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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

$\displaystyle y(n)$ $\displaystyle =$ $\displaystyle b_0 x(n) + b_1 x(n-1) + b_2 x(n-2) + \cdots + b_M x(n-M)$  
  $\displaystyle =$ $\displaystyle \sum_{m=0}^M b_m x(n-m)$  
  $\displaystyle =$ $\displaystyle \sum_{m=-\infty}^{\infty} h(m) x(n-m)$  
  $\displaystyle \isdef$ $\displaystyle (h\ast x)(n)$ (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$.

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FIR impulse response