67

4142

2048

1536

0.75

5010

2412

23.3

7958

30.

0468

0.78

162.

1403

2.96

7887

2048

1638

0.8

5010

2411

19.4

0293

90.

0468

0.78

162.

1406

2.88

2567

2048

1843

0.9

5010

2411

25.0

7796

50.

0468

0.78

162.

1406

2.74

1203

2048

2048

1.0

5010

249

30.9

2454

00.

0468

0.78

152.

1409

2.94

9021

Page 23 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 14

Per

form

ance

ana

lysi

s of

 the

prop

osed

coi

f4 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ra

tio (C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2416

1.26

2131

0.05

670.

9480

0.46

432.

8437

27

2048

410

0.2

5010

2417

2.43

9706

0.05

240.

8756

1.15

382.

8302

43

2048

512

0.25

5010

2417

3.70

4519

0.05

240.

8757

1.15

292.

7916

44

2048

614

0.3

5010

2416

4.09

7077

0.04

910.

8211

1.71

232.

7433

63

2048

849

0.4

5010

2415

6.15

3716

0.04

850.

8105

1.82

482.

7836

60

2048

1024

0.5

5010

2414

9.04

1550

0.04

850.

8103

1.82

672.

8127

75

2048

1229

0.6

5010

2413

10.7

1278

10.

0473

0.79

112.

0354

2.63

3953

2048

1434

0.7

5010

2413

14.2

1377

50.

0468

0.78

242.

1318

2.63

1650

2048

1536

0.75

5010

2412

16.3

6827

80.

0468

0.78

162.

1404

2.78

5540

2048

1638

0.8

5010

2412

17.1

4283

30.

0468

0.78

162.

1406

2.69

0370

2048

1843

0.9

5010

2411

19.4

9450

50.

0468

0.78

162.

1406

2.60

6268

2048

2048

1.0

5010

249

21.4

3224

70.

0468

0.78

152.

1409

2.65

9983

Page 24 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 15

Per

form

ance

ana

lysi

s of

 the

prop

osed

coi

f5 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2418

1.50

3436

0.05

520.

9229

0.69

662.

8660

05

2048

410

0.2

5010

2417

2.49

1811

0.05

090.

8498

1.41

322.

8956

38

2048

512

0.25

5010

2417

3.85

2817

0.05

090.

8500

1.41

143.

0079

31

2048

614

0.3

5010

2416

4.18

8601

0.04

890.

8164

1.76

163.

0094

24

2048

849

0.4

5010

2414

5.97

1268

0.04

820.

8060

1.87

372.

8452

10

2048

1024

0.5

5010

2414

9.49

8384

0.04

820.

8058

1.87

533.

0334

61

2048

1229

0.6

5010

2413

11.0

1206

10.

0472

0.78

892.

0598

3.03

3341

2048

1434

0.7

5010

2413

14.3

4888

40.

0468

0.78

242.

1313

3.10

5129

2048

1536

0.75

5010

2412

17.0

8854

40.

0468

0.78

162.

1404

2.82

1626

2048

1638

0.8

5010

2412

18.2

1105

70.

0468

0.78

162.

1406

3.00

5033

2048

1843

0.9

5010

2411

20.5

3312

00.

0468

0.78

162.

1406

3.09

5259

2048

2048

1.0

5010

249

23.0

4774

00.

0468

0.78

152.

1409

2.93

6779

Page 25 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

It is noted from Fig. 8 that the sym9 wavelet based sensing matrix requires the less reconstruction time compared to all other Symlets wavelet based sensing matrices. Fur-thermore, the sym5 also shows a very close performance to that of the sym9 wavelet based sensing matrix. From Fig. 9, it can be observed that the sym9 and the sym10 wave-let based sensing matrices almost demonstrate similar performance with minimum rela-tive error compared to all other matrices. Also, from Fig. 10, it is observed that the sym9 and the sym10 wavelet based sensing matrices nearly shows similar performance and exhibits the high SNR compared to other sensing matrices.

Thus, it is evident from Figs. 9 and 10 that overall the sym9 wavelet sensing matrix demonstrates the less reconstruction time and the less relative error, and thus exhib-its the good performance compared to other Symlets wavelet based sensing matrices. Moreover, the sym10 may be the second choice of sensing matrix followed by the sym5.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Ti

me

(Sec

onds

) coif1 Matrixcoif2 Matrixcoif3 Matrixcoif4 Matrixcoif5 Matrix

Fig. 5 Effect of compression ratio on signal reconstruction time for different Coiflets wavelet sensing matri-ces

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.75

0.8

0.85

0.9

0.95

1

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Er

ror (

Rel

ativ

e er

ror)

coif1 Matrixcoif2 Matrixcoif3 Matrixcoif4 Matrixcoif5 Matrix

Fig. 6 Effect of compression ratio on relative error for different Coiflets wavelet sensing matrices

Page 26 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Performance analysis of the Beylkin, Vaidyanathan and Battle wavelet family based

sensing matrices

This section shows the performance analysis of the different DWT sensing matri-ces based on Beylkin, Vaidyanathan, and Battle1, Battle3 and Battle5 wavelet families (Tables 23, 24, 25, 26, 27).

Figure 11 shows that the Beylkin wavelet based sensing matrix requires the less recon-struction time compared to all other Symlets wavelet based sensing matrices. From Fig. 12, it can be observed that the Beylkin and the Battle5 wavelet based sensing matri-ces shows a very close performance with minimum relative error compared to all other matrices. Also, from Fig. 13, it can be seen that the Beylkin and the Battle5 wavelet based sensing matrices shows a very comparable performance and exhibits the high SNR com-pared to other sensing matrices.

Thus, it can be noted from Figs. 11, 12 and 13 that overall the Beylkin wavelet sensing matrix demonstrates the less reconstruction time and relative error, and thus exhibits the good performance compared to other wavelet based sensing matrices. However, the Battle5 shows a close performance and may be the second best choice of sensing matrix.

Performance analysis of the best‑proposed DWT based sensing matrices namely: Beylkin,

db10, coif5 and sym9 wavelet family

This section illustrates the performance analysis of the best-proposed DWT sensing matrices namely: Beylkin, db10, coif5 and sym9 wavelet families.

Figure 14 shows that the sym9 wavelet based sensing matrix clearly outperforms the Beylkin, db10, and the coif5 wavelet based sensing matrices in terms of signal recon-struction time. From Fig. 15, it can be observed that the db10 shows the good perfor-mance over CR =  0.3–0.5; however overall the sym9 wavelet based sensing matrices shows the good (from CR =  0.5–1.0) and comparable performance with db10. Also, from Fig. 16, it can be observed that the db10 and sym9 wavelet based sensing matrices shows a comparable performance and exhibits the high SNR compared to other sensing

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

Compression Ratio (CR)

Sign

al-to

-Noi

se R

atio

(SN

R) (

db)

coif1 Matrixcoif2 Matrixcoif3 Matrixcoif4 Matrixcoif5 Matrix

Fig. 7 Effect of compression ratio on signal-to-noise ratio for different Coiflets wavelet sensing matrices

Page 27 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 16

Per

form

ance

ana

lysi

s of

 the

prop

osed

sym

4 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) 

× 1

00 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2415

1.42

6156

0.06

061.

0128

0.11

012.

2199

32

2048

410

0.2

5010

2415

2.46

6473

0.05

500.

9186

0.73

762.

2874

69

2048

512

0.25

5010

2415

4.03

4720

0.05

500.

9188

0.73

602.

4085

24

2048

614

0.3

5010

2415

4.52

0797

0.05

100.

8516

1.39

562.

2620

42

2048

849

0.4

5010

2414

6.88

3726

0.05

030.

8412

1.50

162.

4613

59

2048

1024

0.5

5010

2414

9.27

5989

0.05

030.

8412

1.50

152.

6008

20

2048

1229

0.6

5010

2413

10.6

9936

60.

0479

0.80

011.

9374

2.08

3195

2048

1434

0.7

5010

2413

13.7

0268

30.

0468

0.78

282.

1269

2.11

7664

2048

1536

0.75

5010

2412

16.1

4265

90.

0468

0.78

172.

1393

2.30

3114

2048

1638

0.8

5010

2412

17.0

2474

60.

0468

0.78

162.

1405

2.25

9328

2048

1843

0.9

5010

2412

21.1

1806

40.

0468

0.78

162.

1406

2.36

9993

2048

2048

1.0

5010

249

21.2

2717

90.

0468

0.78

152.

1409

2.24

8962

Page 28 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 17

Per

form

ance

ana

lysi

s of

 the

prop

osed

sym

5 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n

time

(s)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2418

1.56

2108

0.05

870.

9813

0.16

442.

4747

25

2048

410

0.2

5010

2416

2.41

0647

0.05

250.

8767

1.14

282.

4229

95

2048

512

0.25

5010

2415

3.36

3769

0.05

240.

8764

1.14

602.

4149

40

2048

614

0.3

5010

2414

3.63

3501

0.04

870.

8139

1.78

852.

4016

02

2048

849

0.4

5010

2414

5.90

1087

0.04

850.

8100

1.83

072.

2859

70

2048

1024

0.5

5010

2413

8.62

9663

0.04

840.

8094

1.83

702.

2709

79

2048

1229

0.6

5010

2412

10.4

3665

80.

0476

0.79

511.

9910

2.29

5386

2048

1434

0.7

5010

2412

12.9

9750

50.

0468

0.78

252.

1299

2.41

6593

2048

1536

0.75

5010

2412

16.5

4321

70.

0468

0.78

162.

1400

2.22

6996

2048

1638

0.8

5010

2411

15.8

3922

20.

0468

0.78

162.

1407

2.25

5090

2048

1843

0.9

5010

2411

19.8

8718

30.

0468

0.78

162.

1407

2.43

1380

2048

2048

1.0

5010

249

21.3

1945

00.

0468

0.78

152.

1409

2.26

6367

Page 29 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 18

Per

form

ance

ana

lysi

s of

 the

prop

osed

sym

6 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2416

1.33

7284

0.05

680.

9486

0.45

872.

2788

76

2048

410

0.2

5010

2416

2.35

4315

0.05

190.

8665

1.24

502.

2914

21

2048

512

0.25

5010

2415

3.40

3486

0.05

200.

8696

1.21

322.

4955

18

2048

614

0.3

5010

2415

3.90

0882

0.05

060.

8450

1.46

322.

5137

99

2048

849

0.4

5010

2414

5.89

4927

0.04

950.

8274

1.64

552.

4446

08

2048

1024

0.5

5010

2414

9.19

1334

0.04

950.

8274

1.64

522.

2826

27

2048

1229

0.6

5010

2413

10.8

7466

30.

0470

0.78

582.

0939

2.35

4739

2048

1434

0.7

5010

2413

13.9

8655

30.

0468

0.78

252.

1300

2.26

9408

2048

1536

0.75

5010

2412

16.2

8915

50.

0468

0.78

162.

1404

2.32

4362

2048

1638

0.8

5010

2412

17.0

0207

40.

0468

0.78

162.

1406

2.32

5569

2048

1843

0.9

5010

2412

21.6

0399

00.

0468

0.78

162.

1407

2.40

9376

2048

2048

1.0

5010

249

21.4

8859

00.

0468

0.78

152.

1409

2.34

4651

Page 30 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 19

Per

form

ance

ana

lysi

s of

 the

prop

osed

sym

7 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2419

1.60

4935

0.05

490.

9178

0.74

502.

3851

64

2048

410

0.2

5010

2416

2.27

0355

0.05

300.

8860

1.05

142.

3689

00

2048

512

0.25

5010

2416

3.61

1425

0.05

300.

8857

1.05

392.

3900

55

2048

614

0.3

5010

2414

3.71

9657

0.04

940.

8259

1.66

202.

4002

46

2048

849

0.4

5010

2414

5.94

3942

0.04

860.

8126

1.80

252.

3796

69

2048

1024

0.5

5010

2414

9.23

1755

0.04

860.

8125

1.80

372.

6262

92

2048

1229

0.6

5010

2414

11.6

9705

60.

0477

0.79

681.

9732

2.39

5632

2048

1434

0.7

5010

2413

14.0

3804

80.

0468

0.78

242.

1319

2.54

7235

2048

1536

0.75

5010

2412

16.3

7947

80.

0468

0.78

162.

1398

2.54

1034

2048

1638

0.8

5010

2412

17.1

3938

10.

0468

0.78

162.

1406

2.29

9743

2048

1843

0.9

5010

2411

19.8

1361

30.

0468

0.78

162.

1406

2.62

8218

2048

2048

1.0

5010

249

21.3

9374

10.

0468

0.78

152.

1409

2.43

5649

Page 31 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 20

Per

form

ance

ana

lysi

s of

 the

prop

osed

sym

8 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f sig

nal

(N)

Num

ber

of m

easu

re‑

men

ts (m

)

Com

pres

sion

ra

tio (C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

 100

 (%)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al re

con‑

stru

ctio

n tim

e (s

)

RMSE

Rela

tive

erro

rSN

R (d

b)Co

nstr

uctio

n tim

e fo

r sen

sing

m

atri

x (s

)

2048

205

0.1

5010

2416

1.33

7804

0.05

910.

9881

0.10

372.

3989

59

2048

410

0.2

5010

2416

2.28

9342

0.05

250.

8775

1.13

542.

4493

29

2048

512

0.25

5010

2416

3.60

5457

0.05

250.

8774

1.13

612.

4139

14

2048

614

0.3

5010

2415

3.89

9311

0.04

900.

8194

1.72

962.

3671

31

2048

849

0.4

5010

2414

5.83

9426

0.04

890.

8165

1.76

132.

4140

94

2048

1024

0.5

5010

2414

9.21

9034

0.04

890.

8167

1.75

902.

4776

53

2048

1229

0.6

5010

2413

10.9

1677

50.

0476

0.79

531.

9899

2.41

2281

2048

1434

0.7

5010

2413

14.1

0579

00.

0468

0.78

232.

1324

2.41

1482

2048

1536

0.75

5010

2412

16.3

3755

30.

0468

0.78

162.

1403

2.42

5477

2048

1638

0.8

5010

2411

15.8

7626

70.

0468

0.78

162.

1406

2.45

2373

2048

1843

0.9

5010

2411

19.8

3864

40.

0468

0.78

162.

1406

2.43

9456

2048

2048

1.0

5010

249

21.4

1229

60.

0468

0.78

152.

1409

2.48

1033

Page 32 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 21

Per

form

ance

ana

lysi

s of

 the

prop

osed

sym

9 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2417

1.43

3657

0.05

820.

9731

0.23

642.

5660

95

2048

410

0.2

5010

2416

2.36

2390

0.05

200.

8695

1.21

492.

5664

05

2048

512

0.25

5010

2416

3.70

3235

0.05

200.

8690

1.21

912.

5294

53

2048

614

0.3

5010

2415

3.94

7898

0.04

930.

8243

1.67

882.

5802

08

2048

849

0.4

5010

2414

5.82

4850

0.04

810.

8038

1.89

752.

5664

82

2048

1024

0.5

5010

2413

8.52

7638

0.04

810.

8038

1.89

732.

6276

00

2048

1229

0.6

5010

2412

9.96

1474

0.04

700.

7850

2.10

232.

6214

38

2048

1434

0.7

5010

2412

12.9

6303

50.

0468

0.78

222.

1333

2.58

2570

2048

1536

0.75

5010

2412

16.4

6801

30.

0468

0.78

162.

1406

2.62

8998

2048

1638

0.8

5010

2411

15.8

2022

40.

0468

0.78

162.

1406

2.70

5263

2048

1843

0.9

5010

2411

19.9

1521

60.

0468

0.78

162.

1407

2.69

5629

2048

2048

1.0

5010

249

21.4

7796

90.

0468

0.78

152.

1409

2.61

8640

Page 33 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 22

Per

form

ance

ana

lysi

s of

 the

prop

osed

sym

10 w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n

time

(s)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2417

1.47

9121

0.05

600.

9351

0.58

312.

7083

58

2048

410

0.2

5010

2416

2.34

8326

0.05

130.

8578

1.33

232.

6074

75

2048

512

0.25

5010

2416

3.76

2705

0.05

130.

8580

1.32

982.

7293

82

2048

614

0.3

5010

2416

4.32

8904

0.04

900.

8192

1.73

212.

8194

32

2048

849

0.4

5010

2414

6.05

7941

0.04

790.

8003

1.93

532.

7192

36

2048

1024

0.5

5010

2414

9.51

6700

0.04

790.

8010

1.92

732.

7157

42

2048

1229

0.6

5010

2413

11.2

6863

40.

0470

0.78

532.

0992

2.48

4238

2048

1434

0.7

5010

2412

13.6

0722

60.

0468

0.78

222.

1333

2.68

0826

2048

1536

0.75

5010

2412

17.2

2508

60.

0468

0.78

162.

1405

2.63

4090

2048

1638

0.8

5010

2412

18.1

9201

90.

0468

0.78

162.

1406

2.62

8795

2048

1843

0.9

5010

2412

22.7

1672

30.

0468

0.78

162.

1406

2.70

9115

2048

2048

1.0

5010

249

22.9

0914

50.

0468

0.78

152.

1409

2.61

2016

Page 34 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

matrices. In addition, the sym9 wavelet based sensing matrix shows an edge over db10 from the CR = 0.5–1.0.

Thus, it can be evident from Figs. 14, 15 and 16 that overall the sym9 wavelet based sensing matrix shows the superior performance compared to the Beylkin, db10 and the coif5 wavelet based sensing matrices in views of signal reconstruction time and relative error. Furthermore, the db10 may be the second best choice of sensing matrix.

Performance analysis of the best‑proposed sym9 wavelet based sensing matrix

with state‑of‑the‑art random and deterministic sensing matrices

This section illustrates the comparative analysis of the proposed sym9 wavelet based sensing matrix and state-of-the-art random sensing matrices such as Gaussian, Uniform, Toeplitz, Circulant and Hadamard matrix along with deterministic sensing matrices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Ti

me

(Sec

onds

) sym4 Matrixsym5 Matrixsym6 Matrixsym7 Matrixsym8 Matrixsym9 Matrixsym10 Matrix

Fig. 8 Effect of compression ratio on signal reconstruction time for different Symlets wavelet sensing matri-ces

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.75

0.8

0.85

0.9

0.95

1

1.05

1.1

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Er

ror (

Rel

ativ

e er

ror)

sym4 Matrixsym5 Matrixsym6 Matrixsym7 Matrixsym8 Matrixsym9 Matrixsym10 Matrix

Fig. 9 Effect of compression ratio on relative error for different Symlets wavelet sensing matrices

Page 35 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

such as the DCT and the sparse binary sensing matrices for speech signal compression (Tables 28, 29, 30, 31, 32, 33, 34).

It is noted from Fig. 17 that the proposed sym9 wavelet based sensing matrix clearly outperforms the state-of-the-art random sensing matrices such as Gaussian, Uniform, Toeplitz, Circulant and Hadamard sensing matrices as well as the deterministic DCT and sparse binary sensing matrices in terms of signal reconstruction time. It can be observed from Figs.  18 and  19 that the proposed sym9 wavelet based sensing matrix demonstrates a close comparable performance compared to the state-of-the-art random and deterministic sensing matrices.

The overall remark

Thus, it is evident from Figs. 17, 18 and 19 (Tables 28, 29, 30, 31, 32, 33, 34) that the pro-posed sym9 wavelet based sensing matrix exhibits the better performance compared to the state-of-the-art random and deterministic sensing matrices.

Subjective quality evaluation

Simple quality measures like SNR do not provide an accurate measure of the speech quality. Hence, speech quality assessment is performed by highly robust and accurate measures such as the mean opinion score (MOS) and perceptual evaluation of speech quality (PESQ) recommended by International Telecommunication Union Telephony (ITU-T) standards.

In this section, the performance of the proposed sensing matrices is evaluated using mean opinion score (MOS). The MOS is a subjective listening test to perceive the speech quality and one of the widely recommended method by ITU standard (ITU-T P.800) (ITU-T 1996).

Table 35 presents subjective evaluation of the reconstructed speech quality using the mean opinion score (MOS) test. The MOS test is performed on a group of seven male listeners and three female listeners. The listeners are required to train and evaluate the quality of the reconstructed speech signal with respect to the original signal. The speech

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

Compression Ratio (CR)

Sign

al-to

-Noi

se R

atio

(SN

R) (

db)

sym4 Matrixsym5 Matrixsym6 Matrixsym7 Matrixsym8 Matrixsym9 Matrixsym10 Matrix

Fig. 10 Effect of compression ratio on signal-to-noise ratio for different Symlets wavelet sensing matrices

Page 36 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 23

Per

form

ance

ana

lysi

s of

 the

prop

osed

Bey

lkin

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2417

1.35

1917

0.05

510.

9205

0.71

962.

6352

92

2048

410

0.2

5010

2416

2.31

3493

0.05

220.

8726

1.18

372.

5003

19

2048

512

0.25

5010

2416

3.65

5763

0.05

220.

8722

1.18

782.

5827

26

2048

614

0.3

5010

2416

4.26

7978

0.04

900.

8193

1.73

162.

6683

84

2048

849

0.4

5010

2414

5.86

5898

0.04

830.

8064

1.86

862.

5015

77

2048

1024

0.5

5010

2414

9.54

1513

0.04

820.

8056

1.87

742.

6263

48

2048

1229

0.6

5010

2413

11.1

1344

80.

0469

0.78

382.

1163

2.46

4044

2048

1434

0.7

5010

2413

14.5

6769

30.

0468

0.78

222.

1341

2.50

4371

2048

1536

0.75

5010

2412

17.1

9308

0.04

680.

7816

2.14

052.

5276

96

2048

1638

0.8

5010

2412

18.0

8175

50.

0468

0.78

162.

1406

2.51

6479

2048

1843

0.9

5010

2412

22.5

2665

40.

0468

0.78

162.

1407

2.52

3531

2048

2048

1.0

5010

249

22.9

5070

30.

0468

0.78

152.

1409

2.46

6272

Page 37 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 24

Per

form

ance

ana

lysi

s of

 the

prop

osed

Vai

dyan

atha

n w

avel

et b

ased

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2419

1.70

7644

0.07

021.

1724

1.38

122.

9129

96

2048

410

0.2

5010

2419

3.45

9463

0.05

800.

9684

0.27

853.

3328

69

2048

512

0.25

5010

2419

5.66

6164

0.05

800.

9685

0.27

823.

0360

02

2048

614

0.3

5010

2418

5.76

0958

0.05

130.

8578

1.33

203.

0347

69

2048

849

0.4

5010

2415

7.31

6646

0.04

850.

8112

1.81

772.

9690

49

2048

1024

0.5

5010

2414

10.8

5523

90.

0486

0.81

231.

8053

3.07

5962

2048

1229

0.6

5010

2413

13.4

1742

20.

0474

0.79

242.

0214

3.44

1779

2048

1434

0.7

5010

2413

19.0

9362

40.

0468

0.78

222.

1339

3.48

3836

2048

1536

0.75

5010

2412

22.6

0845

30.

0468

0.78

162.

1405

3.02

2221

2048

1638

0.8

5010

2412

34.9

7041

50.

0468

0.78

162.

1406

3.58

6396

2048

1843

0.9

5010

2412

49.3

1445

00.

0468

0.78

162.

1407

3.51

5476

2048

2048

1.0

5010

249

34.9

4370

20.

0468

0.78

152.

1409

3.06

0519

Page 38 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 25

Per

form

ance

ana

lysi

s of

 the

prop

osed

Bat

tle1

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ra

tio (C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n

time

(s)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2415

1.45

3478

0.06

331.

0577

0.48

732.

7945

29

2048

410

0.2

5010

2414

2.11

3667

0.05

560.

9283

0.64

592.

9536

15

2048

512

0.25

5010

2415

3.53

0676

0.05

600.

9363

0.57

132.

7698

43

2048

614

0.3

5010

2414

3.95

7949

0.05

140.

8598

1.31

252.

6216

63

2048

849

0.4

5010

2414

7.28

0928

0.05

130.

8571

1.33

992.

8469

45

2048

1024

0.5

5010

2414

9.77

7801

0.05

140.

8587

1.32

302.

8585

07

2048

1229

0.6

5010

2413

11.8

5049

70.

0480

0.80

131.

9241

2.64

3100

2048

1434

0.7

5010

2413

15.1

9562

80.

0468

0.78

272.

1278

2.62

1185

2048

1536

0.75

5010

2412

17.5

0308

70.

0468

0.78

172.

1387

2.80

4361

2048

1638

0.8

5010

2411

17.4

2154

50.

0468

0.78

162.

1406

2.83

5200

2048

1843

0.9

5010

2411

28.1

7992

90.

0468

0.78

162.

1406

2.86

9269

2048

2048

1.0

5010

249

32.6

4096

50.

0468

0.78

152.

1409

3.44

7333

Page 39 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 26

Per

form

ance

ana

lysi

s of

 the

prop

osed

Bat

tle3

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2421

2.22

9166

0.07

761.

2960

2.25

204.

1843

86

2048

410

0.2

5010

2417

3.00

0747

0.06

191.

0350

0.29

914.

0370

21

2048

512

0.25

5010

2417

6.76

2783

0.06

191.

0350

0.29

844.

2141

51

2048

614

0.3

5010

2417

5.49

5381

0.05

760.

9626

0.33

133.

7794

77

2048

849

0.4

5010

2415

7.92

0600

0.05

380.

8989

0.92

613.

7067

19

2048

1024

0.5

5010

2414

11.3

3121

20.

0538

0.89

950.

9199

3.94

5231

2048

1229

0.6

5010

2413

13.9

0919

00.

0485

0.81

061.

8235

3.76

4331

2048

1434

0.7

5010

2412

19.8

3191

70.

0468

0.78

232.

1321

3.77

7555

2048

1536

0.75

5010

2412

34.0

1038

80.

0468

0.78

162.

1404

4.46

4107

2048

1638

0.8

5010

2412

30.2

3574

40.

0468

0.78

162.

1406

3.85

5705

2048

1843

0.9

5010

2411

40.1

9081

60.

0468

0.78

162.

1406

4.00

3610

2048

2048

1.0

5010

249

45.7

8919

50.

0468

0.78

152.

1409

3.72

6738

Page 40 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 27

Per

form

ance

ana

lysi

s of

 the

prop

osed

Bat

tle5

wav

elet

bas

ed s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ra

tio (C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2416

1.54

7147

0.05

870.

9809

0.16

724.

3558

11

2048

410

0.2

5010

2416

2.77

6326

0.05

170.

8633

1.27

644.

5236

38

2048

512

0.25

5010

2417

4.87

6061

0.05

180.

8648

1.26

194.

2586

86

2048

614

0.3

5010

2416

5.06

8130

0.04

920.

8226

1.69

614.

2160

43

2048

849

0.4

5010

2415

8.17

3479

0.04

790.

7999

1.93

984.

6539

44

2048

1024

0.5

5010

2413

22.0

5622

30.

0480

0.80

131.

9240

4.19

6677

2048

1229

0.6

5010

2413

13.5

7698

40.

0470

0.78

582.

0936

4.17

4972

2048

1434

0.7

5010

2413

18.4

0510

00.

0468

0.78

212.

1350

4.35

5072

2048

1536

0.75

5010

2412

22.4

3025

20.

0468

0.78

162.

1404

4.26

8081

2048

1638

0.8

5010

2411

28.3

2550

10.

0468

0.78

162.

1406

4.26

9212

2048

1843

0.9

5010

2411

38.8

0524

90.

0468

0.78

162.

1407

4.62

3429

2048

2048

1.0

5010

249

41.1

8635

00.

0468

0.78

152.

1409

4.28

3169

Page 41 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

quality is evaluated by rating to a signal within the range of 1–5. The MOS is computed by taking the average score of all the individual listeners and it ranges between 1 (bad speech quality) and 5 (excellent speech quality).

The following conclusions can be drawn from Table 35.

1. Overall, the Symlets wavelet family achieves the good MOS scores compared to other proposed as well as state-of-the-art sensing matrices.

2. The highest MOS score of 4.4 is achieved by the sym9 wavelet family followed by the sym6, sym8, sym10, Battle1, Battle3 (MOS = 4.1) and followed by the db2, coif5 (MOS = 4.0) respectively. Thus, these MOS scores can be considered as an accept-able score for speech quality.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Ti

me

(Sec

onds

) Beylkin MatrixVaidyanathan MatrixBattle1 MatrixBattle3 MatrixBattle5 Matrix

Fig. 11 Effect of compression ratio on signal reconstruction time for Beylkin, Vaidyanathan and Battle wave-let sensing matrices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Er

ror (

Rel

ativ

e er

ror)

Beylkin MatrixVaidyanathan MatrixBattle1 MatrixBattle3 MatrixBattle5 Matrix

Fig. 12 Effect of compression ratio on relative error for Beylkin, Vaidyanathan and Battle wavelet sensing matrices

Page 42 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

3. Moreover, the state-of-the-art DCT sensing matrix (MOS =  4.2) and the random Hadamard sensing matrix (MOS =  4.0) shows the good MOS score compared to other state-of-the-art sensing matrices.

However, MOS test frequently requires a sizeable number of listeners to accomplish stable results, and is also the time-consuming and expensive. Nevertheless, subjective quality measures are still one of the most decisive ways to estimate speech quality.

Objective quality evaluation

The PESQ is a most modern international ITU-T standard (P.862) (ITU-T 2005) for an automated prediction of speech quality by estimating quality scores ranging from −1 to 4.5. In other way, it estimates the MOS (Mean Opinion Score) from both the clean

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

Compression Ratio (CR)

Sign

al-to

-Noi

se R

atio

(SN

R) (

db)

Beylkin MatrixVaidyanathan MatrixBattle1 MatrixBattle3 MatrixBattle5 Matrix

Fig. 13 Effect of compression ratio on signal-to-noise ratio for Beylkin, Vaidyanathan and Battle wavelet sens-ing matrices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Ti

me

(Sec

onds

) Beylkin Matrixdb10 Matrixcoif5 Matrixsym9 Matrix

Fig. 14 Effect of compression ratio on signal reconstruction time for the best DWT based sensing matrices

Page 43 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

signal and its distorted signal. A higher quality score signifies the better speech quality. Moreover, since human listeners are not required; PESQ is less expensive, accurate and less time-consuming;

Table 36 presents the different objective speech quality metrics such as the PESQ, log-likelihood ratio (LLR) and weighted spectral slope (WSS) along with the three subjective rating scales namely: signal distortion, noise distortion, and overall quality. The ratings are based on the five-point (1–5) MOS scale (Hu and Loizou 2008).

The following conclusions can be drawn from Table 36.

1. The Symlets wavelet family shows the higher signal distortion rating (rating between: 3–4) indicating the fairly natural speech signal quality compared to other proposed and state-of-the art sensing matrices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Er

ror (

Rel

ativ

e er

ror)

Beylkin Matrixdb10 Matrixcoif5 Matrixsym9 Matrix

Fig. 15 Effect of compression ratio on relative error for the best DWT based sensing matrices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

Compression Ratio (CR)

Sign

al-to

-Noi

se R

atio

(SN

R) (

db)

Beylkin Matrixdb10 Matrixcoif5 Matrixsym9 Matrix

Fig. 16 Effect of compression ratio on signal-to-noise ratio for the best DWT based sensing matrices

Page 44 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 28

Per

form

ance

ana

lysi

s of

 the

rand

om G

auss

ian

sens

ing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ra

tio (C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2421

1.61

4613

0.05

290.

8846

0.66

671.

0967

33

2048

410

0.2

5010

2422

3.10

1767

0.05

060.

8459

1.45

260.

7480

22

2048

512

0.25

5010

2421

4.63

9830

0.04

880.

8163

1.68

021.

9425

04

2048

614

0.3

5010

2420

5.06

7599

0.04

880.

8149

1.81

492.

1425

37

2048

849

0.4

5010

2421

8.24

0858

0.04

760.

7949

2.05

164.

9093

29

2048

1024

0.5

5010

2420

12.7

9205

70.

0470

0.78

472.

1202

11.6

3578

9

2048

1229

0.6

5010

2421

17.3

7562

60.

0468

0.78

172.

1431

14.0

8166

9

2048

1434

0.7

5010

2422

23.4

7232

30.

0468

0.78

162.

1388

34.6

3169

3

2048

1536

0.75

5010

2424

33.4

7127

90.

0468

0.78

162.

1416

39.1

9467

6

2048

1638

0.8

5010

2427

38.9

0717

60.

0468

0.78

162.

1408

43.4

7618

5

2048

1843

0.9

5010

2423

41.1

5626

10.

0468

0.78

162.

1409

51.0

9675

3

2048

2048

1.0

5010

249

22.1

2998

80.

0468

0.78

152.

1409

57.7

5539

8

Page 45 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 29

Per

form

ance

ana

lysi

s of

 the

rand

om U

nifo

rm s

ensi

ng m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n

time

(s)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2428

2.23

5539

0.06

341.

0600

1.02

960.

2236

25

2048

410

0.2

5010

2425

3.60

0845

0.05

370.

8966

0.81

520.

7385

91

2048

512

0.25

5010

2425

5.74

5004

0.05

160.

8616

0.86

981.

9172

57

2048

614

0.3

5010

2425

6.43

4319

0.04

960.

8281

1.69

512.

1138

97

2048

849

0.4

5010

2423

9.44

3467

0.04

740.

7924

1.98

105.

3320

54

2048

1024

0.5

5010

2423

15.9

8657

80.

0470

0.78

572.

0911

11.9

7539

5

2048

1229

0.6

5010

2424

19.7

2746

40.

0468

0.78

232.

1297

14.1

6544

0

2048

1434

0.7

5010

2424

25.9

2022

90.

0468

0.78

162.

1401

34.9

6894

9

2048

1536

0.75

5010

2423

72.4

8138

30.

0468

0.78

162.

1407

48.4

0863

1

2048

1638

0.8

5010

2420

29.1

9373

70.

0468

0.78

162.

1407

93.4

0706

0

2048

1843

0.9

5010

2417

32.8

2170

00.

0468

0.78

162.

1408

51.4

3767

7

2048

2048

1.0

5010

249

22.3

9544

00.

0468

0.78

152.

1409

57.8

0310

1

Page 46 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 30

Per

form

ance

ana

lysi

s of

 the

rand

om H

adam

ard

sens

ing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2424

2.16

0564

0.05

430.

9068

0.68

160.

1752

90

2048

410

0.2

5010

2421

3.35

2213

0.05

080.

8486

1.50

020.

1956

49

2048

512

0.25

5010

2421

5.24

5114

0.04

880.

8162

1.79

100.

2272

24

2048

614

0.3

5010

2414

4.05

9570

0.04

760.

7959

1.78

400.

2037

26

2048

849

0.4

5010

2414

7.10

3978

0.04

830.

8075

2.15

090.

2385

74

2048

1024

0.5

5010

2417

12.4

2031

20.

0469

0.78

382.

1742

0.31

2728

2048

1229

0.6

5010

2420

18.2

0294

10.

0468

0.78

142.

1341

0.26

2874

2048

1434

0.7

5010

2424

28.3

3006

30.

0468

0.78

162.

1400

0.28

1225

2048

1536

0.75

5010

2421

32.0

1789

30.

0467

0.78

052.

1406

0.40

0085

2048

1638

0.8

5010

2423

47.0

9144

90.

0467

0.78

032.

1407

0.30

4155

2048

1843

0.9

5010

2426

70.6

8479

80.

0467

0.78

052.

1408

0.31

8110

2048

2048

1.0

5010

249

31.9

5310

30.

0468

0.78

152.

1408

0.46

3961

Page 47 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 31

Per

form

ance

ana

lysi

s of

 the

rand

om T

oepl

itz

sens

ing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2421

1.75

6908

0.05

360.

8957

0.91

040.

4096

20

2048

410

0.2

5010

2422

3.23

3689

0.04

980.

8315

1.54

690.

4299

34

2048

512

0.25

5010

2421

4.78

8428

0.04

930.

8232

1.73

660.

4695

36

2048

614

0.3

5010

2420

5.35

5458

0.04

830.

8068

1.80

570.

4501

23

2048

849

0.4

5010

2420

8.42

9285

0.04

740.

7918

2.09

180.

4716

07

2048

1024

0.5

5010

2420

13.4

3943

30.

0469

0.78

412.

1270

0.54

4915

2048

1229

0.6

5010

2421

17.9

6135

40.

0467

0.77

972.

1501

0.50

3870

2048

1434

0.7

5010

2421

23.1

6253

70.

0467

0.78

062.

1523

0.52

4939

2048

1536

0.75

5010

2423

32.3

1277

60.

0467

0.78

062.

1511

0.62

6275

2048

1638

0.8

5010

2424

35.3

2530

30.

0467

0.78

072.

1490

0.54

9642

2048

1843

0.9

5010

2427

49.5

9110

00.

0467

0.78

122.

1453

0.56

2993

2048

2048

1.0

5010

246

18.3

5195

40.

0468

0.78

152.

1409

0.69

5494

Page 48 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 32

Per

form

ance

ana

lysi

s of

 the

rand

om C

ircu

lant

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2426

2.50

7042

0.05

970.

9968

0.42

222.

7571

10

2048

410

0.2

5010

2424

3.72

9785

0.05

100.

8516

1.56

731.

7551

48

2048

512

0.25

5010

2422

5.27

8771

0.04

910.

8206

1.72

781.

8192

34

2048

614

0.3

5010

2419

5.23

7320

0.04

800.

8021

1.95

821.

8136

63

2048

849

0.4

5010

2418

8.22

0705

0.04

700.

7846

2.10

301.

8456

29

2048

1024

0.5

5010

2417

11.9

1656

20.

0468

0.78

252.

1335

1.91

7508

2048

1229

0.6

5010

2416

14.6

5670

40.

0468

0.78

162.

1404

1.85

7957

2048

1434

0.7

5010

2415

17.5

7700

90.

0468

0.78

162.

1408

1.85

2721

2048

1536

0.75

5010

2416

23.8

6419

50.

0468

0.78

162.

1408

2.02

0327

2048

1638

0.8

5010

2415

23.4

5197

10.

0468

0.78

162.

1409

1.89

3177

2048

1843

0.9

5010

2412

25.9

5422

90.

0468

0.78

152.

1409

1.91

4215

2048

2048

1.0

5010

248

24.1

9048

00.

0468

0.78

152.

1409

2.06

5995

Page 49 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 33

Per

form

ance

ana

lysi

s of

 the

Det

erm

inis

tic

DCT

sen

sing

mat

rix

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ratio

(C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2415

1.33

6874

0.05

700.

9532

0.41

650.

0328

36

2048

410

0.2

5010

2416

2.30

9603

0.05

370.

8974

0.94

050.

0604

71

2048

512

0.25

5010

2417

3.77

3514

0.05

110.

8543

1.36

820.

1104

42

2048

614

0.3

5010

2416

4.12

2633

0.05

100.

8525

1.38

580.

0871

57

2048

849

0.4

5010

2414

5.82

2766

0.04

940.

8259

1.66

170.

1177

17

2048

1024

0.5

5010

2414

9.05

8112

0.04

780.

7981

1.95

920.

2130

83

2048

1229

0.6

5010

2415

12.3

6980

90.

0472

0.78

892.

0593

0.17

0241

2048

1434

0.7

5010

2413

14.3

2483

60.

0469

0.78

322.

1224

0.19

8500

2048

1536

0.75

5010

2413

18.2

3687

80.

0468

0.78

252.

1303

0.32

3367

2048

1638

0.8

5010

2413

18.6

0074

50.

0468

0.78

202.

1363

0.22

5549

2048

1843

0.9

5010

2412

21.4

6199

50.

0468

0.78

162.

1409

0.25

3978

2048

2048

1.00

5010

249

21.8

9347

60.

0468

0.78

152.

1409

0.43

8310

Page 50 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Tabl

e 34

Per

form

ance

ana

lysi

s of

 the

Det

erm

inis

tic

Spar

se B

inar

y se

nsin

g m

atri

x

Leng

th o

f  si

gnal

(N)

Num

ber o

f  m

easu

rem

ents

(m

)

Com

pres

sion

ra

tio (C

R =

 m/N

)Sp

arsi

ty

leve

l = (k

/N) ×

  10

0 (%

)

No.

of n

on‑

zero

s (k

)N

o. o

f ite

ratio

ns

requ

ired

Sign

al

reco

nstr

uctio

n tim

e (s

)

RMSE

Rela

tive

er

ror

SNR

(db)

Cons

truc

tion

time

for s

ensi

ng

mat

rix

(s)

2048

205

0.1

5010

2412

0.92

780.

0548

0.91

540.

7678

66.9

190

2048

410

0.2

5010

2413

1.88

710.

0503

0.84

051.

5090

207.

1541

2048

512

0.25

5010

2414

3.14

310.

0497

0.83

081.

6102

380.

2697

2048

614

0.3

5010

2415

4.83

550.

0487

0.81

461.

7809

502.

1809

2048

849

0.4

5010

2419

8.38

950.

0473

0.79

062.

0411

878.

8051

2048

1024

0.5

5010

2420

18.3

753

0.04

690.

7843

2.11

0312

71.5

155

2048

1229

0.6

5010

2423

25.4

419

0.04

680.

7824

2.13

1318

69.4

049

2048

1434

0.7

5010

2430

42.2

472

0.04

680.

7816

2.14

0729

13.4

315

2048

1536

0.75

5010

2428

57.3

386

0.04

680.

7816

2.14

0829

84.3

214

2048

1638

0.8

5010

2424

52.8

342

0.04

680.

7816

2.14

0935

75.2

037

2048

1843

0.9

5010

2417

50.7

545

0.04

680.

7816

2.14

0940

30.6

333

2048

2048

1.0

5010

246

26.8

845

0.04

680.

7815

2.14

0958

56.4

331

Page 51 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

2. The db5, db9, db10, coif3, coif4, coif5 and Symlets wavelet families shows the good background distortion rating (between rating: 2–3) indicating noticeable noise, but not intrusive and are close comparable to state-of-the art sensing matrices.

3. The db5, db9, db10, coif3, coif4, coif5 and Symlets wavelet families shows the higher signal quality rating (between rating: 3–4) indicating the good/fair speech quality compared to state-of-the art sensing matrices.

4. Overall, the sym9 and the sym10 wavelet family based sensing matrices exhibits good/fair overall quality (For db9 and db10 ratings are 3.1843 and 3.1985 respec-tively) compared to other proposed and state-of-the art sensing matrices.

5. In terms of objective measures, the sym9 and the sym10 wavelet family based sensing matrices exhibits the lower values of log-likelihood ratio (LLR) and weighted spectral slope (WSS) metrics, indicating the good speech quality and are close comparable with state-of-the art sensing matrices.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Ti

me

(Sec

onds

) Random Gaussian MatrixRandom Uniform MatrixRandom Toeplitz MatrixRandom Circulant MatrixRandom Hadamard MatrixDeterministic DCT MatrixSparse Binary MatrixProposed sym9 Matrix

Fig. 17 Effect of compression ratio on signal reconstruction time for the proposed sym9 matrix and state-of-the-art random and deterministic sensing matrices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

Compression Ratio (CR)

Sign

al R

econ

stru

ctio

n Er

ror (

Rel

ativ

e er

ror)

Random Gaussian MatrixRandom Uniform MatrixRandom Toeplitz MatrixRandom Circulant MatrixRandom Hadamard MatrixDeterministic DCT MatrixSparse Binary MatrixProposed sym9 Matrix

Fig. 18 Effect of compression ratio on relative error for the proposed sym9 matrix and state-of-the-art ran-dom and deterministic sensing matrices

Page 52 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

6. Finally, in views of PESQ measure, the sym9 and the sym10 wavelet family based sensing matrices exhibits the higher PESQ scores; PESQ  =  2.6003 (sym9) and PESQ =  2.6006 (sym10) respectively, signifying the good/fair speech quality com-pared to other proposed and state-of-the art sensing matrices.

Information based evaluation

Entropy (H) is a measure of an average information content of a signal (x) and widely used in signal processing applications. It is defined as:

where X = {x1, x2,…,xN} is a set of random variable, P(xi) is a probability of random vari-able xi and N is the length of a signal or possible outcomes. It is obvious that the higher signal entropy reflects more information content or more unpredictability of informa-tion content.

Table 37 presents the information based evaluation of speech quality. Furthermore, it also provides insights on the selection of the best basis sensing matrix.

The following observations are evident from Table 37.

1. CS based sensing matrices, including proposed as well as state-of-the-art sensing matrices has the higher entropy (H = 11.0) compared to classical wavelet compres-sion technique (H = 9.7573).

2. It is also evident that for the proposed sensing matrices the entropy of the recon-structed speech signal (H  =  11.0) is very close to the original signal entropy (H = 10.2888).

3. Furthermore, we have computed the entropy of sensing matrices which shows that state-of-the-art random matrices like Gaussian, Uniform, Toeplitz, Circulant attains higher entropy due to its randomness, followed by deterministic DCT matrix.

(22)H(X) = −

N∑

i=1

P(xi) logP(xi)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.5

1

1.5

2

2.5

3

Compression Ratio (CR)

Sign

al-to

-Noi

se R

atio

(SN

R) (

db)

Random Gaussian MatrixRandom Uniform MatrixRandom Toeplitz MatrixRandom Circulant MatrixRandom Hadamard MatrixDeterministic DCT MatrixSparse Binary MatrixProposed sym9 Matrix

Fig. 19 Effect of compression ratio on signal-to-noise ratio for the proposed sym9 matrix and state-of-the-art random and deterministic sensing matrices

Page 53 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

4. The proposed sensing matrices such as the Battle (for Battle5, H = 4.0745) and the Symlets wavelet families (for sym9 and sym10, H = 1.7689 and H = 1.9047, respec-tively) shows the higher entropy compared to the sparse binary (H = 0.0659) and the random Hadamard sensing matrices (H = 1).

Table 35 Subjective evaluation of speech quality using Mean Opinion Score (MOS) test

Sr.no.

Sensingmatrix

Listeners MOSScore

Male listener Female listener

1 2 3 4 5 6 7 8 9 10

1. db1 4 3 3 3 3 3 3 3 4 3 3.2

2. db2 5 5 4 4 4 4 4 4 3 3 4.0

3. db3 4 4 4 3 4 4 3 4 4 3 3.7

4. db4 4 3 3 5 4 4 5 3 3 4 3.8

5. db5 3 3 3 4 4 4 4 4 3 3 3.5

6. db6 4 4 4 4 4 4 3 3 4 4 3.8

7. db7 4 5 3 3 4 4 4 3 4 4 3.8

8. db8 4 4 4 4 4 4 3 4 4 4 3.9

9. db9 3 5 3 3 5 4 4 4 3 4 3.8

10. db10 3 3 4 4 4 4 4 3 4 3 3.6

11. coif1 4 5 4 5 3 3 4 3 4 4 3.9

12. coif2 4 3 5 4 3 3 4 3 4 4 3.7

13. coif3 3 3 3 3 3 3 4 4 3 4 3.3

14. coif4 4 5 4 4 3 3 4 4 3 4 3.8

15. coif5 4 5 3 3 3 4 5 4 5 4 4.0

16. sym4 5 4 4 4 3 3 5 3 3 4 3.8

17. sym5 5 4 4 4 4 3 4 3 3 4 3.8

18. sym6 5 5 3 5 4 4 4 3 4 4 4.1

19. sym7 4 3 4 3 4 4 5 4 3 4 3.8

20. sym8 4 4 4 4 4 4 5 4 4 4 4.1

21. sym9 5 5 5 5 5 4 5 4 3 3 4.4

22. sym10 4 4 4 4 5 4 4 4 4 4 4.1

23. Battle1 5 5 5 3 5 3 4 3 4 4 4.1

24. Battle3 4 4 3 3 5 4 3 4 4 4 4.1

25. Battle5 3 4 4 4 4 4 3 4 4 4 3.8

26. Beylkin 3 3 4 3 3 3 3 3 4 4 3.3

27. Vaidynathan 4 3 4 3 4 3 4 3 3 4 3.5

28. Sparse Binary 4 3 3 3 4 4 5 3 3 3 3.5

29. DCT matrix 4 5 4 4 5 4 4 4 4 4 4.2

30. Random Gaussian 4 4 3 4 4 4 3 4 4 4 3.8

31. Randomuniform

3 4 4 3 4 3 4 3 4 4 3.6

32. RandomToeplitz

4 4 3 3 4 3 4 3 4 4 3.6

33. RandomCirculant

4 4 4 3 4 4 4 4 4 4 3.9

34. RandomHadamard

4 5 4 4 4 3 4 4 4 4 4.0

35. Wavelet compression 3 4 3 3 5 5 5 4 3 4 3.9

Page 54 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048

Spectrographic analysis

The spectrograms are used to visually investigate the joint time–frequency properties of speech signals with intensity or color representing the relative energy of contributing frequencies and it plays an important role in decoding the underlying linguistic massage.

Table 36 Objective evaluation of speech quality using measures such as Perceptual Evalu‑ation of  Speech Quality (PESQ), Log‑Likelihood Ratio (LLR) and Weighted Spectral Slope (WSS)

Sr.no.

Different sensingmatrices

Speech distortion

Background distortion

Overall quality

LLR WSS PESQMOSScore

1. db1 3.4959 2.3958 2.8940 0.618194 45.751366 2.4060

2. db2 3.3878 2.3902 2.8228 0.735509 40.165614 2.3436

3. db3 3.1887 2.2231 2.6572 0.791790 50.476578 2.2632

4. db4 3.5157 2.4790 2.9531 0.701403 37.954092 2.4644

5. db5 3.7339 2.5471 3.0838 0.535443 34.465134 2.4909

6. db6 3.4648 2.3444 2.8207 0.612405 40.395098 2.2646

7. db7 3.4839 2.3110 2.8473 0.619242 39.438455 2.2937

8. db8 3.5478 2.3337 2.8925 0.560800 41.502149 2.3307

9. db9 3.7200 2.5313 3.0680 0.521539 37.389013 2.4879

10. db10 3.7502 2.5574 3.0842 0.508764 34.772191 2.4771

11. coif1 3.4799 2.2788 2.8831 0.646498 42.964582 2.3862

12. coif2 3.5241 2.3374 2.9602 0.705475 36.450763 2.4628

13. coif3 3.8500 2.5787 3.2063 0.527448 29.371499 2.5938

14. coif4 3.7495 2.5243 3.1167 0.551742 34.069359 2.5387

15. coif5 3.7443 2.5377 3.1099 0.551411 34.169205 2.5310

16. sym4 3.2751 2.1626 2.7240 0.658010 63.799345 2.3770

17. sym5 3.7144 2.4995 3.0894 0.550669 38.139942 2.5395

18. sym6 3.6323 2.4225 3.0189 0.603048 36.964962 2.4751

19. sym7 3.8098 2.5829 3.1993 0.569036 31.505778 2.6300

20. sym8 3.8135 2.5483 3.1875 0.548968 31.641938 2.6039

21. sym9 3.8163 2.5878 3.1843 0.534342 32.757831 2.6003

22. sym10 3.8335 2.6147 3.1985 0.536466 30.625240 2.6006

23. Battle1 3.4479 2.3080 2.8621 0.664658 44.171851 2.3821

24. Battle3 3.7297 2.4548 3.0899 0.532391 37.307250 2.5212

25. Battle5 3.6017 2.4841 2.9674 0.578535 39.038937 2.4135

26. Beylkin 3.6053 2.4451 2.9289 0.546456 35.907963 2.3180

27. Vaidynathan 3.8044 2.5202 3.1186 0.461501 35.343833 2.4948

28. Sparse Binary 3.6393 2.7576 3.0467 0.666517 29.706413 2.4868

29. DCT matrix 3.7628 2.5854 3.1092 0.518874 34.672901 2.5138

30. Random Gaussian 2.9737 2.6990 2.6855 1.276865 30.019462 2.4291

31. Randomuniform

3.3255 2.7452 2.8794 0.966803 28.298916 2.4577

32. RandomToeplitz

2.6847 2.6565 2.5249 1.520660 33.026663 2.4108

33. RandomCirculant

3.8147 2.7854 3.1549 0.529008 28.229330 2.5210

34. RandomHadamard

3.8147 2.7854 3.1549 0.529008 28.229330 2.4934

35. Wavelet compres-sion

3.6017 2.4841 2.9674 0.578535 39.038937 2.4135

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Figure 20 shows the spectrographic analysis of the original and the reconstructed speech signal for the proposed sym9 wavelet based sensing matrix (for CR = 0.5). Figure 20a shows the spectrogram of the original input speech signal and Fig. 20b shows the spec-trogram of the reconstructed speech signal.

Thus, the spectrographic analysis from Fig. 20 shows that the time–frequency charac-teristic of the reconstructed spectrogram is a very close to the original speech spectro-gram, preserving most of the signal energy. Moreover, the red color shows energy at the

Table 37 Information based evaluation of  speech quality and  selection of  the best basis sensing matrices

Sr. no. Different sensing matrices Entropy of original speech signal

Entropy of  reconstructed speech signal

Entropy of  sensing matrix

1. Daubechieswavelet family

db1 10.2888 11.0000 0.1191

2. db2 10.2888 11.0000 0.4663

3. db3 10.2888 11.0000 0.6966

4. db4 10.2888 11.0000 0.9066

5. db5 10.2888 11.0000 1.0980

6. db6 10.2888 11.0000 1.2699

7. db7 10.2888 11.0000 1.4416

8. db8 10.2888 11.0000 1.6132

9. db9 10.2888 11.0000 1.7689

10. db10 10.2888 11.0000 1.9047

11. Coiflet wavelet family

coif1 10.2888 11.0000 0.6966

12. coif2 10.2888 11.0000 1.2699

13. coif3 10.2888 11.0000 1.7689

14. coif4 10.2888 11.0000 2.1759

15. coif5 10.2888 11.0000 2.5818

16. Symmlet wavelet family

sym4 10.2888 11.0000 0.9066

17. sym5 10.2888 11.0000 1.0980

18. sym6 10.2888 11.0000 1.2699

19. sym7 10.2888 11.0000 1.4416

20. sym8 10.2888 11.0000 1.6132

21. sym9 10.2888 11.0000 1.7689

22. sym10 10.2888 11.0000 1.9047

23. Battle wavelet family

Battle1 10.2888 11.0000 2.0789

24. Battle3 10.2888 11.0000 3.1632

25. Battle5 10.2888 11.0000 4.0745

26. Other wavelet families

Beylkin 10.2888 11.0000 1.7689

27. Vaidynathan 10.2888 11.0000 2.1759

28. Random sensing matrices

Random Gaussian 10.2888 11.0000 21.0000

29. Random uniform 10.2888 11.0000 21.0000

30. Random Toeplitz 10.2888 11.0000 20.7505

31. Random Circulant 10.2888 11.0000 11

32. Random Hadamard 10.2888 11.0000 1

33. Deterministic sensing matrices

DCT matrix 10.2888 11.0000 19.1415

34. Sparse Binary 10.2888 11.0000 0.0659

35. Classical approach Wavelet compression

10.2888 9.7573 –

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highest frequency followed by the yellow, blue respectively, and the white area shows the absence of frequency components.

Furthermore, Fig. 21 shows the original and the reconstructed speech signal with the DCT basis for CR = 0.5 (N = 2048 and m = 1024). It can be observed that the original speech signal is successfully reconstructed using the proposed sym9 wavelet based sens-ing matrix.

ConclusionsIn this study, an attempt was made to investigate the DWT based sensing matrices for the speech signal compression. This study presents the performance comparison of the different DWT based sensing matrices such as the: Daubechies, Coiflets, Symlets, Bat-tle, Beylkin and Vaidyanathan wavelet families. Further study presents the performance analysis of the proposed DWT based sensing matrices with state-of-the-art random and deterministic sensing matrices. The speech quality is evaluated using subjective and objective measures. The subjective evaluation of speech quality is performed by mean

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 140

60

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Fig. 20 Spectrographic analysis of original and reconstructed speech signal for the proposed sym9 wavelet based sensing matrix (For CR = 0.5). a Spectrogram of original speech signal and b spectrogram of recon-structed speech signal

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opinion sore (MOS). Moreover, the objective speech quality is evaluated using the PESQ and other measures such as the log-likelihood ratio (LLR) and weighted spectral slope (WSS). Besides, an attempt was made to evaluate the speech quality using the informa-tion based measure such as Shannon entropy. In addition, efforts are made to present an insight on the selection of the best basis sensing matrix using the information based measure.

The following major conclusions are drawn based on the investigation:

• Overall, the db10 wavelet based sensing matrix shows the good balance between sig-nal reconstruction error and signal reconstruction time compared to other Daube-chies wavelet based sensing matrices. Moreover, the db9 also shows close perfor-mance to the db10 and may be the second best choice.

• The coif5 wavelet based sensing matrix shows the good performance, since it requires less reconstruction time, minimum relative error and the high SNR com-pared to other Coiflets wavelet based sensing matrices. In addition, the coif4 may be the second choice of sensing matrix.

• Overall, the sym9 wavelet sensing matrix demonstrates the less reconstruction time and the less relative error, and thus exhibits the good performance compared to other Symlets wavelet based sensing matrices. Moreover, the sym10 may be the sec-ond choice of sensing matrix followed by the sym9.

• The Beylkin wavelet sensing matrix demonstrates the less reconstruction time and relative error, and thus exhibits the good performance compared to the Battle and the Vaidyanathan wavelet based sensing matrices. However, the Battle5 shows a close performance and may be the second best choice of sensing matrix.

• When compared for the best of the DWT sensing matrix, the sym9 wavelet based sensing matrix shows the superior performance compared to the db10, coif5 and Beylkin wavelet based sensing matrices, in the views of signal reconstruction time and relative error. Furthermore, the db10 may be the second best choice of sensing matrix.

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Original Speech signalReconstructed Speech signal

Fig. 21 Original and reconstructed speech signal for proposed sym9 wavelet based sensing matrix (for CR = 0.5)

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• Finally, it is revealed that the proposed sym9 wavelet based sensing matrix exhibits the better performance compared to state-of-the-art random and deterministic sens-ing matrices in terms of signal reconstruction time and reconstruction error.

• Overall, the Symlets wavelet family achieves good MOS scores compared to other proposed as well as state-of-the-art sensing matrices.

• The highest MOS score of 4.4 is achieved by the sym9 wavelet family followed by the sym6, sym8, sym10, Battle1, Battle3 (MOS = 4.1) and followed by the db2, coif5 (MOS = 4.0) respectively. Thus, these MOS scores can be considered as an accept-able score for speech quality.

• In terms of the PESQ measure, the sym9 and the sym10 wavelet family based sensing matrices exhibits the higher PESQ scores i.e. PESQ =  2.6003 (sym9) and PESQ =  2.6006 (sym10) respectively; signifying the good/fair speech quality com-pared to other proposed and state-of-the art sensing matrices.

• The sym9 and the sym10 wavelet family based sensing matrices exhibits the lower values of Log-Likelihood Ratio (LLR) and Weighted Spectral Slope (WSS) metrics indicating the good speech quality, and are the close comparable with state-of-the art sensing matrices.

• In views of information based evaluation, CS based sensing matrices, includ-ing the proposed DWT based as well as state-of-the-art sensing matrices, has the higher entropy (H = 11.0) compared to the classical wavelet compression technique (H = 9.7573).

• The proposed sensing matrices such as the Battle (For the Battle5, H  =  4.0745) and the Symlets wavelet families (For the sym9 and the sym10, H =  1.7689 and H =  1.9047 respectively) shows the higher entropy compared to the sparse binary (H = 0.0659) and the random Hadamard sensing matrices (H = 1).

• Finally, the DWT based sensing matrices exhibits the good promise for speech signal compression.

Thus, this study shows the effectiveness of the DWT based sensing matrices for speech signal processing applications. The scope of this study can be further expanded by inves-tigating the use of the DWT based sensing matrices in other application areas such as music signal processing, under water acoustics and the biomedical signal processing such as the ECG and EEG analysis.

AbbreviationsDWT: discrete wavelet transform; CS: compressed sensing; CR: compression ratio; RMSE: root mean square error; SNR: signal to noise ratio; MOS: mean opinion score; PESQ: perceptual evaluation of speech quality.

Authors’ contributionsYVP have made substantial contributions to design and development of DWT based sensing matrices and their applica-tion to speech signal processing. YVP formulated the problem with objective, performed the experimentation and wrote the paper. SLN has been involved in the critical testing and analysis of proposed DWT based sensing matrices, manuscript preparation and proof reading. Both authors read and approved the final manuscript.

AcknowledgementsThe authors wish to acknowledge the Dr. Babasaheb Ambedkar Technological University, Lonere, Maharashtra, India for providing infrastructure for this research work. The authors would like to thank the anonymous reviewers for their constructive comments and questions which greatly improved the quality of article.

Competing interestsThe authors declare that they have no competing interests.

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Availability of data and materialsAll datasets on which the conclusions of the manuscript are rely and the data supporting their findings are presented in the main paper.

FundingThe authors declare that they have no funding provided for the research reported in this paper.

Received: 25 June 2016 Accepted: 25 November 2016

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