Short-time Fourier Transform 2.13 2001/12/30 2005/10/05 15:21:14.374 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Elizabeth Chan lizychan@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu Gabar Transform STFT Time-frequency analysis 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 the frequency components defined by the Fourier transform have infinite (i.e., global) time support. For example, if we have a signal with periodic components plus a glitch at time t0 , we might want accurate knowledge of both the periodic component frequencies and the glitch time ().

The Short-Time Fourier Transform (STFT) provides a means of joint time-frequency analysis. The STFT pair can be written XSTFT Ω τ t x t w t τ Ω t x t 1 2 t Ω XSTFT 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 XSTFT Ω τ , slide the center of window w t to time τ, window the input signal, and compute the CTFT of the result ().

"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 at time 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 a particular basis element ():

The height and width of a tile represent the spectral and temporal widths of the basis element, respectively, and the position 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 from a continuum of time/frequency shifts. Note that we can decrease spectral width ΔΩ at the cost of increased temporal width Δt by stretching basis waveforms in time, although the time-bandwidth product Δt ΔΩ (i.e., the area of each tile) will remain constant ().

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 window will give good time resolution but poor frequency resolution. (When tiles are stretched in one direction they shrink in the other.) The combined time-frequency resolution of the basis, proportional to 1 Δt ΔΩ , 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 to implement. (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 Ω0 t 2 . From casual observation, we might expect an instantaneous frequency of Ω0 τ at time τ since t τ Ω0 t 2 Ω0 τ t The STFT, however, will indicate a time-τ instantaneous frequency of t τ t Ω0 t 2 2 Ω0 τ The phase-derivative interpretation of instantaneous frequency only makes sense for signals containing exactly 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!