Miscellaneous Exercises - Digital Signal Processing: An Introduction with MATLAB and Applications - page 296

Q7: Where do you expect the poles of a digital resonator to be located in the z-
plane?
Q8: (a) Draw a block diagram for a generic FIR filter of order N.
(b) Plot the impulse response of this filter.
(c) What is the condition on this filter to have a linear phase transfer
function?
(d) Plot the impulse response of a linear FIR filter and draw an
efficient block diagram assuming N is odd.
Q9: Two digital filters have impulse responses given by h 1 (n)={1,2,-1, -1, 2, 1}
(starting at n =9)andh 2 (n) = {1, 2, -1, 5, -1, 2, 1} (starting at n =0).
(a) Which one is an FIR filter? (Ans. both)
(b) Which one have a linear phase transfer function? (Ans. both, as they
have symmetric impulse response.)
(c) Plot the above impulse responses.
(d) Draw the efficient implementation diagrams of these filters.
Q10: (a) Plot the transfer function of an ideal digital LPF with cutoff frequency
10 Hz and sampling frequency 50 Hz. Use the frequency range [-100,
100 Hz].
(b) Roughly plot the impulse response of this filter.
(c) repeat part (a) for a high-pass filter with cutoff 10 Hz.
Q11: Explain Gibbs phenomenon. How can we alleviate the effect of this
phenomenon? (Hint: when we truncate the infinite impulse response of an
ideal digital filter using a time window, we get magnitude ripples in the
filter transfer function, with maxima at the ends of the transfer function.
This ripple is significant when we use a rectangular widow.)
Q12: Explain why we need just one matched filter at the receiver of a binary
communication system with antipodal signals (symbols).
Q13: Explain the basic operation of the optimal receiver in a binary communication
system with orthogonal signals. [Hint: it consists of two matched filters, one
matched to the symbol that represents logic ''0'', the other to ''1'' (explain how
we choose the impulse responses). If ''0'' was transmitted, the output of the
first matched filter (which is matched to ''0'') will be higher than that of the
second (which is matched to ''1''). This is because the matched filter is a
correlator. Hence, the comparator will decide that ''0'' was transmitted.
Similar reasoning if ''1'' was transmitted, where the output of the second filter
will be higher this time. Draw a block diagram.]
Q14: A signal x(t) = sin(x t) was corrupted by a white Gaussian noise n(t) with
variance 0.1. Find the statistical mean, time mean, and variance of the
signal s(t)=x(t)+n(t). [Ans. Statisticalmean = mean {x(t)+n(t)} = mean
{x(t)} + mean {n(t)} = x(t)+0=x(t), thetimemean = mean{x(t)} + mean
{n(t)} = 0 + 0 = 0. Note that since n(t) is ergodic, its time mean = its
statistical mean.]