Short-Time Fourier Transformation

Given the limitations of the FT, some approximations are needed to handle nonstationary signals. The discrete FT (DFT) and the short-time FT (STFT, a.k.a. the Gabor transform) are two alternative transformation methods that address this issue. 3 jn the mid-20th century, Jean Ville pointed out that 

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Recently, Darowicki [29, 30] has presented a new mode of electrochemical impedance measurements. This method employed a short time Fourier transformation to impedance evaluation. The digital harmonic analysis of cadmium-ion reduction on mercury electrode was presented [31]. A modern concept in nonstationary electrochemical impedance spectroscopy theory and experimental approach was described [32]. The new investigation method allows determination of the dependence of complex impedance versus potential [32] and time [33]. The reduction of cadmium on DM E was chosen to present the possibility of these techniques. Figure 2 illustrates the change of impedance for the Cd(II) reduction on the hanging drop mercury electrode obtained for the scan rate 10 mV s k... 

Analysis In standard applications, the short-time Fourier transform analysis is performed at a constant rate the analysis time-instants / are regularly spaced, i.e. tua =uR where R is a fixed integer increment which controls the analysis rate. However, in pitch-scale and time-scale modifications, it is usually easier to use regularly spaced synthesis time-instants, and possibly non-uniform analysis time-instants. In the so-called band-pass convention, the short-time Fourier transform X (t",Q.k) is defined by ... 

Fourier channel k. The phase is known up to a multiple of 27t(since only exp /( ),(/")) is known). Time-scale modifications also require the knowledge of the instantaneous frequency G) (f ). 0), Uua) can also be estimated from successive short-time Fourier transforms for a given value of k, computing the backward difference of the short-time Fourier transform phase yields... 

Time scaling. Because the phase-vocoder (the short-time Fourier transform) gives access to the implicit sinusoidal model parameters, the ideal time-scale operation described by Eq. (7.3) can be implemented in the same framework. Synthesis time-instants / are usually set at a regular interval / +1 - / = R. From the series of synthesis time-instants / analysis time-instants / are calculated according to the desired time warping function tua = T l(t ). The short-time Fourier transform of the time-scaled signal is then ... 

Puckette in [Puckette, 1995] proposes an alternate way of computing the phases and the amplitudes of the short-time Fourier transform at the synthesis instants, replacing the calculation of the arc tangent and the phase-unwrapping stage by another Fourier transform. Essentially, in Eq. 7.15 the phase increment can also be estimated if the phase < >fc(t ) of the input signal at time ta = t + t - t 1 is known ... 

Nawab and Quatieri, 1988a] Nawab, H. and Quatieri, T. (1988a). Short-time fourier transform. In Lim, J. S. and Oppenheim, A. V., editors, Advanced Topics in Signal Processing. Prentice Hall, Englewood Cliffs, New Jersey. 

Nawab et al., 1983] Nawab, S., Quatieri, T., and Lim, J. (1983). Signal Reconstruction from Short-Time Fourier Transform Magnitude. IEEE Trans. Acoust., Speech, Signal Processing ASSP-31(4) 986-998. 

In most STSA techniques the short-time analysis of the signal is performed by use of the Short-Time Fourier Transform (STFT) [Lim and Oppenheim, 1979, Boll, 1991, Ephraim and Malah, 1984, Moorer and Berger, 1986], or with a uniform filter-bank that can be implemented by STFT [Sondhi et al., 1981, Vary, 1985, Lagadec and Pelloni, 1983], Note that in such cases the two interpretations (multirate filter-... 

Suppression rules. Let X(p,Qk) denote the short-time Fourier transform of x[ri, where p is the time index, and Qk the normalized frequency index (0t lies between 0 and 1 and takes N discrete values for k = 1,N, Wbeing the number of sub-bands). Note that the time index p usually refers to a sampling rate lower than the initial signal sampling rate (for the STFT, the down-sampling factor is equal to hop-size between to consecutive short-time frames) [Crochiere and Rabiner, 1983]. 

The Phase Vocoder. The Phase Vocoder [Flanagan and Golden, 1966][Gordon and Strawn, 1985] is a common analysis technique because it provides an extremely flexible method of spectral modification. The phase vocoder models the signal as a bank of equally spaced bandpass filters with magnitude and phase outputs from each band. Portnoff s implementation of the Short Time Fourier Transform (STFT) provides a time-efficient implementation of the Phase Vocoder. The STFT requires a fast implementation of the Fast Fourier Transform (FFT), which typically involves bit addressed arithmetic. 

Methods based on the short-time Fourier transform... 

The reader can refer to chapter 9.2.2 for an alternative presentation of the short-time Fourier transform, in the context of sinusoidal modeling. 

Perfect reconstruction One can show that the short-time Fourier transform yields perfect reconstruction in the absence of modification (i.e. a synthesis signal y(n) exactly similar to the original x(n) when / = / and Y(t , lk) = X (t , lk)) if... 

Short-time Fourier transform of a sinusoidal signal. When the signal corresponds to the model of Eq. (7.1), the short-time Fourier transform can be expressed in terms of the model parameters. Substituting Eq. (7.1) into Eq. (7.6) yields... 

Eq. (7.11) shows that the short-time Fourier transform gives access to the instantaneous amplitude A i(t ), and the instantaneous phase c u (t ) of the sinusoid i which falls into... 

Compute the short-time Fourier transform at next analysis time-instant +1 and calculate the instantaneous frequency in each channel according to Eq. (7.14). 

A short-time Fourier transform analysis is first carried out on the signal to be resampled, at regularly spaced analysis time-instants / = Ru. 

An analysis/synthesis system based on a filter bank representation of the signal can be derived from the time-dependent short-time Fourier transform (STFT) [Nawab and Quatieri, 1988a]... 

Allen, 1982] Allen, J. (1982). Application of the Short-Time Fourier Transform to Speech Processing and Spectral Analysis. Proc. IEEE ICASSP-82, pages 1012— 1015. 

Technische Universitat Wien Does latent class analysis, short-time Fourier transform, fuzzy clustering, support vector machines, shortest path computation, bagged clustering, naive Bayes classifier, etc. (http //cran.r-project.org/ web/packages/el071/index.html)... 

In this paper, we utilized the method of short-time Fourier transform (spectrogram). The time-frequency ( -a>) distributions P(, at) of the time-dependent signal s(t) is given as follows ... 

Figure 2.6 shows the results of short-time Fourier transform corresponding to the modulation frequency power spectra of these two instantaneous... 

When localisation is an issue, the intuitive solution still making use of the Fourier transform would be to cut up the signal and to transform the pieces. This approach is called the short-time Fourier transform, it adds a dimension to the Fourier transform, namely time, as it allows following frequencies over time. Where the Fourier transform is a frequency analysis, the short-time Fourier transform is a time-frequency analysis. Instead of describing the signal in either the time or the frequency domain, we describe it in both, a joint time-frequency domain. When we do this, we are faced with a fundamental limitation we cannot localise in the time domain and the frequency domain at the same time. 

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