Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 588

where

f s ¼ sampling rate in the oversampling DSP system
f max ¼ maximum frequency of the analog signal
m ¼ number of bits per sample in the oversampling DSP system
n ¼ number of bits per sample in the regular DSP system using the minimum sampling rate

From Equation (12.20) and given the number of bits ( m ) used in the oversampling scheme, we can
determine the number of bits per sample equivalent to the regular ADC. On the other hand, given the
number of bits in the oversampling ADC, we can determine the required oversampling rate so that the
oversampling ADC is equivalent to the regular ADC with the larger number of bits per sample ( n ). Let
us look at the following examples.

EXAMPLE 12.7
Given an oversampling audio DSP system with maximum audio input frequency of 20 kHz and ADC resolution of
14 bits, determine the oversampling rate to improve the ADC resolution to 16-bit resolution.
Solution:
Based on the specifications, we have
f max ¼ 20 kHz; m ¼ 14 bits

and n ¼ 16 bits

Using Equation (12.21) leads to
f s ¼ 2f max 2 2ðnmÞ ¼ 2 20 2 2ð1614Þ ¼ 640 kHz
Since fs=ð2f max Þ¼2 4 , we see that each doubling of the minimum sampling rate (2f max ¼ 40 kHz) will increase
the resolution by a half bit.

EXAMPLE 12.8
Given an oversampling audio DSP system with a maximum audio input frequency of 4 kHz, and ADC resolution of 8
bits, an a sampling rate of 80 MHz, determine the equivalent ADC resolution.
Solution:
Since f max ¼ 4 kHz, f s ¼ 80 kHz, and m ¼ 8 bits, appyling Equation (12.20) yields

f s
2f max

80;000 kHz
2 4 kHz

n ¼ m þ 0:5 log 2 ¼ 8 þ 0:5 log 2

z 15 bits

The MATLAB program shown in Program 12.7 validates the oversampling technique. We consider
the following signal,

xðtÞ¼ 1 : 5 sin ð 2 p 150 tÞþ 0 : 9 sin ð 2 p 175 t þ p= 6 Þþ 0 : 6 sin ð 2 p 200 t þ p= 4 Þ (12.22)

with a regular sampling rate of 1 kHz. The oversampling rate is 4 kHz and each sample is quantized
using a 3-bit code. The anti-aliasing lowpass filter is designed with a cutoff frequency of
U ¼ 2 pf max T ¼ 2 p 500 = 4 ; 000 ¼ 0 : 25 p radians. Figure 12.25 shows the frequency responses of