Short-Time Fourier Analysis (Mertins - Signal Analysis (Revised Edition))

Signal Analysis: Wavelets,Filter Banks, Time-Frequency Transforms andApplications. Alfred MertinsCopyright 0 1999 John Wiley & Sons Ltdprint ISBN 0-471-98626-7 Electronic ISBN 0-470-84183-4Chapter 7Short-TimeFourier AnalysisA fundamental problem in signal analysis is to find the spectral componentscontainedin a measuredsignal z ( t ) and/or to provide information aboutthe time intervals when certain frequencies occur. An example of what weare looking for is a sheet of music, which clearly assigns time to frequency,see Figure 7.1. The classical Fourier analysis only partly solves the problem,because it does not allow an assignment of spectralcomponents to time.Therefore one seeks other transforms which give insight into signal propertiesin a different way.

The short-time Fourier transform is such a transform. Itinvolves both time and frequency and allows a time-frequency analysis, or inother words, a signal representation in the time-frequency plane.7.17.1.1Continuous-Time SignalsDefinitionThe short-time Fouriertransform (STFT) is the classical method of timefrequency analysis. The concept is very simple. We multiply z ( t ) ,which is tobe analyzed, with an analysis window y*(t - T) and then compute theFourier196197Signals 7.1. Continuous-TimeIFigure 7.1.

Time-frequency representation.Figure 7.2. Short-timeFouriertransform.transform of the windowed signal:ccF ~ ( WT) ,=z ( t )y * ( t - T) ,-jutJdt.-ccThe analysis window y*(t - T) suppresses z ( t ) outside a certain region,and the Fourier transform yields a local spectrum. Figure 7.2 illustrates theapplication of the window.

Typically, one will choose a real-valued window,which may be regarded as the impulse response of a lowpass. Nevertheless,the following derivations will be given for the general complex-valued case.Ifwe choose the Gaussian function to be the window, we speak of theGabor transform, because Gabor introduced the short-time Fourier transformwith this particular window [61].Shift Properties. As we see from the analysis equation (7.1), a time shiftz ( t ) + z(t - t o ) leads to a shift of the short-time Fourier transform by t o .Moreover, a modulation z ( t ) + z ( t ) ejwot leads to a shift of the short-timeFourier transform by W O .

As we will see later, other transforms, such as thediscrete wavelet transform, do not necessarily have this property.198Chapter 7. Short-Time Fourier Analysis7.1.2Time-Frequency ResolutionApplying the shift and modulation principle of the Fourier transform we findthe correspondence~ ~ ; , ( t:=) ~ (- rt ) ejwt:=r7;Wk)S__r(t- 7) e- j ( v(7.2)-w)tdt = r ( v - W ) e-j(v-From Parseval's relation in the formJ -03we concludeThat is, windowing in the time domain with y * ( t - r ) simultaneously leadsto windowing in the spectral domain with the window r*(v- W ) .Let us assume that y*(t - r ) and r*(v - W ) are concentrated in the timeand frequency intervalsand[W + W O- A,, W +WO+ A,],respectively.

Then Fz(r,W ) gives information on a signalz ( t )and its spectrumX ( w ) in the time-frequency window[7+t0 -At,r+to+At]X[W + W O-A,, W +WO+A,].(7.7)The position of the time-frequency window is determined by the parameters rand W . The form of the time-frequency window is independent of r and W , sothat we obtain a uniform resolution in the time-frequency plane, as indicatedin Figure 7.3.7.1. Continuous-TimeSignals199........mAO A;, . . . .

. . . ~W 2 ............*tto-&totto+&(4z1z2z(b)Figure 7.3. Time-frequency window of the short-time Fourier transform.Let us now havea closer look at the size and position of the time-frequencywindow. Basic requirements for y*(t) tobe called a time window are y*(t)EL2(R) and t y*(t)E L2(R). Correspondingly, we demand that I'*(w) E L2(R)and W F*( W ) E Lz(R) for I'*(w) being a frequency window. The center t o andthe radius A, of the time window y*(t) are defined analogous to the meanvalue and the standard deviation of a random variable:Accordingly, the centerWOand the radius A, of the frequency windowr * ( w ) are defined as(7.10)(7.11)The radius A, may be viewed as half of the bandwidth of the filter y*(-t).Intime-frequencyanalysisoneintendsto achieve both high timeandfrequency resolution if possible. In other words, one aims at a time-frequencywindow that is as small aspossible.

However, the uncertainty principle applies,giving a lower bound for the areaof the window. Choosing a short timewindowleads to good time resolution and, inevitably, to poor frequency resolution.On the other hand, a long time window yields poor time resolution, but goodfrequency resolution.2007.1.3Chapter 7.

Short-Time Fourier AnalysisThe Uncertainty PrincipleLet us consider the term (AtAw)2,whichis the square of the area in thetime-frequency plane beingcovered by the window. Without loss of generalitywe mayassume J t Ir(t)12dt = 0 and S W l r ( w ) I 2 dw = 0, becausetheseproperties are easily achieved for any arbitrary window by applying a timeshift and a modulation.

With (7.9) and (7.11) we haveFor the left term in the numerator of (7.12), we may write(7.13)J -ccwith [ ( t )= t y ( t ) .Using the differentiation principle of the Fourier transform,the right term in the numerator of (7.12) may be written asmLccw2 lP(W)I2 dw= ~ m l F { r ’ (l2t )dw}=%T(7.14)llY‘ll2111112where y’(t) = $ y ( t ) . With (7.13),(7.14) and= 27r lly112 we get for(7.12)1ll-Y‘1I2(AtA,)2 = ll-Y114 1 1 t 1 1 2(7.15)Applying the Schwarz inequality yields(AtAJ2 22&f I (t,-Y’)l2&f lR{(t,-Y’)l l2(7.16)By making use of the relationship18 { t y ( t )y’*(t)} = 5 tdIr(t)12(7.17)7which can easily be verified, we may write the integralin (7.16) ast y ( t ) y’*(t) d t } =i/_”,tg Ir(t)12dt.(7.18)7.1. Continuous-TimeSignals201Partial integration yieldsThe propertylim t Iy(t)12= 0,(7.20)1tl-wwhich immediately follows from t y ( t ) E La, implies that(7.21)so that we may conclude that14’2that isAtAw 215’(7.22)(7.23)The relation (7.23) is known as the uncertainty principle.

It shows that thesize of a time-frequency windows cannot be made arbitrarily small and thata perfect time-frequency resolution cannot be achieved.In (7.16) we see that equality in (7.23) is only given if t y ( t ) is a multipleof y’(t). In other words, y(t) must satisfy the differential equationt d t )= c(7.24)whose general solution is given by(7.25)Hence, equality in (7.23) isachieved only if y ( t ) is the Gaussian function.If we relax the conditions on the center of the time-frequency window ofy ( t ) ,the general solution with a time-frequency window of minimum size is amodulated and time-shifted Gaussian.7.1.4 The SpectrogramSince the short-time Fourier transform is complex-valued in general, we oftenuse the so-called spectrogrum for display purposes or for further processingstages.

This is the squared magnitude of the short-time Fourier transform:202Chapter 7.FourierShort-Timev v v v v v vUUYYYYYYYYIVYYYYYYYVAnalysisv v v v v v v vt"IFigure 7.4. Example of a short-time Fourieranalysis;(a) test signal; (b) idealtime-frequency representation; (c) spectrogram.Figure 7.4 gives an example of a spectrogram; the values S, (7,W ) are represented by different shades of gray. The uncertainty of the STFT in both timeand frequency can be seen by comparing the result in Figure 7.4(c) with theideal time-frequency representation in Figure 7.4(b).A second example that shows the applicationin speech analysis is picturedin Figure 7.5. The regular vertical striations of varying density are due to thepitch in speech production.

Each striation correspondsto a single pitch period.A high pitch is indicated by narrow spacing of the striations. Resonances inthe vocal tract in voiced speech show up as darker regions in the striations.The resonance frequencies are known as the formantfrequencies. We see threeof them in the voiced section in Figure 7.5. Fricative or unvoiced sounds areshown as broadband noise.7.1.5ReconstructionA reconstruction of z(t) from FJ(T,W ) is possible in the form(7.28)Signals 7.1.

Continuous-Timet -@lFigure 7.5. Spectrogram of a speech signal; (a) signal; (b) spectrogram.We can verify this by substituting (7.1) into (7.27) and by rewriting theexpression obtained:z(t) = L / / / z ( t ’ )27ry*(t’ - r ) e-iwt‘ dt’ g ( t - r ) ejwt d r dw= / x ( t ‘ ) / y * ( t ‘ - T) g ( t - T)27rejw(t-t’) dw d r dt‘(7.29)= /z(t’)/y*(t’ - r ) g(t - r ) 6 ( t - t’) dr dt’.For (7.29) to be satisfied,L006 ( t - t’) =y*(t’ - T) g ( t - T) 6 ( t - t ’ ) d r(7.30)must hold, which is true if (7.28) is satisfied.Therestriction (7.28)is not very tight, so that an infinite number ofwindows g ( t ) can be found which satisfy (7.28). The disadvantage of (7.27) isof course that the complete short-time spectrum must beknown and must beinvolved in the reconstruction.2047.1.6Chapter 7.FourierShort-TimeAnalysisReconstructionvia Series ExpansionSince the transform (7.1) represents aone-dimensional signal in the twodimensional plane, the signal representation is redundant. For reconstructionpurposes this redundancy can be exploited by using only certain regions orpoints of the time-frequency plane.

Reconstruction from discrete samples inthe time-frequency plane is of special practical interest. For this we usuallychoose a grid consisting of equidistant samples as shown in Figure 7.6........................f .............................................* TM....................3W . . . . . . .

. . . . . . . . . . . . . . .O/it -Figure 7.6. Sampling the short-time Fourier transform.Reconstruction is given by0000The sample values F . ( m T ,I ~ u A ) ,m, Ic E Z of the short-time Fouriertransform are nothing but the coefficients of a series expansion of x ( t ) .

In(7.31) we observe that the set of functions used for signal reconstruction isbuilt from time-shifted and modulated versions of the same prototype g@).Thus, each of the synthesis functions covers adistinctareaof the timefrequency plane of fixed size and shape. This type of series expansion wasintroduced by Gabor [61] and is also called a Gabor expansion.Perfect reconstruction according to (7.31) is possible if the condition2T-c00w A m=-mg ( t - mT) y * ( t - mT -2Te-)UA= de0Vt(7.32)is satisfied [72], where de0 is the Kronecker delta.