DISCRETE-SIGNAL ANALYSIS AND DESIGN- P34 ppt

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 ha

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 trcpep@aol.com

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, http://mathworld.wolfram.com/AnalyticFunction.html

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

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

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

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 = 0 to N − 1 that decays as

x(n)= exp



−n N



The decay of this function from n = 0 to N 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= 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 29, 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,