Digital Signal Processing 22 (2012) 463–470

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Digital Signal Processing

www.elsevier.com/locate/dsp

A novel optimization based method for separation of periodic signals

Tamás Kovács

Department of Informatics, Kecskemét College, Faculty of Technology, Kecskemét, Hungary

a r t i c l e i n f o a b s t r a c t

Article history:Available online 31 January 2012

Keywords:Separation of periodic signalsFIR filtersOptimization

In the present paper a new method is proposed for separating the individual periodic components of amixed signal. The method is capable to extract not only a harmonic but an anharmonic signal componentfrom the mixture. To achieve this, the component is extracted by an FIR narrowband filter, which canmodulate the output harmonic signal by an appropriate time-shift function. The search for this functionis based on the minimization of a functional, which is calculated as the sum of the unsigned differencesof the separated signal in a certain time window. The theoretical basis of this optimization method isthat the functional above has global minimum if the separation is complete. The introduced tests showthat the proposed method is more robust than the matrix algebraic separation (MAS) system in the caseof a slightly frequency-modulated test signal.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

In most of the signal analysis and processing problems sep-arating the main components of a multi-component signal is acrucial and often difficult part of the process. These componentsare mostly periodic and produced by different sources, which havetheir own dynamical characteristics. Usually these characteristicsof the sources are not known and the only available informationis given in the measured signal, which is a sum of the indepen-dent signal components (Blind Source Separation problems, seee.g. [1,2]). Because of this lack of information the general separa-tion methods do not make any assumption regarding the individualcomponents except that they are periodic and their lengths of pe-riod are different, which is generally true for the measured signalsin various areas.

Almost two decades ago Z. Mouyan et al. [3] proposed a matrixalgebraic separation (MAS) method to solve the problem. With thehelp of this method two periodic signals can be separated fromeach other exactly, provided that the lengths of period are co-primes. Later Santhanam and Maragos [4], with some additionalconditions, extended the MAS system to non-co-prime cases too.The MAS applies the least square solution of the derived matrixequation, however, the method is still very sensitive to the noise,especially to the fluctuations of the amplitude or length of period(i.e. frequency) modulations.

The empirical mode decomposition (EMD), which is known as apart of the Hilbert–Huang Transform (HHT) and invented by Huanget al. [5], is also considered to be a general separation algorithm.It does not assume any specific property of the individual compo-

E-mail address: kovacs.tamas@gamf.kefo.hu.

1051-2004/$ – see front matter © 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2011.12.002

nents and, moreover, it is very robust against the noise and anykind of modulation of the signal components. Yet its applicabilityis limited. Recently Rilling and Flandrin [6] investigated the theo-retical limitations of the EMD and they obtained that this methodin its present form can separate two harmonic signals only if

f2

f1< 0.67 and

A2

A1

(A2

A1

)2

< 1, (1.1)

where A1, A2, and f1, f2, are the amplitudes and the frequenciesof the two signals. In other words, if the frequencies are close toeach other (0.67 f1 < f2 < f1) or the lower frequency componenthas considerably higher amplitude than the higher one, then thepure EMD is useless for the separation task. Unfortunately, indus-trial or other measured signals often show both of these excludingproperties. This problem, though not in such an exact form, wasrealized earlier, thus many proposals were made to circumvent it.

Most of these improving methods applied some band filteringof the original signal before the EMD so as to get rid of the dis-turbing frequencies and focus on only the extracted band. H. Liet al. [7] and later Z.K. Peng et al. [8,9] used consecutive discretewavelet filter banks in order to divide the frequency spectrum intonon-overlapping bands and then the EMD was applied on the sep-arate bands. This raised the selectivity of these hybrid methods.Q. Pinle et al. [10] achieved further improvement by making use ofa translation invariance algorithm to suppress the artifacts causedby the wavelets. The most straightforward hybrid method was pro-posed recently by W.X. Yang [11]. He applied simply narrowbandFIR filtering to extract the mono-component parts of the signal.The dominant frequencies ( f i ) were obtained from the Fourierspectrum and then the signal was consecutively filtered by theband filters with band boundaries [ f i − 0.05 f s , f i + 0.05 f s] corre-sponding to the dominant frequencies, where f s is the sampling

464 T. Kovács / Digital Signal Processing 22 (2012) 463–470

frequency. However, this method and all other approaches (in-cluding the wavelet methods), using a single narrowband filterfor a specific mono-component, produces approximately harmonicfunctions as separated components, since they neglect the higherfrequency harmonics (if there are any) of the periodic signal com-ponent. Therefore the above hybrid EMD methods in their presentform cannot retrieve an anharmonic component in one functionbut divide it into separate approximately harmonic functions cor-responding to its main Fourier components.

In order to solve the problem of anharmonic signal compo-nents there are multi-band filtering approaches as well [12–14].These methods try to identify all of the considerable peaks in theFourier spectrum that belong to a specific periodic component, andthen construct a multi-band filter corresponding to the peaks. Themethod is also known as harmonic selection. Regarding the math-ematical details there are two different ways to accomplish sucha harmonic selection. One is to take the Fourier transform of thecomposite signal and select the peaks of the spectrum as belong-ing to either of the components [13]. The other implementation isto apply a multi-band FIR filter that passes the frequency bandsaround the peaks belonging to the chosen component. Such a nar-rowband filtering can be seen in [11]. The problem with theseprocedures is that the frequency bands of the different compo-nents often overlap, and in these regions one cannot distinguishclearly the higher harmonics of the components. Beside the MASmethod and the harmonic selection method, Santhanam and Mara-gos [14] also propose a harmonic cancellation based feed-forwardtwo-prong comb filter with impulse response:

h(t) = 1

2

[δ(t) − δ(t − τ )

], (1.2)

for signal separation. This comb filter eliminates the spectral con-tent of the component with the period τ . Since it is an FIR filter ofthe order τ , the required time window is small (only one period)compared to the time window required by the multi-band FIR fil-ter mentioned above. However, corresponding to this, the combfilter is not so robust and cannot produce as good results as themulti-band FIR filter in the case of random frequency modulation,as it will be demonstrated in Section 4.

Regarding its purposes, the problem of separating periodic sig-nals resembles the Independent Component Analysis (ICA) ap-proach of the Blind Source Separation problems, but in these lattercases the basic task is to identify the sources in multi-channelmixtures, where the mixing matrix is of full rank [2,15,16]. Q. Heet al. [16] applied the ICA for the multi-channel signal obtainedby filtering the original single-channel signal by a series of Con-tinuous Wavelet filters, and this way solved a problem similar tothose discussed in the present paper. They used a Negetropy max-imization based ICA method, which involves the optimization of afunctional and is based on statistical laws. The first stage of themethod presented in this paper is based on an FIR filtering, butin the second stage it also applies a functional optimization step.(This maximization, however, is based on number theory and notstatistical laws.) Considering this, the presented method may beapplied as a part of a novel ICA algorithm, though the discussionof such an application is out of the scope of the present paper.Since the method involves an integration of the signal in a timewindow, it may be easily made compatible with adaptive controlmethods too, that consist of an integrating part and searches foran optimum (see for example [17]).

2. Formulation of the proposed separation method

In the present work, partly starting out from Yang’s idea [11],an FIR filtering method will be constructed that is able to extract

an anharmonic periodic signal component from the composite sig-nal. The construction is based on the minimization of a functional,of the residual signal after the extraction.

In Yang’s method the transmitting function and the action ofthe filter in the z-domain can be given as

G(z) =N−1∑k=0

akzk, Xout(z) = G(z)Xin(z), (2.1)

where Xin and Xout are the z-transforms of the original (in) andthe filtered (out) signal, and N is the order of the filter. In thetime domain (2.1) has the form of

x(h)out(t) =

N−1∑k=0

akxin(t − k). (2.2)

As it was emphasized before, if this is a narrowband filter, thenxout(t) is close to harmonic. The upper index (h) refers to thisproperty. Presently, our first purpose is to change the filter givenin (2.1) and (2.2) so as to extend the set of possible output (xout(t))signals to anharmonic functions. To achieve this, the transmittingfunction will be rewritten as

G(a)(z) =N−1∑k=0

akzk−ϕ(t), (2.3)

where φ(t) is an arbitrary but periodic function with the samelength of period as the one of the x(h)

out function. With this trans-mitting function xout(t) is given as

x(a)out(t) =

N−1∑k=0

akxin(t − k − ϕ(t)

), (2.4)

with the upper index (a) referring to the anharmonic extension.Indeed, x(a)

out(t) can be anharmonic as well since using (2.2)

x(a)out(t) = x(h)

out

(t − ϕ(t)

). (2.5)

Note that the degree and the coefficients ak of the filter are notaltered, they are the same in both forms of G(z). So the act of themodified filter can be given as a filtering by a conventional filterof the form of (2.1) and then time-shifting the output signal by afunction ϕ(t).

For example, if we have the anharmonic mono-componentx(t) = A sin(ω0t +ε sin(2ω0t)+π/2) to extract then the ϕ(t) func-tion should be −ε/ω0 sin(2ω0t). It is obvious that the specificϕ(t) function must be known for each component. The problemof finding ϕ(t) will be solved here for the simple case when theoriginal signal consists of two real periodic components, with dif-ferent lengths of period τ1 and τ2, and these components can beexpressed in the form of (2.5). That is, the signal to be decomposedis given by

x(t) = x1(t) + x2(t), (2.6)

where

xi(t) = A sin

(2π

τit + ϕi(t) + φi

), i = 1,2, (2.7)

and

ϕi(t + τi) = ϕi(t), i = 1,2. (2.8)

The length of period of ϕi(t) must be also τi to make xi(t) periodicwith the length of period τi .

In order to extract x2(t) from the signal the output of the filtermust be x2(t) and, therefore, one must find ϕ2(t) to construct thefilter.

T. Kovács / Digital Signal Processing 22 (2012) 463–470 465

3. The theoretical basis of the filter construction

The solution of the problem of finding ϕ2(t), addressed in theprevious section, will be based on the following mathematicalstatements:

If x1(t) and x2(t) are two discrete periodic functions (t = 0,1,

2, . . .) with different lengths of period of τ1 and τ2, and gcd(τ1, τ2) = 1(i.e. τ1 and τ2 are co-primes) then

τ−1∑t=0

∣∣x1(t)∣∣� τ−1∑

t=0

∣∣x1(t) + x2(t)∣∣, (3.1)

where τ = lcm(τ1, τ2) = τ1τ2 (the least common multiple of the peri-ods), and x(t) = x(t) − x(t − 1).

Equality holds if and only if∣∣x2(t)∣∣� ∣∣x1(t)

∣∣ for every t ∈ [0, τ − 1]. (3.2)

Here a relatively simple proof of these statements will be pre-sented. The case when τ1 and τ2 are not co-primes will be treatedafter the proof.

Proof of the statements (3.1) and (3.2). Let A denote the union ofthe time intervals, where x1(t) and x2(t) have the same sign(more precisely: both are nonnegative or both are negative), anddenote D the union of the time intervals, where they have differ-ent signs (again, the zero is counted to the positive side), that isD = [0, τ − 1] − A. So the inequality can be written as∑t∈A

∣∣x1(t)∣∣ +

∑t∈D

∣∣x1(t)∣∣ �∑

t∈A

(∣∣x1(t)∣∣ + ∣∣x2(t)

∣∣)+

∑t∈D

∣∣x1(t) + x2(t)∣∣. (3.3)

The second sum on both sides can be further decomposed bydecomposition of D into two disjoint sets: D = D(1,2) ∪ D(2,1),where D(1,2) and D(2,1) denote the subsets of D above which|x1(t)| < |x2(t)| and |x2(t)| � |x1(t)|, respectively. After thisdecomposition:∑t∈A

∣∣x1(t)∣∣ +

∑t∈D(1,2)

∣∣x1(t)∣∣ +

∑t∈D(2,1)

∣∣x1(t)∣∣

�∑t∈A

(∣∣x1(t)∣∣ + ∣∣x2(t)

∣∣) +∑

t∈D(1,2)

(∣∣x2(t)∣∣ − ∣∣x1(t)

∣∣)+

∑t∈D(2,1)

(∣∣x1(t)∣∣ − ∣∣x2(t)

∣∣). (3.4)

After rearranging the inequality as

0 �∑t∈A

∣∣x2(t)∣∣ +

∑t∈D(1,2)

(∣∣x2(t)∣∣ − 2

∣∣x1(t)∣∣)

−∑

t∈D(2,1)

∣∣x2(t)∣∣, (3.5)

and by adding a null term∑

t∈D(1,2) |x2(t)| − ∑t∈D(1,2) |x2(t)|

to the right-hand side, we obtain:

0 �∑t∈A

∣∣x2(t)∣∣ + 2

∑t∈D(1,2)

(∣∣x2(t)∣∣ − ∣∣x1(t)

∣∣)

−( ∑

t∈D(1,2)

∣∣x2(t)∣∣ +

∑t∈D(2,1)

∣∣x2(t)∣∣)

︸ ︷︷ ︸∑t∈D |x2(t)|

. (3.6)

Since the sum of the last two terms is∑

t∈D |x2(t)|, the right-hand side appears in the simple form of

0 �(∑

t∈A

∣∣x2(t)∣∣ −

∑t∈D

∣∣x2(t)∣∣)

︸ ︷︷ ︸=0

+ 2∑

t∈D(1,2)

(∣∣x2(t)∣∣ − ∣∣x1(t)

∣∣)︸ ︷︷ ︸

�0

. (3.7)

The second term is obviously nonnegative, since in D(1,2) the in-equality |x1(t)| < |x2(t)| holds. This term is zero if and only ifD(1,2) is empty, that is |x2(t)| � |x1(t)| for every t ∈ [0, τ − 1].In the following it will be shown that the first (bracketed) termin the right-hand side is zero, and with this the statement will beproven.

Let I(+) and I(−) denote the unions of the intervals (in [0, τ −1]) where x2(t) is nonnegative or negative, respectively. The signof x1(t) will be represented by a function p(t), defined as

p(t) ={

1 if x1(t)� 0,

0 otherwise.(3.8)

With the help of these notations the sums in the first bracketof (3.7) can be decomposed by the means of the signs of x1(t)and x2(t). After this decomposition and some rearrangements weobtain:∑t∈A

∣∣x2(t)∣∣ −

∑t∈D

∣∣x2(t)∣∣

=( ∑

t∈I(+)

p(t)∣∣x2(t)

∣∣ +∑

t∈I(−)

(1 − p(t)

)∣∣x2(t)∣∣)

−( ∑

t∈I(+)

(1 − p(t)

)∣∣x2(t)∣∣ +

∑t∈I(−)

p(t)∣∣x2(t)

∣∣)

=∑

t∈I(+)

(2p(t) − 1

)∣∣x2(t)∣∣ +

∑t∈I(−)

(2p(t) − 1

)(−∣∣x2(t)∣∣)

=τ−1∑t=0

(2p(t) − 1

)x2(t) = 2

τ−1∑t=0

p(t)x2(t) −τ−1∑t=0

x2(t)

= 2τ−1∑t=0

p(t)x2(t) − (x2(τ − 1) − x2(−1)

)︸ ︷︷ ︸=0

= 2τ−1∑t=0

p(t)x2(t). (3.9)

By considering that τ1 and τ2 are co-primes and using the linearcongruency theorem we obtain:

2τ−1∑t=0

p(t)x2(t) = 2τ−1∑t=0

p(t mod τ1)x2(t mod τ2)

= 2τ1−1∑t1=0

τ2−1∑t2=0

p(t1)x2(t2)

= 2

[τ1−1∑t1=0

p(t1)

][τ2−1∑t2=0

x2(t2)

]

= 2

[τ1−1∑t1=0

p(t1)

](x2(τ − 1) − x2(−1)

)︸ ︷︷ ︸=0

= 0. � (3.10)

466 T. Kovács / Digital Signal Processing 22 (2012) 463–470

Let us examine the case when τ1 and τ2 are not co-primes,that is gcd(τ1, τ2) = q > 1. In the proof the co-prime condition wasused in the final step (3.10). Without this condition the time ar-guments (t mod τ1) and (t mod τ2) are congruent modulo q, andtherefore (3.10) is written as

2τ−1∑t=0

p(t)x2(t) = 2τ−1∑t=0

p(t mod τ1)x2(t mod τ2)

= 2τ1−1∑t1=0

[p(t1)

τ2/q−1∑n=0

x2(t1 mod q + nq)

].

(3.11)

In order to solve the not co-prime case with MAS, Santhanam andMaragos [4] made additive conditions (the dc conditions of thesub-sampled signal):

τ2/q−1∑n=0

x2(t1 + nq) = 0, t1 = 0,1, . . . , (q − 1). (3.12)

Note that by making these additive conditions, the initial state-ments will be true for the not co-prime case too, since the secondsum of the final form of (3.11) becomes zero. Therefore, the sit-uation seems to be similar to that of the MAS: the additive dcconditions help to solve the not co-prime case.

It is important to remark that the proven statement is useful forthe search of ϕ2(t) only if the right-hand side of (3.1) is definitelybigger than the left-hand side. Regarding this problem it can besaid that if the original signal before sampling is a continuous anddifferentiable function and the sampling frequency is high enough,then the definite inequality is guaranteed, since under such cir-cumstances at the local extrema of x1(t) the difference x1(t) isapproximately zero, so the criterion for equality (3.2) cannot betrue.

How do the proven statements help us to find ϕ2(t)? By themeans of (3.1) the functional

Fτ

[x(t)

] :=τ−1∑t=0

∣∣x(t)∣∣ (3.13)

is minimal if and only if the x2(t) component is completely re-moved from the composite signal. The relationship (2.5) gives theoutput of the modified filter as x(a)

out(t) = x(h)out(t − ϕ(t)), where the

ak filter coefficients of (2.4) belong to the narrowband filter withpass-band around the frequency 1/τ2 (i.e. the basic frequency ofx2(t)) and ϕ(t) is the time-shift function of the modified filter. Ifϕ(t) = ϕ2(t) then x(h)

out(t − ϕ(t)) = x2(t). Therefore the functional

Hτ

[ϕ(t)

] := Fτ

[x(t) − x(h)

out

(t − ϕ(t)

)](3.14)

of the ϕ(t) time-shift function has global minimum at ϕ(t) =ϕ2(t). So the original problem of finding ϕ2(t) is transformed intoa minimum search problem of a functional.

ϕ2(t) is supposed to be periodic with the length of period τ2,therefore it is straightforward to write ϕ(t) in a Fourier expansionas:

ϕ(t) =M∑

m=1

fm sin

(2mπ

τ2t + gm

). (3.15)

The m = 0 term is neglected because the dc term is taken intoconsideration by the constant phase shift φ2 in (2.7). With thisexpansion the ϕ(t) function is represented by a 2M-dimensionalvector of ( f1, . . . , f M , g1, . . . , gM), and the functional Hτ [ϕ(t)] isrewritten as a function Hτ ( f1, . . . , f M , g1, . . . , gM) defined in the

2M-dimensional vector-space. From numerical point of view, weend up with an unconstrained global optimization problem ina high-dimensional vector-space. To solve this, here the gradientmethod was used with bisection backtracking line search, which isdescribed for example in [18] or [19].

4. Experiments and results

In order to get a picture about the performance of the pro-posed separation method two test signals of the form (2.6) and(2.7) were used as subjects of separation. Both of them are mix-ture of two periodic component signals, but one (xtest(t)) consistsof ‘clear’ periodic components, that satisfy the (2.8) periodicitycondition exactly, and the other (x̃test(t)) consists of quasi-periodiccomponents, that are randomly frequency-modulated. The two testfunctions are formulated as

xtest(t) = x1(t) + x2(t)

= A sin

(2π

77t + ϕ1(t)

)+ A sin

(2π

50t + ϕ2(t)

),

x̃test(t) = x̃1(t) + x̃2(t)

= A sin

(2π

τ1(t)t + ϕ̃1(t)

)+ A sin

(2π

τ2(t)t + ϕ̃2(t)

), (4.1)

where

ϕ1(t) = 1

2sin

(2π

77t + π

2

)+ 1

4sin

(4π

77t + π

4

)

+ 1

8sin

(8π

77t − π

4

),

ϕ2(t) = 1

2sin

(2π

50t − π

2

)+ 1

4sin

(4π

50t + π

3

)

+ 1

8sin

(8π

50t − π

3

), (4.2)

and

ϕ̃1(t) = 1

2sin

(2π

τ1(t)t + π

2

)+ 1

4sin

(4π

τ1(t)t + π

4

)

+ 1

8sin

(8π

τ1(t)t − π

4

),

ϕ̃2(t) = 1

2sin

(2π

τ2(t)t − π

2

)+ 1

4sin

(4π

τ2(t)t + π

3

)

+ 1

8sin

(8π

τ2(t)t − π

3

). (4.3)

The random frequency modulations of x̃test(t) are given by theτ1(t) and τ2(t) time-dependent periods in the form of

τi(t) = τi0

(1 + ri(t)

t

), i = 1,2 (τ10 = 77, τ20 = 50), (4.4)

where ri(t) is a random modulator function, which is generated bythe following simple algorithm:

– Generate {R1, R2, . . . , R K } random scalar values drawn from auniform distribution on the interval [−2,2].

– Draw an interpolated spline curve through the points {(2kτi0,

Rk) | k = 1,2, . . . , K }.

If the length of the examined signal is T , then the number of ran-dom numbers is K = T /(2τi0). With this formulation of x̃test(t) theperiod changes in the timescale of 2τi0 (the double of the average

T. Kovács / Digital Signal Processing 22 (2012) 463–470 467

Fig. 1. Fourier spectra of the separate components (thick and thin lines) of the con-stant period (a) and the random frequency-modulated (b) test signals.

period), so the signal at hand is close to an intra-wave frequency-modulated signal. (The factor t in the denominator in (4.4) ensuresthe constant average fluctuation of the phase of the signal.) In thenon-modulated case the lengths of period are clearly defined, butin the modulated case only average values for them can be in-terpreted. This construction of the modulation implies that theseaverage values are equal to τ10 and τ20. The lengths of period werealso determined with the help of the double difference function(DDF) algorithm [12], which produced the same result.

The present test components are constructed so that they haveinteger periods, however it is generally not true in the case of ameasured signal. It is possible to round the periods, which willcause some inaccuracy, but it is more accurate to employ a Sam-pling Rate Conversion by a CIC filter, which is recently developedto render an automatic anti-aliasing property [20].

Fig. 1 shows the Fourier spectra of the separate components ofthe clear and the random frequency-modulated test signals. Sincethe two lengths of period (77 and 50) are co-primes, the peaksthat belong to different components do not overlap considerablyin the case of the non-modulated signal, therefore a harmonic se-lection method [12–14] can be successfully used for separation.However, in the modulated case the peaks are widened in such ameasure that a harmonic selection method encounters problemsbecause of the overlapping peaks. Beside the proposed separationmethod, the MAS [3,4], the multi-band filtering harmonic selec-tion [11] and the harmonic canceling comb filter [4] methods arealso applied for separation of the test signal components, and theseparation results serve as a basis of comparison.

By the means of the theoretical description, the proposed sep-aration process will follow the main steps below:

1) Construct a constant coefficient FIR band filter with centralfrequency f2 = 1/τ2 and pass-band boundaries f2 ± 1

3 ( f2 − f1).This is the start-out filter corresponding to (2.2). Here the fil-ter order was chosen to be N = 1000.

2) Calculate x(h)out(t) with the filter.

3) Construct the ϕ(t) phase-shift function corresponding to(3.15). Here the number of the Fourier components was cho-sen to be M = 8, and the starting values of the ( f1, . . . , f8, g1,

. . . , g8) vector were (0.1, . . . ,0.1, rnd1, . . . , rnd8), where rndi

are random real scalars drawn from a uniform distribution onthe interval [0,2π ].

4) Calculate the functional Hτ [ϕ(t)] corresponding to (3.14) atthe necessary points of the 16-dimensional space of the vec-tors ( f1, . . . , f8, g1, . . . , g8), while proceeding with the gradi-ent line search method in this 16-dimensional space.

5) If the value of Hτ [ϕ(t)] does not change more than 0.1% com-pared to the value of the previous algorithmic step of thesearch method, then stop the procedure and store the finalvalues of ( f1, . . . , f8, g1, . . . , g8).

6) Calculate x(a)out(t) with the modified FIR filter given in (2.4).

The FIR filtering in step 1 was accomplished by the firls()function of the MATLAB Signal Processing ToolBox. In order toget the value of Hτ [ϕ(t)] in step 4 at the points of the 16-dimensional space visited by the line search method, the wholefunction x(a)

out(t) = x(h)out(t − ϕ(t)) should be calculated at each such

point. This was done by approximating x(h)out(t − ϕ(t)) by the linear

interpolation formula:

x(a)out(t) = x(h)

out

(t − ϕ(t)

)= (

1 − {ϕ(t)

}) · x(h)out

(t + ⌊

ϕ(t)⌋)

+ {ϕ(t)

} · x(h)out

(t + ⌊

ϕ(t)⌋ + 1

), (4.5)

where �ϕ(t)� and {ϕ(t)} are the integer and the fractional partsof ϕ(t). This approximation is needed because the signal valuesare given in only at the integer values of t .

In the following some details about the applied reference meth-ods are given.

The MAS method was accomplished as it is given in [3] or [14],with the time window τ1 + τ2 + 1 = 128. In the case of the non-modulated test signal (xtest(t)) this method gives an exact result,that is, the estimated x2(t) component coincides exactly with theoriginal one.

The comb filter was constructed according to (1.2), i.e. with thetransmitting function:

G(z) = 1

2

[1 − z−τ1

]. (4.6)

Since the x̃2(t) is to be produced, the spectral content of the com-ponent x̃1(t) should be canceled, which has the average period τ1.Therefore the order (and the required time window) of this filteris 77.

For the harmonic selection method, the transmitting functionof the multi-band filter was simply constructed as a sum of thoseof single band filters:

G(z) =8∑

i=1

Gi(z), (4.7)

where Gi(z) are the transmitting functions of the band filters cen-tered around the multiples nf2 of the frequency f2 = 1/τ2 withpass-band boundaries f2 ± 1

3 ( f2 − f1). As it can be seen from (4.6),only the first eight harmonics were filtered. In the case of the testsignal at hand, including more harmonics does not yield measur-able improvement.

Here the performance of the proposed and the referencemethod is given as the Peak Signal to Noise Ratio (PSNR) mea-sured between the original and the separated x̃2(t) component.This quantity is given as

PSNR = 10 · log10

n∑t=1

max[(x̃2(t))original]2

[(x̃2(t))original − (x̃2(t))separated]2. (4.8)

The time window (n) for calculating the PSNR values was 1000in the case of every method independently of the time windowapplied by the method itself.

468 T. Kovács / Digital Signal Processing 22 (2012) 463–470

In the present tests only the random frequency-modulated testsignal (x̃1(t) + x̃2(t)) will be used, because in the non-modulatedcase (x1(t)+ x2(t)) the MAS method gives a perfect solution to theseparation problem with a quite small time window (τ1 + τ2 + 1 =128). This means that if a composite signal is free (or almost free)of random perturbations, then the MAS solution seems to be anideal one.

The PSNR values and the graphs of the components separatedby the proposed and the reference methods can be seen in Fig. 2.The original and the separated x̃2(t) components are drawn withthick or thin lines, respectively. In the frequency-modulated casethe MAS system fails to give exact result as is demonstrated byFig. 2(a). Compared to the original x̃2(t) component the estimatedone shows strong fluctuations, which is obviously the effect of thefrequency modulation. The strong fluctuations indicate that the ac-curacy of MAS system is sensitive to the random perturbations ofthe period. Nevertheless, the PSNR value produced by this method(15.2) is not very low compared to the other methods. This is dueto that the separated signal, aside from the high frequency fluctu-ations, follows well the original one.

With the comb filtering results the case is quite different (seeFig. 2(b)). Here no high frequency fluctuations can be seen, but atsome parts the separated signal falls far from the original one, andthis causes a quite low PSNR value (9.2). This poor performancecan be expected in the case of a cancellation comb filter, since thismethod removes spectral content of x1(t) (with constant periodof 77), but fails to remove completely that of x̃1(t) when period isfluctuated around a mean value.

In the case of the multi-band FIR filtering method, however,better results can be expected. It is partly because the order of themulti-band filter can be chosen much longer than that of the two-prong comb filter, and, this filter passes not a single frequency atthe harmonics, but a wider frequency band. Therefore, this methodis more robust against the frequency perturbations. The multi-bandfiltering was tested with two different filter orders (i.e. time win-dows): 200 and 1000, and the corresponding results are seen inFigs. 2(c) and 2(d). In the case of time window = 200, the perfor-mance is as poor as that of the comb filter, but when the timewindow is raised to 1000, the PSNR value (17.5) is better thanthat of even the MAS method. So, the performance of this methodseems to be very sensitive to the order of the filter.

The method proposed in the present paper is tested with thesame time windows 200 and 1000 (see Figs. 2(e) and 2(f)). In thecase of time window = 200, the produced PSNR value (14.2) is abit lower than that given by the MAS method. This means that theproposed method performs somewhat worse than the MAS systemwhen a relatively short time window is used. However, with thetime window of 1000, the method gives a quite high PSNR value(21.8), which is far the best among the introduced test results.

In order to obtain a more detailed picture about the effect ofthe time window, the PSNR values produced by the multi-bandfiltering and the proposed method are graphed as a function ofthe time window (see Fig. 3). The PSNR value 15.2 given by theMAS method is also indicated in the figure. (When applying timewindows longer than 128, the performance of the MAS methoddoes not get better, since the longer time window includes morefluctuations the periods, which have a bad influence on the per-formance.) It can be seen that for time window values around 500the three methods give similar PSNR values, and for time win-dow values under and above 500 the examined methods performworse and better than the MAS system, respectively. In the regionof time windows under 1000, the performances of both methodsrise quickly, but above 1000, the curves have a saturation character.This means that only a very small improvement can be reached byincreasing the time window, the PSNR functions are approximatelyconstant. In this region, the proposed method shows considerably

Fig. 2. The results of separation regarding the component x̃2(t) accomplished by theMAS (a), the comb filtering (b), the multi-band filtering ((c) and (d)) and the pro-posed ((e) and (f)) method. The original and the separated components are drawnby thick or thin lines, respectively. The applied time windows and the achievedPSNR values are shown above the axis boxes.

better performance than the harmonic selection method (and theMAS method).

As it was discussed in the introduction, the methods above havethe advantage over the EMD method that they are applicable when

T. Kovács / Digital Signal Processing 22 (2012) 463–470 469

Fig. 3. The PSNR values produced by the multi-band filtering and the proposedmethod as functions of the applied time window. The PSNR value produced by theMAS method with time window of 128 is indicated by dashed line.

Fig. 4. The PSNR values produced by the MAS, multi-band filtering and the proposedmethod as functions of the frequency ratio.

the frequencies of the components are relatively close to eachother (see (1.1)). However, the frequency resolution of these meth-ods is limited as well; their applicability depends on the distancebetween the frequencies of the components. Fig. 4 characterizesthis dependence. The MAS, the multi-band filtering and the pro-posed method were tested for various frequency ratios. The formof the test signal was the same (4.1) but the value of the longer pe-riod τ1 varied between 50.5 and 131, and the PSNR values of theexamined methods were given as functions of log10(

f2− f1f1

) (wheref i = 1/τi ). In the cases of the multi-band filtering and the pro-posed method the time windows 200 and 1000 were used. Forτ1 periods above 77 (above 0.5 abscissa value) there is no im-provement in the performances, but as the period approaches 50.5(−2 abscissa value) the PSNR functions decrease to very low val-ues (under 10). As it can be expected, the frequency resolution ismuch better in the cases when the time window is 1000. At thepair of periods τ1 = 53 and τ2 = 50, when there is only 6% differ-ence between the frequencies (−1.25 abscissa value) the proposedand the multi-band method (with time window = 1000) producePSNR values around 15, which can be considered as a good resultcompared to the other methods.

Although the present paper focuses on the separation of pe-riodic signals, a test with a quasi-periodic damped signal is alsoincluded in this section in order to demonstrate the behavior ofthe presented method in the case of slowly varying amplitude. Thedamped signal is generated in a way that the random frequency-modulated test signal x̃1(t) + x̃2(t) has been multiplied by an ex-ponential damping factor

x̃damped(t) = exp(−0.001t) · [x̃1(t) + x̃2(t)]. (4.9)

Fig. 5 shows the separation results of the MAS and the pro-posed method with time windows of 200 and 1000. As in theother tests so far, the x̃2(t) component was separated, which hasslowly decreasing amplitude in this case. The MAS separation wasperformed in each consecutive non-overlapping time window oflength 128. It can be seen, that this moderate but clearly observ-able variation of the amplitude affects badly the MAS method; its

Fig. 5. The results of separation regarding the exponentially damped mixture andcomponent x̃2(t) accomplished by the MAS (a) and the proposed ((b) and (c))method. The original and the separated components are drawn by thick or thinlines, respectively. The applied time windows and the achieved PSNR values areshown above the axis boxes.

PSNR result is 9.2. The presented method, however, gives good sep-aration results; the PSNR values are 14.4 and 23.8 in the cases ofthe shorter and longer time windows. This good behavior is dueto that this method, in the first step, is based on an FIR filter,which transmits the exponential decreasing of the amplitude intothe filtered (and finally separated) signal. At the first sight, it seemsstrange that the PSNR value of the proposed method is higher inthe case of the damped signal compared to the original tests. Thisfollows from the definition of the PSNR value: the peak signal ismeasured at the beginning of the time window, and then the sig-nal decreases exponentially together with the measured error inthe rest of the time window.

In the final part of this section, we apply the MAS, multi-band filtering and the proposed method to separate the twocomponents of the mixture of natural signals, which are chosento be voice samples of a peregrine falcon. (The test signal wasdownloaded from the electronic library of www.findsounds.com orwww.dinosoria.com.) If we want to see clearly the performanceof the separation methods, the two components of the mixturemust be known exactly. Therefore, the two components were theoriginal voice sample with measured period of τ1 = 19 and anartificially elongated form of the original voice sample with pe-riod of τ2 = 23. (Note that the two periods are co-primes.) In theFourier spectrum of the original signal (Fig. 6) it can be seen thatthe peaks of the harmonics are widened, which indicates a mod-erate random frequency modulation, similarly to the spectrum ofthe previous modulated test signal (see Fig. 1). In addition to this,clearly observable fluctuations of the amplitude are also present inthe signal. A part of the test signal with longer periods is shownin Fig. 7 (thick lines). The thin lines in Fig. 7 show the signals sep-arated from the mixture by the methods at hand. The multi-bandfiltering and the proposed methods achieves similar PSNR values(17.3 and 17.8, respectively), while the MAS system produces a low

470 T. Kovács / Digital Signal Processing 22 (2012) 463–470

Fig. 6. Fourier spectrum of the peregrine falcon’s voice test sample.

Fig. 7. The results of separation regarding the mixture of the peregrine falcon’s voicetest samples accomplished by the MAS (a), the multi-band filtering (b), and the pro-posed (c) method. The original and the separated components are drawn by thickor thin lines, respectively. The applied time windows and the achieved PSNR valuesare shown above the axis boxes.

(10.4) value. This big difference between the filter based and theMAS method is due to mostly the random variation of the ampli-tude, as it was demonstrated in the previous section. In the case ofthe present test signal the random fluctuations of the periods (i.e.random frequency modulation) is not so intensive as in the caseof the first test signal given by (4.1), as it can be seen from theFourier spectra. In the present case the multi-band filtering andthe proposed method show similar performance, while in the caseof the first test signal with more pronounced frequency modula-tion the proposed method gives better PSNR values.

5. Conclusions

The conclusions of the present paper can be summarized asfollows:

– A constant coefficient narrowband FIR filter, from mathemat-ical point of view, can be easily modified so as to produceanharmonic outputs.

– The minimization of the functional defined in Section 3 is ap-plicable for separating the individual periodic components ofthe signal, when the lengths of period of the components aredifferent.

– The method derived from the modified FIR filtering and theminimization of the functional seems to be more robust thanthe MAS system in the case of a random frequency-modulatedor amplitude-modulated test signal.

– In the case of moderate random frequency modulation theproposed method and the multi-band filtering (i.e. harmonicselection) method achieve similar performance in signal sep-aration. If the random frequency modulation is more pro-nounced the proposed method performs definitely better.

Appendix A. Supplementary material

The online version of this article contains additional supple-mentary material.

Please visit doi:10.1016/j.dsp.2011.12.002.

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Tamás Kovács received M.S. (1992) and Ph.D. (1997) in StatisticalPhysics at University of Debrecen (Hungary). He is currently an AssociateProfessor in the Department of Informatics at Kecskemét College (Hun-gary).