Short-Time Fourier Analysis (779447), страница 2
Текст из файла (страница 2)
For a given window y ( t ) ,(7.32) representsa linear set of equations for determining g ( t ) . However,here, as with Shannon’s sampling theorem,a minimal sampling rate mustbe guaranteed, since (7.32) can be satisfied only for [35, 7212057.2. Discrete-Time SignalsUnfortunately, for critical sampling, that is for T W A = 27r, and equal analysisand synthesis windows, it is impossible to have both a goodtimeandagoodfrequency resolution. If y ( t ) = g ( t ) is a window that allows perfectreconstruction with critical sampling, then either A, or A, is infinite.
Thisrelationship isknown as the Balian-Low theorem[36]. It shows that it isimpossible to construct an orthonormal short-time Fourier basis where thewindow is differentiable and has compact support.7.2Discrete-Time SignalsThe short-time Fourier transform of a discrete-time signal x(n) is obtainedby replacing the integration in (7.1) by a summation. It is then givenby[4, 119, 321Fz(m,ejw)= C x ( n )r*(n- m ~e-jwn.)(7.34)nHere we assume that the sampling rate of the signal is higher (by the factorN E W) than the rate used for calculating the spectrum.
The analysis andsynthesis windows are denoted as y* and g, as in Section 7.1; in the followingthey aremeant to be discrete-time. Frequency W is normalized to thesamplingfrequency.In (7.34) we must observethat the short-time spectrumis a function of thediscrete parameter m and the continuous parameter W . However, in practiceone would consider only the discrete frequencieswk= 2nIc/M,k = 0 , .
. . , M -(7.35)1.Then the discrete values of the short-time spectrum can be given byX ( m ,Ic) =c).(Xy*(n - m N ) W E ,(7.36)nwhereX ( m ,k ) = F:(,,2Q)(7.37)andW M = e- j 2 ~ / M(7.38)Synthesis. As in (7.31), signal reconstruction from discrete values of thespectrum can be carried out in the formg(.)=ccccM-lm=-mk=OX ( m ,Ic) g(. - m N ) WGkn.(7.39)206Chapter 7. Short-Time Fourier AnalysisThe reconstruction is especially easy for the case N = 1 (no subsampling),because then all PR conditions are satisfied for g(n) = dnO t)G ( e J w )= 1and any arbitrary length-M analysis window ~ ( nwith) $0) = l / M [4, 1191.The analysis and synthesis equations (7.36) and (7.39) then becomeX ( m ,k ) =cr*(n).(X-m) WE(7.40)nandcM-lqn)=X ( n , k ) W&?(7.41)k=OThis reconstruction method is known as spectral summation.
The validity of?(n)= z(n) provided y(0) = 1/M can easily be verified by combining theseexpressions.Regarding the design of windows allowing perfect reconstruction in thesubsampled case, the reader is referred to Chapter 6. As wewill see below,the STFT may be understood as a DFT filter bank.Realizations using Filter Banks. The short-timeFourier transform, whichhas beendefined as theFourier transform of a windowed signal, can berealizedwith filter banks as well.
The analysis equation (7.36) can be interpreted asfiltering the modulated signals z(n)W& with a filterh(n) = r*(-n).(7.42)The synthesis equation (7.39) can be seen as filtering the short-time spectrumwith subsequent modulation. Figure 7.7 shows the realization of the shorttime Fourier transform by means of a filter bank. The windows g ( n ) and r(n)typically have a lowpass characteristic.Alternatively, signal analysis and synthesis can be carried out by meansof equivalent bandpass filters. By rewriting (7.36) aswe see that the analysis can also be realized by filtering the sequence ).(Xwith the bandpass filtershk(lZ) = y*(-n) WGk",and by subsequent modulation.k = 0,. .
. , M - 1(7.44)2077.3. Spectral Subtraction based on the STFTFigure 7.7. Lowpass realization of the short-time Fourier transform.Rewriting (7.39) asccM-I--k(n-mN)(7.45)m=-cc k=Oshows that synthesis canbeachievedwithmodulatedfilters as well. Toaccomplish this, first the short-time spectrum is modulated, then filteringwith the bandpass filtersgk(n) = g ( n )wi-kn,L = 0,.
. . , M-1,(7.46)takes place; see Figure 7.8.We realize thattheshort-timeFourier transformbelongs to the classof modulated filter banks. On the other hand, it has been introduced as atransform, which illustrates the close relationship between filter banks andshort-time transforms.The most efficient realization of the STFTis achieved when implementingit as a DFT polyphase filter bank as outlined in Chapter 6.7.3 Spectral Subtraction based on the STFTInmanyreal-wordsituationsoneencounterssignals distorted by additivenoise. Several methods areavailable for reducing the effect of noise in a more orless optimal way. For example, in Chapter 5.5 optimal linear filters that yielda maximum signal-to-noise ratio were presented.
However, linear methods areChapter 7. Short-Time Fourier Analysis208Figure 7.8. Bandpass realization of the short-time Fourier transform.not necessarily the optimal ones, especially if a subjective signal quality withrespect to human perception is of importance. Spectral subtraction is a nonlinear method for noise reduction, which is very well suited for the restorationof speech signals.We start with the modelwhere we assume that the additive noise process n(t) is statistically independent of the signal ~ ( t Assuming).that the Fourier transform of y ( t ) exists, wehaveY ( w )= X ( w ) N(w)(7.48)+in the frequency domain.
Due to statistical independence between signal andnoise, the energy density may be written as(7.49)If we now assume that E { IN(w)12} is known, the least squares estimate forIX(w)I2 can be obtained aslX(412= IY(w)I2 - E { I N ( w ) 1 2 )*(7.50)Inspectralsubtraction,oneonlytriesto restorethemagnitudeof thespectrum, while the phase is not attached.
Thus, the denoised signal is givenin the frequency domain asX ( w ) = IR(w)I L Y ( w ) .(7.51)7.3. Spectral Subtraction based on the STFT209Keeping the noisy phase is motivated by the fact that the phase is of minorimportance for speech quality.So far, the time dependence of the statistical properties of the signaland the noise process hasnotbeenconsidered. Speech signals are highlynonstationary, but within intervals of about 20 msec, the signal propertiesdo not change significantly, and the assumption of stationarity is valid on ashort-time basis. Therefore, one replaces the above spectra by the short-timespectra computed by the STFT. Assuming a discrete implementation, thisyieldsY ( m ,k ) = X ( m ,k ) N ( m ,k ) ,(7.52)+is the STFT ofwhere m is the time and k is the frequency index. Y(m,k)Y (m).Instead of subtracting an average noise spectrum E { I N ( w ) I 2 } ,one triesto keep track of the actual (time-varying) noise process.
This can for instancebe done by estimating the noise spectrum in the pauses of a speech signal.Equations (7.50) and (7.51) are then replaced byl X ( m 7 k ) 1 2= IY(m,k)12 - p v ( G k ) l 2(7.53)X ( m , k ) = I X ( m ,k)l L Y ( m ,k ) ,(7.54)andhwhere IN(m,k)I2 is the estimated noise spectrum.Y (m,k ) l2 Since it cannot beassured that the short-time spectrasatisfy IIN(m,k)I2 > 0, V m , k , one has to introduce further modifications such asclipping. Several methods for solving this problem and for keeping track ofthe time-varying noise have been proposed. For more detail, the reader isreferred to [12, 50, 51, 60, 491.
Finally, note that a closely related technique,known as wavelet-based denoising, will be studied in Section 8.10.h.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
