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90 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
Using Euler’s identity on the given x a(t) and the properties, the CTFT
X a(jΩ) is given by
X a(jΩ) = 8πδ(Ω) + 2πe jπ/3 δ(Ω − 150π)+2πe −jπ/3 δ(Ω + 150π)
+4jπδ(Ω − 350π) − 4jπδ(Ω + 350π). (3.39)
It is informative to plot the CTFT X a(jΩ) as a function of the cyclic frequency
F in Hz using Ω = 2πF. Thus the quantity X a(j2πF) from (3.39) is given by
X a(j2πF) =4δ(F )+e jπ/3 δ(F − 75) + e −jπ/3 δ(F + 75)
+2jδ(F − 175) − 2jδ(F + 175). (3.40)
where we have used the identity δ(Ω) = δ(2πF) = 1 δ(F ). Similarly, the CTFT
2π
Y a(j2πF) isgiven by
Y a(j2πF) =4δ(F )+e jπ/3 δ(F − 75) + e −jπ/3 δ(F + 75)
+2jδ(F − 25) − 2jδ(F + 25). (3.41)
Figure 3.15a shows the CTFT of the original signal x a(t) asafunction of
F . The DTFT X ( e jω) of the sampled sequence x(n) isshown as a function of
ω in Figure 3.15b, in which the impulses due to shifted replicas are shown in
gray shade for clarity. The ideal D/A converter response is also shown in gray
shade. The CTFT of the reconstructed signal y a(t) isshown in Figure 3.15c
which clearly shows the aliasing effect.
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Practical D/A converters In practice we need a different approach
than (3.37). The two-step procedure is still feasible, but now we replace
the ideal lowpass filter by a practical analog lowpass filter. Another interpretation
of (3.37) is that it is an infinite-order interpolation. We want
finite-order (and in fact low-order) interpolations. There are several approaches
to do this.
• Zero-order-hold (ZOH) interpolation: In this interpolation a given
sample value is held for the sample interval until the next sample is
received.
ˆx a (t) =x(n),
nT s ≤ n