Time Varying Modifications in FBS

Consider now applying a time varying modification.

$$Y_m(\omega_k) = X_m(\omega_k)H_m(\omega_k) \qquad \hbox{($R=1$)}$$ (10.32)

where

$$H_m(\omega_k) \;\longleftrightarrow\;h_m(n) = \frac{1}{N} \sum_{k=0}^{N-1} H_m(\omega_k) e^{j\omega_kn}$$ (10.33)

$h_m(n)$ refers to the $n^{th}$ tap of the FIR filter at time $m$ .

$$\begin{eqnarray*} y(m) &=& \frac{1}{N} \sum_{k=0}^{N-1} Y_m(\omega_k) e^{j\omega_k m} \\ &=& \frac{1}{N} \sum_{k=0}^{N-1} X_m(\omega_k)H_m(\omega_k) e^{j\omega_k m} \\ &=& \frac{1}{N} \sum_{k=0}^{N-1} \left\{ \sum_{n=-\infty}^\infty x(n)w(n-m)e^{-j\omega_kn} \right\} H_m(\omega_k) e^{j\omega_k m} \\ &=& \frac{1}{N} \sum_{n=-\infty}^\infty x(n)w(n-m) \sum_{k=0}^{N-1} H_m(\omega_k) e^{j\omega_k(m-n)} \\ &=& \sum_{n=-\infty}^\infty x(n) [ w(n-m) h_m(m-n)] \\ &=& \sum_{n=-\infty}^\infty x(n) [\tilde{w}(m-n)h_m(m-n)] \\ &=& (x*[\tilde{w} \cdot h_m])(m) \\ \end{eqnarray*}$$

Hence, the result is the convolution of $x$ with the windowed $h_m$ .