Page 126 - DSP Integrated Circuits

Problems 111

3.4 Determine the 2-transform of

(a) The autocorrelation function:

(b) The convolution:

3.5 Prove Equation (1.5) assuming that the size of the subproblems is a power of
m
m
m
c, i.e., n = c . Hint: Use the substitution x(m) = T(c )/b .
3.6 Determine the transfer function for a filter that has the following impulse
response:

3.7 Determine the step response of the filter in Problem 3.6.
3.8 The essential part of the power spectrum of an analog signal x(t) lies in the
range 0 to 25 kHz. The signal is corrupted by additive wide-band noise which
is to be removed by a digital lowpass filter. Select a suitable sample
frequency when a third-order Butterworth filter is used as the anti-aliasing
filter. The attenuation of unwanted images shall be > 40 dB. The ripple in the
passband of the anti-aliasing filter shall be < 1 dB.
3.9 Show that h(n) = 0 for n < 0 for a causal LSI system.
3.10 Show that the ordering of two cascaded LSI systems may be interchanged.

3.11 Determine the transfer function for a system that is described by the
difference equation

3.12 A stable, causal digital filter has the following transfer function

(a) Determine the impulse response.
(b) Determine the region of convergence.
(c) Plot the pole-zero configuration in the z-plane.
(d) Plot the magnitude response.
(e) Determine the step response.

3.13 Determine the transfer function of the following difference equation:

Determine also the magnitude and phase responses as well as the group delay.
3.14 A digital filter has the pole-zero configuration shown in Figure P3.14a.
(a) Sketch the magnitude response between coT = 0 and coT = 2n using a
logarithmic scale (dB).
(b) Sketch the impulse and step responses.