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Sampling and Reconstruction in the z-Domain 153
Consider
Let
Then
1 − e −jωN
N { 1 − e j2πk/N e −jω} =
2πk
1 − e−j(ω−
{
N )N
}
2πk
N 1 − e−j(ω− N )
= e−j N 2πk
2 (ω− N )
e − 1 2πk
2 j(ω− N
N ) N 2
{ [
sin (ω −
2πk
) N sin [ (ω − 2πk
]
N ) 1 2
Φ(ω) = △ sin( ωN 2 )
N−1
N sin( ω 2 ) :aninterpolating function (5.18)
2
)e−jω(
X(e jω )=
N−1
∑
k=0
(
˜X(k)Φ ω − 2πk )
N
]
}
(5.19)
This is the DTFT interpolation formula to reconstruct X(e jω ) from its
samples ˜X (k). Since Φ(0) = 1, we have that X(e j2πk/N )= ˜X(k), which
means that the interpolation is exact at sampling points. Recall the
time-domain interpolation formula (3.33) for analog signals:
∞∑
x a (t) = x(n) sinc [F s (t − nT s )] (5.20)
n=−∞
The DTFT interpolating formula (5.19) looks similar.
However, there are some differences. First, the time-domain formula
(5.20) reconstructs an arbitrary nonperiodic analog signal, while the
frequency-domain formula (5.19) gives us a periodic waveform. Second, in
(5.19) we use a sin(Nx) interpolation function instead of our more familiar
sin x
x
N sin x
(sinc) function. The Φ(ω) function is a periodic function and hence
is known as a periodic-sinc function. It is also known as the Dirichlet
function. This is the function we observed in Example 5.2.
5.2.3 MATLAB IMPLEMENTATION
The interpolation formula (5.19) suffers the same fate as that of (5.20)
while trying to implement it in practice. One has to generate several interpolating
functions (5.18) and perform their linear combinations to obtain
the discrete-time Fourier transform X(e jω ) from its computed samples
˜X(k). Furthermore, in MATLAB we have to evaluate (5.19) on a finer grid
over 0 ≤ ω ≤ 2π. This is clearly an inefficient approach. Another approach
is to use the cubic spline interpolation function as an efficient approximation
to (5.19). This is what we did to implement (5.20) in Chapter 3.
However, there is an alternate and efficient approach based on the DFT,
which we will study in the next section.
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