592
Practical MATLAB
® Applications for Engineers
H(e
jW ) = h(0)e
−2jW 2cos(2W) + h(1)e
−jW2 2cos(W) + h(2)e
−2jW
H(e
jW ) = e
−2jW [h(0)2cos(2W) + 2h(1)cos(W) + h(2)]
͉H(e
jW )͉ = ͉2h(0)cos(2W) + 2h(1)cos(W) + h(2)͉ for 0 ≤ n ≤ 4
then
∠H(e
jW ) = −2W
Observe that the magnitude of H(e
−jW ) is a real function of W, whereas the phase is a
linear function of W over the range of −π ≤ W ≤ +π.
R.6.125 Now consider the sequence
h(n) = −0.09(n) + 0.29(n − 1) + 0.53(n − 2) + 0.29(n − 3) − 0.09(n − 4)
which represents the impulse response of a type-1 FIR fi lter with N = 5.
ANALYTICAL Solution
The magnitude and phase response is then given by (using R.6.124)
͉H(e
jW )͉ = 2(−0.09)cos(2W) + 2 * (0.29)cos(W) + (0.53)
for n = 0, 1, 2, 3, 4 and ∠H(e
jW ) = −2W
R.6.126 The example below illustrates the analysis of a transfer function of an FIR type-2
fi lter with N = 8.
ANALYTICAL Solution
H(z) = h(0) + h(1)z
−1
+ h(2)z
−2
+ h(3)z
−3
+ h(4)z
−4
+ h(5)z
−5
+ h(6)z
−6
+ h(7)z
−7
Then applying symmetry conditions, the following relations hold:
h(0) = h(7)
h(1) = h(6)
h(2) = h(5)
h(3) = h(4)
Then, its spectrum is given by
H(e
jW ) = [2h(0)cos(7W/2) + 2h(1)cos(5W/2) + 2h(2)cos(3W/2)
+ 2h(3)cos(W/2)]e
−jW7/2
Clearly, the (phase) ∠H(e
jW ) = −(7/2)W and the quantity inside the brackets
represents its magnitude spectrum.
R.6.127 Let us analyze an FIR type-3 fi lter with N = 9 given by
H(z) = h(0) + h(1)z
−1
+ h(2)z
−2
+ h(3)z
−3
+ h(4)z
−4
+ h(5)z
−5
+ h(6)z
−6
+ h(7)z
−7
+ h(8)z
−8
CRC_47760_CH006.indd 592
CRC_47760_CH006.indd 592
7/23/2008 2:26:39 PM
7/23/2008 2:26:39 PM