Time Varying OLA Modifications
In the preceding sections, we assumed that the spectral modification
did not vary over time. We will now examine the implications of
time-varying spectral modifications. The derivation below
follows [9], except that we'll keep our previous
notation:

Using
in our OLA formulation with a hop size
results in
![\begin{eqnarray*}
y(n) &=& \sum_{m=-\infty}^\infty y_m(n) \\
&=& \sum_{m=-\infty}^\infty \frac{1}{N}\sum_{k=0}^{N-1} X_m(\omega_k) H_m(\omega_k) e^{j\omega_kn} \\
&=& \sum_{m=-\infty}^\infty \frac{1}{N}\sum_{k=0}^{N-1}
\left[ \sum_{l=-\infty}^\infty x(l) w(l-m)e^{-j\omega_kl} \right]
H_m(\omega_k) e^{j\omega_kn} \\
&=& \sum_{l=-\infty}^\infty x(l) \sum_{m=-\infty}^\infty w(l-m)
\frac{1}{N}\sum_{k=0}^{N-1} H_m(\omega_k)
e^{j\omega_k(n-l)} \\
&=& \sum_{l=-\infty}^\infty x(l)
\sum_{m=-\infty}^\infty w(l-m) h_m(n-l) \\
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img1499.png)
Define
to get
![]() |
(9.42) |
Let's examine the term

describes the time variation of the
tap.
-
is a filtered version of the
tap
. It is lowpass-filtered by w and delayed by
samples.
- Denote the
th time-varying, lowpass-filtered, delayed-by-
filter tap by
. This can be interpreted as the weighting in the output at time
of an impulse entering the time-varying filter at time
.

This is a superposition sum for an arbitrary linear, time-varying filter
.
Block Diagram Interpretation of Time-Varying STFT Modifications
Assuming
is causal gives

This is depicted in Fig.8.17.
The term
can be interpreted as the FIR filter tap
at time
. Note how each tap is lowpass filtered by the FFT window
. The window thus enforces bandlimiting each filter tap to
the bandwidth of the window's main lobe. For an
-term length-
Blackman-Harris window, for example, the main-lobe reaches zero at
frequency
(see Table 5.2 in §5.5.2
for other examples). This bandlimiting places a limit on the bandwidth expansion
caused by time-variation of the filter coefficients, which in turn places a limit
on the maximum STFT hop-size that can be used without frequency-domain aliasing.
See Allen and Rabiner 1977
[9] for further details on the bandlimiting
property.
Length L FIR Frame Filters
To avoid time aliasing, we restrict the filter length to a maximum of
samples. Since
is an arbitrary multiplicative
weighting of the
th spectral frame, the frame filter need not be
causal. For odd
, the filter impulse response indices may run from
to
, where
![]() |
(9.43) |
This gives

This is the general length
time-varying FIR filter convolution sum for
time
, when
is odd.
Next Section:
Weighted Overlap Add
Previous Section:
Overlap-Save Method