Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 35

From these two equations, we have to satisfy
ð3; 000=f c Þ 2n ¼ 10 0:154 1

ð13; 000=f c Þ 2n ¼ 10 2 1

Taking the ratio of these two equations yields
13; 000
3; 000

2n

10 2 1
10 0:154 1

¼

Then

n ¼ 1

2 logðð10 2 1Þ=ð10 0:154 1ÞÞ=logð13; 000=3; 000Þ¼1:86 z 2

Finally, the cutoff frequency can be computed as

13; 000
ð10 2 1Þ 1=ð2nÞ ¼

13; 000
ð10 2 1Þ 1=4 ¼ 4; 121:30 Hz

f c ¼

3; 000
ð10 0:154 1Þ 1=ð2nÞ ¼

3; 000
ð10 0:154 1Þ 1=4 ¼ 3; 714:23 Hz

f c ¼

We choose the smaller one, that is,

f c ¼ 3; 714:23 Hz

With the filter order and cutoff frequency, we can realize the anti-image (reconstruction) filter using the second-
order unit gain Sallen-Key lowpass filter described in Figure 2.17 .

Note that the specifications for anti-aliasing filter designs are similar to anti-image (reconstruction)
filters, except for their stopband edges. The anti-aliasing filter is designed to block the frequency
components beyond the folding frequency before the ADC operation, while the reconstruction filter
is designed to block the frequency components beginning at the lower edge of the first image after
the DAC.

2.3 ANALOG-TO-DIGITAL CONVERSION, DIGITAL-TO-ANALOG CONVERSION,
AND QUANTIZATION
During the ADC process, amplitudes of the analog signal to be converted have infinite precision. The
continuous amplitude must be converted to digital data with finite precision, which is called quanti-
zation . Figure 2.26 shows quantization as a part of ADC.
There are several ways to implement ADC. The most common ones are

• Flash ADC
• Successive approximation ADC
• Sigma-delta ADC.