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Page 2

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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Page 2

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