The Short-Time Fourier Transform

Fourier analysis has been called the most important mathematical tool in the history of engineering. The Fast Fourier Transform makes Fourier Analysis economical and much more useful. In the next chapter, weTl look at the spectra of different kinds of signals, and some tools for better analyzing, displaying, and modeling the spectra of sound source objects and systems. 

Ronald Bracewell. The Fourier Transform and its Applications. Boston McGraw Hill, 2000. 

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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 ... 

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). 

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. 

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. 

The windowed Fourier transform [2,3] (also called the short time Fourier transform) was introduced so that the frequency information about a signal could be localised with respect to time. Instead of analysing the function f(t) as a whole, the windowed Fourier transform performs a Fourier transform on pieces of the function. The pieces are obtained by using a windowing function G(t) which slides across the function. The windowed Fourier transform of f(t) is defined as... 

Some of the commonly used algorithms for time-frequency representations of spectral estimates include the short-time Fourier transform (STFT), the Wigner-Ville transform, wavelet transforms, etc. A good summary of the mathematical basis of each of these techniques can be found in Ref. 11. 

To address the aforementioned issues of ICA, it is proposed by Yang and Nagarajaiah (2013a) to transform the time-domain modal expansion Eq. 11 to the time-frequency domain where the target modal responses have sparse representations, using the short-time Fourier transform (STFT) prior to the ICA estimation. 

A system identification method is considered parametric if a mathematical dynamic model (often formulated in state-space) is realized in a first step and the dynamic properties of the system estimated from the realized model in the second step. Nonparametric system identification methods directly estimate the dynamic parameters of a system from transformation of data, e.g., Fourier transform or power-spectral density estimation. Time-domain identification methods estimate the dynamic parameters of a system by directly using the measured response time histories, while frequency-domain methods use the Fourier transformation or power-spectral density estimation of the measured time histories. There is also a class of time-frequency methods such as the short-time Fourier transform and the wavelet transform. These methods are commonly used for identification of time-varying systems in which the dynamic properties are time-variant Linear system identification methods are based mi the assumption that the system behaves linearly and... 

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