DIGITAL SINUSOIDAL OSCILLATORS

8.2.5 DIGITAL SINUSOIDAL OSCILLATORS

A digital sinusoidal oscillator can be viewed as a limiting form of a 2-pole resonator for which the complex-conjugate poles are located on the unit

Magnitude Response

Phase Response

ω in π units FIGURE 8.10 Magnitude and phase responses for 1-pole (solid line) and 2-pole

ω in π units

(dotted line) allpass filters

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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± jω 0 is

H(z) =

1 − (2r cos ω 0 )z +r z When we set r = 1 and select the gain parameter b 0 as

b 0 = A sin ω 0 (8.35)

The system function becomes

A sin ω 0

(8.36) and the corresponding impulse response of the system becomes

H(z) =

1 − (2 cos ω 0 )z − 1 +z − 2

h(n) = A sin(n + 1)ω 0 u(n)

(8.37) Thus, this system generates a sinusoidal signal of frequency ω 0 when ex-

cited 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) = (2 cos 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 equa- tion 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 zero- input response to the 2nd-order system described by the homogeneous difference equation

y(n) = (2 cos ω 0 ) y(n − 1) − y(n − 2)

FIGURE 8.11 Digital sinusoidal oscillator

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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 homo- geneous difference equation in (8.39) can be obtained directly from the trigonometric identity

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 car-

rier 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 gener- ated 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

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 il-

lustrated 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.