A Realization of Adaptive Compressive Sensing System

K. Sever*, T. Vlašić** and D. Seršić** * Ericsson Nikola Tesla d.d. Research and Development Centre, Zagreb, Croatia

** University of Zagreb Faculty of Electrical Engineering and Computing, Zagreb, Croatia E-mails: karlo.sever@ericsson.com, {tin.vlasic, damir.sersic}@fer.hr

Abstract - There are numerous papers analyzing the theoretical background of compressive sensing (CS), but a few practical implementations exist due to the realization complexity. In this paper, a realization of a system for CS of analog one-dimensional signals in a shift-invariant subspace is proposed. We use statistical compressive sensing that allows efficient sampling of signals that follow some statistical distribution. In fact, principal component analysis is used to obtain a small number of principal components in which the majority of signal’s energy is compacted. The signal is demodulated using the obtained principal components, which provides optimal measurement procedure in the mean-square error sense. Orthonormality allows for a linear reconstruction which is perfectly suited for a real-time embedded system implementation. Due to hardware realization constraints, the adaptive demodulating signal is quantized. We provide qualitative and quantitative validations based on simulations of the proposed CS system realization. Furthermore, we validate performance of the system when additive white Gaussian noise is present in the measurements.

Keywords – sparse representation; statistical compressive sensing; inverse problems; signal reconstruction

I. INTRODUCTION The conventional bridge between continuous and

discrete-time signals is the Nyquist-Shannon sampling theorem [1], [2], which states that the sampling frequency must be at least twice the maximum frequency present in a signal. There are several problems associated with the theorem, that provide motivation to seek for a different sampling technique such as the fact that real world signals are seldomly bandlimited and often better represented in other bases than the Fourier basis [3]. Furthermore, the Nyquist-Shannon approach results in a high sampling rate whenever a signal has a wide bandwidth, even if the information rate is low. However, the Nyquist-Shannon theorem can be seen as the projection of the signal onto the bandlimited subspace and only the signals included within the reconstruction space can be reconstructed perfectly [4]. This formulation of the classical theorem leads to more realistic sampling and reconstruction that are obtained by

filtering operations in a more general class of shift-invariant (SI) spaces [3], [4]. Alternatively, information content of a signal can be leveraged to create a more efficient sampling and reconstruction technique that overcomes the limitations of the classical sampling theorem. Signal sparsity in a transform domain is a type of information description that inspired the development of compressive sensing (CS) [5], [6], a sampling and reconstruction technique that aims to go beyond the Nyquist limit.

Compressive sensing is a non-adaptive signal acquisition process that aims to completely recover the input signal from an incomplete set of linear measurements. Consider a discrete signal 𝒙 ∈ ℝ𝑁 and a transform basis 𝚿 in which we can represent the signal as the inner product of the transform basis and a vector 𝒔 ∈ ℝ𝑁:

𝒙 = 𝚿𝒔. (1)

Signal 𝒙 that has a sparse representation in the transform basis 𝚿 is represented by 𝒔 that has only K nonzero entries, where 𝐾 ≪ 𝑁. Signal acquisition in CS is a process of measuring the signal through linear projections by a measurement matrix 𝚽 ∈ ℝ𝑀×𝑁 , where 𝑀 < 𝑁. Signal measurements 𝒚 ∈ ℝ𝑀 can be written as:

𝒚 = 𝚽𝒙 + 𝒘 = 𝚽𝚿𝒔 + 𝒘, (2)

where 𝒘 is a disturbance term usually assumed to be small and independent of 𝒔 . Being that the number of measurements 𝑀 is less than the length of the signal, the problem in (2) is ill-posed and 𝒔 is recovered by solving an 𝑙1 optimization problem.

Compressive sensing framework has focus primarily set on discrete-time signals. However, there were several frameworks developed that leverage CS to the analog domain. In [7], Tropp et al. introduce the random demodulator (RD), an analog-to-information conversion (AIC) system for acquiring of sparse bandlimited signals. The signal is demodulated by a high-rate pseudo-random sequence of ±1s, known as chipping sequence; integrated, and sampled at a low rate. Another efficient hardware implementation of an AIC system front-end, named modulated wideband converter (MWC), is introduced in [8]. The MWC consists of 𝑝 channels, where 2𝐾 ≤ 𝑝 ≪𝑁, that mix the signal with a chipping sequence prior to low-pass filtering and following by low-rate sampling. There are a few other realizations of the CS front-end that are basically modifications of the RD and the MWC, and

This research was supported in part by the Croatian Science Foundation under the project IP-2019-04-6703, in part by the European Regional Development Fund under the grant KK.01.1.1.01.0009 (DATACROSS), and in part by Ericsson Nikola Tesla d.d. and University of Zagreb Faculty of Electrical Engineering and Computing under the project ILTERA.

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are summarized in [9]. A method for low rate sampling of analog signals in a union of shift-invariant subspaces is proposed in [10]. The sparsity is modeled by treating the case in which only 𝑘 out of 𝑁 signal generators are active. The signal is filtered by a bank of only 𝑝 filters, where 2𝑘 ≤ 𝑝 ≪ 𝑁, and is uniformly sampled at the sub-Nyquist rate before being recovered. In [11], [12], [13], the authors introduce and develop one-bit CS that extends the scope of sparse recovery by showing that sparse signals can be accurately reconstructed when their linear measurements are subject to the extreme quantization of binary samples (only the sign of each measurement is maintained). In all of the mentioned CS systems the reconstruction is based on solving an 𝑙1 minimization problem.

In this paper, a realization of CS system for acquisition and reconstruction of signals in a shift-invariant subspace that follow some statistical distribution is proposed. The front-end consists of parallel channels that mix, integrate, and sample an input signal. The signal is mixed by functions that lie in an SI subspace characterized by an arbitrary sequence of coefficients. A known statistical distribution and subspace priors of the signal allow for reconstruction from a reduced set of measurements by statistical compressive sensing (SCS) [14], [15]. A small number of principal components obtained by the principal component analysis (PCA) on a training set of signals determine coefficients of the mixing functions. Thus, the measurements obtained by the proposed front-end are optimal in the mean-square error (MSE) sense. Statistical CS allows for a linear recovery of the coefficients, which is equivalent to knowing the orthogonal projection of the signal onto the mixing functions’ subspace.

Experimental results of a system realization are presented in the paper. Due to the hardware realization constraints and contrarily to quantization of measurements in one-bit CS, we quantize piecewise constant mixing functions. Qualitative and quantitative validations of the experimental results are provided based on quantization resolution. We, as well, validate the performance of the system when additive white Gaussian noise (AWGN) is present in the measurements.

This paper is organized as follows. In Section II we give a short review of sampling in SI spaces. In Section III the SCS is explained. We propose a SCS measurement model for sampling and reconstruction of signals in B0-spline space in Section IV. Finally, we provide experimental results in Section V and conclude the paper in section VI.

II. SAMPLING AND RECONSTRUCTION IN SI SPACES Let us assume that a signal 𝑥(𝑡) lies in an appropriate

SI subspace 𝒜 of 𝐿2 which can be expressed as a linear

combination of the integer shifts of a generator [3]. Any 𝑥(𝑡) ∈ 𝒜 is of the form [4]

𝑥(𝑡)  = ∑ 𝑑[𝑛]𝑎(𝑡 − 𝑛𝑇)

𝑛∈𝑍

, (3)

where 𝑎(𝑡) is the generator of 𝒜. Any signal 𝑥(𝑡) is uniquely characterized by a sequence of coefficients 𝑑[𝑛] which are not necessarily samples. For bandlimited signals, 𝑎(𝑡) corresponds to the sinc function. However, one can define a much broader class of generators that are easier to handle.

In order to guarantee a unique signal representation in 𝒜 and a stable sampling framework, the generator 𝑎(𝑡) is typically chosen to form a Riesz basis or a frame [16]. A countable set of vectors {𝑎(𝑡 − 𝑛𝑇)} is a Riesz basis for 𝐿2 if it is complete and there exist constants α > 0 and β < +∞ such that [4]:

α‖𝒅‖𝑙22 ≤ ‖∑ 𝑑[𝑛]𝑎(𝑡 − 𝑛𝑇)

𝑛∈𝑍

‖

2

≤ β‖𝒅‖𝑙22 , (4)

where ‖𝒅‖𝑙22 = ∑ |𝑑[𝑛]|2𝑛 is the squared 𝑙2 -norm (or

energy) of 𝑑[𝑛]. This ensures that a small modification of coefficients 𝑑[𝑛] results in a small distortion of the signal.

Fig. 1. shows a scheme of sampling and reconstruction of signals in an SI subspace. Generally, the signal is prefiltered prior to uniform sampling with a prefilter 𝑠(−𝑡) [3], [4], [10]. The samples 𝑐[𝑛] can be expressed as

𝑐[𝑛] = ∫ 𝑥(𝑡)𝑠(𝑡 − 𝑛𝑇)∞

−∞

≜ ⟨𝑥(𝑡), 𝑠(𝑡 − 𝑛𝑇)⟩, (5)

where ⟨⋅,⋅⟩ denotes the 𝐿2 inner product. Let us denote the sampled cross-correlation sequence by

𝑟𝑠𝑎[𝑛] = ⟨𝑠(𝑡 − 𝑛𝑇), 𝑎(𝑡)⟩ (6)

and its discrete time Fourier transform (DTFT) pair by 𝜑𝑆𝐴. The signal 𝑥(𝑡) ∈ 𝒜 can be perfectly reconstructed from samples 𝑐[𝑛] if [3]

|𝜑𝑆𝐴(𝑒𝑗ω)| > α > 0, (7)

where the DTFT of 𝑟𝑠𝑎[𝑛] is given by

𝜑𝑆𝐴(𝑒𝑗ω) =1

𝑇∑𝑆∗ (

ω

𝑇−

2π

𝑇𝑘) 𝐴 (

𝜔

𝑇−

2𝜋

𝑇𝑘) .

𝑘∈𝑍

(8)

Here, ( ∙ )∗ denotes the complex conjugate, and 𝐴(ω) and 𝑆(ω) are the continuous-time Fourier transform of 𝑎(𝑡) and 𝑠(𝑡), respectively. Coefficients 𝑑[𝑛] are recovered by filtering the samples 𝑐[𝑛] using a discrete-time filter [3], [4], [17]:

Fig. 1. Sampling and reconstruction of a signal in shift-invariant spaces.

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𝐻(𝑒𝑗ω) =1

𝜑𝑆𝐴(𝑒𝑗ω), (9)

since:

𝐶(𝑒𝑗ω) = 𝐷(𝑒𝑗ω)𝜑𝑆𝐴(𝑒𝑗ω). (10)

In the case when basis functions 𝑎(𝑡) and 𝑏(𝑡) are orthonormal, the cross-correlation sequence 𝑟𝑠𝑎[𝑛] =δ[𝑛]. Thus, the filter 𝐻(𝑒𝑗ω) = 1 and the samples 𝑐[𝑛] do not have to be filtered in the discrete time before the reconstruction. Obtained coefficients 𝑑[𝑛] are modulated by an impulse train ∑ δ(𝑡 − 𝑛𝑇)𝑛∈𝑍 prior to filtering with the corresponding analog filter 𝑎(𝑡), for the purpose of reconstructing 𝑥(𝑡). Efficient implementation of (9) is particularly possible in spline spaces, when generator 𝑎(𝑡) and prefilter 𝑠(𝑡) are both B-splines [4], [18], [19].

III. STATISTICAL COMPRESSIVE SENSING Statistical compressive sensing uses statistical models

in order to develop an efficient measurement strategy for signals that follow some statistical distribution [14], [20]. The statistical CS framework introduces an optimal reconstruction algorithm implemented through linear filtering that is computationally less complex than the reconstruction algorithms based on optimization. This paper employs the SCS framework by extracting statistical features from collections of signals in B0-spline subspace. The coefficients follow a Gaussian distribution, thus enabling development of an optimal measurement technique that leads to linear signal reconstruction.

Signals are divided into 𝐾 intervals of length 𝑁 characterized by 𝑁 expansion coefficients. Each of the signal intervals {𝒙𝑘}𝑘=1

𝐾 ∈ ℝ𝑁 are assumed to follow a Gaussian distribution 𝒩(𝝁, 𝚺), where 𝝁 and 𝚺 denote the mean vector and covariance matrix, respectively. In order to induce sparsity, the PCA is applied to the Gaussian model to obtain a small number of principal components in which the majority of signal’s energy is compacted. Energy compaction property of the PCA can be useful if observed as a special case of structured sparsity [20]. For a given set of signals, mean vector 𝝁 and covariance matrix 𝚺 are calculated. Applying the eigenvalue decomposition on the covariance matrix results in obtaining the principal components in 𝑽 and a diagonal matrix 𝚲 whose diagonal elements are the sorted eigenvalues of the covariance matrix:

𝚺 = 𝑽𝚲𝑽𝑇 . (11)

A measurement matrix 𝚽 ∈ ℝ𝑀×𝑁, where 𝑀 < 𝑁, consists of 𝑀 principal components corresponding to the largest eigenvalues:

𝚽 = [𝐕𝑁−𝑀+1:𝑁𝑇 ]. (12)

Under the assumption that the variance reflects the informational content of the signal, the obtained SCS measurement matrix 𝚽 optimally describes the observed signal in terms of the MSE [20]. Rows in 𝚽 can be denoted as a set of vectors {𝜙𝑖}𝑖=1

𝑀 . Compressive sensing measurement is a linear process that computes inner products between the signal and a set of measurement vectors, 𝑦𝑖[𝑘] = ⟨𝒙𝑘, 𝜙𝑖⟩. In statistical CS, measurement

procedure can be seen as the projection of the signal onto the principal components. The signal 𝒙𝑘 is reconstructed from the measurements 𝒚[𝑘] = {𝑦𝑖[𝑘]}𝑖=1

𝑀 by a decoder that minimizes the MSE, which is obtained by a linear maximum a posteriori (MAP) estimator [14], [20], [21]:

𝒙𝑘 = (𝚽𝑻𝚽)−𝟏𝚽𝑇𝒚[𝑘]. (13)

Instead of computationally complex, time consuming solving of the 𝑙1 optimization problem in conventional CS, Gaussian SCS has an optimal decoder (13) calculated via a closed-form linear filtering for any 𝚽.

IV. A SCS MEASUREMENT MODEL IN 𝐵0-SPLINE SPACE

A single front-end channel of the proposed system consists of a mixer, an ideal integrator, and an ADC. Typically, the mixing (or demodulating) signal is a piecewise constant function with pseudo-random amplitudes. Such demodulating functions are frequent in the conventional CS frameworks [7], [8], [22]; in which measurement precedes the 𝑙1 optimization. This paper focuses on mixing of the signal by a deterministic piecewise constant function, where both the signal and mixing function lie in a B0-spline subspace. We define a generator in the B0-spline subspace as a constant causal signal with a duration 𝑇. The mixing function in the 𝑖-th channel is characterized by the 𝑖 -th row of the SCS measurement matrix 𝚽 in (12) and can be written as:

𝑧𝑖(𝑡) = ∑ 𝜙𝑖[𝑚]∑ 𝑠(𝑡 − 𝑚𝑇 − 𝑘𝑁𝑇)

𝑘∈𝑍

𝑁

𝑚=1

, (14)

where 𝑠(𝑡) is a generator of the B0-spline subspace. The mixing function is periodic with period 𝑁𝑇, and values 𝜙𝑖[𝑚] are cyclically repeated. The mixed signal passes through an ideal integrator before being sampled with period 𝜏 = 𝑁𝑇. Note that the sampling rate is 𝑁 times lower than the rate of the subspace basis functions. Fig. 2. illustrates the acquisition scheme.

A single measurement in the 𝑖 -th channel and 𝑘 -th interval is given by:

𝑦𝑖[𝑘] = ∫ 𝑧𝑖(𝑡)𝑥(𝑡)𝑑𝑡.

(𝑘+1)𝜏

𝑘𝜏

(15)

By applying (3) and (14), (15) can be expanded to:

𝑦𝑖[𝑘] = ∑ 𝜙𝑖[𝑚]

𝑁

𝑚=1

∑ 𝑑[𝑛]

𝑛∈𝑍

⋅ ∫ 𝑠(𝑡 − 𝑚𝑇 − 𝑘𝜏)𝑎(𝑡 − 𝑛𝑇)𝑑𝑡

(𝑘+1)𝜏

𝑘𝜏

= ∑ 𝜙𝑖[𝑚]

𝑁

𝑚=1

∑ 𝑑[𝑛 + 𝑘𝑁]

𝑁

𝑛=1

⋅ ∫ 𝑠(𝑡 − 𝑚𝑇 − 𝑘𝜏)𝑎(𝑡 − 𝑛𝑇 − 𝑘𝜏)𝑑𝑡.

(𝑘+1)𝜏

𝑘𝜏

(16)

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The integration in (16) can be arranged in a matrix 𝑹. By introducing 𝑹, (16) becomes

𝑦𝑖[𝑘] = ∑ 𝜙𝑖[𝑚] ∑ 𝑑[𝑛 + 𝑘𝑁]𝑅𝑚,𝑛

𝑁

𝑛=1

𝑁

𝑚=1

. (17)

Since the basis functions of the B0-spline subspace are orthonormal, the integral in (16) is equal to δ𝑚,𝑛 . It follows that 𝑹 is an 𝑁 × 𝑁 identity matrix. A single CS measurement in the 𝑖-th channel is then given by:

𝑦𝑖[𝑘] = ∑ 𝜙𝑖[𝑚]𝑑𝑘[𝑚]

𝑁

𝑚=1

, (18)

where 𝑑𝑘[𝑚] = 𝑑[𝑚 + 𝑘𝑁].

After finding the expression for a single measurement, we can write the measurement procedure of the whole system in a matrix form:

[ 𝜙1[1] ⋯ 𝜙1[𝑁]

⋮ ⋮ ⋮𝜙𝑖[1] ⋯ 𝜙𝑖[𝑁]

⋮ ⋮ ⋮𝜙𝑀[1] ⋯ 𝜙𝑀[𝑁]]

[𝑑𝑘[1]

⋮𝑑𝑘[𝑁]

] = [𝑦1[𝑘]

⋮𝑦𝑀[𝑘]

] (19)

or

𝚽𝒅𝑘 = 𝒚[𝑘]. (20)

Once we obtained the measurement vector 𝒚[𝑘], the optimal linear decoder (13) is applied in order to recover the expansion coefficients 𝒅𝑘 . The recovered 𝒅𝑘 are projections of signal 𝑥(𝑡) onto the SI subspace spanned by B0-spline basis functions {𝑠(𝑡 − 𝑚𝑇)}. An optimal measurement matrix for the coefficients 𝒅𝑘 is then obtained by applying the PCA to a training set of expansion coefficients which follow a Gaussian distribution. That way the decoder for coefficients 𝒅𝑘 is optimal in the MSE sense.

The signal reconstruction procedure is illustrated in Fig. 2. Since B0-spline basis functions are orthonormal, there is no need for discrete-time filtering when compared to the scheme in Fig. 1., i.e. ℎ[𝑛] = 𝛿[𝑛]. The obtained

coefficients 𝑑[𝑛] are modulated by an impulse train with period 𝑇 and filtered by the B0-spline analog filter 𝑎(𝑡). Due to the properties of SCS, the reconstructed signal has the information rate same as the input signal, despite being sampled at a far lower rate.

V. EXPERIMENTAL RESULTS The proposed CS system was simulated by acquiring

and reconstructing signals from the multi-speaker speech Sinhala TTS dataset [23]. We assume that the samples represent coefficients of a signal that lies in the B0-spline subspace. Next, the coefficients are assumed to follow a Gaussian distribution. In the preprocessing phase, the signal is centered with respect to the mean so that the expansion coefficients 𝑑[𝑛] follow Gaussian distribution 𝒩(0, 𝚺). The PCA is applied to the training dataset in order to obtain principal components, which construct a measurement matrix as described in Section III. Test signals are divided into intervals of length 𝑁 , i.e. the sampling rate in a single channel is 𝑁 times lower than the rate of the B0-spline basis functions. This leads to a system with the sampling rate of 𝑀/(𝑁𝑇), where 𝑀 represents the number of measurement channels in the front-end of the system. The compressive sensing measurement ratio is defined by 𝑟 = 𝑀/𝑁. We simulate an efficient low-cost front-end with a small number of measurement channels, from 𝑀 = 2 to 𝑀 = 12. Such a front-end can be designed as an extended circuit connected to an embedded system development board. Thus, we treat cases where test signals are divided into rather small intervals of length 𝑁 = 10 and 𝑁 = 20.

We quantize values of the measurement matrix entries to simulate a real-world implementation. The measurement matrix is quantized for three different cases where the number of bits used in quantization is 𝑛𝑞 ={1, 2, 4} , in order to provide 2𝑛𝑞 levels of a mixing function. For 𝑛𝑞 = 1, the mixing function is a chipping sequence of ±1𝑠, which is particularly practical to implement. Alternatively, when the number of quantization bits are 𝑛𝑞 > 1, the mixers can be realized by an off-the-shelf multiplying digital-to-analog converter, e.g. [24].

Fig. 2. Statistical compressive sensing of a signal in B0-spline space. The signal is multiplied by 𝑴 mixing functions generated by analog filters 𝒔(𝒕) and sequences {𝝓𝒊}𝒊=𝟏

𝑴 corresponding to the principal components. The sequences {𝝓𝒊}𝒊=𝟏𝑴 are modulated by an impulse train before being

filtered by 𝒔(𝒕).

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Validating the impact of quantization on the success of a reconstruction is done by using four different measurement matrices, specifically the ideal measurement matrix and three quantized matrices for a given number of bits 𝑛𝑞 = {1, 2, 4}. In the simulations, the measurement ratio is set to 𝑟 = {0.2, 0.3, 0.4, 0.5, 0.6}. Signal-to-noise ratio (SNR) is used to assess the quality of the linear reconstruction algorithm described in Section IV. The SNR is defined as the ratio of powers of the reconstructed signal and the reconstruction error. Table I. shows reconstruction results for different scenarios based on the measurement ratio and the quantization resolution. The left-hand side of Table I. shows calculated SNRs of reconstructions that include the effects of quantization errors without the presence of noise. Notice that the SNR values for 𝑛𝑞 = 4 are close to the ideal case. Thus, the proposed SCS system shows robustness to the quantization errors. Furthermore, the resolution dependency effect [25] is visible in the reconstruction results with respect to the number 𝑁 of coefficients being recovered in a single SCS reconstruction process. The reconstruction quality for the same measurement ratio gets better as 𝑁 increases. Note that the reconstruction is obtained by multiplying the pseudoinverse of the PCA basis (13) with the measurement vector. Even better results are achievable by applying the 𝑙1 optimization as in the conventional CS setting, but that way the computational complexity is much higher.

In order to simulate the system more realistically, we include additive noise in the simulations. We model the disturbance term in (2) by AWGN and add it to the measurement vector 𝒚[𝑘]. We add AWGN so that the SNR of the measurement vector is 20 dB. The right-hand side of Table I. shows calculated SNRs of reconstructions that include the effects of the quantization and additive noise. The results show that after introducing additive noise to our CS system, we can still expect reconstructions with similar SNRs. The system remains robust to the effect of quantization even in the presence of additive noise. The resolution dependency effect is visible for reconstructions from noisy measurements, as well. A small segment of the original and reconstructed signal is shown in Fig. 3.

VI. CONCLUSION We proposed a realization of a compressive sensing

system for acquisition of signals that follow a Gaussian distribution and lie in a B0-spline subspace. An input signal is mixed by functions that are characterized by the principal components of a Gaussian distribution, that way minimizing the mean-square error. The signal is reconstructed from measurements that are linear projections of the signal onto the mixing functions by linear filtering. The realization is perfectly suited for an implementation of the reconstruction on an embedded device. We provided extensive qualitative validations based on simulations of a realistic realization of the proposed system. The simulations showed that the proposed system can be efficiently realized with a small number of quantization bits and is robust to the additive noise.

REFERENCES [1] H. Nyquist, "Certain Topics in Telegraph Transmission Theory,"

Transactions of the American Institute of Electrical Engineers, vol. 47, no. 2, pp. 617-644, 1928.

[2] C. E. Shannon, "Communication in the Presence of Noise," Proc. IRE, vol. 37, no. 1, pp. 10-21, 1949.

[3] Y. C. Eldar and T. Michaeli, "Beyond Bandlimited Sampling," IEEE Signal Process. Mag. , vol. 26, no. 3, pp. 48-68, May 2009.

[4] M. Unser, "Sampling-50 years after Shannon," Proc. IEEE, vol. 88, no. 4, pp. 569-587, April 2000.

TABLE I. SIGNAL-TO-NOISE RATIOS OF RECONSTRUCTED SIGNALS IN DECIBELS

Measurements done without additive noise Measurements done in presence of additive noise N = 10 r = 0.2 r = 0.3 r = 0.4 r = 0.5 r = 0.6 N = 10 r = 0.2 r = 0.3 r = 0.4 r = 0.5 r = 0.6 𝒏𝒒 = 𝟏 0.69 3.94 7.05 7.79 9.82 𝒏𝒒 = 𝟏 0.67 3.87 6.87 7.57 9.44 𝒏𝒒 = 𝟐 2.61 5.50 7.80 8.79 9.87 𝒏𝒒 = 𝟐 2.57 5.36 7.53 8.48 9.42 𝒏𝒒 = 𝟒 2.47 7.44 10.47 12.47 14.49 𝒏𝒒 = 𝟒 2.43 7.22 10.04 11.81 13.45 ideal 2.48 7.61 10.77 12.93 15.14 ideal 2.43 7.38 10.28 12.15 13.88

N = 20 r = 0.2 r = 0.3 r = 0.4 r = 0.5 r = 0.6 N = 20 r = 0.2 r = 0.3 r = 0.4 r = 0.5 r = 0.6 𝒏𝒒 = 𝟏 3.90 6.98 8.24 9.51 10.36 𝒏𝒒 = 𝟏 3.83 6.79 7.98 9.12 9.88 𝒏𝒒 = 𝟐 5.47 8.44 7.85 8.65 10.16 𝒏𝒒 = 𝟐 5.34 8.15 7.64 8.39 9.78 𝒏𝒒 = 𝟒 6.94 11.67 13.83 15.56 17.15 𝒏𝒒 = 𝟒 6.75 11.05 12.90 14.23 15.31 ideal 6.98 11.84 14.15 16.03 17.65 ideal 6.79 11.19 13.11 14.52 15.57

Fig. 3. A segment of a signal from Sinhala TTS dataset represented by B0-spline basis functions and its reconstruction. A visualization for N=10, M=4, 𝒏𝒒=4, with additive noise and SNR=17.08 dB for this particular segment.

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[5] E. J. Candes, J. Romberg and T. Tao, "Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information," IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489-509, Feb. 2006.

[6] D. L. Donoho, "Compressed sensing," IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289-1306, April 2006.

[7] J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg and R. G. Baraniuk, "Beyond Nyquist: Efficient sampling of sparse bandlimited signals," IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 520-544, Jan. 2010.

[8] Y. C. Eldar and M. Mishali, "From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals," IEEE J. Sel. Topics Signal Process., vol. 4, no. 2, pp. 375-391, April 2010.

[9] M. Rani, S. B. Dhok and R. B. Deshmukh, "A Systematic Review of Compressive Sensing: Concepts, Implementations and Applications," IEEE Access, vol. 6, pp. 4875-4894, 2018.

[10] Y. C. Eldar, "Compressed Sensing of Analog Signals in Shift-Invariant Spaces," IEEE Trans. Signal Process., vol. 57, no. 8, pp. 2986-2997, Aug. 2009.

[11] P. T. Boufounos and R. G. Baraniuk, "1-Bit compressive sensing," in Proc. of the 42nd Annual Conference on Information Sciences and Systems (CISS), IEEE, 2008.

[12] L. Jacques, J. N. Laska, P. T. Boufounos and R. G. Baraniuk, "Robust 1-bit compressive sensing via stable embeddings of sparse vectors," IEEE Trans. Inform. Theory, vol. 59, pp. 2082-2102, 2013.

[13] K. Knudson, E. Saab and R. Ward, "One-bit compressive sensing with norm estimation," IEEE Trans. Inform. Theory, vol. 62, pp. 2748-2758, 2016.

[14] G. Yu and G. Sapiro, "Statistical Compressed Sensing of Gaussian Mixture Models," IEEE Trans. Signal Process., vol. 59, no. 12, pp. 5842-5858, Dec. 2011.

[15] J. M. Duarte-Carvajalino, G. Yu, L. Carin and G. Sapiro, "Task-Driven Adaptive Statistical Compressive Sensing of Gaussian Mixture Models," IEEE Trans. Signal Process., vol. 61, no. 3, pp. 585-600, Feb. 2013.

[16] I. Daubechies, "The wavelet transform, time-frequency localization and signal analysis," IEEE Trans. Inf. Theory, vol. 36, pp. 961-1005, Sept. 1990.

[17] M. Unser and A. Aldroubi, "A general sampling theory for nonideal acquisition devices," IEEE Trans. Signal Process., vol. 42, no. 11, pp. 2915-2925, Nov. 1994.

[18] M. Unser, "Splines: A perfect fit for signal and image processing," IEEE Signal Process. Mag., vol. 16, no. 6, pp. 22-38, Nov. 1999.

[19] M. Unser, A. Aldroubi and M. Eden, "B-spline signal processing. I. Theory," IEEE Trans. Signal Process., vol. 41, no. 2, pp. 821-833, Feb. 1993.

[20] I. Ralašić, A. Tafro and D. Seršić, "Statistical compressive sensing for efficient signal reconstruction and classification," in 2018 4th International Conference on Frontiers of Signal Processing (ICFSP), Poitiers, 2018.

[21] S. M. Kay, Fundamentals of Statistical Signal Processing, vol. I, Englewood Cliffs, NJ: Prentice-Hall, 1998.

[22] T. Vlašić, J. Ivanković, A. Tafro and D. Seršić, "Spline-Like Chebyshev Polynomial Representation for Compressed Sensing," in 2019 11th International Symposium on Image and Signal Processing and Analysis (ISPA), Dubrovnik, Croatia, 2019.

[23] Google, Inc., "Open Speech and Language Resources," [Online]. Available: http://openslr.org/52/. [Accessed 15 January 2020].

[24] Analog Devices, "AD5424," [Online]. Available: https://www.analog.com/en/products/ad5424. [Accessed 28 January 2020].

[25] B. Roman, A. Hansen and B. Adcock, "On asymptotic structure in compressed sensing," arXiv:1406.4178, 2014.

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