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### Mathematical Definition of the STFT

The usual mathematical definition of the STFT is
[9]

where

If the window has the
*Constant OverLap-Add (COLA) property* at hop-size , *i.e.*, if

then the sum of the successive

DTFTs over time equals the DTFT of the
whole

signal :

We will say that windows satisfying
(or some
constant) for all
are said to be
. For example,
the length rectangular window is clearly
(no overlap).
The Bartlett window and all windows in the generalized Hamming family
(Chapter 3) are
(50% overlap),
when the endpoints are handled correctly.^{7.1} A
example
is depicted in
Fig.7.10. Any window that is
is also
,
for
, provided is an
integer.^{7.2} We will explore COLA windows
more completely in Chapter 7.

When using the short-time Fourier transform for signal processing, as
taken up in Chapter 7, the COLA requirement is
important for avoiding artifacts. For usage as a spectrum analyzer
for measurement and display, the COLA requirement can often be
relaxed, as doing so only means we are not weighting all information
equally in our analysis. Nothing disastrous happens, for example, if
we use 50% overlap with the Blackman window in a short-time spectrum
analysis over time--the results look fine; however, in such a case,
data falling near the edges of the window will have a slightly muted
impact on the results relative to data falling near the window center,
because the Blackman window is not COLA at 50% overlap.

**Previous:** The Short-Time Fourier
Transform**Next:** Practical Computation of the STFT

**About the Author: ** Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at

Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See

http://ccrma.stanford.edu/~jos/ for details.