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TitleUnderstandig Digital Signal Processing
TagsSignal (Electrical Engineering) Discrete Fourier Transform Digital Signal Processing Hertz Fast Fourier Transform
File Size984.9 KB
Total Pages81
Table of Contents
                            CONTENTS
PREFACE
ABOUT THE AUTHOR
1 DISCRETE SEQUENCES AND SYSTEMS
	1.1 Discrete Sequences and Their Notation
	1.2 Signal Amplitude, Magnitude, Power
	1.3 Signal Processing Operational Symbols
	1.4 Introduction to Discrete Linear Time-Invariant Systems
	1.5 Discrete Linear Systems
	1.6 Time-Invariant Systems
	1.7 The Commutative Property of Linear Time-Invariant Systems
	1.8 Analyzing Linear Time-Invariant Systems
	References
	Chapter 1 Problems
INDEX
	A
	B
	C
	D
	E
	F
	G
	H
	I
	J
	K
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	V
	W
	Z
                        
Document Text Contents
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Understanding Digital
Signal Processing

Third Edition

Page 40

which is shown as the all-positive sequence in the center of Figure 1–8(b). Be-
cause Eq. (1–19) results in a frequency sum (α + β) and frequency difference
(α – β) effect when multiplying two sinusoids, the y1(n) output sequence will
be a cosine wave of 2 Hz and a peak amplitude of –0.5, added to a constant
value of 1/2. The constant value of 1/2 in Eq. (1–20) is interpreted as a zero
Hz frequency component, as shown in the Y1(m) spectrum in Figure 1–8(b).
We could go through the same algebraic exercise to determine that when a
3 Hz sinewave x2(n) sequence is applied to this nonlinear system, the output
y2(n) would contain a zero Hz component and a 6 Hz component, as shown
in Figure 1–8(c).

System nonlinearity is evident if we apply an x3(n) sequence comprising
the sum of a 1 Hz and a 3 Hz sinewave as shown in Figure 1–8(d). We can

1.5 Discrete Linear Systems 15

Figure 1–8 Nonlinear system input-to-output relationships: (a) system block dia-
gram where y(n) = [x(n)]2; (b) system input and output with a 1 Hz
sinewave applied; (c) with a 3 Hz sinewave applied; (d) with the sum
of 1 Hz and 3 Hz sinewaves applied.

(b)

(c)

0

0.5

1

1.5

2

2.5

0

0.5

1

0

0.2

0.4

0.6

0.8

1

0

0.5

1

0

0.2

0.4

0.6

0.8

1

–1

2

4 6 8 10 12 14

0

1

2

(d)

x (n)
y (n) Y (m)

1
1

1

x (n)
y (n) Y (m)

2
2 2

x (n)
y (n) Y (m)

3
3 3

Time

Time

Time

Time

Time

Time

1

0.5

0

–0.5

(a) Input x(n) Output y(n) = [x(n)]
2

Nonlinear
Discrete
System

zero Hz component

0

–1

2 4

6

8 10 12 14

1

0
0

–1

2

4 6

8 10 12 14

1

0
0

Freq
(Hz)

Freq
(Hz)

Freq
(Hz)

0.5

–0.5

0.5

–0.5

–1

–0.5

–1

–2

–1

–0.5

Page 41

predict the frequency content of the y3(n) output sequence by using the alge-
braic relationship

(a+b)2 = a2+2ab+b2, (1–21)

where a and b represent the 1 Hz and 3 Hz sinewaves, respectively. From Eq.
(1–19), the a2 term in Eq. (1–21) generates the zero Hz and 2 Hz output sinu-
soids in Figure 1–8(b). Likewise, the b2 term produces in y3(n) another zero Hz
and the 6 Hz sinusoid in Figure 1–8(c). However, the 2ab term yields addi-
tional 2 Hz and 4 Hz sinusoids in y3(n). We can show this algebraically by
using Eq. (1–19) and expressing the 2ab term in Eq. (1–21) as

Equation (1–22) tells us that two additional sinusoidal components will be
present in y3(n) because of the system’s nonlinearity, a 2 Hz cosine wave
whose amplitude is +1 and a 4 Hz cosine wave having an amplitude of –1.
These spectral components are illustrated in Y3(m) on the right side of Figure
1–8(d).

Notice that when the sum of the two sinewaves is applied to the nonlin-
ear system, the output contained sinusoids, Eq. (1–22), that were not present
in either of the outputs when the individual sinewaves alone were applied.
Those extra sinusoids were generated by an interaction of the two input sinu-
soids due to the squaring operation. That’s nonlinearity; expression (1–13)
was not satisfied. (Electrical engineers recognize this effect of internally gen-
erated sinusoids as intermodulation distortion.) Although nonlinear systems are
usually difficult to analyze, they are occasionally used in practice. References
[2], [3], and [4], for example, describe their application in nonlinear digital fil-
ters. Again, expressions (1–13) and (1–14) state that a linear system’s output
resulting from a sum of individual inputs is the superposition (sum) of the in-
dividual outputs. They also stipulate that the output sequence y1(n) depends
only on x1(n) combined with the system characteristics, and not on the other
input x2(n); i.e., there’s no interaction between inputs x1(n) and x2(n) at the
output of a linear system.

2 2 2 1 2 3

2 2 1 2 3
2

2 2 1 2 3
2

2 2 2 4

ab nt nt

nt nt nt nt

nt nt

s s

s s s s

s s

= π π

=
π − π


π + π

= π − π

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅

sin( ) sin( )

cos( ) cos( )

cos( ) cos( ) .†

16 Discrete Sequences and Systems

† The first term in Eq. (1–22) is cos(2π ⋅ nts – 6π ⋅ nts) = cos(–4π ⋅ nts) = cos(–2π ⋅ 2 ⋅ nts). However, be-
cause the cosine function is even, cos(–α) = cos(α), we can express that first term as cos(2π ⋅ 2 ⋅nts).

(1–22)

Page 80

by ± π/8, 809–810
rotational symmetries, 807

Vector-magnitude approximation,
679–683

von Hann windows. See Hanning
windows.

W
Warping, frequency, 319, 321–325, 328–330
Weighted overlap-add spectrum analysis,

755
Weighting factors, coherent signal

averaging, 608, 789
Wideband compensation, 564
Wideband differentiators, 367–370
Willson, A., 386
Window design method, FIR filters,

186–194
Windowed-presum FFT spectrum

analysis, 755
Windows

Blackman, 195–201, 686, 733
Blackman-Harris, 686, 733
exact Blackman, 686
FFTs, 139
in the frequency domain, 683–686
magnitude response, 92–93
mathematical expressions of, 91
minimizing DFT leakage, 89–97
processing gain or loss, 92
purpose of, 96
rectangular, 89–97, 686
selecting, 96
triangular, 89–93

Windows, Hamming
description, 89–93
DFT leakage reduction, 89–93
in the frequency domain, 683–686
spectral peak location, 733

Windows, Hanning
description, 89–97
DFT leakage, minimizing, 89–97
in the frequency domain, 683–686
spectral peak location, 733

Windows used in FIR filter design
Bessel functions, 198–199
Blackman, 195–201

Index 953

Chebyshev, 197–201, 927–930
choosing, 199–201
Dolph-Chebyshev, 197
Kaiser, 197–201
Kaiser-Bessel, 197
Tchebyschev, 197

Wingless butterflies, 156
Wraparound leakage, 86–88
Wrapping, phase, 209, 900

Z
z-domain expression for Mth-order IIR

filter, 275–276
z-domain transfer function, IIR filters,

282–289
Zero padding

alleviating scalloping loss, 97–102
FFTs, 138–139
FIR filters, 228–230
improving DFT frequency granularity,

97–102
spectral peak location, 731

Zero stuffing
interpolation, 518
narrowband lowpass filters, 834–836

Zero-overhead looping
DSP chips, 333
FSF (frequency sampling filters),

422–423
IFIR filters, 389

Zero-phase filters
definition, 902
techniques, 725

Zeros
IIR filters, 284–289
on the s-plane, Laplace transform,

263–270
Zoom FFT, 749–753
z-plane, 270–273
z-plane pole / zero properties, IIR filters,

288–289
z-transform. See also Laplace transform.

definition, 270
description of, 270–272
FIR filters, 288–289
IIR filters, 270–282
infinite impulse response, definition, 280

Page 81

z-transform (cont.)
polar form, 271
poles, 272–274
unit circles, 271
zeros, 272–274

z-transform, analyzing IIR filters
digital filter stability, 272–274, 277

954 Index

Direct Form 1 structure, 275–278
example, 278–282
frequency response, 277–278
overview, 274–275
time delay, 274–278
z-domain transfer function, 275–278,

279–280

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