Digital Signal Processing and Digital Filters
Practice Sheet 4
1) Let h [n] be an ideal low pass filter given by
(a) Compute its frequency response function H (ejω).
(b) Calculate |H (ejω0)| and ∠H (ejω0) for ω0 = 63.6.
(c) Design a window with a length of 20 sampling times (containing 21 samples) for h [n] and choose N that would guarantee a linear phase response. Write the frequency response of the filter.
2) A finite-length complex exponential signal is given by x [n] = ejωn, n ∈ [0, N − 1]. The DFT of x [n] satisfies
By using the approximation sin θ ≈ θ, |θ| < 0.2 rad, show that |X [k]| is approximately bounded by for a suitable range of k. Give the range for k for which the bound applies and explain the significance of the term
3) Given a maximum transition width of ∆ω = 0.2 rad/s, compute the stop band gain for a filter with a number of M = 50 taps.
4) Let denote the number of taps of a low pass filter, and let ∆ω1 = 2∆ω0 .
(a) Give general expressions of M1 and ϵ1 as function of M0, ϵ0 and ∆ω0 that correspond to the number of taps and stop band gain corresponding to a filter with transition bandwidth ∆ω1 = 2∆ω0 .
(b) What can be said about the change in number of taps and stop band gain? What are the conditions to guarantee this change?