DISCRETE-SIGNAL ANALYSIS AND DESIGN- P34 ppt

THE HILBERT TRANSFORM 151
expensive. Receivers very often combine the phasing and Þlter methods
in the same or different signal frequency ranges to get greatly improved
performance in difÞcult-signal environments.
The comments for the SSB transmitter section also apply to the receiver,
and no additional comments are needed for this chapter, which is intended
only to show the Hilbert transform and its mathematical equivalent in
a few speciÞc applications. Further and more complete information is
available from a wide variety of sources [e.g., Sabin and Schoenike, 1998],
that cannot be pursued adequately in this introductory book, which has
emphasized the analysis and design of discrete signals in the time and
frequency domains.
REFERENCES
Bedrosian, S. D., 1963, Normalized design of 90
◦
phase-difference networks,
IRE Trans. Circuit Theory, vol. CT-7, June.
Carlson, A. B., 1986, Communication Systems, 3rd ed., McGraw-Hill, New York.
Cuthbert, T. R., 1987, Optimization Using Personal Computers with Applications
to Electrical Networks, Wiley-Interscience, New York. See
or used-book stores.
Dorf, R. C., 1990, Modern Control Systems, 5th ed., Addison-Wesley, Reading,
MA, p. 282.
Krauss H. L., C. W. Bostian, and F. H. Raab, 1980, Solid State Radio Engineer-
ing, Wiley, New York.
Mathworld, />Sabin, W. E., and E. O. Schoenike, 1998, HF Radio Systems and Circuits,
SciTech, Mendham, NJ.
Schwartz, M., 1980, Information Transmission, Modulation and Noise, 3rd ed.,
McGraw-Hill, New York.
Van Valkenburg, M. E., 1982, Analog Filter Design, Oxford University Press,
New York.

Williams, A. B., and F. J. Taylor, 1995, Electronic Filter Design Handbook , 3rd
ed., McGraw-Hill, New York.

APPENDIX
Additional Discrete-Signal
Analysis and Design
Information
†
This brief Appendix will provide a few additional examples of how Math-
cad can be used in discrete math problem solving. The online sources and
Mathcad User Guide and Help (F1) are very valuable sources of infor-
mation on speciÞc questions that the user might encounter in engineering
and other technical activities. The following material is guided by, and is
similar to, that of Dorf and Bishop [2004, Chap. 3].
DISCRETE DERIVATIVE
We consider Þrst Fig. A-1, the discrete derivative, which can be a useful
tool in solving discrete differential equations, both linear and nonlinear.
We consider a speciÞc example, the exponential function exp(·) from
†
Permission has been granted by Pearson Education, Inc., Upper Saddle River, NJ, to use
in this appendix, text and graphical material similar to that in Chapter 3 of [Dorf and
Bishop, 2004].
Discrete-Signal Analysis and Design, By William E. Sabin
Copyright 2008 John Wiley & Sons, Inc.
153

154 DISCRETE-SIGNAL ANALYSIS AND DESIGN
N := 256 n := 0,1 N
−
x(n) := e

n
N
T := 1
0 50 100 150 200 250
0
0.5
1
y(n)
n
(a)
(b)
+
(c)
y(n):= x(0) if n = 0
y(n−1)
x(n + T) − x(n)
T
if n > 0
0 50 100 150 200 250
0
0.5
x(n)
1
n
x(N) = 36.79%
y(N) − x(N)
x(N)
= 0.67%
y(N) = 37.03%
Error for the

discrete derivative
Figure A-1 Discrete derivative: (a) exact exponential decay; (b) deÞ-
nition of the discrete derivative; (c) exponential decay using the discrete
derivative.
n =0toN −1 that decays as
x(n) = exp

−n
N

,0<n<N−1(A-1)
The decay of this function from n =0toN is from 1.0 to 0.3679,
corresponding to a time constant of 1.0. Figure A-1 shows the exact
decay.
ADDITIONAL DISCRETE-SIGNAL ANALYSIS AND DESIGN INFORMATION 155
Now consider the discrete approximation to this derivative, called y(n),
and deÞne y(n)/n as an approximation to the true derivative, as fol-
lows:
y(n) =

x(0) if n = 0
y(n −1) +
x(n+T)−x(n)
T
if n>0
(A-2)
T =1 in this example.
In this equation the second additive term is derived from an incre-
ment of x (n). In other words, at each step in this process, y(n) hopefully
does not change too much (in some situations with large sudden transi-

tions, it might). The advantage that we get is an easy-to-calculate discrete
approximation to the exact derivative.
Figure A-1c shows the decay of x (n) using the discrete derivative.
In part (b) the accumulated error in the approximation is about 0.67%,
which is pretty good. Smaller values of T can improve the accuracy; for
example, T =0.1 gives an improvement to about 0.37%, but values of
T smaller than this are not helpful for this example. A larger number of
samples, such as 2
9
, is also helpful. The discrete derivative can be very
useful in discrete signal analysis and design.
STATE-VARIABLE SOLUTIONS
We will use the discrete derivative and matrix algebra to solve the two-
state differential equation for the LCR network in Fig. A-2. There are
two energy storage elements, L and C , in the circuit. There is a voltage
across and a displacement current through the capacitor C , and a voltage
across and an electronic current through the inductor L. We want all of
these as a function of time t. There are also possible initial conditions at
t =0, which are a voltage V
C0
on the capacitor and a current I
L0
through
the inductor, and a generator (u) (in this case, a current source) is con-
nected as shown. The two basic differential equations are, in terms of v
C
and i
L
,