Convolution Representation of FIR Filters

Note that the output of the $k$th delay element in Fig.5.5 is $x(n-k)$, $k=0,1,2,\ldots,M$, where $x(n)$ is the input signal amplitude at time $n$. The output signal $y(n)$ is therefore

$$y(n) = b_0 x(n) + b_1 x(n-1) + b_2 x(n-2) + \cdots + b_M x(n-M)$$

$$= \sum_{m=0}^M b_m x(n-m)$$

$$= \sum_{m=-\infty}^{\infty} h(m) x(n-m)$$

$$\isdef (h\ast x)(n)$$ (6.7)

where we have used the convolution operator ``$\ast$'' to denote the convolution of $h$ and $x$, as defined in Eq. (5.4). An FIR filter thus operates by convolving the input signal $x(n)$ with the filter's impulse response $h(n)$.