Signal Sampling and Quantization - Digital Signal Processing: Fundamentals and Applications - page 26

Dig ital value

Anti-aliasing
LP filter

Sample and
hold

ADC
coding

Analog signal spectrum
(wo r s t case)

Xf

s ()

X ()

X a

f a

f f c f f c f s

2

f

f s

f − f

s a

Aliasing level X a
at f a (image from f s −

f a )

FIGURE 2.16

Spectrum of the sampled analog signal with a practical anti-aliasing filter.

For a second-order Butterworth lowpass filter with unit gain, the transfer function (which will be
discussed in Chapter 8) and its magnitude frequency response are given by

2

ð 2 pf c Þ

HðsÞ¼ (2.8)

2

2

s þ 1 : 4141 ð 2 pf c Þs þð 2 pf c Þ

1

s

jHðf Þj ¼ (2.9)

f
f c

4

1 þ

A unit gain second-order lowpass filter using a Sallen-Key topology is shown in Figure 2.17 . Matching
the coefficients of the circuit transfer function to that of the second-order Butterworth lowpass transfer
function in Equation (2.10) gives the design formulas shown in Figure 2.17 , where for a given cutoff

C 2

V in

R 1 R 2

V o

+

− Choose

C 2

1 4142
(2

.

RR Cf c ) ==

C 1 1 2

π

2

1
(2

C =

1 2

RRC π f c )

12 2

FIGURE 2.17

Second-order unit gain Sallen-Key lowpass filter.