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Some Special Filter Types 399
circle. From our previous discussion of resonators, the system function for
a resonator with poles at re± jω0 is
H(z) =
b 0
1 − (2r cos ω 0 )z −1 + r 2 z −2 (8.34)
When we set r =1and select the gain parameter b 0 as
The system function becomes
H(z) =
b 0 = A sin ω 0 (8.35)
A sin ω 0
1 − (2 cos ω 0 )z −1 + z −2 (8.36)
and the corresponding impulse response of the system becomes
h(n) =A sin(n +1)ω 0 u(n) (8.37)
Thus, this system generates a sinusoidal signal of frequency ω 0 when excited
by an impulse δ(n) =1.
The block diagram representation of the system function given by
(8.36) is illustrated in Figure 8.11. The corresponding difference equation
for this system is
y(n) =(2cos ω 0 ) y(n − 1) − y(n − 2) + b 0 δ(n) (8.38)
where b 0 = A sin ω 0 .
Note that the sinusoidal oscillation obtained from the difference equation
in (8.38) can also be obtained by setting the input to zero and setting
the initial conditions to y(−1) =0, y(−2) = −A sin ω 0 .Thus, the zeroinput
response to the 2nd-order system described by the homogeneous
difference equation
y(n) =(2cos ω 0 ) y(n − 1) − y(n − 2) (8.39)
FIGURE 8.11
Digital sinusoidal oscillator
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