Relationships represented by Eqs. (6.67) and (6.68) are depicted in Fig. 6-3. Let in VS .NET

Drawing QR Code 2d barcode in VS .NET Relationships represented by Eqs. (6.67) and (6.68) are depicted in Fig. 6-3. Let

Relationships represented by Eqs. (6.67) and (6.68) are depicted in Fig. 6-3. Let
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As in the continuous-time case, the function H(R) is called the frequency response of the system, I H(R)l the magnitude response of the system, and BH(R)the phase response of the system.
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y[n]=x[n] * h[n]
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Y(n)=X(R)H(n)
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Fig. 6-3 Relationships between inputs and outputs in an LTI discrete-time system.
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Consider the complex exponential sequence
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Then, setting z
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= ejRo
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 DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6
B. LTI Systems Characterized by Difference Equations:
As discussed in Sec. 2.9, many discrete-time LTI systems of practical interest are described by linear constant-coefficient 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 time-shifting property (6.43), we have
C a, e-jkRY(R)= C bk e-Jkb'X( a )
or, equivalently,
The result (6.77) is the same as the 2-transform 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 continuous-time systems, that of all discrete-time 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 CONTINUOUS-TIME 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. 6-4). 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. 6-4 System responses to elnn, cos Rn, and sin Rn.
CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
When a sinusoid cos R n is obtained by sampling a continuous-time 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 continuous-time 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 ever-increasing frequency. O n the other hand, the discrete-time 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 discrete-time 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 continuous-time sinusoid. Then from Eq. (6.81) the condition for a sampled discrete-time 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 continuous-time sinusoid by a discrete-time 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.
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