Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 563

x () wm () y ()

Interpolation
filter H(z)

L

f s Lf s Lf s

x ()

n

5

0

1 2 3 4

w ()

m

0 2 4 6 8 10 12 14 16

y ()

m

0 2 4 6 8 10 12 14 16

FIGURE 12.5A

Block diagram for the upsampling process with L ¼ 3.

After filtering via the interpolation filter, we will achieve the desired spectrum for yðnÞ , as shown in
Figure 12.5B . Note that since the interpolation is to remove the high-frequency images that are aliased
by the upsampling operation, it is essentially an anti-aliasing lowpass filter.
To verify the upsampling principle, we generate the signal xðnÞ with 1 kHz and 2.5 kHz as follows:
x n ¼ 5 sin 2 p 1 ; 000 n
8 ; 000

þ cos 2 p 2 ; 500 n
8 ; 000

with a sampling rate of f s ¼ 8 ; 000 Hz. The spectrum of xðnÞ is plotted in Figure 12.6 . Nowwe
upsample xðnÞ by a factor of 3, that is, L ¼ 3. We know that the sampling rate is increased to be
3 8,000 ¼ 24,000 Hz. Hence, without using the interpolation filter, the spectrum would contain the
image frequencies originally centered at the multiple frequencies of 8 kHz. The top plot in Figure 12.6
shows the spectrum for the sequence after upsampling and before applying the interpolation filter.
Now we apply an FIR lowpass filter designed with a length of 53, a cutoff frequency of 3,250 Hz,
and a new sampling rate of 24,000 Hz as the interpolation filter, whose normalized frequency should be

1
24 ; 000

U c ¼ 2 p 3 ; 250 ¼ 0 : 2708 p