A

Tables of Fourier Series and TransformProperties

320 A Tables of Fourier Series and Transform Properties

Table A.1 Properties of the continuous-time Fourier series

x(t) =∞∑

k=−∞CkejkΩt Ck =

1T

T/2∫

−T/2

x(t)e−jkΩtdt

Property Periodic function x(t)with period T = 2π/Ω Fourier series Ck

Time shifting x(t ± t0) Cke±jkΩt0

Time scaling x(αt), α > 0 Ck with period Tα

Differentiation ddtx(t) jkΩCk

Integrationt∫

−∞x(t)dt < ∞ 1

jkΩCk

Linearity∑i

αixi(t)∑i

αiCik

Conjugation x∗(t) C∗−k

Time reversal x(−t) C−k

Modulation x(t)ejKΩt Ck−K

Multiplication x(t)y(t)∞∑

i=−∞CxiCy(k−i)

Periodicconvolution

∫T

x(θ)y(t − θ)dθ TCxkCyk

Symmetry x(t) = x∗(t) real

⎧⎪⎪⎨⎪⎪⎩

Ck = C∗−k , |Ck| = |C−k| ,

Re Ck = Re C−k ,ImCk = −Im C−k ,arg Ck = − arg C−k

x(t) = x∗(t) = x(−t)real and even

Ck = C−k , Ck = C∗

k ,real and even

x(t) = x∗(t) = −x(−t)real and odd

Ck = −C−k , Ck = −C∗

k ,imaginary and odd

Parseval’s theorem 1T

T/2∫−T/2

|x(t)|2dt =∞∑

k=−∞|Ck|2

A Tables of Fourier Series and Transform Properties 321

Table A.2 Properties of the continuous-time Fourier transform

x(t) =12π

∞∫

−∞

X(jω)ejωtdω X(jω) =

∞∫

−∞

x(t)e−jωtdt

Property Nonperiodicfunction x(t)

Fourier transformX(jω)

Time shifting x(t ± t0) e±ωt0X(jω)

Time scaling x(αt) 1|α|X

(jωα

)

Differentiation ddtx(t) jωX(jω)

Integrationt∫

−∞x(t)dt 1

jω X(jω) + πX(j0)δ(ω)

∞∫−∞

x(t)dt X(j0)

Frequencyintegration 2πx(0)

∞∫−∞

X(jω)dω

Linearity∑i

αixi(t)∑i

αiXi(jω)

Conjugation x∗(t) X∗(−jω)

Time reversal x(−t) X(−jω)

Modulation x(t)ejω0t X(jω − jω0)

Multiplication x(t)y(t) 12πX(jω) × Y (jω)

Convolution x(t) ∗ y(t) X(jω)Y (jω)

Symmetry x(t) = x∗(t) real

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

X(jω) = X∗(−jω) ,|X(jω)| = |X(−jω)| ,

Re X(jω) = Re X(−jω) ,Im X(jω) = −Im X(−jω) ,arg X(jω) = − arg X(−jω)

x(t) = x∗(t) = x(−t)real and even

⎧⎨⎩

X(jω) = X(−jω) ,X(jω) = X∗(jω) ,

real and even

x(t) = x∗(t) = −x(−t)real and odd

⎧⎨⎩

X(jω) = −X(−jω) ,X(jω) = −X∗(jω) ,imaginary and odd

Rayleigh’s theorem Ex =∞∫

−∞|x(t)|2dt = 1

2π

∞∫−∞

|X(jω)|2dω

B

Tables of Fourier Series and Transform of BasisSignals

324 B Tables of Fourier Series and Transform of Basis Signals

Table B.1 The Fourier transform and series of basic signals

Signal x(t) Transform X(jω) Series Ck

1 2πδ(ω) C0 = 1 , Ck =0 = 0

δ(t) 1 Ck = 1T

u(t) 1jω + πδ(ω) —

u(−t) − 1jω + πδ(ω) —

ejΩt 2πδ(ω − Ω) C1 = 1 , Ck =1 = 0∞∑

k=−∞CkejkΩt 2π

∞∑k=−∞

Ckδ(ω − kΩ) Ck

cos Ωt π[δ(ω − Ω) + δ(ω + Ω)] C1 = C−1 = 12 , Ck =±1 = 0

sinΩt πj [δ(ω − Ω) − δ(ω + Ω)] C1 = −C−1 = 1

2j , Ck =±1 = 0

1α2+t2 e−α|ω| 1

T e−2πα|k|

T

Rectangular(Fig. 2.16c) Aτ sin(ωτ/2)

ωτ/2Aq

sin(kπ/q)kπ/q

Triangular(Fig. 2.21a)

Aτ2

sin2(ωτ/4)(ωτ/4)2

A2q

sin2(kπ/2q)(kπ/2q)2

Trapezoidal(Fig. 2.30) Aτ sin(ωτ/2)

ωτ/2sin(ωτs/2)

ωτs/2Aq

sin(kπ/q)kπ/q

sin(kπ/qs)kπ/qs

Ramp(Fig. 2.34b)

Ajω

[sin(ωτ/2)

ωτ/2 ej ωτ2 − 1

]A

j2πk

[sin(kπ/q)

kπ/q ej kπq − 1

]

Ramp(Fig. 2.34c)

Ajω

[1 − sin(ωτ/2)

ωτ/2 e−j ωτ2

]A

j2πk

[1 − sin(kπ/q)

kπ/q e−j kπq

]

sin αtαt

πα , |ω| < α0, |ω| > α

π

αT , |k| < αT2π

0, |k| > αT2π

e−αtu(t) ,Re α > 0

1α+jω

1αT+j2πk

te−αtu(t) ,Re α > 0

1(α+jω)2

T(αT+j2πk)2

B Tables of Fourier Series and Transform of Basis Signals 325

Table B.1 The Fourier transform and series of basic signals (Contd.)

tn−1

(n−1)!e−αtu(t) ,

Re α > 01

(α+jω)nT n−1

(αT+j2πk)n

e−α|t| , α > 0 2αα2+ω2

2αTα2T 2+4π2k2

e−α2t2√

πα e−

ω2

4α2√

παT e−

π2k2

α2T2

Ck corresponds to x(t) repeated with period T , τ and τs are durations, q = Tτ ,

and qs = Tτs

.

Table B.2 The Fourier transform and series of complex signals

Signal y(t) Transform Y (jω) Series Ck

Burst of Npulses withknown X(jω)

X(jω) sin(ωNT/2)sin(ωT/2)

1T1

X(j 2kπ

T1

)sin(kπ/q2)

sin(kπ/Nq2)

Rectangularpulse-burst(Fig. 2.47)

Aτ sin(ωτ/2)ωτ/2

sin(ωNT/2)sin(ωT/2)

Aq1

sin(kπ/q1)kπ/q1

sin(kπ/q2)sin(kπ/Nq2)

Triangularpulse-burst

Aτ2

sin2(ωτ/4)(ωτ/4)2

sin(ωNT/2)sin(ωT/2)

A2q1

sin2(kπ/2q1)(kπ/2q1)2

sin(kπ/q2)sin(kπ/Nq2)

Sinc-shapedpulse-burst

Aπα

sin(ωNT/2)sin(ωT/2) , |ω| < α

0, |ω| > α

AπαT1

sin(kπ/q2)sin(kπ/Nq2)

, |k| < αT12π

0, |k| > αT12π

Ck corresponds to y(t) repeated with period T1, τ is pulse duration, T isperiod of pulse in the burst, T1 is period of pulse-bursts in the train, q1 = T1

τ ,and q2 = T1

NT .

C

Tables of Hilbert Transform and Properties

328 C Tables of Hilbert Transform and Properties

Table C.1 Properties of the Hilbert transform

y(t) =1π

∞∫

−∞

y(θ)θ − t

dθ y(t) =1π

∞∫

−∞

y(θ)t − θ

dθ

Property Function y(t) Transform y(t)

Filtering y(t) is constant y(t) is zero

is real is real

is even (odd) is odd (even)

Causality

If y(t) is causal withknown transformY (jω) = Yr(ω) + jYi(ω),then:Yr(ω) Yi(ω)

Yi(ω) −Yr(ω)

Linearity∑i

αiyi(t)∑i

αiyi(t)

Time shifting y(t ± θ) y(t ± θ)

Time reversal y(−t) −y(−t)

Scaling y(at) y(at)

y(−at) −y(−at)

Multipletransform y(t) −y(t)

H3y(t) = H−1y(t)

H4y(t) = y(t)

Hny(t) F⇔ [−j sgn(ω)]nY (jω)

Differentiation ddty(t) d

dt y(t)

Integration

b∫a

y(t)dt,

a and b are constants0

Convolution y1(t) ∗ y2(t) y1(t) ∗ y2(t)

Autocorrelation y(t) ∗ y(t) y(t) ∗ y(t)

Multiplication ty(t) ty(t) + 1π

∞∫−∞

y(t)dt

C Tables of Hilbert Transform and Properties 329

Table C.2 Useful relations between y(t) and its Hilbert transform y(t)

Property Relation

Orthogonality∞∫

−∞y(t)y(t)dt = 0

Integration∞∫

−∞y1(t)y2(t)dt =

∞∫−∞

y1(t)y2(t)dt

∞∫−∞

y1(t)y2(t)dt =∞∫

−∞y1(t)y2(t)dt

∞∫−∞

y2(t)dt =∞∫

−∞y2(t)dt

Energy∞∫

−∞|y(t)|2(t)dt =

∞∫−∞

|y(t)|2dt

Autocorrelation∞∫

−∞y(t)y(t−θ)dt =

∞∫−∞

y(t)y(t−θ)dt

Table C.3 The Hilbert transform of analytic signals

Property Signal Transform

Analytic signal ya(t) = y(t) + jy(t) −jya(t) = y(t) − jy(t)

Multiplication ya1(t)ya2(t) −jya1(t)ya2(t)

= ya1(t)ya2(t)

= ya1(t)ya2(t)

Power n yna (t) −jyn

a (t)

Real product Re[ya1(t)ya2(t)] Im[ya1(t)ya2(t)]

Imaginary product Im[ya1(t)ya2(t)] −Re[ya1(t)ya2(t)]

Table C.4 Products of the analytic signals

Product Relation

ya1(t)ya2(t) = jya1(t)ya2(t) = jya1(t)ya2(t)

ya1(t)ya2(t) = −jya1(t)ya2(t) = ya1(t)ya2(t)

330 C Tables of Hilbert Transform and Properties

Table C.5 The Hilbert transform of basic signals

Signal Transform

δ(x) 1πx

δ′(x) − 1πx2

cos x sinx

sinx − cos x

cos(ax)Jn(bx) [0 < b < a] − sin(ax)Jn(bx)

sin xx

sin2 x/2x/2

ejx jejx

ax2+a2 [a > 0] x

x2+a2

ab−x2

(ab−x2)2+x2(a+b)2x(a+b)

(ab−x2)2+x2(a+b)2

δ(x − a) − δ(x + a) 2π

ax2−a2

δ(x − a) + δ(x + a) 2π

xx2−a2

e−x [x > 0] 1π e−xEi(x)

sign xe−|x| 1π [e−x Ei (x) + exEi(−x)]

e−|x| 1π [e−xEi(x) − exEi(−x)]

|x|v−1 [0 <Rev < 1] cot( πv

2

)|x|v−1 sign x

|x|v−1 sign x [0 <Rev < 1] − tan( πv

2

)|x|v−1

sign x −∞

Even rectangular pulseu (x + τ) − u (x − τ)

1π ln∣∣∣x+τx−τ

∣∣∣Odd rectangular pulse−u(x + τ) + 2u(x) − u(x − τ) − 1

π

[ln∣∣x+τ

x

∣∣+ ln∣∣x−τ

x

∣∣]

Ramp pulse1τ (τ − x)[u(x) − u(x − τ)]

1π

[x−τ

τ ln∣∣x−τ

x

∣∣+ 1]

Triangular pulse1τ (τ − |x|)[u(x + τ) − u(x − τ)]

1π

[x+τ

τ ln∣∣x+τ

x

∣∣+ x−ττ ln

∣∣x−τx

∣∣]

Sawtooth pulse1τ (τ − |x|)[−u(x + τ) + 2u(x) − u(x − τ)]

1π

[x−τ

τ ln∣∣x−τ

x

∣∣− x+ττ ln

∣∣x+τx

∣∣+ 2]

D

Mathematical Formulas

Limits:

limx→a

f(x)g(x) = lim

x→a

∂f(x)/∂x∂g(x)/∂x (L’Hospital’s rule)

limx→0

sin xx = 1

limx→0

sin Nxsin x = N

∞∫0

sin bxdx = limα→1

∞∫0

xα−1 sin bxdx = Γ(α)bα sin απ

2

∣∣∣α=1

= 1b

Trigonometric identities:

ejx = cos x + j sinx (Euler’s formula)

e(α+jx) = eα(cos x + j sin x)

cos x = ejx+e−jx

2

sinx = ejx−e−jx

2j

cos2 x + sin2 x = 1

cos2 x − sin2 x = cos 2x

2 cos x sin x = sin 2x

cos2 x = 12 (1 + cos 2x)

sin2 x = 12 (1 − cos 2x)

cos x cos y = 12 [cos (x + y) + cos (x − y)]

sinx sin y = 12 [cos (x − y) − cos (x + y)]

332 D Mathematical Formulas

sinx cos y = 12 [sin (x + y) + sin (x − y)]

cos (x ± y) = cos x cos y ∓ sinx sin y

sin (x ± y) = sin x cos y ± cos x sin y

cos x + cos y = 2 cos x+y2 cos x−y

2

cos x − cos y = −2 sin x+y2 sin x−y

2

sinx ± sin y = 2 sin x±y2 cos x∓y

2

a cos x + b sin x = r sin (x + ϕ) = r cos (x − ψ),r =

√a2 + b2, sin ϕ = a

r , cos ϕ = br , sin ψ = b

r , cos ψ = ar

ddx arcsin x = 1√

1−x2

ddx arccos x = − 1√

1−x2

ddx arctan x = 1

1+x2

ddxarccot x = − 1

1+x2

Hyperbolic identities:

sinhx = − sinh(−x) = ±√

cosh2 x − 1 = ±√

12 (cosh 2x − 1) = ex−e−x

2

cosh x = cosh(−x) =√

sinh2 x + 1 =√

12 (cosh 2x + 1) = 2 cosh2 x

2 −1 = ex+e−x

2

tanhx = sinh xcosh x = ex−e−x

ex+e−x

coth x = cosh xsinh x = ex+e−x

ex−e−x

cosh2 x − sinh2 x = 1

cosh x + sinhx = ex

cosh x − sinhx = e−x

Exponents:

eln x = x

ex

ey = ex−y

exey = ex+y

(ex)α = eαx

D Mathematical Formulas 333

Logarithms:

ln ex = x

ln xy = lnx − ln y

ln αx = lnα + lnx

ln xα = α ln x

Extension to series:

sinx = x − x3

3! + x5

5! − . . . + (−1)n x2n+1

(2n+1)! + . . .

cos x = 1 − x2

2! + x4

4! −x6

6! + . . . + (−1)n x2n

(2n)! + . . .

ex = 1 + x + x2

2! + x3

3! + . . . + xn

n! + . . .

ejz cos φ =∞∑

n=−∞jnJn(z)ejnφ

Series:

N−1∑n=0

xn = 1−xN

1−x , x = 1 (by geometric progression)

N−1∑n=0

eαn = 1−eαN

1−eα

∞∑

n=0xn = 1

1−x , |x| < 1

∞∑

k=1

sin2(kπ/q)k2 = π2(q−1)

2q2

Indefinite integrals:

∫

f ′(x)g(x)dx = f(x)g(x) −∫

f(x)g′(x)dx (integration by parts)

∫

f(x)dx =∫

f [g(y)]g′(y)dy [x = g(y)] (change of variable)

∫

dxx = ln |x|

∫

exdx = ex

∫

axdx = ax

ln a

∫

sin xdx = − cos x

∫

cos xdx = sin x

334 D Mathematical Formulas

∫

x

sin xcos x

dx =

sinxcos x

∓ x

cos xsinx

∫

dxx(axr+b) = 1

rb ln∣∣∣ xr

axr+b

∣∣∣

∫sin2 xdx = − 1

4 sin 2x + x2

∫

cos2 xdx = 14 sin 2x + x

2

∫

xeαxdx = eαx(

xα − 1

α2

)

∫

x2eαxdx = eαx(

x2

α − 2xα2 + 2

α3

)

∫

xλeαxdx = 1αxλeαx − λ

α

∫xλ−1eαxdx

∫

e−(ax2+bx+c)dx = 12

√ πae

b2−4ac4a erf

(x√

a + b2√

a

)

∫

1xeαxdx = Ei(αx) [α = 0]

∫

1√xe−αxdx =

√ πα erf (

√αx) [α > 0]

∫

ex

x2+α2 dx = 1α Im[ejα Ei(x − jα)]

Definite integrals:

∞∫

−∞

sin αxx dx = π

∞∫

−∞e−αx2

dx =√ π

α

∞∫

−∞x2e−αx2

dx =√

πα−3/2

∞∫0

sin αxx dx = π

2 sgn α

∞∫0

sin2 αxx2 dx = πα

2

∞∫

−∞

sin4 αxx2 dx = πα

2

∞∫0

dxα2+x2 = π

2α

∞∫0

xα−1 sin bxdx = Γ(α)bα sin απ

2

D Mathematical Formulas 335

∞∫0

sin bxdx = 1b

∞∫

−∞

sin xx(x−α)dx = π

α (cos α − 1) , α is real

∞∫

−∞

cos(ax)b2−x2 dx = π

2b sin(ab), a, b > 0

Special functions:

erf (x) = 2√π

x∫0

e−t2dt (Error function)

Ei (x) = −∞∫−x

e−t

t dt =x∫

−∞

et

t dt [x < 0] (Exponential-integral

function)

Ei (x) = ex

[1x +

∞∫0

e−t

(x−t)2 dt

][x > 0] (Exponential-integral

function)

S(x) = 2√2π

x∫0

sin t2dt =x∫0

sin πt2

2 dt (Fresnel integral)

C(x) = 2√2π

x∫0

cos t2dt =x∫0

cos πt2

2 dt (Fresnel integral)

Jn(z) = 12π

π∫−π

e−j(nθ−z sin θ)dθ (Bessel function of the first kind)

References

1. Anderson, J. B., Aulin, T., and Sundberg, C.-E. Digital Phase Modulation,New York, Plenum Press, 1986.

2. Baskakov, S. I. Radio Circuits and Signals, 2nd edn., Moscow, Vysshaya Shkola,1988 (in Russian).

3. Bracewell, R. N. The Fourier Transform and its Applications, 3rd edn.,New York, McGraw-Hill, 1999.

4. Chen, C.-T. Signals and Systems, 3rd edn., New York, Oxford University Press,2004.

5. Cracknell, A. and Hayes, L. Introduction to Remote Sensing, London, Taylor& Francis, 1990.

6. Gonorovsky, I. S. Radio Circuits and Signals, 2nd edn., Moscow, Radio i sviaz,1986 (in Russian).

7. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products,San Diego, CA, Academic Press, 1980.

8. Elali, T. and Karim, M. A. Continuous Signals and Systems with MATLAB,Boca Raton, FL, CRC Press, 2001.

9. Frerking, M. Digital Signal Processing in Communication Systems, Boston,MA, Kluwer, 1994.

10. Haykin, S. and Van Veen, B. Signals and Systems, 2nd edn., New York, Wiley,2002.

11. Hsu, H. P. Signals and Systems, New York, McGraw-Hill, 1995.12. Komarov, I. V. and Smolskiy, S. M. Fundamentals of Short-Range FM Radars,

Boston, MA, Artech, 2003.13. Lathi, B. P. Linear Systems and Signals, New York, Oxford University Press,

2002.14. Le Chevalier, F. Principles of Radar and Sonar Signal Processing, Norwood,

MA, Artech, 2002.15. Logston, T. Understanding the Navstar GPS, GIS, and IVHS, 2nd edn., New

York, Chapman & Hall, 1995.16. Nathanson, F. E., Reilly, J. P., and Cohen, M. N. Radar Design Principles,

2nd edn., Mendham, NJ, SciTech Publishing, 1999.17. Oppenheim, A. V., Willsky, A. S., and Hamid Nawab, S., Signals and Systems,

2nd edn., New York, Prentice-Hall, 1994.

338 References

18. Poularikas, A. D. The Handbook of Formulas and Tables for Signal Processing,Boca Raton, FL, CRC Press, 1999.

19. Roberts, M. J. Signals and Systems Analysis Using Transform Methods andMATLAB, New York, McGraw-Hill, 2004.

20. Schwartz, M. Mobile Wireless Communications, Cambridge University Press,2005.

21. Soliman, S. S. and Srinath, M. D. Continuous and Discrete Signals andSystems, New York, Prentice-Hall, 1990.

22. Tetley, L. and Calcutt, D. Electronic Navigation Systems, 3rd edn., Oxford,Elsevier, 2001.

23. Taylor, F. J. Principles of Signals and Systems, New-York, McGraw Hill, 1994.24. Tikhonov, V. I. and Mironov, M. A, Markov Processes, 2nd edn., Moscow,

Sovetskoe Radio, 1977 (in Russian).25. Trakhtman, A. M. and Trakhtman, V. A. Tables of Hilbert transform,

Radiotekhnika, 25, 3, 1970.26. Vig, J. Introduction to Quartz Frequency Standards, Fort Monmouth, US Army

SLCET-TR-92-1 (Rev. 1), 1992.27. Shannon, C. E. A mathematical theory of communication, The Bell System

Technical Journal, 25, 379–423, July 1948.

Index

Absolute value, 31, 279

Accuracy, 211, 212, 279, 285

ADC

delta-coded, 303

direct conversion, 303

pipeline, 303

ramp-compare, 303

structure, 303

successive-approximation, 303

Aliasing, 299, 301, 304

distortions, 299, 302

effect, 299

Aliasing distortions, 303

AM signal, 149

by periodic pulse-burst, 150

non periodic RF, 150

Amage, 299

Ambiguity, 193

Amplitude, 47, 51, 76, 98, 132, 135

complex, 58, 62, 97

continuous range, 303

in-phase, 258, 259, 282

low-pass, 260

quadrature phase, 259, 282

quadrature-phase, 258

squared, 274

unit, 138

variable, 172

Amplitude detection, 279

Amplitude deviation, 135

Amplitude function, 133

Amplitude modulation, 134, 174

Amplitude modulation (AM), 133

Amplitude sensitivity, 135

Amplitude shift keying (ASK), 134, 167

Amplitude spectral density, 12

Analog circuit, 306

Analog FM

scheme, 173

Analog signal, 11

Analog technique, 239

Analog waveform, 134

Analog-to-digital converter, 11, 255

Analysis

correlation, 211

Fourier, 37

frequency, 57

harmonic

symbolic, 25

mathematical, 49

problem, 57, 87, 90, 97, 98, 100, 106,116, 141, 145, 192

relation, 61

Analysis problem, 53, 55, 85

Analytic signal, 255, 263, 279, 283, 285

complex, 284

complex conjugate, 264

complex valued, 284

envelope, 284

imaginary part, 265

instantaneous frequency, 284

instantaneous value, 264

narrowband, 286

phase, 284

real part, 265

Angle, 132

340 Index

Angular measure, 168Angular modulation, 133, 134, 136, 168,

174, 185, 188Angular variable, 296Antenna, 131, 211Applied theory, 296Asymmetry, 95Attenuation, 299Attenuation coefficient, 103, 143Autocorrelation, 217Autocorrelation function, 211, 213, 222,

225, 235, 236, 239, 277structure, 239width, 249

Autocorrelation width, 251frequency domain, 250time domain, 249

Average energy, 189, 209Average power, 140, 189, 209, 210, 217,

218maximum, 141minimum, 141

Bandlimited signal, 296analog, 255digital, 255

Bandlimited signals, 255, 290Bandlimited spectrum, 256Bandwidth, 47, 122, 230, 256, 283

absolute, 122effective, 118, 119halfpower, 118of main lobe, 120

Bandwidth efficient, 163Barker code, 192

length, 239Barker codes, 193, 239Barker phase-coding, 239Baseband spectral density, 74Basic functions, 296Basis

harmonic, 37orthogonal, 36

combined, 38harmonic, 50

orthonormalized, 49harmonic functions, 49

Bell, 131Bernoulli, 48

Bernstein polynomial, 290Bessel functions, 170, 186–188Bezier splines, 290BFSK transmitter

block diagram, 183Bi-linearity, 35Binary amplitude shift keying (BASK),

165, 167Binary coding, 192Binary frequency shift keying (BFSK),

183Binary FSK, 184Bit duration, 167Bit-pulse, 77Bode plot, 208Boundary frequency, 287, 296Bounds, 201BPSK demodulation, 283Brownian motion, 259Burst, 224, 236Burst duration, 190Burst-train, 113

Calculuscontinuous-time, 239

Capacitance, 173Cardinal function, 295Carrier, 9, 133, 234, 239

frequency, 256Carrier frequency, 135, 142, 174, 214,

285Carrier phase, 134, 135Carrier recovery circuit, 196Carrier signal, 132, 133, 138, 140, 189,

255, 281in-phase, 282

Carrier signal vector, 136Carrier vector, 136Cauchy-Bunyakovskii inequality, 35,

216, 248Cauchy-Schwartz inequality, 35Cauchy-Schwarz inequality, 219Causality, 8Central frequency, 120Channel, 282

in-phase, 262optical, 132quadrature phase, 262radio, 132

Index 341

Channel utilization efficiency, 196Charge, 306Clock

atomic, 7Code, 192, 303

digital, 306binary, 306

Code length, 191, 238Coefficient

energy autocorrelation, 216Coherent detection, 281Commercial broadcast station, 156Communication channel, 201Communication system, 263Communications, 133, 190, 217, 255

analog, 3, 134digital, 3, 134two-way satellite, 4

Commutativity, 35Comparator, 184, 212, 303Complex amplitude, 209Complex carrier signal, 138Complex conjugate, 201, 249Complex envelope, 258Complex phase-coded signal, 235Complex signal, 217

with LFM, 217with PSK, 217

Complex single pulse, 87Complex vector, 264, 284Conditions

Dirichlet, 50, 60, 66, 73, 80periodicity, 50

Conjugate symmetry property, 247Continuous signal, 255, 309Continuous time, 296Continuous-time

real physical world, 302Continuous-time signal, 11Continuous-to-discrete converter, 303Control voltage, 283Conventional AM, 153Conversion, 302, 303

analog-to-digital, 303digital-to-analog, 306impulse-to-sample (I/S), 305

Converteranalog-to-digital, 255continuous-to-discrete, 303

digital-to-analog, 255, 302sample-to-impulse (S/I), 307single, 306

Convolution, 70, 163, 166, 248, 267Convolution property, 166Coordinate basis, 29Coordinate vector, 136Correlation, 201, 214, 216Correlation analysis, 201, 211, 217Correlation function, 174, 234Correlation time, 11Correlator, 184Cosine multiplier, 205Costas loop, 283Counter

up-down, 303Covariance function, 11Cross ESD, 203Cross-correlation, 241

geometrical interpretation, 242Cross-correlation function, 211

odd, 248periodic, 248

Cross-power spectral density function,246

Cubic splines, 290Current, 306Cut-off frequency, 206

d’Alembert, 48DAC

R2R Ladder, 306binary weighted, 306

four-bit, 306delta-sigma, 306hybrid, 306lower resolution, 306oversampling, 306pulse width modulator, 306segmented, 306with zero-order hold, 308

Damping factor, 285Decoded voltage range, 303Delay, 169, 211Delayed signal, 212Delta function, 18, 80Delta-function, 18, 61, 66, 73, 79, 138,

188, 274, 308delta-function, 20

342 Index

Delta-pulse, 79Delta-spectrum, 22Demodulation, 279, 282

asynchronous, 155coherent, 154square-law, 155synchronous, 154, 157

Demodulation scheme, 152, 282QAM, 162

Demodulator, 155for VSB, 163

Detection, 279envelope

noncoherent, 163synchronous, 157, 161, 164

Detector, 279envelope, 156, 164, 168

Deviation, 174Device, 306Differential phase-shift keying (DPSK),

196Differentiator, 185Digital M -ry QAM, 163Digital bit stream, 190Digital code, 11, 190, 255Digital communications, 77Digital PAM, 134Digital processing, 255Digital signal, 11Digital signal processing, 308Digital-to-analog converter, 255Digital-to-analog converter (DAC), 302,

306Dimensionality, 10Diode, 173Diode-varactor, 173Dirac

delta-function, 18impulse, 18

Dirac delta function, 27, 40, 74Dirac delta-function, 166Dirichlet conditions, 37Discontinuity, 50, 76Discrete correlation function, 238Discrete time, 296Discrete values

neighboring, 239Discrete-time

digital computer-based world, 302

Discrete-time points, 291Discrete-time sample, 290Discrete-time signal, 11Distance, 211, 295Distortion, 131, 140, 154, 300

aliasing, 300Distortions, 309Divergency, 213Diversity, 241Divided differences, 290, 294

interpolation polynomial, 290, 294Doppler shift, 2Double sideband large carrier (DSB-

LC), 153Double sideband reduced carrier

(DSB-RC), 157Double sideband suppressed carrier

(DSB-SC), 156Drift, 186Duality property, 256, 296duality property, 82Duration, 81, 90, 103, 142, 175, 191, 217

bit, 167Duty cycle, 98

Edge frequency, 123Effective bandwidth, 118Effective duration, 11, 118Efficiency, 72, 279Electric current, 140, 189Electric voltage, 140, 189Electricity, 306Electronic image, 11Electronic system, 259

wireless, 104Elementary pulse, 87Elements, 211Encoder, 303Energu autocorrelation function, 213Energy, 30, 71, 201, 206, 245

electrical, 131finite, 72, 202infinite, 202joint, 71, 202total

signal, 71Energy autocorrelation coefficient, 216Energy autocorrelation function, 211,

212, 214, 217, 220, 221

Index 343

Energy correlation function, 228Energy cross spectral density, 203Energy cross-correlation, 241, 245Energy signal, 14, 202, 211, 227, 241,

250real, 34

Energy signalscomplex, 36

Energy spectra, 208Energy spectral density, 203, 219Energy spectral density (ESD)

equivalent width, 251Envelope, 11, 24, 78, 111, 133, 146, 152,

237, 239, 255, 257, 279, 284complex, 258, 261physical, 258, 259, 262positive-valued, 258rectangular, 142sinc-shape, 234triangular, 236

Envelope detection, 279, 285Envelope shape, 178Equivalent width, 118, 120, 121, 249Error, 103, 305ESP function, 220

logarithmic, 208Essential bandwidth, 201Euler

formula, 52Euler formlula, 88Euler formula, 175, 188, 231Euler’s formula, 25External resources, 309

Fading, 185Filter

compensating, 309compensation, 309high-pass, 158, 306ideal, 309low-pass, 154, 156, 161, 163, 164, 166

analog, 306reconstruction, 309transfer function, 164

Filter bandwidth, 206Filtering, 206Filtering property, 18, 272Final voltage range, 303Finite energy, 202, 209, 213, 227

Finite energy signal, 14Finite impulse response (FIR), 279Finite power, 227Finite power signal, 14Finite signal, 97FIR filter, 279, 281Fluctuations, 206FM modulation index, 188Folding, 299

frequency, 299Fourier

series, 11transform, 12

Fourier eriesHarmonic form, 50

Fourier series, 48, 97, 99, 113, 138, 172,209, 225, 297

exponential, 52generalized, 36, 49property

conjugation, 58linearity, 58modulation, 59time-differentiation, 57time-integration, 58time-reversal, 59time-scaling, 56time-shifting, 56

symbolic form, 52trigonometric, 50

Fourier transform, 21, 38, 60, 72, 77, 80,87, 106, 138, 139, 144, 167, 175,188, 203, 219, 244, 246, 249, 256,276

direct, 61inverse, 61, 203, 257, 296property

conjugation, 65convolution, 70duality, 68integration, 67linearity, 67, 77modulation, 68multiplication, 69scaling, 64time shifting, 77time-differentiation, 66time-reversal, 66time-shifting, 63, 142

344 Index

Fourier transform(Continued)short-time, 39windowed, 39

Frequency, 133angular, 22, 38, 51carrier, 5, 40, 133, 172, 234

angular, 133central, 121continuous

angular, 97current angular, 61cut-off, 112, 148, 206, 283, 308deviation, 169folding, 299fundamental, 25, 48, 60, 97

angular, 25, 50natural, 50

instantaneous, 259modulated, 169natural, 217negative, 78, 269Nyquist, 297positive, 78, 269principle, 97sampling, 299, 304

Frequency content, 255Frequency deviation, 175, 178, 181, 229,

235Frequency domain, 11, 47, 72, 119Frequency modulation (FM), 133, 168Frequency range, 139

infinite, 203Frequency resolution, 40, 42Frequency sensitivity, 169, 172Frequency shift keying (FSK), 134, 182Frequency variation, 186Fresnel functions, 179, 229, 233Fresnel integrals, 176Function

absolutely integrable, 50AM, 174antisymmetric, 62autocorrelation, 211, 223, 242

even, 213auxiliary, 267continuous, 295correlation, 174

discrete, 238cross correlation, 247

cross ESD, 204, 205, 246cross PSD, 246cross-correlation, 211, 243cross-power spectral density, 246delta, 18delta-shaped, 235discrete-time, 238energy autocorrelation, 211, 212ESD, 205–207, 221, 223, 229, 234even, 62exponential, 177, 232finite, 27Gaussian, 85generalized, 19, 26Haar, 42Lagrange, 298low-pass, 260modulating, 169, 185narrowband

weighted, 271non periodic, 72normalized, 219odd, 62, 204, 269ordinary, 26periodic, 48PM, 174power autocorrelation, 211, 217PSD, 224, 225ramp

truncated, 96real

positive-valued, 85singular, 27somplex, 203staircase, 305, 309symmetric, 62, 78testing, 19unit step, 18unit-step

reverse, 95weighted, 77

weighting, 39Functional, 26Functions

Haar, 36harmonic, 38, 50orthogonal, 34, 37, 39, 49, 275

periodic, 244sinc, 298

Index 345

orthogonal periodicharmonic, 36

orthonormal, 49Fundamental frequency, 209

Gabor complex signal, 263Gain factor, 282Gaussian pulse, 84, 85Gaussian RF pulse, 143Gaussian waveform, 279Generalized Fourier coefficients, 36Generalized function, 19, 27

regular, 27singular, 27

Generalized spectrum, 138Generic relation, 309Geometrical progression, 106Gibbs phenomenon, 55, 84, 104, 111,

117, 148Global navigation system (GLONASS),

134Global positioning system (GPS), 5,

134Goodness, 212Group delay, 279, 281

Hadamard MFSK, 185Halfpower bandwidth, 118Harmonic, 49, 51, 172harmonic, 55Harmonic series, 171Harmonic wave, 256

periodic, 209Heaviside unit step function, 16HF signal, 279High precision, 306High speed, 306Hilbert modulator, 281Hilbert transform, 255, 263, 268, 275,

276, 279, 283–285analytic signal, 288direct, 267inverse, 267property

autocorrelation, 277causality, 271convolution, 275filtering, 269linearity, 271

multiple transform, 273orthogonality, 274scaling, 272time-derivative, 273time-shifting, 272

Hilbert transformer, 158, 269, 279,281–283

Human voice, 139

I/S converter, 305IIR filter, 279Imaginary, 203Imaginary version, 263Impulse AM, 174

periodic, 149Impulse response, 18, 248, 285, 309

infinite, 308rectangular, 166

Impulse signal, 108rectangular, 268

Impulse signals, 77Impulse symbol, 18In-phase amplitude, 258In-phase component, 279In-phase component I, 284Inaccuracy, 55Inductance, 173Infinite bounds, 259Infinite energy, 202Infinite impulse response (IIR), 279Infinite suppression, 281Information, 203, 262, 299

loss, 295information quantization, 132Informative phase, 258, 286Inner product, 34, 36, 49, 201, 203, 213,

241, 244integral, 61

inner product spaces, 33Input signal, 282Instantaneous

angle, 133frequency, 133, 186, 255phase, 186power, 140, 189

Instantaneous frequency, 257, 286Instantaneous power, 71Instantaneous signal power, 201

346 Index

IntegralCauchy sense, 268Lebesque, 27Riemann, 19, 27

Integration bounds, 233Integration property, 82, 91, 93Integration range

finite, 148Integrator, 185, 283Interpolant, 295Interpolated signal, 297, 308Interpolating polynomial, 291Interpolation, 290, 295, 298, 308

Lagrange form, 290Lagrange method, 255Newton form, 290Newton method, 255

Interpolation error, 294Interpolation formula, 298Interpolation function, 290Interpolation polynomial

Lagrange form, 291Newton method, 294

Interrogating pulse, 201, 212Interrogating signal

transmitted, 211Interval, 296Inverse problem, 255

Joint energy, 34, 201, 202, 212, 214,216, 244

Kotelnikov, 295Kronecker

impulse, 18symbol, 18

Kronecker symbol, 166

L’Hosupital rule, 107Lagrange, 48Lagrange function, 298Lagrange interpolation, 298Lebesque integral, 31Length, 103LFM pulse, 229, 234

rectangular, 230ESD function, 229with LFM, 231

LFM signal, 235, 239

Linespectral, 47, 60

Linear amplification, 163Linear frequency modulation (LFM),

134, 174Linear phase, 279Linear phase drift, 144Linearity, 77Linearity property, 77Lobe, 78, 92

main, 78, 84, 93, 96peak value, 147

negative, 78positive, 78side, 78, 82, 93, 95

maximum, 147Lobes

side, 207Logarithmic measure, 208Logic circuit, 303Long pulse, 192Low bound, 235Low cost, 306Low-pass (LP) filter, 154Low-pass signal

ideal, 256noncausal, 256

Lower sideband, 139, 281Lower-sideband, 54, 63LP filter, 206, 263, 282, 283, 307

first order, 294

M -ry amplitude shift keying (M-ASK),165, 168

M -ry frequency shift keying (MFSK),183

Magnitude, 28, 88, 98Magnitude spectrum, 11, 97–99, 116Main lobe, 84, 93, 103, 205, 224, 235

width, 209Main lobes, 109, 112Main side lobe, 146Mapping, 272Mark, 134Marking frequency, 183Mathematical idealization, 77Maxima, 50Maximum, 78, 82Mean-square value, 120

Index 347

Mean-square width, 118, 120Measurements, 212, 220Memory, 299

size, 302Message, 133

navigation, 5Message signal, 135, 164, 169, 186, 191Miltiplication, 153, 212Minima, 50Minimum, 97Model

narrowband, 262Modulated signal, 139, 141, 168, 255,

256narrowband, 255with SSB, 281

Modulating signal, 132, 133, 136, 138,146, 166, 255

differentiated, 186energy, 139impulse, 141periodic, 149rectangular, 142

Modulating waveform, 149Modulation, 131, 132, 279

amplitude, 2, 133angular, 133, 134frequency, 3, 133phase, 133undistorted, 136

Modulation factor, 135, 136Modulation frequency, 169

angular, 135maximum, 173

Modulation index, 169, 172Modulation process, 186, 257Modulation scheme, 152, 282

QAM, 162Modulation theory, 68Modulator

for SSB, 163Mono-pulse radar, 211Multi-frequency shift keying (MFSK),

183Multiple Hilbert transform, 273Multiplication property, 138Multiplication rule, 72Multiplier, 192Music, 255

Narrow band, 142Narrowband channel, 143Narrowband model, 257Narrowband signal, 258, 259, 266

energy, 255Narrowness, 216, 235Navigation, 217Negative feedback, 306Newton interpolation formula, 295Newton interpolation polynomial, 294Noise, 241, 255, 306

large, 306measuring, 306

Noise intensity, 306Non negativity, 35Non periodic pulse train, 72Non periodic train, 104Non-periodic rate, 299Noncausality, 256Nonstationary, 186Norm, 29Number of samples, 299

redundant, 299Nyquist, 295

frequency, 297, 299rate, 299, 302, 304sample rate, 297–299

Nyquist rate, 303

One-sided spectrum, 54Operation time, 299Origin, 212Orthogonal basis, 34Orthogonality, 14, 34, 244, 298Oscillating envelope, 104Oscillation, 24, 49, 89

FM, 169Oscillation amplitude, 285Oscillations, 111, 230Oscillator, 173

local, 154quartz, 7

Oscillatory system, 285Overlapping effect, 175Overmodulation, 136

π/2 shifter, 282π/2-shifter, 281, 283

348 Index

Parsevalrelation, 59theorem, 59

Parseval relation, 72Parseval theorem, 210Parsevale theorem, 274Part

imaginary, 93Pass-band, 306PCR, 234

large, 234negligible, 234

Peak amplitude, 11, 169Peak hold, 279Peak modulating amplitude, 135Peak phase deviation, 169, 186Peak value, 31, 96, 109, 229

intermediate, 96Pear value, 186Period, 140, 189, 218Period of repetition, 11, 23, 48, 98, 209,

285Period-to-pulse duration ratio, 98, 190,

270Periodic Gaussian pulse, 102Periodic pulse, 97, 103Periodic pulse sequence, 72Periodic pulse signal, 97Periodic pulse train, 72Periodic pulse-burst, 113Periodic pulse-train, 97Periodic signal, 202, 210Periodic signals, 209Periodic sinc pulse-burst, 116Periodic sinc-shape pulse, 103Periodicity, 9, 190, 209, 296Perturbation, 131PF pulse, 174Phase, 51, 78, 85, 88, 133, 255, 257

carrier, 6informative, 258, 262

slowly changing, 259initial, 169modulated, 169modulo 2π, 259

Phase angle, 138, 235Phase channel

in-phase, 283quadrature phase, 283

Phase coding, 237Phase deviation, 185, 187, 188Phase difference shift keying (PDSK),

134Phase error, 281Phase locked loop (PLL), 157Phase manipulation, 193Phase modulation, 185Phase modulation (PM), 133, 134, 168Phase sensitivity, 185, 186Phase shift, 190, 191

90 degree, 279Phase shift keying (PSK), 134Phase shift method, 160Phase shifter

π/2 lag, 281Phase spectral density, 12Phase spectrum, 11, 97–99, 152Phase-coded signal, 239Phase-coding, 235

Barker, 239Phase-locked loop (PLL), 283Phase-shift filter, 281Physical periodicity, 259Physical spectrum, 54Physically realizable, 256Plancherel identity, 72Polynomial of nth order, 291Popov, 132Positioning, 255Postprocessing, 309Power, 201, 232, 245

carrier signal, 140instantaneous, 71transmitted, 164

Power autocorrelation function, 211,217–219

Power cross-correlation function, 244Power efficient, 163Power signal, 14, 209, 210, 227, 241Power signals

joint power, 247Power spectral density, 211Power spectral density (PSD), 211, 224,

225Pre-envelope, 264Preprocessing, 309Principle

superposition, 67

Index 349

Probability, 305Process

quantization, 305Processing system, 211Propagation time, 211Property

addition, 67Pseudo random code, 193Pulse, 205

acoustic, 3burst, 3cosine, 165delta-shape, 76elementary, 87, 236, 239envelope, 236exponential

truncated, 73Gaussian, 84, 121noncausal, 82periodic, 96

Gaussian, 101radio frequency (RF), 9ramp

truncated, 95received, 71rectangular, 77, 79, 82, 204

causal, 271periodic, 98single, 98, 250

rectangular with LFM, 175sinc, 165sinc-shape, 82, 277single, 2, 96, 236

complex, 87train, 2train-rectangular, 47transmitted, 71trapezoidal, 89triangular, 80

asymmetric, 93symmetric, 80, 206

truncated exponentialreal, 74

unit-step, 74, 95rectangular, 165Pulse amplitude modulation (PAM),

134, 165Pulse asymmetry, 89Pulse burst, 72

Pulse compression, 192Pulse density conversion, 306Pulse duration, 84, 85, 98, 102, 176,

205, 216Pulse radar, 77Pulse radars, 134Pulse time modulation (PTM), 134Pulse train

non periodic, 72periodic, 72

Pulse waveform, 72, 107, 114Pulse-burst, 104, 112, 235, 237

periodic, 113rectangular, 108, 109, 112, 239RF rectangular, 145RF sinc, 147sinc, 111

symmetric, 111single, 106, 113triangular, 110

symmetric, 110Pulse-burst-train, 113

rectangular, 115sinc, 116symmetric, 113triangular, 116

Pulse-bust-trainrectangular, 114

Pulse-code modulator (PCM), 303Pulse-compression ratio, 229Pulse-compression ratio (PCR), 178Pulse-train, 98, 112

Gaussian, 101, 103rectangular, 55, 99, 149, 210RF rectangular, 150sinc-shape, 103triangular, 100

symmetric, 100Pulse-width modulator, 306

Quadrature amplitude modulation(QAM), 134, 161

Quadrature amplitude shift keying(QASK), 134

Quadrature amplitudes, 258Quadrature channel, 283Quadrature demodulator, 281–283Quadrature filter, 281

350 Index

Quadrature phase-shift keying (QPSK),197

Quadrature pulse amplitude modulation(QPAM), 167

Quadrature-phase amplitude, 258Quadrature-phase component, 279Quadrature-phase component Q, 284Quality factor, 285Quantization, 305

level, 305Quantization error, 11Quantizer, 303, 305

uniform, 305Quartz crystal resonator, 173

Radar, 1, 201bistatic, 2monostatic, 2pulse compensation, 192

Radars, 143, 190, 217, 255Radian, 296Radio wave, 132Ramp pulse, 95, 242Ramp voltage, 303Range, 300Range information, 134Rayleigh theorem, 202, 250, 276Real component, 204Real physical process, 255Real physical signal, 97Real signal, 262, 284Received signal, 212Receiver, 2, 7, 154, 162, 184, 189, 192,

201, 211, 262coherent, 262

Reconstructionerrorless, 309

Reconstruction filter, 307, 308ideal, 307

Rectangular envelope, 178Rectangular pulse, 77, 80, 91, 120, 205,

208, 214complex, 87even, 88odd, 88

Rectangular pulse waveform, 77Rectangular pulse-burst, 108Rectifier, 155Reflected copy, 211

Regularity, 8Remote control, 255Remote sensing, 11, 143

active, 3passive, 3

Replications, 301Resistance, 189, 201Resistor, 306Resolution, 305

ADC voltage, 305RF pulse

Gaussian, 279Gaussian with LFM, 180rectangular, 230

with LFM, 234rectangular with LFM, 174, 175, 177trapezium with LFM, 179triangular with LFM, 180

RF pulse burst, 144RF pulse-burst, 190RF pulse-burst-train, 149RF pulse-train, 149, 165, 190RF signal, 149

symmetric, 262Riemann integral, 31Root-mean-square (RMS), 31

Sample, 255Sample time, 290Sampled signal, 296, 304Sampler, 303Samples, 295, 296, 303

equidistantly placed, 295spectral band, 295

Sampling, 255, 295, 298bandpass signal, 300frequency, 298, 299, 302period, 299periodic, 303rate, 298Shannon definition, 299theorem, 299time, 299

Sampling frequency, 300Sampling interval, 295Sampling process, 304Sampling property, 18Sampling theorem, 255, 295–297, 308Sampling theory, 295

Index 351

Sawtooth signal, 303Scalar product, 34Scaling property, 272Schwartz, 26Sensing, 255Series

finite, 55Shannon, 295, 297

theorem, 299Shape, 212

smoothed, 96uniform, 103

Sharp bound, 299Shifted copy, 213Short subpulses, 192Side lobe, 93Side lobes, 78, 82, 109, 205, 209, 235,

236, 239level, 96

minimum, 96multiple, 224

Side vector, 136Side-line vector, 136Sideband, 281Sideband power, 141Sifting, 18Sifting property, 27, 61, 308Signal, 1

4-ASK, 168absolutely integrable, 73analog, 11, 166analytic, 29, 255bandlimited, 13, 255, 294, 300bandpass, 256, 258, 300baseband, 14, 123, 131, 138BASK, 167BFSK, 183binary, 168BPSK, 190, 192, 282broadband, 13carrier, 5, 9, 164

radio frequency (RF), 135causal, 8, 284classification, 7

causality, 8dimensionality, 10periodicity, 9presentation form, 11regularity, 8

complexexponential, 56harmonic, 40

complex exponential, 23conjugate, 267continuous

sampling, 298continuous time

narrowband, 303continuous-time, 11

bandlimited, 290correlation function, 239cosine, 25critically sampled, 299delayed, 212delta-shaped, 61, 274demodulated, 154, 156deterministic, 8digital, 11discrete time, 304discrete-time, 11DSB, 163DSB-LC, 160DSB-RC, 157DSB-SC, 156, 157, 161–163, 167elementary, 26exponential, 23FM, 169general complex exponential, 23generalized exponential, 23harmonic, 25, 47, 217

periodic, 225high-hass, 251impulse, 2, 9, 262input, 282interpolated, 297, 308interrogating, 211inverted, 236low-pass, 156, 256, 265, 276, 296M-ASK, 168message

frequency, 173harmonic, 186

MFSK, 185MFSK4, 185modulated, 132modulating, 132, 169, 185multi-dimensional, 10narrowband, 13, 123, 251, 256, 259

352 Index

negative, 242non periodic, 60, 61, 71non-periodic, 9noncausal, 9one-dimensional, 10original, 213oversampled, 299PAM, 166parameter, 132periodic, 9, 12, 53, 63, 96, 202

causal, 97noncausal, 97

phaseinitial, 236

phase keying (manipulation), 190phase-coded, 235, 237, 239phase-manipulated, 191physical, 26PM, 186PSK, 190, 191

Barker codes, 192QAM, 163QPSK, 197, 283radio frequency pulse, 9random, 8real exponential, 24received, 212reconstructed, 308reference, 156

coherent, 283reflected version

attenuated, 262delayed, 262

sampled, 296, 304sawtooth, 303scalar, 10scaled version, 296shifted in time, 241sine, 25single, 9spectra, 300SSB, 160, 163stationary, 39synthesized, 117test, 98time, 7undersampled, 299uniform, 80video pulse, 9

VSB, 163demodulated, 164

waveform, 55Signal autocorrelation, 211Signal bandwidth, 122Signal cross-correlation, 241Signal detection, 217Signal energy, 12, 30, 142, 201, 203, 205,

209, 211, 238, 239, 277total, 201

Signal ESP function, 211Signal frequency, 131Signal identification, 217Signal immunity, 132signal jamming immunity, 132Signal length, 30Signal phase, 203Signal power, 12

instantaneous, 201Signal processing (SP), 284Signal reconstruction, 255Signal spectrum, 131, 135Signal spectrum conversion, 132Signal total energy, 236Signal total resources, 30Signal waveform, 71, 251, 279, 299Signal widths, 118Signal-to-noise ratio, 144, 185Signaling, 306Signals

bandlimited nature, 295electrical, 30energy cross-correlation, 241joint energy, 204, 242, 247mutually orthogonal, 36non orthogonal, 204orthogonal, 14, 34, 36, 204orthonormal, 15scalar-valued, 30vector, 32

Signals orthogonality, 244Signum function, 267, 274Simplest AM, 135, 171Simplest FM, 169, 186Simplest PM, 186Sinc pulse-burst, 111Sinc-function, 78, 82, 210Sinc-pulse, 85, 221

finite, 84

Index 353

noncausal, 84

Sinc-shape pulse, 82

Single pulse, 60, 106, 112, 150, 190

Single pulse with LFM, 229

Single side band (SSB), 139

Single sideband (SSB), 158

Slope, 73, 80, 205

negative, 95

positive, 95

Sonar, 3

SP block, 284, 290

computer aided, 284

digital, 295

Space

Hilbert, 36

linear, 28

complex, 29

normed, 29

real, 28

metric, 32

nonlinear

complex, 29

real, 29

Spaces

inner product, 33

linear

normed, 33

metric, 33

Spacing, 302

Spectra, 201

energy, 208

Spectral

characteristic

phase, 62

Spectral analysis, 97

Spectral band, 295

Spectral bounds, 86

Spectral characteristic, 11, 61

Spectral characteristics, 48

Spectral components, 62

Spectral content, 174

Spectral density, 12, 62, 65, 69, 71, 78,84, 86, 93, 97, 98, 175, 180, 204,239, 246, 251, 296, 300

amplitude, 62

baseband, 74

energy, 203

magnitude, 62, 63, 74, 75, 78, 82, 84,85, 88, 89, 95, 108, 112, 144, 176,180

nulls, 146one-side

symmetric, 286one-sided, 265phase, 62, 63, 74, 75, 78, 82, 108, 144,

176pulse-burst, 104rectangular, 82replications, 301sinc-shape, 82uniform, 77, 79

Spectral envelope, 109Spectral function, 225Spectral line, 55, 97, 135, 150Spectral lines, 101

neighboring, 152Spectral nulls, 109Spectral shape, 12Spectral width, 12, 47, 83, 84, 255Spectrum, 51, 53, 57, 96, 103

baseband, 14continuous, 60discrete, 60, 113double-sided, 54envelope, 101magnitude, 11, 53, 56, 97, 99, 150

double-sided, 54one-sided, 54

mathematical, 54of periodic signal, 62one-sided, 98phase, 11, 53, 56, 97, 99

double-sided, 54one-sided, 54

right-hand side, 170Spectrum envelope, 98, 104Spectrum width, 173, 181Speech signal, 255Speed of change, 11Square amplitude, 72Squaring device, 155SSB signal, 281Step, 305

constant, 305Structure, 212, 281

PAM system, 166

354 Index

Subranging, 303Superposition, 229Superposition principle, 58, 106Suppression, 92Symmetry, 20Synchronization, 154Synchronous demodulation, 153Synthesis, 55, 116

error, 55problem, 55, 57, 97, 98, 101, 115

continuous-frequency, 61discrete-frequency, 61

relation, 61synthesized signal, 117System, 284

broadcasting, 4Doppler radar, 7electronic

wireless, 133LTI, 248, 249navigation, 5, 7, 134positioning, 5radio electronic, 132

System channel, 241System efficiency, 284System operation, 279System problems, 278Systems

wireless, 47

Target, 201, 211Taylor polynomial, 29Technique, 306Telegraphy

wireless, 132Telephony, 131Television information, 255Test function, 26Test signal, 98Testing function, 19Theorem

addition, 67conjugation, 65differentiation, 66Kotelnikov, 297modulation, 68Nyquist-Shannon sampling, 297Parseval, 38Rayleigh, 71, 72

sampling, 297scaling, 65Shannon sampling, 297similarity, 65, 119Whittaker-Nyquist-Kotelnikov-

Shannon sampling, 297Time, 7

continuous, 48precise, 7propagation, 211

Time bounds, 82infinite, 85, 217

Time delay, 201, 211, 281Time domain, 10Time duration, 11

infinite, 227Time interval

infinite, 209Time resolution, 40, 42Time scale, 10Time scaling, 20Time shift, 205, 212, 236Time unit, 295Time-invariant system (LTI), 248Time-shift, 100Time-shift theorem, 272Time-shifting, 106, 138, 192, 204time-shifting, 77Time-shifting property, 77, 81, 308Timekeeping, 7Timer, 303Total energy, 30, 201, 236

infinite, 209Total phase, 175, 259, 285Total signal power, 141Tracking, 279Transfer function, 166, 284, 307, 309Transform, 61, 66, 113

analysis, 39continuous

wavelet, 40continuous wavelet

inverse, 40Fourier, 21synthesis, 39

Transition, 286Transition band, 281Transmission, 131, 295Transmitter, 2, 154

Index 355

Trapezoidal pulse, 89Triangular inequality, 30, 33Triangular pulse, 80, 91, 96, 120, 208

asymmetric, 93symmetric, 96

Triangular pulse-burst, 110Triangular pulse-burst-train, 116Triangular pulse-train, 100Triangular waveform, 81

Uncertainty, 93, 107Uncertainty principle, 119, 251Uniform signal, 80Unit amplitude, 79Unit impulse, 18Unit step, 16, 18Unit-pulse train, 303Unit-step function, 73Unit-step pulse, 95

reverse, 95Unshifted copy, 212Upper bound, 300Upper sideband, 139, 142, 281Upper-sideband, 54, 63

Valuediscrete-time, 305

Vandermonde matrix, 291Variable, 10, 232

angular, 296Vector, 10, 28, 29, 51

FM, 172rotated, 136

Vectorslinearly independent, 29

Velocity, 131Very short pulse burst, 134Vestigial sideband (VSB) modulation,

163Vicinity, 299, 301Video pulse, 220

Gaussian, 222rectangular, 215sinc-shape, 221

Voltage, 201, 303, 306Voltage controlled oscillator (VCO), 173Voltage-controlled capacitor, 173

Voltage-controlled oscillator (VCO),283

Waring-Euler-Lagrange interpolation,291

Waring-Lagrange interpolation, 291Wasted power, 140Wave, 134

acoustic, 3continuous, 134electromagnetic, 3harmonic, 47radio, 1

Wave radars, 134Waveform, 9, 26, 28, 30, 47, 68, 82, 97,

180, 201, 220analog, 134complex, 138continuous-time

periodic, 47exponential truncated

real , 73Gaussian, 143, 299

single, 102non periodic, 72pulse, 165rectangular, 55

complex, 93squared, 206

single, 72, 109trapezoidal, 93triangular, 81undersampled, 299

Wavelength, 131Wavelet transforms, 244Wavelets, 40, 244Weighted integration, 288Whittaker, 295Width

equivalent, 118mean-square, 118, 120, 122

Willard sequences, 193Window, 39

rectangular, 40Window function, 39Wireless applications, 255

Zero-order hold, 305, 309