TIME-FREQUENCY SIGNAL ANALYSIS 923 AWindowWidthOptimized … · TIME-FREQUENCY SIGNAL ANALYSIS 923...

TIME-FREQUENCY SIGNAL ANALYSIS 923

A Window Width OptimizedS-transform

Ervin Sejdic, Igor Djurovic and Jin Jiang

Abstract– Energy concentration of the S-

transform in the time-frequency domain has

been addressed in this paper by optimizing the

width of the window function used. A new

scheme is developed and referred to as a win-

dow width optimized S-transform. Two opti-

mization schemes have been proposed, one for

a constant window width, the other for time-

varying window width. The former is intended

for signals with constant or slowly varying fre-

quencies, while the latter can deal with sig-

nals with fast changing frequency components.

The proposed scheme has been evaluated us-

ing a set of test signals. The results have indi-

cated that the new scheme can provide much

improved energy concentration in the time-

frequency domain in comparison with the stan-

dard S-transform. It is also shown using the

test signals that the proposed scheme can lead

to higher energy concentration in comparison

with other standard linear techniques, such as

short-time Fourier transform and its adaptive

forms. Finally, the method has been demon-

strated on engine knock signal analysis to show

its effectiveness.

I. I��

In the analysis of the nonstationary signals,one often needs to examine their time-varyingspectral characteristics. Since time-frequencyrepresentations (TFR) indicate variations ofthe spectral characteristics of the signal as afunction of time, they are ideally suited fornonstationary signals [1], [2]. The ideal time-frequency transform only provides informationabout the frequency occurring at a given timeinstant. In other words, it attempts to com-bine the local information of an instantaneousfrequency spectrum with the global informa-tion of the temporal behavior of the signal [3].The main objectives of the various types oftime-frequency analysis methods are to obtaintime-varying spectrum functions with high res-olution and to overcome potential interferences

Journal of Advances in Signal Processing, Vol. 2008,Article ID 672941, Feb. 2008

[4].The S-transform can conceptually be viewed

as a hybrid of short-time Fourier analysis andwavelet analysis. It employs variable windowlength. By using the Fourier kernel, it canpreserve the phase information in the decom-position [5]. The frequency dependent windowfunction produces higher frequency resolutionat lower frequencies, while at higher frequen-cies sharper time localization can be achieved.In contrast to wavelet transform, the phase in-formation provided by the S-transform is ref-erenced to the time origin, and therefore pro-vides supplementary information about spec-tra which is not available from locally refer-enced phase information obtained by the con-tinuous wavelet transform [5]. For these rea-sons, the S-transform has already been consid-ered in many fields such as geophysics [6], [7],[8], cardiovascular time series analysis [9], [10],[11], signal processing for mechanical systems[12], [13], power system engineering [14], andpattern recognition [15].

Even though the S-transform is becominga valuable tool for the analysis of signals inmany applications, in some cases, it suffersfrom poor energy concentration in the time-frequency domain. Recently, attempts to im-prove the time-frequency representation of theS-transform have been reported in the litera-ture. A generalized S-transform, proposed in[12], provides greater control of the windowfunction, and the proposed algorithm also al-lows nonsymmetric windows to be used. Sev-eral window functions are considered, includ-ing two types of exponential functions, am-plitude modulation and phase modulation bycosine functions. Another form of the gener-alized S-transform is developed in [7], wherethe window scale and shape are a function offrequency. The same authors introduced a bi-Gaussian window in [8], by joining two non-

924 TIME-FREQUENCY SIGNAL ANALYSIS

symmetric half-Gaussian windows. Since thebi-Gaussian window is asymmetrical, it alsoproduces an asymmetry in the time-frequencyrepresentation, with higher time resolution inthe forward direction. As a result, the pro-posed form of the S-transform has better per-formance in detection of the onset of the sud-den events. However, in the current litera-ture, none have considered optimizing the en-ergy concentration in the time-frequency do-main directly, i.e., to minimize the spread ofthe energy beyond the actual signal compo-nents.

The main approach used in this paper isto optimize the width of the window usedin the S-transform. The optimization is per-formed through the introduction of a newparameter in the transform. Therefore, thenew technique is referred to as a windowwidth optimized S-transform (WWOST). Thenewly introduced parameter controls the win-dow width, and the optimal value can be de-termined in two ways. The first approachcalculates one global, constant parameter andits use is recommended for signals with con-stant or very slowly varying frequency com-ponents. The second approach calculates thetime-varying parameter, and it is more suit-able for signals with fast varying frequencycomponents.

The proposed scheme has been tested usinga set of synthetic signals and its performanceis compared with the standard S-transform.The results have shown that the WWOST en-hances the energy concentration. It is alsoshown that the WWOST produces the time-frequency representation with a higher con-centration than other standard linear tech-niques, such as the short-time Fourier trans-form and its adaptive forms. The proposedtechnique is useful in many applications whereenhanced energy concentration is desirable.As an illustrative example, the proposed algo-rithm is used to analyze knock pressure signalsrecorded from a Volkswagen Passat engine inorder to determine the presence of several sig-nal components.

This paper is organized as follows. In Sec-tion II, the concept of ideal time-frequencytransform is introduced, which can be used

to compare with other time-frequency repre-sentations including transforms proposed here.The development of the WWOST is coveredin Section III. Section IV evaluates the per-formance of the proposed scheme using testsignals, and also the knock pressure signals.Conclusions are drawn in Section V.

II. E�� C��

T��-F�� D��

The ideal TFR should only be distributedalong frequencies for the duration of signalcomponents. Thus, the neighboring frequen-cies would not contain any energy and theenergy contribution of each component wouldnot exceed its duration [3].

For example, let’s consider two sim-ple signals: an FM signal, x1(t) =A(t) exp(jφ(t)), where |dA(t)/dt| � |dφ(t)/dt|and an instantaneous frequency is definedas f(t) = (dφ(t)/dt)/2π; and a sig-nal with the Fourier transform given asX(f) = G(f) exp(j2πχ(f)), where the spec-trum is slowly varying in comparison to phase|dG(f)/df | � |dχ(f)/df |. Further, A(t) andG(t) are real functions. The ideal TFRs forthese signals are given respectively as [16]:

ITFR(t, f) = 2πA(t)δ

(f − 1

2π

dφ(t)

dt

)(1)

ITFR(t, f) = 2πG(f)δ

(t+dχ(f)

df

)(2)

where ITFR stands for an ideal time-frequency representation. These two represen-tations are ideally concentrated along the in-stantaneous frequency, (dφ(t)/dt)/2π, and ongroup delay −dχ(f)/df . Simplest examples ofthese signals are a sinusoid with A = const.and dφ(t)/dt = const. depicted in Fig. 1(a)and a Dirac pulse x2(t) = δ(t − t0) shown inFig. 1(b). The ideal time-frequency represen-tations are depicted in Figs. 1(c) and 1(d).These two graphs are compared with the TFRsobtained by the standard S-transform in Figs.1(e) and 1(f).

For the sinusoidal case, the frequencies sur-rounding (dφ(t)/dt)/2π also have a strong con-tribution, and from (1) it is clear that theyshould not have any contributions. Similarly,

A WINDOW WIDTH OPTIMIZED S-TRANSFORM 925

0 0.2 0.4 0.6 0.8 1-1

0

1

Am

plit

ud

e

Time (s)

(a)

30 Hz sinusoid

0 0.2 0.4 0.6 0.8 10

0.5

1

Am

plit

ud

e

Time (s)

(b)

Dirac function

0 0.2 0.4 0.6 0.8 10

50

100

Fre

qu

en

cy (

Hz)

Time (s)(c)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

qu

en

cy (

Hz)

Time (s)(d)

Fre

qu

en

cy (

Hz)

Time (s)

(e)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

qu

en

cy (

Hz)

Time (s)

(f)

0 0.2 0.4 0.6 0.8 10

50

100

Fig. 1. Comparison of the ideal time-frequency representation and S-transform for the two simple signal forms:(a) 30 Hz sinusoid; (b) sample Dirac function; (c) ideal TFR of a 30 Hz sinusoid; (d) ideal TFR of a Diracfunction (e) TFR by standard S-transform for a 30 Hz sinusoid; (f) TFR by standard S-transform of theDirac delta function.

for the Dirac function, it is expected that allthe frequencies have the contribution but onlyfor a single time instant. Nevertheless, it israther clear that the frequencies are not onlycontributing during a single time instant asexpected from (2), but the surrounding timeinstants also have rather strong energy contri-bution.

The examples presented here are for illus-trations only, since a priori knowledge aboutthe signals is assumed. In most practical situ-ations, the knowledge about a signal is limitedand the analytical expressions similar to (1)and (2) are often not available. However, theexamples illustrate a point that some modifi-cations to the existing S-transform algorithm,which do not assume a priori knowledge about

the signal, may be useful to achieve improvedperformance in time-frequency energy concen-tration. Such improvements only become pos-sible after modifications to the width of thewindow function are made.

III. T�� P�� S��

A. Standard S-transform

The standard S-transform of a function x(t)is given by an integral as [5], [7], [12]:

Sx(t, f) =

∫ +∞

−∞

x(τ)

×w(t− τ , σ(f)) exp(−j2πfτ)dτ (3)


with a constraint∫ +∞

−∞

w(t− τ , σ(f))dτ = 1. (4)

A window function used in S-transform is ascalable Gaussian function defined as

w(t, σ(f)) =1

σ(f)√2πexp

(− t2

2σ2(f)

). (5)

The advantage of the S-transform over theshort-time Fourier transform (STFT) is thatthe standard deviation σ(f) is actually a func-tion of frequency, f , defined as

σ(f) =1

|f | . (6)

Consequently, the window function is also afunction of time and frequency. As the widthof the window is dictated by the frequency, itcan easily be seen that the window is widerin the time domain at lower frequencies, andnarrower at higher frequencies. In other words,the window provides good localization in thefrequency domain for low frequencies whileproviding good localization in time domain forhigher frequencies.

The disadvantage of the current algorithm isthe fact that the window width is always de-fined as a reciprocal of the frequency. Somesignals would benefit from different windowwidths. For example, for a signal containing asingle sinusoid, the time-frequency localizationcan be greatly improved if the window is verynarrow in the frequency domain. Similarly,for signals containing only a Dirac impulse,it would be beneficial for good time-frequencylocalization to have very wide window in thefrequency domain.

B. Window Width Optimized S-transform

A simple improvement to the existing al-gorithm for the S-transform can be made bymodifying the standard deviation of the win-dow to

σ(f) =1

|f |p . (7)

Based on the above equation, the new S-transform can be calculated represented as:

Spx(t, f) =|f |p√2π

∫ +∞

−∞

x(τ)

× exp(−(t− τ)

2f2p

2

)exp(−j2πfτ)dτ. (8)

The parameter p can control the width of thewindow. By finding an appropriate value ofp, an improved time-frequency concentrationcan be obtained. The window functions withthree different values of p are plotted in Fig.2, where p = 1 corresponds to the standardS-transform window. For p < 1, the windowbecomes wider in the time domain, and forp > 1, the window narrows in the time domain.Therefore, by considering the example fromSection II, for the single sinusoid a small valueof p would provide almost perfect concentra-tion of the signal, whereas, for the Dirac func-tion, a rather large value of p would produce agood concentration in the time-frequency do-main. It is important to mention that in thecase of 0 < f < 1, the opposite is true.

The optimal value of p will be found basedon the concentration measure proposed in [17],which has some favorable performance in com-parison to other concentration measures re-ported in [18][19], [20]. The measure is de-signed to minimize the energy concentrationfor any time-frequency representation basedon the automatic determination of some time-frequency distribution parameter. This mea-sure is defined as

CM(p) =1

∫ +∞−∞

∫+∞−∞

|Spx(t, f)| dtdf(9)

where CM stands for a concentration measure.There are two ways to determine the optimal

value of p. One is to determine a global, con-stant value of p for the entire signal. The otheris to determine a time-varying p(t), which de-pends on each time instant considered. Thefirst approach is more suitable for the signalswith the constant or slowly varying frequencycomponents. In this case one value of p willsuffice to give the best resolution for all com-ponents. The time-varying parameter is moreappropriate for signals with fast varying fre-quency components. In these situations, de-pending on the time duration of the signalcomponents, it would be beneficial to use lowervalue of p (somewhere in the middle of theparticular component’s interval), and to use


-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

No

rma

lize

d A

mp

litu

de

p=0.5

p=1

p=2

Fig. 2. Normalized Gaussian window for different values of p.

higher values of p for beginning and end ofthe component’s interval, so the component isnot smeared in the time-frequency plane. Itis important to mention that both proposedschemes for determining the parameter p arethe special cases of the algorithm which wouldevaluate the parameter on any arbitrary subin-terval, rather than over the entire duration ofthe signal.

B.1 Algorithm for determining the time-invariant p

The algorithm for determining the opti-mized time-invariant value of p is definedthrough the following steps:1. For p selected from a set 0 < p ≤ 1, com-pute S-transform of the signal, Spx(t, f) using(8).2. For each p from the given set, normalizethe energy of the S-transform representation,so that all of the representations have the equalenergy:

Spx(t, f) =Spx(t, f)√∫+∞

−∞

∫ +∞−∞

|Spx(t, f)|2 dtdf. (10)

3. For each p from the given set, compute theconcentration measure according to (9), thatis

CM(p) =1

∫ +∞−∞

∫+∞−∞

∣∣∣Spx(t, f)∣∣∣ dtdf

. (11)

4. Determine the optimal parameter popt by

popt = maxp[CM(p)] . (12)

5. Select Spx(t, f) with popt to be the WWOST:

Spx(t, f) = Spoptx (t, f). (13)

As it can be seen, the proposed algorithmcomputes the S-transform for each value ofp and, based on the computed representa-tion it determines the concentration measure,CM(p), as an inverse of L1 norm of thenormalized S-transform for a given p. Themaximum of the concentration measure cor-responds to the optimal p which provides leastsmear of Spx(t, f).

It is important to note that in the first step,the value of p is limited to the range 0 < p ≤ 1.


Any negative value of p corresponds to an nth

root of a frequency which would make the win-dow wider as frequency increases. Similarly,values greater than 1 provide a window whichmay be too narrow in the time domain. Unlessthe signal being analyzed is a superposition ofDelta functions, the value of p should not ex-ceed unity. As a special case, it is importantto point out that for p = 0, the WWOST isequivalent to STFT with a Gaussian windowwith σ2 = 1.

B.2 Algorithm for determining p(t)

The time-varying parameter p(t) is requiredfor signals with components having greater orabrupt changes. The algorithm for choosingthe optimal p(t) can be summarized throughthe following steps:1. For p selected from a set 0 < p(t) ≤ 1, com-pute S-transform of the signal, Spx(t, f) using(8).2. Calculate the energy, E1, for p = 1. Foreach p from the set, normalize the energy of theS-transform representation to E1, so that allof the representations have the equal energy,and the amplitude of the components is notdistorted:

Spx(t, f) =√E1

Spx(t, f)√∫ +∞−∞

∫+∞−∞

|Spx(t, f)|2 dtdf.

(14)3. For each p from the set and a time instantt compute

CM(t, p) =1

∫+∞−∞

∣∣∣Spx(t, f)∣∣∣df. (15)

4. Optimal value of p for the consideredinstant t maximizes concentration measureCM(t, p)

popt(t) = argmaxp[CM(t, p)] . (16)

5. Set the WWOST to be:

Spx(t, f) = Spopt(t)x (t, f). (17)

The main difference between the two tech-niques lies in step 3. For the time invariantcase a single value of p is chosen, whereas in

the time-varying case, an optimal value of p(t)is a function of time. As it is demonstrated inSection IV, the time-dependent parameter isbeneficial for the signals with the fast varyingcomponents.

B.3 Inverse of the WWOST

Similarly to the standard S-transform, theWWOST can be used as both an analysis anda synthesis tool. The inversion procedure forthe WWOST resembles that of the standard S-transform, but with one additional constraint.The spectrum of the signal obtained by aver-aging Spx(t, f) over time must be normalized byW (0, f), whereW (α, f) represents the Fouriertransform (from t to α) of the window func-tion, w(t, σ(f)). Hence, the inverse WWOSTfor a signal, x(t), is defined as

x(t) =

∫ +∞

−∞

∫ +∞

−∞

1

W (0, f)

×Spx(τ, f) exp(j2πft)dτdf. (18)

In a case of the time-invariant p, it can beshown that W (0, f) = 1. In a general case,the Fourier transform of the proposed modifiedwindow can only be determined numerically.

IV. WWOST P�� A��!��

In this section, the performance of the pro-posed scheme is examined using a set of syn-thetic test signals first. Furthermore, theanalysis of signals from an engine is also given.The first part includes several cases: (1) a sim-ple case involving three slowly varying frequen-cies and (2) more complicated cases involvingmultiple time-varying components. The goalis to examine the performance of WWOSTin comparison to the standard S-transform.The proposed algorithm is also compared toother time-frequency representations, such asthe short-time Fourier transform (STFT) andadaptive STFT (ASTFT), to highlight the im-proved performance of the S-transform withthe proposed window width optimization tech-nique. In particular, the proposed algorithmcan be used for some classes of the signals forwhich the standard S-transform would not besuitable.


As for the synthetic signals, the samplingperiod used in the simulations is Ts = 1/256seconds. Also, the set of p values, used in thenumerical analysis of both test and the knockpressure signals, is given by p = {0.01n : n ∈ Nand 1 ≤ n ≤ 100}. The ASTFT is calculatedaccording to the concentration measure givenby (9). In the definition of the measure a nor-malized STFT is used instead of the normal-ized WWOST. The standard deviation of theGaussian window, σgw, is used as the optimiz-ing parameter, where the window is defined as

wSTFT (t) =1

σgw√2πexp

(− t2

2σ2gw

). (19)

The optimization for synthetic signals is per-formed on the set of values defined by

σgw = {n/128 : n ∈ N and 1 ≤ n ≤ 128}(20)

and both the time-invariant and time-varyingvalue of σgw are calculated.

A. Synthetic Test Signals

Example 1

The first test signal is shown in Fig. 3(a).It has the following analytical expression:

x1(t) = cos(132πt+ 14πt2)

+ cos(10πt− 2πt2) + cos(30πt+ 6πt2) (21)

where the signal exists only on the interval 0 ≤t < 1. The signal consists of three slowly vary-ing frequency components. It is analyzed us-ing the STFT (Fig. 3(b)), ASTFT with time-invariant optimum value of σgw (Fig. 3(c)),standard S-transform (Fig. 3(d)) and the pro-posed algorithm (Fig. 3(f)). A Gaussian win-dow is also used in the analysis by the STFT,with standard deviations equal to 0.05. Theoptimum value of standard deviation for theASTFT is calculated to be σopt = 0.094. Thecolormap used for plotting the time-frequencyrepresentations in Fig. 3 and all the subse-quent figures is a linear grayscale with valuesfrom 0 to 1.

The standard S-transform, shown in Fig.3(d), depicts all three components clearly.

However, only first two components have rela-tively good concentration, while the third com-ponent is completely smeared in frequency. Asshown in Fig. 3(b), the STFT provides bet-ter energy concentration than the standard S-transform. The ASTFT, depicted in Fig. 3(c),shows a noticeable improvement for all threecomponents. The results with the proposedscheme is shown in Fig. 3(f) for p = 0.57.The value of p is found according to (12). Forthe determined value of p, the first two com-ponents have higher concentration even thanthe ASTFT, while the third component hasapproximately the same concentration.

In Fig. 3(e), the normalized concentrationmeasure is depicted. The obtained results ver-ify the theoretical predictions from Section III-B. For this class of signals, i.e., the signalswith slowly varying frequencies, it is expectedthat smaller values of p will produce the bestenergy concentration. In this example, the op-timal value, found according to (12), is deter-mined numerically to be 0.57.

Based on the visual inspection of the time-frequency representations shown in Fig. 3, itcan be concluded that the proposed algorithmachieves higher concentration among the con-sidered representations. To confirm this fact,a performance measure given by

ΞTF =

(∫ +∞

−∞

∫ +∞

−∞

∣∣∣TF (t, f)∣∣∣dtdf

)−1

(22)is used for measuring the concentration of the

representation, where∣∣∣TF (t, f)

∣∣∣ is a normal-

ized time-frequency representation. The per-formance measure is actually the concentra-tion measure proposed in (9). A more con-centrated representation will produce a highervalue of ΞTF . Table I summarizes the perfor-mance measure for the STFT, the ASTFT, thestandard S-transform and the WWOST.

The value of the performance measure forthe standard S-transform is the lowest, fol-lowed by the STFT. The WWOST producesthe highest value of ΞTF , and thus achievesa TFR with the highest energy concentrationamongst the transforms considered.


0 0.2 0.4 0.6 0.8 1-5

0

5

Time (s)

(a)

Am

plit

ud

e

Fre

qu

en

cy (

Hz)

Time (s)

(b)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

qu

en

cy (

Hz)

Time (s)(c)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

qu

en

cy (

Hz)

Time (s)(d)

0 0.2 0.4 0.6 0.8 10

50

100

0.2 0.4 0.6 0.8 10

0.5

1

p

(e)

Am

plit

ud

e o

f C

M

Fre

qu

en

cy (

Hz)

Time (s)

(f)

0 0.2 0.4 0.6 0.8 10

50

100

Fig. 3. Test signal x1(t): (a) Time-domain representation; (b) STFT of x1(t); (c) ASTFT of x1(t) with σopt;(d) Spx(t, f) of x1(t) with p = 1 (standard S-transform); (e) Concentration measure CM(p); (f) Spx(t, f) ofx1(t) with the optimal value of p = 0.57.

TABLE I

P��

��-�� .

TFR ΞTFSTFT 0.0119ASTFT 0.0131Standard S-transform 0.0080WWOST 0.0136

Example 2

The signal in the second example containsmultiple components with faster time-varyingspectral contents. The following signal is used:

x2(t) = cos[40π(t− 0.5) arctan(21t− 10.5)

−20π ln((21t− 10.5)2 + 1)/21 + 120πt]

+ cos(40πt− 8πt2

)(23)

where x2(t) exists only on the interval 0 ≤t < 1. This signal consists of two components.The first has a transition region from lowerto higher frequencies, and the second is a lin-ear chirp. In the analysis, the time-frequencytransformations that employ a constant win-dow exhibit a conflicting issue between goodconcentration of the transition region for thefirst component versus good concentration forthe rest of the signal. In order to numeri-cally demonstrate this problem, the signal isagain analyzed using the STFT (Fig. 4(a)),ASTFT with the optimal time-invariant valueof σgw (Fig. 4(c)), ASTFT with the optimaltime-varying value of σgw (Fig. 4(e)), stan-dard S-transform (Fig. 4(b)), the proposed al-gorithm with both time-invariant (Fig. 4(d)),


and time-varying p (Fig. 4(f)). A Gaussianwindow is used for the STFT, with σ = 0.03.The optimum time-invariant value of the stan-dard deviation for the ASTFT is determinedto be σopt = 0.055.

The standard deviation of the Gaussian win-dow used should be small in order for theSTFT to provide relatively good concentra-tion of the transition region. However, as thevalue of the standard deviation decreases, sois the concentration of the rest of the signal.To a certain extent, the standard S-transformis capable of producing a good concentrationaround the instantaneous frequencies at thelower frequencies and also in the transition re-gion for the first component. However, at thehigh frequencies the standard S-transform ex-hibits poor concentration for the first compo-nent. The WWOST with a time-invariant penhances the concentration of the linear chirp,as shown in Fig. 4(d). However the concentra-tion of the transition region of the first compo-nent is deteriorated in comparison to the stan-dard S-transform. The concentration obtainedwith the WWOST with the time-invariant pfor this transition region is equivalent to thepoor concentration exhibited by the STFT.Even though the ASTFT with both time-invariant and time-varying optimum value ofstandard deviation provide good concentrationof the linear FM component and the stationaryparts of the second component, the transitionregion of the second component is smeared intime.

Fig. 4(f) represents the signal optimized S-transform obtained by using p(t). A signifi-cant improvement in the energy concentrationis easily noticeable in comparison to the stan-dard S-transform. All components show im-proved energy concentration in comparison tothe S-transform. Further, a comparison of therepresentations obtained by the proposed im-plementation of the S-transform and the STFTshows both components have higher energyconcentration in the representation obtainedby the WWOST with p(t).

As mentioned previously, for this type of sig-nals it is more appropriate to use the time-varying p(t) rather than a single constant pvalue in order to achieve better concentration

TABLE II

P�� -��

�� E"��!� 2.

TFR ΞTFSTFT 0.0108ASTFT with σopt 0.0115ASTFT with σopt(t) 0.0119WWOST with p 0.0116WWOST with p(t) 0.0124

of the nonstationary data. By comparing Fig.4(d) and 4(f), the component with the fastchanging frequency has better concentrationwith p(t) than a fixed p, which is calculatedaccording to (12), while the linear chirp hassimilar concentration in both cases.

It would be beneficial to quantify the re-sults by evaluating the performance measureagain. The performance measure is given by(22) and the results are summarized in TableII. A higher value of the performance mea-sure for WWOST with p(t) reconfirms thatthe time-varying algorithm should be used forthe signals with fast changing components.Also, it is worthwhile to examine the valueof (22) for the STFT and the ASTFT. Thetime-frequency representations of the signalobtained by the STFT and ASTFT algorithmsachieve smaller values of the performance mea-sure than WWOST. This supports the earlierconclusion that the WWOST produces moreconcentrated energy representation than theSTFT and the ASTFT. The WWOSTwith thetime-invariant value of p produces higher con-centration than the ASTFT with the optimumtime-invariant value of σgw, and the WWOSTwith p(t) produces higher concentration thanthe ASTFT with the optimum time-varyingvalue of the σgw.

Example 3

Another important class of signals are thosewith crossing components that have fast fre-quency variations. A representative signal asshown in Fig. 5(a) is given by

x3(t) = cos(20π ln(10t+ 1))


Fre

que

ncy (

Hz)

Time (s)

(a)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

que

ncy (

Hz)

Time (s)

(b)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

quen

cy (

Hz)

Time (s)(c)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

quen

cy (

Hz)

Time (s)(d)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

qu

en

cy (

Hz)

Time (s)

(e)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

qu

en

cy (

Hz)

Time (s)

(f)

0 0.2 0.4 0.6 0.8 10

50

100

Fig. 4. Comparison of different algorithms: (a) STFT of x2(t); (b) Spx(t, f) of x2(t) with p = 1 (standardS-transform); (c) ASTFT of x2(t) with σopt = 0.055; (d) Spx(t, f) of x2(t) with p = 0.73; (e) ASTFT ofx2(t) with σopt(t); (f) S

px(t, f) of x2(t) with the optimal p(t).

+cos(48πt+ 8πt2) (24)

with x3(t) = 0 outside the interval 0 ≤ t < 1.For this class of signals, similar conflicting is-sues occur as in the previous example; how-ever, here exists an additional constraint, i.e.,the crossing components. The time-frequencyanalysis is performed using the STFT (Fig.5(b)), the ASTFT with the time-varying σgw(Fig. 5(c)), the standard S-transform (Fig.5(d)) and the proposed algorithm for the S-transform (Fig. 5(f)). In the STFT, aGaussian window with a standard deviationof 0.02 is used. Due to the time-varying na-ture of the frequency components present inthe signal, the time-varying algorithm is usedin the calculation of the WWOST in order todetermine the optimal value of p.

The representation obtained by the STFT

depicts good concentration of the higher fre-quencies, while having relatively poor concen-tration at the lower frequencies. An improve-ment in the concentration of the lower frequen-cies is obtained with the ASTFT algorithm.The standard S-transform is capable of provid-ing better concentration for the high frequen-cies, but for the linear chirp the concentrationis equivalent to that of the STFT.

From the time-frequency representation ob-tained by the WWOST, it is clear that theconcentration is preserved at high frequencies,while the linear chirp has significantly higherconcentration in comparison to the other rep-resentations. It is also interesting to note howp(t) varies between 0.6 and 1.0 as a functionof time as shown in Fig. 5(e). In particular,p(t) is close to 1 at the beginning of the sig-nal in order to achieve good concentration of


0 0.2 0.4 0.6 0.8 1-2

0

2

Time (s)

(a)

Am

plit

ud

e

Fre

qu

en

cy (

Hz)

Time (s)

(b)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

qu

en

cy (

Hz)

Time (s)(c)

0 0.2 0.4 0.6 0.8 10

50

100

Fre

qu

en

cy (

Hz)

Time (s)(d)

0 0.2 0.4 0.6 0.8 10

50

100

0 0.2 0.4 0.6 0.8 10

0.5

1

Time (s)

(e)

Am

plit

ud

e

Fre

qu

en

cy (

Hz)

Time (s)

(f)

0 0.2 0.4 0.6 0.8 10

50

100

Fig. 5. Time-frequency analysis of signal with fast variations in frequency: (a) Time-domain representation; (b)STFT of x3(t); (c) ASTFT of x3(t) with σopt(t); (d) Spx(t, f) of x3(t) with p = 1 (standard S-transform);(e) p(t); (f) Spx(t, f) of x3(t) with the optimal p(t).

the high frequency component. As time pro-gresses, the value of p(t) decreases in order toprovide a good concentration at the lower fre-quencies. Towards the end of the signal, p(t)increases again to achieve a good time local-ization of the signal.

In Section III, it was stated that for thecomponents with faster variations, it is recom-mended that the time-varying algorithm withthe WWOST be used. In order to substan-tiate that statement, the performance mea-sure implemented in the previous examples isused again and the results are shown in TableIII. The optimized time-invariant value of theparameter popt for this signal, found accord-ing to (12), is determined numerically to be0.71. These performance measures verify thatthe time-varying algorithm should be used forthe faster varying components. For compar-

ison purposes, the performance measures forthe representations given by the STFT andits time-invariant (σopt = 0.048) and time-varying adaptive algorithms are calculated aswell. By comparing the values of the per-formance measure for different time-frequencytransforms, these values confirm the earlierstatement that the each algorithm for theWWOST produces more concentrated time-frequency representation in its respective classthan the ASTFT.

In the analysis performed so far, it was as-sumed that the signal-to-noise ratio (SNR) isinfinity, i.e., the noise-free signals were consid-ered. It would be beneficial to compare theperformance of the considered algorithms inthe presence of additive white Gaussian noisein order to understand whether the proposedalgorithm is capable of providing the enhanced


TABLE III

P�� -�� E"��!� 3.

TFR ΞTF (Noise-free) ΞTF (SNR = 25 dB)STFT 0.0106 0.0100ASTFT with σopt 0.0121 0.0114ASTFT with σopt(t) 0.0122 0.0113WWOST with p 0.0122 0.0110WWOST with p(t) 0.0126 0.0116

performance in noisy environment. Hence, thesignal x3(t) is contaminated with the additivewhite Gaussian noise and it is assumed thatSNR = 25 dB. The results of such an analy-sis are summarized in Table III. Even though,the performance has degraded in comparisonto the noiseless case, the WWOST with p(t)still outperforms the other considered repre-sentations.

B. Demonstration Example

In order to illustrate the effectiveness of theproposed scheme, the method has been appliedto the analysis of engine knocks. A knock isan undesired spontaneous auto-ignition of theunburned air-gas mixture causing a rapid in-crease in pressure and temperature. This canlead to serious problems in spark ignition carengines, e.g., environment pollution, mechani-cal damages, and reduced energy efficiency [21][22]. In this paper, a focus will be on the analy-sis of knock pressure signals.

It has been previously shown that high-pass filtered pressure signals in the presenceof knocks can be modeled as multi-componentFM signals [22]. Therefore, the goal of thisanalysis is to illustrate how effectively theproposed WWOST can decouple these com-ponents in time-frequency representation. Aknock pressure signal recorded from a 1.81Volkswagen Passat engine at 1200 rpm is con-sidered. Note that the signal is high-pass fil-tered with a cut-off frequency of 3000 Hz. Thesampling rate is fs = 100 kHz and the signalcontains 744 samples.

The performance of the proposed schemein this case is evaluated by comparing itwith that of the STFT, the ASTFT, andthe standard S-transform. The results are

shown in Figs. 6 and 7. These resultsrepresent two sample cases from fifty tri-als. For the STFT, a Gaussian window,with a standard deviation of 0.3 millisec-onds, is used for both cases. The optimiza-tion of the standard deviation for the ASTFTis performed on the set of values definedby σgw = {0.01n : n ∈ N and � ≤ � ≤ ��}milliseconds.

A comparison of these representations showthat the WWOST performs significantly bet-ter than the standard S-transform. The pres-ence of several signal components can be easilyidentified with the WWOST, but rather diffi-cult with the standard S-transform. In addi-tion, both proposed algorithms produce higherconcentration than the STFT and the corre-sponding class of the ASTFT. This is accu-rately depicted through the results presentedin Table IV. The best concentration is achivedwith the time-varying algorithm, while thetime invariant value p produces slightly higherconcentration than the ASTFT with the time-invariant value of σgw (σopt = 0.2 millisecondsfor the signal in Fig. 6 and σopt = 0.19 mil-liseconds for the signal in Fig. 7).

The direct implication of the results is thatthe WWOST could potentially be used for theknock pressure signal analysis. A major ad-vantage of such an approach in comparison tosome existing methods is that the signals couldbe modeled based on a single observation, in-stead of multiple realizations required by someother time-frequency methods such as Wigner-Ville distribution [23], since the WWOST doesnot suffer from the cross-terms present in thebilinear transforms.


0 2 4 6

-1

0

1

Time (ms)

(a)

Am

plit

ud

e (

kP

a)

Time (ms)

(b)

Fre

qu

en

cy (

kH

z)

0 2 4 60

1

2

3

Time (ms)(c)

Fre

qu

en

cy (

kH

z)

0 2 4 60

1

2

3

Time (ms)(d)

Fre

qu

en

cy (

kH

z)

0 2 4 60

1

2

3

Time (ms)

(e)

Fre

que

ncy (

kH

z)

0 2 4 60

1

2

3

Time (ms)

(f)

Fre

que

ncy (

kH

z)

0 2 4 60

1

2

3

Fig. 6. Time-frequency analysis of engine knock pressure signal (17th trial): (a) Time-domain representation; (b)STFT; (c) ASTFT with σopt(t); (d) S

px(t, f) with p = 1 (standard S-transform); (e)Spx(t, f) with p = 0.86;

(f) Spx(t, f) with the optimal p(t).

TABLE IV

C�� %� ��!� ��!�.

Trial ΞSTFT ΞASTFTσopt ΞASTFTσopt(t) Ξp=1 Ξpopt Ξpopt(t)17th trial 0.0057 0.0059 0.0068 0.0054 0.0065 0.007448th trial 0.0052 0.0054 0.0060 0.0053 0.0058 0.0069

C. Remarks

It should be noted that, in some cases, whenimplementing the proposed algorithm, it maybe beneficial to window the signal before eval-uating Spx(t, f) in step 1. This additional stepdiminishes the effects of a discrete implemen-tation. As shown in [24], wideband signalsmight lead to some irregular results unless theyare properly windowed.

The STFT and the ASTFT are valublesignal decomposition based representations,

which can achieve good energy concentra-tion for a wide variety of signals. How-ever, throughout this manuscript, it is shownthat the proposed optimization of the windowwidth used in the S-transform is beneficial, andin presented cases, outperforms other standardlinear techniques, such as the STFT and theASTFT. It is also crucial to mention that theWWOST is designed to achieve better concen-tration in the class of the time-frequency repre-sentations based on the signal decomposition.

In comparison to the standard S-transform


0 2 4 6

-1

0

1

Time (ms)

(a)

Am

plit

ude (

kP

a)

Time (ms)

(b)

Fre

que

ncy (

kH

z)

0 2 4 60

1

2

3

Time (ms)(c)

Fre

qu

en

cy (

kH

z)

0 2 4 60

1

2

3

Time (ms)(d)

Fre

qu

en

cy (

kH

z)

0 2 4 60

1

2

3

Time (ms)

(e)

Fre

quen

cy (

kH

z)

0 2 4 60

1

2

3

Time (ms)

(f)

Fre

quen

cy (

kH

z)

0 2 4 60

1

2

3

Fig. 7. Time-frequency analysis of engine knock pressure signal (48th trial): (a) Time-domain representation; (b)STFT; (c) ASTFT with σopt(t); (d) S

px(t, f) with p = 1 (standard S-transform); (e)Spx(t, f) with p = 0.87;

(f) Spx(t, f) with the optimal p(t).

or the STFT, the WWOST does have a highercomputational complexity. The algorithm forthe WWOST is based on an optimization pro-cedure and requires a parameter tuning. How-ever, when compared to the transforms ofsimilar group algorithms (e.g. ASTFT), theWWOST has almost the same degree of com-plexity.

The sampled data version of the standardS-transform and their MATLAB implementa-tions have been discussed in several publica-tions [5][7], [25]-, [27]. The WWOST is astraightforward extension of the standard S-transform. Therefore, the sampled data ver-sion of the WWOST follows the steps pre-sented in earlier publications.

V. C��!��

In this paper, a scheme for improvement ofthe energy concentration of the S-transformhas been developed. The scheme is based onthe optimization of the width of the windowused in the transform. The optimization is car-ried out by means of a newly introduced pa-rameter. Therefore, the developed techniqueis referred to as a window width optimizedS-transform (WWOST). Two algorithms forparameter optimization have been developed:one for finding an optimal constant value ofthe parameter p for the entire signal; whilethe other is to find a time-varying parameter.The proposed scheme is evaluated and com-pared with the standard S-transform by usinga set of synthetic test signals. The results haveshown that the WWOST can achieve better


energy concentration of the signals in compari-son with the standard S-transform. As demon-strated, the WWOST is capable of achievinghigher concentration than other standard lin-ear methods, such as the STFT and its adap-tive form. Furthermore, the proposed tech-nique has also been applied to engine knockpressure signal analysis, and the results haveindicated that the proposed technique providesa consistent improvement over the standard S-transform.

A&��%!��

Ervin Sejdic and Jin Jiang would like tothank the Natural Sciences and EngineeringResearch Council of Canada (NSERC) for fi-nancially supporting this work.

R��

[1] L. Cohen, Time-Frequency Analysis. EnglewoodCliffs, N.J: Prentice Hall PTR, 1995.

[2] S. G. Mallat, A Wavelet Tour of Signal Process-ing, 2nd ed. San Diego: Academic Press, 1999.

[3] K. Gröchenig, Foundations of Time-FrequencyAnalysis. Boston: Birkhäuser, 2001.

[4] I. Djurovic and LJ. Stankovic, “A virtual instru-ment for time-frequency analysis,” IEEE Trans-actions on Instrumentation and Measurement,vol. 48, no. 6, pp. 1086—1092, Dec. 1999.

[5] R. G. Stockwell, L. Mansinha, and R. P.Lowe, “Localization of the complex spectrum:The S-transform,” IEEE Transactions on SignalProcessing, vol. 44, no. 4, pp. 998—1001, Apr.1996.

[6] S. Theophanis and J. Queen, “Color display of thelocalized spectrum,” Geophysics, vol. 65, no. 4,pp. 1330—1340, Jul./Aug. 2000.

[7] C. R. Pinnegar and L. Mansinha, “The S-transform with windows of arbitrary and varyingshape,” Geophysics, vol. 68, no. 1, pp. 381—385,Jan./Feb. 2003.

[8] ––, “The bi-Gaussian S-transform,” SIAMJournal on Scientific Computing, vol. 24, no. 5,pp. 1678—1692, 2003.

[9] M. Varanini, G. De Paolis, M. Emdin, A. Macer-ata, S. Pola, M. Cipriani, and C. Marchesi, “Spec-tral analysis of cardiovascular time series by theS-transform,” in Computers in Cardiology 1997,Lund, Sweden, Sep. 7—10, 1997, pp. 383—386.

[10] G. Livanos, N. Ranganathan, and J. Jiang, “Heartsound analysis using the S-transform,” in Com-puters in Cardiology 2000, Cambridge, MA, Sep.24—27, 2000, pp. 587—590.

[11] E. Sejdic and J. Jiang, “Comparative study ofthree time-frequency representations with appli-cations to a novel correlation method,” in Proc.of IEEE International Conference on Acoustics,Speech, and Signal Processing (ICASSP 2004),vol. 2, Montréal, Canada, May 17—21, 2004, pp.633—636.

[12] P. D. McFadden, J. G. Cook, and L. M. Forster,“Decomposition of gear vibration signals by thegeneralized S-transform,” Mechanical Systemsand Signal Processing, vol. 13, no. 5, pp. 691—707,Sep. 1999.

[13] A. G. Rehorn, E. Sejdic, and J. Jiang, “Faultdiagnosis in machine tools using selective re-gional correlation,” Mechanical Systems and Sig-nal Processing, vol. 20, no. 5, pp. 1221—1238, Jul.2006.

[14] P. K. Dash, B. K. Panigrahi, and G. Panda,“Power quality analysis using S-transform,” IEEETransactions on Power Delivery, vol. 18, no. 2,pp. 406—411, Apr. 2003.

[15] E. Sejdic and J. Jiang, “Selective regional cor-relation for pattern recognition,” IEEE Transac-tions on Systems, Man and Cybernetics - Part A,vol. 37, no. 1, pp. 82—93, Jan. 2007.

[16] LJ. Stankovic, “An analysis of some time-frequency and time-scale distributions,” Annalesdes Telecommunications, vol. 49, no. 9/10, pp.505—517, Sep./Oct. 1994.

[17] ––, “A measure of some time-frequency distrib-utions concentration,” Signal Processing, vol. 81,no. 3, pp. 621—631, Mar. 2001.

[18] D. L. Jones and T. W. Parks, “A high reso-lution data-adaptive time-frequency representa-tion,” IEEE Transactions on Acoustics, Speech,and Signal Processing, vol. 38, no. 12, pp. 2127—2135, Dec. 1990.

[19] T. H. Sang and W. J. Williams, “Rényi infor-mation and signal dependent optimal kernel de-sign,” in Proc. of IEEE International Confer-ence on Acoustics, Speech, and Signal Processing(ICASSP 1995), vol. 2, Detroit, MI, USA, May8—12, 1995, pp. 997—1000.

[20] W. J. Williams, M. L. Brown, and A. O. Hero,“Uncertainty, information and time-frequencydistributions,” in Proc. of SPIE Conference onAdvanced Signal Processing Algorithms, Archi-tectures, and Implementations II, vol. 1566, SanDiego, CA, USA, Jul. 24, 1991, pp. 144—156.

[21] M. Urlaub and J. F. Böhme, “Evaluation ofknock begin in spark ignition engines by leastsquares,” in Proc. of IEEE International Confer-ence on Acoustics, Speech, and Signal Processing(ICASSP 2005), vol. 5, Philadelphia, PA, USA,Mar. 18—23, 2005, pp. 681—684.

[22] I. Djurovic, M. Urlaub, LJ. Stankovic, and J. F.Böhme, “Estimation of multicomponent signalsby using time-frequency representations with ap-plication to knock signal analysis,” in Proc.of 2004 European Signal Processing Conference(EUSIPCO 2004), Vienna, Austria, Sep. 6—10,2004, pp. 1785—1788.

[23] D. König and J. F. Böhme, “Application of cyclo-stationary and time-frequency signal analysis tocar engine diagnosis,” in Proc. of IEEE Interna-tional Conference on Acoustics, Speech, and Sig-nal Processing (ICASSP 1994), vol. 4, Adelaide,SA, Australia, Apr. 19—24, 1994, pp. 149—152.

[24] F. Gini and G. B. Giannakis, “Hybrid FM-polynomial phase signal modeling: Parameter es-timation and Cramér-Rao bounds,” IEEE Trans-actions on Signal Processing, vol. 47, no. 2, pp.363—377, Feb. 1999.

[25] R. G. Stockwell, “S-transform analysis of gravity


wave activity from a small scale network of air-glow imagers,” Ph.D. dissertation, The Universityof Western Ontario, London, Ontario, Canada,Sep. 1999.

[26] C. R. Pinnegar and L. Mansinha, “Time-localFourier analysis with a scalable, phase-modulatedanalyzing function: the S-transform with a com-plex window,” Signal Processing, vol. 84, no. 7,pp. 1167—1176, Jul. 2004.

[27] C. R. Pinnegar, “Time-frequency and time-timefiltering with the S-transform and TT-transform,”Digital Signal Processing, vol. 15, no. 6, pp. 604—620, Nov. 2005.

TIME-FREQUENCY SIGNAL ANALYSIS 923 AWindowWidthOptimized … · TIME-FREQUENCY SIGNAL ANALYSIS 923...

Documents

Transcript of TIME-FREQUENCY SIGNAL ANALYSIS 923 AWindowWidthOptimized … · TIME-FREQUENCY SIGNAL ANALYSIS 923...