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Unformatted text preview: 12.2a)
(12.2b) A ddition of these two equations yields
2 cos O k = =} H [eifljeiflk + H [e -iflje - iflk = 2Re ( H [ eiflje iflk ) IH[eifljl cos (Ok + 0+ LH[eiflj ) + 0) is (12.6b) This result applies only t o a symptotically stable systems because Eq. (12.1) is valid
only for values of z lying in t he region of convergence of H [zj. For z = e lfl,.z lies
on t he u nit circle (Izl = 1). T he region of convergence for unstable a nd m argmally
stable systems does n ot include t he u nit circle.
This i mportant r esult shows t hat t he response of a n a symptotically stable LTID
system t o a d iscrete-time sinusoidal i nput of frequency 0 is also a discrete-time sinusoid of t he s ame frequency. T he a mplitude o f t he o utput sinusoid is I H[elfl)1 t imes
t he i nput a mplitude, a nd t he p hase o f t he o utput sinusoid is shifted by L H [elflj w ith
r espect to t he i nput p hase. Clearly IH[eiflJl is t he a mplitude gain, a nd ~ p~ot of
IH[eifl]1 versus 0 is t he a mplitude response of t he discrete-time sys~m. SimIlarly,
L H [e ifl ] is t he p hase response of t he s ystem, a nd a p lot of L H [e l ] vs 0 shows
how t he s ystem modifies or shifts t he p hase o f t he i nput sinusoid. Note t hat H [e ]
i ncorporates t he i nformation of b oth a mplitude a nd phase response a nd t herefore
is called t he f requency r esponse of t he s ystem.
These results, although parallel t o t hose for continuous-time systems, differ
from t hem i n one significant aspect. I n t he continuous-time case, t he frequency response is H (jw). A p arallel result for t he discrete-time case would lead t o frequency
response H [jO]. I nstead, we found t he frequency response t o b e H [elflj. T his deviation causes some interesting differences between t he behavior of continuous-time
a nd d iscrete-time systems. S teady-State Response t o Causal Sinusoidal Input
As in t he case of continuous-time systems, we c an show t hat t he response of a n
LTID system t o a c ausal sinusoidal i nput cos O k u [k] is y[kJ in Eq. (12.6a), plus a
n atural c omponent consisting o f t he c haracteristic modes (see Prob. 12.1-4). For a
stable system, all t he modes decay exponentially, a nd only t he sinusoidal component
in Eq. (12.6a) persists. For this reason, this component is called t he sinusoidal
s teady-state response of t he s ystem. Thus, yss[kJ, t he s teady-state response of a
system t o a causal sinusoidal i nput cos O k u[kJ, is (12.3) E xpressing H [einj in the polar form
H [eiflj = I H[eifllleiLH[ei"J E q. (12.3) can be expressed as
716 (12.4) S ystem Response t o Sampled Continuous-Time Sinusoids
So far we have considered t he s ystem response of a discrete-time system t o a
d iscrete-time sinusoid cos O k (or exponential eiflk ). I n practice, t he i nput may be a 12 7 18 F requency R esponse a nd D igital F ilters 12.1 719 F requency R esponse o f D iscrete-Time S ystems jwt s ampled c ontinuous-time s inusoid c os w t ( or a n e xponential e ). W hen a s inusoid
c os w t i s s ampled w ith s amplin...
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