Example

• Let the filter coefficients be given by $a_i=\frac{1}{N+1}$ for $i=0,\ldots,N$.
• Then, we can write
  
  $$y_n = \frac{1}{N+1} \sum_{i=0}^N x_{n-i}.$$
  
  

• Hence, the n-th output is the average of the N+1 most recent inputs.
• Let us consider what happens if a sinusoid of frequency f is input to this filter.
• Note: remember that f must be less than $\frac{1}{2T_s}$, where T_s denotes the sampling interval, to meet the Nyquist criterion.
• Here, we assume T_s=1 and, hence, f must be less than 0.5.
• Example: Let N=9 and f=0.01.
• The resulting input and output are shown below.

  

Figure 4.8: Input (top) and output (bottom) when f=0.01 and N+1=10