Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Digital Signal Processing (Ohio State EE700) » Short-time Fourier Transform

Navigation

Table of Contents

Recently Viewed

This feature requires Javascript to be enabled.
 

Short-time Fourier Transform

Module by: Phil Schniter. E-mail the author

Summary: 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 t0 , we might want accurate knowledge of both the periodic component frequencies and the glitch time (Figure 1).

Figure 1
Figure 1 (stft.png)

The Short-Time Fourier Transform (STFT) provides a means of joint time-frequency analysis. The STFT pair can be written XSTFT Ωτ=xtwtτe(jΩt)d t XSTFT Ω τ t x t w t τ Ω t xt=12π XSTFT ΩτwtτejΩtdΩd t x t 1 2 t Ω XSTFT Ω τ w t τ Ω t assuming real-valued wt w t for which |wt|2dt=1 t w t 2 1 . The STFT can be interpreted as a "sliding window CTFT": to calculate XSTFT Ωτ XSTFT Ω τ , slide the center of window wt w t to time ττ, window the input signal, and compute the CTFT of the result (Figure 2).

Figure 2
"Sliding Window CTFT"
"Sliding Window CTFT" (sliding_window_ctft.png)

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

Essentially, the STFT uses the basis elements b Ω , τ t=wtτejΩt b Ω , τ t w t τ Ω t over the range t t and Ω Ω . This can be understood as time and frequency shifts of the window function wt w t . The STFT basis is often illustrated by a tiling of the time-frequency plane, where each tile represents a particular basis element (Figure 3):

Figure 3
Figure 3 (time_frequency_tiling.png)

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 ΔΩ at the cost of increased temporal width Δt Δt by stretching basis waveforms in time, although the time-bandwidth product Δt ΔΩ Δt ΔΩ (i.e., the area of each tile) will remain constant (Figure 4).

Figure 4
Figure 4 (wndow_scaling.png)

Our observations can be summarized as follows:

  • the time resolutions and frequency resolutions of every STFT basis element will equal those of the window wt 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 1 Δt ΔΩ , is determined not by window width but by window shape. Of all shapes, the Gaussian 1 wt=12πe(12t2) 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 xt=sin Ω0 t2 x t Ω0 t 2 . From casual observation, we might expect an instantaneous frequency of Ω0 τ Ω0 τ at time ττ since sin Ω0 t2=sin Ω0 τt  ,   t=τ    t τ Ω0 t 2 Ω0 τ t The STFT, however, will indicate a time-ττ instantaneous frequency of ddt Ω0 t2| t=τ =2 Ω0 τ t τ t Ω0 t 2 2 Ω0 τ

Caution:

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!

Footnotes

  1. The STFT using a Gaussian window is known as the Gabor Transform (1946).

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks