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Consider applying a fixed (time-invariant) filter
to
each
before resynthesizing the signal:
 |
(10.28) |
where,
is the sampled frequency response of a filter with
impulse response
 |
(10.29) |
Let's examine the result this has on the signal in the time domain:
We see that the result is 
convolved with a windowed version
of the impulse response 
. This is in contrast to the OLA technique
where the result gave us a windowed
filtered by
without the
window having any effect on the filter, provided it obeys the COLA
constraint and sufficient zero padding is used to avoid time aliasing.
In other words, FBS gives
![$\displaystyle y = x * [\tilde{w} \cdot h] \;\longleftrightarrow\;X \cdot [{\tilde W}\ast H]$](img1695.png) |
(10.30) |
while OLA gives (for 
)
![$\displaystyle y = x * [W(0)\cdot h] \;\longleftrightarrow\;X \cdot [W(0)\cdot H]$](img1696.png) |
(10.31) |
- In FBS, the analysis window
smooths the filter frequency response by time-limiting the corresponding impulse response.
- In OLA, the analysis window can only affect scaling.
For these reasons, FFT implementations of FIR filters normally use the
Overlap-Add method.
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![\begin{eqnarray*}
y(m) &=& \frac{1}{N} \sum_{k=0}^{N-1} Y_m(\omega_k) e^{j\omega_k m} \\
&=& \frac{1}{N} \sum_{k=0}^{N-1} X_m(\omega_k)H(\omega_k) e^{j\omega_k m} \\
&=& \frac{1}{N} \sum_{k=0}^{N-1} \left\{ \sum_{n=-\infty}^\infty x(n)w(n-m)e^{-j\omega_kn} \right\} H(\omega_k) e^{j\omega_k m} \\
&=& \frac{1}{N} \sum_{n=-\infty}^\infty x(n)w(n-m) \sum_{k=0}^{N-1} H(\omega_k) e^{j\omega_k(m-n)} \\
&=& \sum_{n=-\infty}^\infty x(n) [ w(n-m) h(m-n)] \\
&=& \sum_{n=-\infty}^\infty x(n) [\tilde{w}(m-n)h(m-n)] \\
&=& (x*[\tilde{w} \cdot h])(m) \\
\end{eqnarray*}](img1694.png)
