189
CHAPTER 6
SIGNAL PROCESSING
John Turnbull
Case Western Reserve University
Cleveland, Ohio
1 FREQUENCY-DOMAIN
ANALYSIS OF LINEAR
SYSTEMS
189
2 BASIC ANALOG FILTERS
191
2.1 Butterworth
193
2.2 Tchebyshev
194
2.3
Inverse Tchebyshev
195
2.4 Elliptical
195
2.5 Arbitrary Frequency Response
Curve Fitting by Method of
Least Squares
196
2.6 Circuit Prototypes for Pole
and Zero Placement for
Realization of Filters Designed
from Rational Functions
197
3 BASIC DIGITAL FILTER
197
3.1
z-Transforms
198
3.2 Design of FIR Filters
198
3.3 Design of IIR Filters
201
3.4 Design of Various Filters from
Low-Pass Prototypes
203
3.5 Frequency-Domain Filtering
205
4 STABILITY AND PHASE
ANALYSIS
206
4.1 Stability Analysis
206
4.2 Phase Analysis
206
4.3 Comparison of FIR and IIR
Filters
208
5 EXTRACTING SIGNAL FROM
NOISE
208
REFERENCES
208
1 FREQUENCY-DOMAIN ANALYSIS OF LINEAR SYSTEMS
Signals are any carriers of information. Our objective in signal processing involves the en-
coding of information for the purpose of transmission of information or decoding the infor-
mation at the receiving end of the transmission. Unfortunately, the signal is often corrupted
by noise during our transmission, and hence it is our objective to extract the information
from the noise. The standard method most commonly used for this involves filters that exploit
some separation of the signal and noise in the frequency domain. To this end, it is useful to
use frequency-domain tools such as the Fourier transform and the Laplace transform in
designing and analyzing various filters. The Fourier transform of a function of a time is
1
jt
2
F{ƒ(t)} F()
ƒ(t)e
dt
j 1
(1)
2
For continuous systems, the transfer characteristics of a filter system is a function that gives
information of the gain versus frequency. The Laplace transform for a given time-domain
function is
st
L{ƒ(t)} F(s) ƒ(t)e
dt
(2)
0
The steady-state Laplace transform (i.e., neglecting transients) for the derivative and integral
of a given function is
Mechanical Engineers’ Handb