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Unformatted text preview: e a notational change by noting
t hat F (lnz) is a function of z . Let us denote it by a simpler n otation F[zJ. Thus,
Eqs. (10.89) become
I[k] = ~f
21TJ F[zJ z k-l d z (10.90) 662 10 Fourier Analysis of Discrete-Time Signals a nd 00 F [z] = L j [k] z -k (10.91) k =-oo T his is t he (bilateral) z-transform pair. Equation (10.90) expresses j [k] as a continuous sum o f exponentials of t he form zk = e(o+jfl)k = r k ejflk . T hus, by selecting
a proper value for r (or a ), we c an make t he e xponential grow (or decay) a t any
exponential r ate we desire.
I f we l et a = 0, we have z = e jfl a nd
F [z] = F (lnz) = F un) = F (n) (10.92) Thus, t he familiar D TFT is j ust a special case of t he z -transform F [z] o btained by
letting z = e jfl . 1 0.8 Summary T his chapter deals with analysis a nd processing o f discrete-time signals. For
analysis, o ur approach is parallel t o t hat used in continuous-time signals. We first
represent a periodic j [k] as a Fourier series formed by a discrete-time exponential
and its harmonics. Later we e xtend this representation t o a n aperiodic signal j [k]
by considering j [k] as a limiting case of a periodic signal with period approaching
infinity. Periodic signals are represented by discrete-time Fourier series (DTFS)j
aperiodic signals are represented by t he discrete-time Fourier transform ( DTFT).
T he development, although similar t o t hat of continuous-time signals, also reveals
some significant differences. T he basic difference in t he two cases arises because
a continuous-time exponential e jwt has a unique waveform for every value of w in
the range - 00 t o 0 0. I n contrast, a discrete-time exponential e jflk h as a unique
waveform only for values of n in a continuous interval of 21r. Therefore, if n o is
t he fundamental frequency, t hen a t most ~: n umber o f exponentials in t he Fourier
series are independent. Consequently, t he discrete-time exponential Fourier series
has only No = ~: terms.
T he discrete-time Fourier transform ( DTFT) o f a n aperiodic signal is a continuous function of n a nd is periodic with period 21r. We can synthesize F (n) from
its spectral components in any b and of width 21r. L inear time-invariant discretetime (LTID) systems can be analyzed using D TFT i f t he i nput signals are DTFtransformable a nd if t he s ystem is stable. Analysis of unstable (or marginally stable) systems a nd/or exponentially growing inputs can b e performed by z-transform,
which is a generalized D TFT. T he relationship of D TFT t o z -transform is similar to
t hat o f t he Fourier transform t o t he Laplace transform. Whereas t he z -transform
is superior t o D TFT for analysis of LTID systems, D TFT is preferable in signal
I f H (n) is t he D TFT of t he s ystem's impulse response h[k], t hen IH(n)1 is t he
amplitude response, and L H(n) is t he phase response of t he system. Moreover,
if F (n) a nd Y (n) are t he D TFTs of t he i nput j [k] a nd t he corresponding o utput
y[k], t hen Y (n) = H (n)F(n). Therefore the o utput s pectrum is t he p roduct of t he
i nput spectrum a nd t he s ystem's frequency response.
T he numerical computations in modern digi...
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