Fundamentals of Digital Signal Processing

Coursework Assignment 2

1.  Consider an impulse response h[n] such that h[n] = 0 for n < 0 and n > M , and h[n] = -h[M - n] for 0 ≤ n ≤ M where M is an odd integer.

a) Express the Fourier transform. of h[n] in the form.

H(ejω ) = ejf (ω)A(ω) ,

where f (ω) and A(ω) are real-valued functions of ω . Determine f (ω) and A(ω).

b) Provide an example of such an impulse response h[n] for M = 7 and find the corresponding f (ω) and A(ω).

2. Read Section 7.2.2 from the textbook and pay particular attention to Example 7.7. We wish to design a generalized linear phase filter satisfying the specifications

0.95 < jH(ejω )j < 1.05, jH(ejω )j < 0.15,

0 ≤ jωj ≤ 0.5π 0.6π ≤ jωj ≤ π                                        (1)

by applying a Kaiser window to the impulse response hd [n] of the ideal low-pass filter with cut- off frequency ωc   = 0.55π .   Find the value β and the window length M required to satisfy the specifications. Plot the corresponding Kaiser window and the impulse response of the designed low-pass filter. Plot the magnitude response

20 log10 jH(ejω )j

of the designed filter in the range ω ∈ (0, π) with resolution 2π/1024 or higher.

3. Suppose that we are given a continuous-time low-pass filter with frequency response Hc (jΩ) such that

1 - δ1  ≤ jHc (jω)j ≤ 1 + δ1 , jHc (jΩ)j ≤ δ2 ,

0 ≤ jΩj ≤ Ωp     jΩj ≥ Ωs             . (2)

A set of discrete-time low-pass filters can be obtained from Hc (s) by using the bilinear transfor- mation,i.e.

H(z) = Hc (s)js=(2/Td)(1-z-1)/(1+z-1)  ,

with Td  variable.

a) Assuming that Ωp  is fixed, find Td  such that the corresponding pass-band cut-off frequency of the discrete-time system is ωp  = π/2.

b) With Ωp  fixed, sketch ωp , the cut-off frequency of the discrete-time filter, as a function of Td , for 0 < Td < ∞ .

c) With Ωp  and Ωs  fixed sketch the width of the transition region, △ω = ωs  - ωp  as a function of Td , for Td  in the range 0 < Td < ∞ .

4.  Suppose that H1 (z), H2 (z) and H(z) are transformed versions of Hc1(s), Hc2(s) and Hc (s), respectively, obtained using impulse invariance or the bilinear transformation.  Which of the two methods will guarantee that H(z) = H1 (z) + H2 (z) whenever Hc (s) = Hc1(s) + Hc2(s).

5. Suppose that we are given an ideal low-pass discrete-time filter with frequency response

H(e jω) = ( 1 0 , , π/ |ω| 4 < π/ < |ω 4 | ≤ π .

We wish to derive new filters from this prototype by manipulating its impulse response h[n].

a) Plot the frequency response for the filter whose impulse response is h1 [n] = h[2n].

b) Plot the frequency response of the filter whose impulse response is

h2[h] = ( h 0, [n/2], n otherwise = 0, ±2, ±4, . . . .

c) Plot the frequency response of the filter whose impulse response is

h3 [n] = ejπnh[n] = (-1)nh[n] .

There is no need to plot these frequency responses in matlab, a sketch would be sufficient.