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 nonzero 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 nonzero 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.)
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Related Book For
Discrete Time Signal Processing
ISBN: 9780137549207
2nd Edition
Authors: Alan V. Oppenheim, Rolan W. Schafer
Posted Date:
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