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Interpolation by a Factor I 489
Ideal Interpolator
x(n)
↑I
v(m)
IDEAL
LPF
y(m)
Rate: F x IF x IF x
FIGURE 9.11
Ideal interpolation by a factor I
Finally, we indicate that the output sequence y(m) can be expressed
as a convolution of the sequence v(n) with the unit sample response h(n)
of the lowpass filter. Thus
y(m) =
∞∑
k=−∞
h(m − k)v(k) (9.34)
Since v(k) = 0 except at multiples of I, where v(kI) = x(k), (9.34)
becomes
∞∑
y(m) = h(m − kI)x(k) (9.35)
k=−∞
The ideal interpolator is shown in Figure 9.11.
MATLAB Implementation MATLAB provides the function [y,h] =
interp(x,I) that resamples the signal in array x at I times the original
sampling rate. The resulting resampled array y is I times longer—i.e.,
length(y) = I*length(x). The ideal lowpass filter given in (9.30) is
approximated by a symmetric filter impulse response, h, which is designed
internally. It allows the original samples to pass through unchanged and
interpolates between so that the mean square error between them and
their ideal values is minimized. The third optional parameter L specifies
the length of the symmetric filter as 2*L*I+1, and the fourth optional
parameter cutoff specifies the cutoff frequency of the input signal in π
units. The default values are L = 5 and cutoff = 0.5. Thus, if I = 2,
then the length of the symmetric filter is 21 for the default L = 5.
□ EXAMPLE 9.4 Let x(n) =cos(πn). Generate samples of x(n) and interpolate them using I =2,
4, and 8 to show the results of interpolation.
Solution
We will plot the middle segments of the signals to avoid end-effects due to
the default lowpass filter in the interp function. The following MATLAB
script shows details of these operations, and Figure 9.12 shows the plots of the
sequences.
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