In the design of FIR filters, an alternative to using windows which tapers to zero is...

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In the design of FIR filters, an alternative to using windows which tapers to zero is to introduce a smooth transition region in the ideal frequency response. Consider now the ideal frequency response shown in Figure 2, and let ha[n] be the impulse response coefficients of this filter. Let we = (wp + ws)/2 and Aw = Ws - Wp. ولما لا - We π 0 ha[n] Wp Figure 2: Lowpass filter = ولنا (i) Show that ha[0]= (ii) Use the results in d) and e) to show that for n 0 71 2 sin(n) sin wen Awn πη (The formula cosa - cos y = -2 sinsin may come in useful.) (iii) The filter in part (ii) above has an infinite number of non-zero filter coefficients. Explain what you need to do in order to obtain an Nth order (N even) causal FIR filter, i.e. a filter with transfer function N H(z) = Sh[n]== -Σ* n=0 -n and find an expression for the filter coefficients h[n]. (There is no need to use a window which tapers to zero.) In the design of FIR filters, an alternative to using windows which tapers to zero is to introduce a smooth transition region in the ideal frequency response. Consider now the ideal frequency response shown in Figure 2, and let ha[n] be the impulse response coefficients of this filter. Let we = (wp + ws)/2 and Aw = Ws - Wp. ولما لا - We π 0 ha[n] Wp Figure 2: Lowpass filter = ولنا (i) Show that ha[0]= (ii) Use the results in d) and e) to show that for n 0 71 2 sin(n) sin wen Awn πη (The formula cosa - cos y = -2 sinsin may come in useful.) (iii) The filter in part (ii) above has an infinite number of non-zero filter coefficients. Explain what you need to do in order to obtain an Nth order (N even) causal FIR filter, i.e. a filter with transfer function N H(z) = Sh[n]== -Σ* n=0 -n and find an expression for the filter coefficients h[n]. (There is no need to use a window which tapers to zero.)