
(d) A first-order low-pass digital filter with z-domain transfer function
H(z) ¼
^
T(z)
T(z)
¼
(1 a)z
z a
¼
(1 a)
1 az
1
(4:693)
is used to filter the monthly temperature variations and pass any low-frequency lake
temperature variation with time. Plot the discrete-time frequency response magnitude and
phase for a ¼0.99. Find the cutoff frequency and determine whether it is less or greater than
v
0
¼p=6 rad=month. In addition, find jH(e
jv0T
)j.
(e) Find the difference equation relating
^
T
k
and T
k
. Solve it recursively for
^
T
k
, k ¼1,2,...,300
and plot both input and output on the same graph.
(a) Taking the z-transform of Equation 4.692 gives
^
T(z) ¼
1
12
[z
1
T(z) þ