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This module introduces short-time Fourier transform.

We saw earlier that Fourier analysis is not well suited to describing local changes in "frequency content" because thefrequency components defined by the Fourier transform have infinite ( i.e. , global) time support. For example, if we have a signal with periodic components plus aglitch at time t 0 , we might want accurate knowledge of both the periodic component frequencies and the glitch time ( [link] ).

The Short-Time Fourier Transform (STFT) provides a means of joint time-frequency analysis. The STFT pair can be written X STFT Ω τ t x t w t τ Ω t x t 1 2 t Ω X STFT Ω τ w t τ Ω t assuming real-valued w t for which t w t 2 1 . The STFT can be interpreted as a "sliding window CTFT": to calculate X STFT Ω τ , slide the center of window w t to time τ , window the input signal, and compute the CTFT of the result ( [link] ).

"sliding window ctft"

The idea is to isolate the signal in the vicinity of time τ , then perform a CTFT analysis in order to estimate the "local" frequency content attime τ .

Essentially, the STFT uses the basis elements b Ω , τ t w t τ Ω t over the range t and Ω . This can be understood as time and frequency shifts of the window function w t . The STFT basis is often illustrated by a tiling of the time-frequency plane, where each tile represents aparticular basis element ( [link] ):

The height and width of a tile represent the spectral and temporal widths of the basis element, respectively, and theposition of a tile represents the spectral and temporal centers of the basis element. Note that, while the tiling diagram suggests that the STFT uses a discrete set of time/frequency shifts, the STFT basis is really constructed froma continuum of time/frequency shifts.

Note that we can decrease spectral width Δ Ω at the cost of increased temporal width Δ t by stretching basis waveforms in time, although the time-bandwidth product Δ t Δ Ω ( i.e. , the area of each tile) will remain constant ( [link] ).

Our observations can be summarized as follows:

  • the time resolutions and frequency resolutions of every STFT basis element will equal those of the window w t . (All STFT tiles have the same shape.)
  • the use of a wide window will give good frequency resolution but poor time resolution, while the use of a narrow windowwill give good time resolution but poor frequency resolution. (When tiles are stretched in one direction theyshrink in the other.)
  • The combined time-frequency resolution of the basis, proportional to 1 Δ t Δ Ω , is determined not by window width but by window shape. Of all shapes, the Gaussian The STFT using a Gaussian window is known as the Gabor Transform (1946). w t 1 2 1 2 t 2 gives the highest time-frequency resolution, although its infinite time-support makes it impossible toimplement. (The Gaussian window results in tiles with minimum area.)
Finally, it is interesting to note that the STFT implies a particular definition of instantaneous frequency . Consider the linear chirp x t Ω 0 t 2 . From casual observation, we might expect an instantaneous frequency of Ω 0 τ at time τ since t τ Ω 0 t 2 Ω 0 τ t The STFT, however, will indicate a time- τ instantaneous frequency of t τ t Ω 0 t 2 2 Ω 0 τ
The phase-derivative interpretation of instantaneous frequency only makes sense for signals containingexactly one sinusoid, though! In summary, always remember that the traditional notion of "frequency"applies only to the CTFT; we must be very careful when bending the notion to include, e.g. , "instantaneous frequency", as the results may be unexpected!

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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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