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Relationships represented by Eqs. (6.67) and (6.68) are depicted in Fig. 63. Let in VS .NET
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in Eq. (4.1), we obtain
which indicates that the complex exponential sequence ejRnnis an eigenfunction of the LTI system with corresponding eigenvalue H(Ro), as previously observed in Chap. 2 (Sec. 2.8). Furthermore, by the linearity property (6.42), if the input x [ n ] is periodic with the discrete Fourier series then the corresponding output y [ n ] is also periodic with the discrete Fourier series
If x [ n ] is not periodic, then from Eqs. (6.68) and (6.28) the corresponding output y [ n ] can
be expressed as
FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
B. LTI Systems Characterized by Difference Equations: As discussed in Sec. 2.9, many discretetime LTI systems of practical interest are described by linear constantcoefficient difference equations of the form C a k y [ n  k ] = C b,x[n k=O
 k] (6.76) with M I N. Taking the Fourier transform of both sides of Eq. (6.76) and using the linearity property (6.42) and the timeshifting property (6.43), we have C a, ejkRY(R)= C bk eJkb'X( a ) or, equivalently, The result (6.77) is the same as the 2transform counterpart H ( z ) = Y(z)/X(z) with z = eJ" [Eq. (4.4411; that is, C. Periodic Nature of the Frequency Response: From Eq. (6.41) we have H ( R ) = H ( n + 27r) Thus, unlike the frequency response of continuoustime systems, that of all discretetime LTI systems is periodic with period 27r. Therefore, we need observe the frequency response of a system only over the frequency range 0 I R R 27r or 7r I I R T . SYSTEM RESPONSE TO SAMPLED CONTINUOUSTIME SINUSOIDS System Responses: We denote by y,[n], y,[n], and y[n] the system responses to cos R n , sin R n , and eJRn, ' respectively (Fig. 64). Since e ~ ~ ="cos R n + j sin R n , it follows from Eq. (6.72) and the linearity property of the system that y [ n ] = y,[n] +jy,[n] = H ( R ) eJRn
= ) ~ , [ n ] Re{y[n]) = R ~ ( H ( ReJRn) y,[n] I ~ {[nl ) Y
= Im{H(R) Fig. 64 System responses to elnn, cos Rn, and sin Rn.
CHAP. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
When a sinusoid cos R n is obtained by sampling a continuoustime sinusoid cos w t with sampling interval T,, that is, cos R n
= cos
= cos
wT,n
(6.80) all the results developed in this section apply if we substitute wT, for R : R
= oT, (6.81) For a continuoustime sinusoid cos wt there is a unique waveform for every value of o in the range 0 to w. Increasing w results in a sinusoid of everincreasing frequency. O n the other hand, the discretetime sinusoid cos R n has a unique waveform only for values of R in the range 0 to 27r because COS[(R+ 27rm)nI = cos(Rn
+ 27rmn) = cos R n
= integer
(6.82) This range is further restricted by the fact that cos(7r f R ) n
= cos
.rrn cos R n T sin 7rn sin R n
Therefore, Equation (6.84) shows that a sinusoid of frequency (7r + R ) has the same waveform as one with frequency (.rr  R). Therefore, a sinusoid with any value of R outside the range 0 to 7r is identical to a sinusoid with R in the range 0 to 7r. Thus, we conclude that every discretetime sinusoid with a frequency in the range 0 I R < .rr has a distinct waveform, and we need observe only the frequency response of a system over the frequency range OsR<7r. B. Sampling Rate: Let w, ( = 27rfM) be the highest frequency of the continuoustime sinusoid. Then from Eq. (6.81) the condition for a sampled discretetime sinusoid t o have a unique 7r waveform is wMTs< 7 r + Ts< or f , > 2fM (6.85) where f, = l/T, is the sampling rate (or frequency). Equation (6.85) indicates that to process a continuoustime sinusoid by a discretetime system, the sampling rate must not be less than twice the frequency (in hertz) of the sinusoid. This result is a special case of the sampling theorem we discussed in Prob. 5.59.

