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92 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
to higher orders. One particularly useful interpolation employed by
MATLAB is the following.
• Cubic spline interpolation: This approach uses spline interpolants
for a smoother, but not necessarily more accurate, estimate of the analog
signals between samples. Hence this interpolation does not require
an analog postfilter. The smoother reconstruction is obtained by using
a set of piecewise continuous third-order polynomials called cubic
splines, given by [3]
x a (t) =α 0 (n)+α 1 (n)(t − nT s )+α 2 (n)(t − nT s ) 2
+ α 3 (n)(t − nT s ) 3 , nT s ≤ n> n = n1:n2; t = t1:t2; Fs = 1/Ts; nTs = n*Ts; % Ts is the sampling interval
>> xa = x * sinc(Fs*(ones(length(n),1)*t-nTs’*ones(1,length(t))));
Note that it is not possible to obtain an exact analog x a (t) inlight of the
fact that we have assumed a finite number of samples. We now demonstrate
the use of the sinc function in the following two examples and also
study the aliasing problem in the time domain.
□ EXAMPLE 3.21 From the samples x 1(n) inExample 3.19a, reconstruct x a(t) and comment on
the results.
Solution
Note that x 1(n) was obtained by sampling x a(t) atT s =1/F s =0.0002 sec. We
will use the grid spacing of 0.00005 sec over −0.005 ≤ t ≤ 0.005, which gives
x(n) over−25 ≤ n ≤ 25.
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