1. Consider the moving-average filter given in Example 5.4. What is the 3-dB bandwidth of this
filter if the sampling rate is 8 kHz?
2. Consider the FIR filter with the impulse response hn f1, 1, 1g. Calculate the magnitude
and phase responses and show that the filter has linear phase.
3. Given a linear time-invariant filter with an exponential impulse response
show that the output due to a unit-step input, u(n), is
1 À a
1 À a
, n ! 0:
4. A rectangular function of length L can be expressed as
xn un À un À L 1, 0 n L À 1
(a) rn un Ã un, where Ã denotes linear convolution and rn n 1un is called the
(b) tn xn Ã xn rn À 2rn À L rn À 2L is the triangular pulse.
5. Using the graphical interpretation of linear convolution given in Figure 5.4 to compute the
linear convolution of hn f1, 2, 1g and x(n), n 0, 1, 2 defined as follows:
(a) xn f1, À 1, 2g,
(b) xn f1, 2, À 1g, and
(c) xn f1, 3, 1g:
6. The comb filter also can be described as
yn xn xn À L,
find the transfer function, zeros, and the magnitude response of this filter and compare the
results with Figure 5.6.
7. Show that at ! 0, the magnitude of the rectangular window function is 2M 1.
8. Assuming hn has the symmetry property hn hÀn for n 0, 1, . . . M, show that H!
can be expressed as
DESIGN AND IMPLEMENTATION OF FIR FILTERS