DISCRETE-SIGNAL ANALYSIS AND DESIGN- P36 pdf

In positive-only frequency systems, a signal in part of the positive-frequency region is invaded by a second signal that is in an adjacent part of the positive-frequency region.. An X k,

ADDITIONAL DISCRETE-SIGNAL ANALYSIS AND DESIGN INFORMATION 161

V C

Vc

•

I L

•

−R/L

−1/C 1/L

I L (0)

I L

V C (0)

R

(a)

V O

1.0 1.0

X1 (s)

X 1(s)

•

−R/L

(b)

R VO

−1/C

1/s

Figure A-4 Flow-chart for the network of Fig A-2: (a) with no input

(u) but with initial values of V C and I L; (b) with no initial conditions but

with a sine-wave input signal u(t).

The book by [Dorf and Bishop] explores this problem using several different methods that are very instructional but that we do not pursue

in this book The reader is encouraged to become more familiar with the network analysis methods described in this appendix It is good practical engineering

Finally, Fig A-4 illustrates the two varieties of ßow graph for the network discussed in this appendix We can understand Fig A-4a by referring to Eq (A-5) with u set to zero (no external inputs) and with

initial values of V C (0) and I L(0), as shown also in Fig A-2 In Fig A-4b,

V C , I L, and their derivatives correspond to those in Eq (A-5) with initial

conditions V C and I L set to zero, as shown in Fig A-3, and the input u drives the network from a zero start with a sine wave that starts at zero

value The output peak amplitude V O (t) ßuctuates for at least the 1000

time increments illustrated

It is also an interesting exercise for the reader to calculate and plot the inductor voltage and current and the capacitor voltage and current as functions of time n in Figs A-2 and A-3

162 DISCRETE-SIGNAL ANALYSIS AND DESIGN

REFERENCES

Dorf, R C., and R H Bishop, 2004, Modern Control Systems, 10th ed., Prentice

Hall, Upper Saddle River, NJ, Chap 3

Zwillinger, D., Ed., 1996, CRC Standard Mathematical Tables and Formulae,

30th ed., CRC Press, Boca Raton, FL

Adjacent channel interference One or more adjacent channel signals

create interference in a desired channel by aliasing or wideband emissions

Aliasing (classical) In positive-only frequency systems, a signal in part

of the positive-frequency region is invaded by a second signal that is

in an adjacent part of the positive-frequency region

Aliasing The overlapping (invasion) from one 0 to N − 1 time or

fre-quency sequence to an adjacent 0 to N− 1 time or frequency sequence

Amplitude noise Noise created by variations in the amplitude of a signal.

Analytic signal (sequence) An X (k), its Hilbert transform ˆ X(k) and

the ±j operator combine to create a phasor sequence that is one-sided in the positive- or negative-frequency domain The phasor A

exp(±jθ) is an analytic signal The analytic phasor sequence is used to

construct SSB signals digitally or discretely It is synthesized to design analog SSB systems

Auto-covariance The ac component of an autocorrelation.

Average value The time average of a signal between two time limits,

often minus inÞnity to plus inÞnity

Discrete-Signal Analysis and Design, By William E Sabin

Copyright 2008 John Wiley & Sons, Inc.

163



164 GLOSSARY

Boltzmann’s constant 1.38× 10−23 joules per Kelvin Used in noise calculations

Coherent Two time signals x1(n) and x2(n) are coherent if their x(n) values add together algebraically at each (n) In the frequency domain the X (k)s add in a similar manner.

Complex frequency domain Values of X (k) phasors contain a real part,

an imaginary part, an amplitude value, a frequency value, and a phase value relative to some reference phase value The domain has a positive-frequency side and an equal-length negative-positive-frequency side

Complex plane The two-dimensional rectangular plane of the real axis

(x) and the imaginary axis (jy) (see Fig 1-5).

Complex signal A signal that is deÞned as part real and part imaginary

on the complex plane In the time domain, sequences can be complex.

In the frequency domain, a single phasor can be complex

Convolution A fold, slide, and multiply operation to obtain an overlap

area between two geometric or mathematical regions

Correlation A measure of the similarity of a function and a time- or

frequency-shifted copy of the function (auto correlation) or the similar-ity of two different functions, one of which is shifted (cross-correlation)

Correlation coefÞcient A measure of the “relatedness” in some sense,

from −1 to +1, of two nondeterministic or deterministic processes

Cross-covariance The ac component of a cross-correlation.

Cross power spectrum The commonality of power spectrum in two

associated signals

Discrete derivative An approximate implementation of a time-derivative

that uses the discrete sequence x(n).

Discrete Fourier series In discrete-signal length-N analysis, a periodic

repeating waveform can be deÞned as a useful set of positive-frequency

harmonics from k = 1 to k = N /2 − 1.

Discrete Fourier transform (DFT) Converts the time domain x(n) to

the frequency domain X (k).

Discrete Fourier transform of convolution Converts a convolution of

two time sequences to the product of two frequency sequences: the system function Used in linear system analysis

GLOSSARY 165

Discrete frequency Signals X (k) in the frequency domain occur at

dis-crete values of frequency (k) from 0 to N− 1

Discrete time Signals x(n) in the time domain occur at discrete values

of time (n) from 0 to N − 1

Digital signal processing (DSP) Signal processing in which signal

amplitudes are also discrete (quantized)

Even symmetry The two sides, X (k) and X (N − k), of a phasor

spec-trum have the same phase

Expected value The sum of products of a signal amplitude at time T

and the probability of occurrence of the signal at time T [Eq (6-1)] Also known as the Þrst moment.

Fast Fourier transform (FFT) A high-speed algorithm for the DFT Flow graph A graphical method of tracing the ßow of signals within a

network

Fourier, Joseph French mathematician who originated the trigonometric

series method of analysis and design of mathematical and physical phenomena

Frequency domain Signals are classiÞed according to their occurrence

in frequency (f ) continuous or discrete X (k).

Frequency scaling A sequence of frequency values have a certain

sequential relationship from low end to high end The maximum fre-quency minus the minimum frefre-quency, divided by the number of frequencies, is the frequency scale factor

Gaussian noise Random electrical noise, perhaps thermally generated

noise, that has the Gaussian (normal) amplitude probability density function

Hermitian symmetry A spectral property such that positive- and

negative-frequency values are complex conjugates The sine and cosine-wave phasors are Hermitian

Hilbert transform In RF work, an algorithm that modiÞes a two-sided

phasor spectrum so that positive-frequency phasors are phase shifted

−90◦and negative-frequency phasors are phase shifted+90◦ This idea

is useful in many applications, especially in SSB

Integer A collection of whole numbers: such as±(1, 2, 3, ).