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Unformatted text preview: nge will
be m times F (mn) [ F(n) frequencycompressed by factor m] r epeating periodically
with period 21r. 10.5 Discrete Time L inear System Analysis by D TFT 639 k
Zk = ejOk , a nd t he response t o a n everlasting exponential input ej O o f a n LTID
system w ith t ransfer function H[z] is H[ejO]ejOk. T his result can be represented as
t he i nputoutput p air with a directed arrow n otation as usual
ejOk ==> H [ejO]ejOk
Also, according t o Eq. (9.57b) with z = ejO , i t follows t hat H[e j o] = 00 L h[k]e jOk k =oo Observe t hat t he r ighthand side of t he above equation is H (n), t he D TFT of h[k].
T herefore
a nd ejOk ==> H (n)e jOk Clearly H ( n) is t he LTID system frequency response. Following t he a rgument in
Sec. 7.1, t he a mplitude response of t he s ystem is I H(n)l, a nd t he p hase response is
L H(n). We shall discuss this topic again in greater details in C hapter 12.
E quation (10.57) s tates t hat t he frequency s pectrum of t he o utput signal is t he
p roduct of t he frequency s pectrum o f t he i nput signal a nd t he frequency response
o f t he s ystem. From Eq. (10.57), we have ly(n)1 = I F(n)IIH(n)1 (10.58) a nd 10.5 DiscreteTime Linear System Analysis by DTFT
L y(n) = L F(n) Consider a linear timeinvariant discretetime system with t he u nit impulse
response h[k]. We shall find t he (zerostate) system response y[k] for t he i nput f [k].
Because y[k] = f [k] * h[k] (10.56) According t o Eq. (1O.50a) i t follows t hat Y (n) = F (n)H(n) (10.57) where F (n), Y (n), a nd H (n) are D TFTs of f [k], y[k], a nd h[k], respectively; t hat
is, f [k] ¢ =} F (n), y[k] ¢ =} Y (n), a nd h[k] ¢ =} + L H(n) (10.59) This result shows t hat t he o utput a mplitude s pectrum is t he p roduct o f t he i nput
a mplitude s pectrum a nd t he a mplitude response of t he system. T he o utput p hase
s pectrum is t he s um o f t he i nput phase s pectrum a nd t he p hase response o f t he
s ystem.
We c an also interpret Eq. (10.57) in terms o f t he frequencydomain viewpoint,
which sees a system in terms of its frequency response (system response t o various
exponential o r sinusoidal components). Frequencydomain views a signal as a s um
o f various exponential o r sinusoidal components. Transmission o f a signal through
a (linear) system is viewed as transmission o f various exponential o r sinusoidal
components of t he i nput signal t hrough t he s ystem. T his concept can b e u nderstood
by displaying t he i nputoutput relationships by a directed arrow as follows: H (n)
t he s ystem response t o T his result is similar t o t hat o btained for continuoustime systems.
T he Frequency Response o f an LT ID System E quation (9.57a) s tates t hat t he response t o a n everlasting exponential input
zk of a n LTID system with transfer function H[z] is H[z]zk. I f we let z = e j O, t hen
t Here, Fc(w) should b e i nterpreted a s t he first cycle (centered a t w = 0) o f T Fc(w) f [k] = ~ y[k] = ~ jOk dn
{ F (n)e
21r
a nd from Eq. (10.57) 21r i2"
{ i2rr d nk is H (o.)d Ok shows I [k] a s a s um o f e verlasting exponential components F (n)H(n...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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