Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 572

before DAC, followed by application of the digital interpolation filter (assume L ¼ 4). Certainly,
the interpolation filter design must satisfy the specifications studied in the previous section on
increasing the sampling rate by an integer factor. Again, after digital interpolation, the audio band
is kept the same, while the sampling frequencyisincreasedbyfourfold( L ¼ 4), that is, 44.1 4 ¼
176.4 kHz.
Since the audio band of 22.05 kHz is now relatively low compared with the new folding frequency
(176.4/2 ¼ 88.2 kHz), the use of a simple first-order or second-order analog anti-image filter may be
sufficient. Let us look the following simulation.
A test audio signal with a frequency of 16 kHz and a sampling rate of 44.1 kHz is generated using
the formula

x n ¼ sin 2 p 16 ; 000 n
441 ; 000

If we use an upsampling factor of 4, then the bandwidth would increase to 88.2 kHz. Based on the
audio frequency of 16 kHz, the original Nyquist frequency of 22.05 kHz, and the new sampling rate of
176.4 kHz, we can determine the filter length as

22 : 05 16
176 : 4

Df ¼ ¼ 0 : 0343

Using the Hamming window for FIR filter design leads to

3 : 3
Df ¼ 96 : 2

N ¼

We choose N ¼ 97. The cutoff frequency therefore is

16 þ 22 : 05
2

f c ¼ ¼ 19 : 025 kHz

The spectrum of the interpolated audio test signal is shown in Figure 12.12 , where the top plot
illustrates that after the upsampling, the audio test signal has a frequency of 16 kHz, along with image
frequencies coming from 44.1 - 16 ¼ 28.1 kHz, 44.1 þ 16 ¼ 60.1 kHz, 88.2 - 16 ¼ 72.2 kHz, and so
on. The bottom graph describes the spectrum after the interpolation filter. From lowpass FIR filtering,
an interpolated audio signal with a frequency of 16 kHz is observed.
Let us examine the corresponding process in the time domain, as shown in Figure 12.13 . The upper
left plot shows the original samples. The upper right plot describes the upsampled signals. The lower
left plot shows the signals after the upsampling process and digital interpolation filter. Finally, the lower
right plot shows the sample-and-hold signals after DAC. Clearly, we can easily design a reconstruction
filter to smooth the sample-and-hold signals and obtain the original audio test signal. The advantage of
reducing hardware is illustrated. The MATLAB implementation can be seen in Program 12.4.
Program 12.4. MATLAB program for CD player example.

close all; clear all; clc
% Generate the 2048 samples with fs ¼ 44100 Hz
fs ¼ 44100;

% Original sampling rate

T ¼ 1/fs; % Sampling period