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400 Chapter 8 IIR FILTER DESIGN
with initial conditions y(−1) =0,y(−2) = −A sin ω 0 is exactly the same
as the response of (8.38) to an impulse excitation. In fact, the homogeneous
difference equation in (8.39) can be obtained directly from the
trigonometric identity
( ) ( )
α + β α − β
sin α + sin β =2sin cos
(8.40)
2
2
where, by definition, α =(n +1)ω 0 ,β=(n − 1)ω 0 , and y(n) =sin(n +
1)ω 0 .
In practical applications involving modulation of two sinusoidal carrier
signals in phase quadrature, there is a need to generate the sinusoids
A sin ω 0 n and A cos ω 0 n. These quadrature carrier signals can be generated
by the so-called coupled-form oscillator, which can be obtained with
the aid of the trigonometric formulas
cos(α + β) =cos α cos β − sin α sin β (8.41)
sin(α + β) =sin α cos β + cos α sin β (8.42)
where by definition, α = nω 0 ,β= ω 0 ,y c (n) =cos(n +1)ω 0 , and y s (n) =
sin(n +1)ω 0 .Thus, with substitution of these quantities into the two
trigonometric identities, we obtain the two coupled difference equations.
y c (n) =(cos ω 0 ) y c (n − 1) − (sin ω 0 ) y s (n − 1) (8.43)
y s (n) =(sin ω 0 ) y c (n − 1) + (cos ω 0 ) y s (n − 1) (8.44)
The structure for the realization of the coupled-form oscillator is illustrated
in Figure 8.12. Note that this is a 2-output system that does
not require any input excitation, but it does require setting the initial
conditions y c (−1) = A cos ω 0 and y s (−1) = −A sin ω 0 in order to begin
its self-sustaining oscillations.
8.3 CHARACTERISTICS OF PROTOTYPE ANALOG FILTERS
IIR filter design techniques rely on existing analog filters to obtain digital
filters. We designate these analog filters as prototype filters. Three prototypes
are widely used in practice. In this section we briefly summarize
the characteristics of the lowpass versions of these prototypes: Butterworth
lowpass, Chebyshev lowpass (Type I and II), and Elliptic lowpass.
Although we will use MATLAB functions to design these filters, it is necessary
to learn the characteristics of these filters so that we can use proper
parameters in MATLAB functions to obtain correct results.
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