Convolution Representation of FIR Filters

Notice 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 $x$ with the filter's impulse response $h$ .