Mathematical Tables and Algorithms

Appendix A Mathematical Tables and Algorithms

This appendix provides lists with some definitions, integrals, trigonometric relations and the like. It is not exhaustive, presenting only that information used throughout the book and a few more just for complementation purposes. The reader can find extensive tables available in the literature. We recommend the works by Abramowitz and Stegun [1], Bronshtein, Semendyayev, Musiol and Muehlig [2], Gradshteyn and Ryzhik [3], Polyanin and Manzhirov [4] and Zwillinger [7]. We also recommend the book by Poularikas [5], though its major application is in the signal processing area. Nevertheless, it complements the tables of the other recommended books. In terms of online resources, the Digital Library of Mathematical Functions (DLMF) is intended to be a reference data for special functions and their applications [6]. It is also intended to be an update of [1].

A.1 Trigonometric Relations

∞ x3 x5 x7 (−1)n x2n+1 sin x = x − + − +···= (A.1) 3! 5! 7! (2n + 1)! n=0 ∞ x2 x4 x6 (−1)n x2n cos x = 1 − + − +···= (A.2) 2! 4! 6! (2n)! n=0 1 sin2 θ = (1 − cos 2θ)(A.3) 2 1 cos2 θ = (1 + cos 2θ)(A.4) 2 1 sin2 θ cos2 θ = (1 − cos 4θ)(A.5) 8 1 cos θ cos ϕ = [cos(θ − ϕ) + cos(θ + ϕ)] (A.6) 2 1 sin θ sin ϕ = [cos(θ − ϕ) − cos(θ + ϕ)] (A.7) 2

D.A. Guimaraes,˜ Digital Transmission, Signals and Communication Technology, 841 DOI 10.1007/978-3-642-01359-1, C Springer-Verlag Berlin Heidelberg 2009 842 A Mathematical Tables and Algorithms

1 sin θ cos ϕ = [sin(θ + ϕ) + sin(θ − ϕ)] (A.8) 2 sin(θ ± ϕ) = sin θ cos ϕ ± cos θ sin ϕ (A.9) cos(θ ± ϕ) = cos θ cos ϕ ∓ sin θ sin ϕ (A.10) tan θ ± tan ϕ tan(θ ± ϕ) = (A.11) 1 ∓ tan θ tan ϕ sin 2θ = 2sinθ cos θ (A.12) cos 2θ = cos2 θ − sin2 θ (A.13) 1 sin θ = (eiθ − e−iθ ) (A.14) 2i 1 cos θ = (eiθ + e−iθ ). (A.15) 2

A.2 Gaussian Error Function and Gaussian Q-Function

Error function:  x 2 2 erf(x) = √ e−t dt. (A.16) π 0

Complementary error function:

 ∞ 2 2 erfc(x) = 1 − erf(x) = √ e−t dt. (A.17) π x

Properties:

erf(−x) =−erf(x) (A.18) erf(x∗) = erf(x)∗ (A.19)  1 ∞ u2 Q(x) = √ exp − du (A.20) 2π x 2 % √ & erfc(x) = 2Q x 2 (A.21) 1 x Q(x) = erfc √ . (A.22) 2 2

The Taylor series expansion of the error function, which provides a good approx- imation of erf(x) for 60 terms in the summation and for x up to 4, is given by

∞ 2 (−1)n x2n+1 erf(x) = √ . (A.23) π n!(2n + 1) n=0 A Mathematical Tables and Algorithms 843

The complementary error function has an alternative representation which is par- ticularly useful in error probability analysis of communication systems. It is known as the desired form of the complementary error function and is given by

 π/2 2 = 1 − 2x θ. erfc(x) exp 2 d (A.24) π 0 2sin θ

A.2.1 Some Tabulated Values of the Complementary Error Function x erfc(x) x erfc(x) x erfc(x) x erfc(x) 0.025 0.971796 1.025 0.147179 2.025 4.19E-03 3.025 1.89E-05 0.05 0.943628 1.05 0.137564 2.05 3.74E-03 3.05 1.61E-05 0.075 0.91553 1.075 0.128441 2.075 3.34E-03 3.075 1.37E-05 0.1 0.887537 1.1 0.119795 2.1 2.98E-03 3.1 1.16E-05 0.125 0.859684 1.125 0.111612 2.125 2.65E-03 3.125 9.90E-06 0.15 0.832004 1.15 0.103876 2.15 2.36E-03 3.15 8.40E-06 0.175 0.804531 1.175 0.096573 2.175 2.10E-03 3.175 7.12E-06 0.2 0.777297 1.2 0.089686 2.2 1.86E-03 3.2 6.03E-06 0.225 0.750335 1.225 0.0832 2.225 1.65E-03 3.225 5.09E-06 0.25 0.723674 1.25 0.0771 2.25 1.46E-03 3.25 4.30E-06 0.275 0.697344 1.275 0.071369 2.275 1.29E-03 3.275 3.63E-06 0.3 0.671373 1.3 0.065992 2.3 1.14E-03 3.3 3.06E-06 0.325 0.645789 1.325 0.060953 2.325 1.01E-03 3.325 2.57E-06 0.35 0.620618 1.35 0.056238 2.35 8.89E-04 3.35 2.16E-06 0.375 0.595883 1.375 0.05183 2.375 7.83E-04 3.375 1.82E-06 0.4 0.571608 1.4 0.047715 2.4 6.89E-04 3.4 1.52E-06 0.425 0.547813 1.425 0.043878 2.425 6.05E-04 3.425 1.27E-06 0.45 0.524518 1.45 0.040305 2.45 5.31E-04 3.45 1.07E-06 0.475 0.501742 1.475 0.036982 2.475 4.65E-04 3.475 8.91E-07 0.5 0.4795 1.5 0.033895 2.5 4.07E-04 3.5 7.43E-07 0.525 0.457807 1.525 0.031031 2.525 3.56E-04 3.525 6.19E-07 0.55 0.436677 1.55 0.028377 2.55 3.11E-04 3.55 5.15E-07 0.575 0.416119 1.575 0.025921 2.575 2.71E-04 3.575 4.29E-07 0.6 0.396144 1.6 0.023652 2.6 2.36E-04 3.6 3.56E-07 0.625 0.376759 1.625 0.021556 2.625 2.05E-04 3.625 2.95E-07 0.65 0.357971 1.65 0.019624 2.65 1.78E-04 3.65 2.44E-07 0.675 0.339783 1.675 0.017846 2.675 1.55E-04 3.675 2.02E-07 0.7 0.322199 1.7 0.01621 2.7 1.34E-04 3.7 1.67E-07 0.725 0.305219 1.725 0.014707 2.725 1.16E-04 3.725 1.38E-07 0.75 0.288844 1.75 0.013328 2.75 1.01E-04 3.75 1.14E-07 0.775 0.273072 1.775 0.012065 2.775 8.69E-05 3.775 9.36E-08 844 A Mathematical Tables and Algorithms

x erfc(x) x erfc(x) x erfc(x) x erfc(x) 0.8 0.257899 1.8 0.010909 2.8 7.50E-05 3.8 7.70E-08 0.825 0.243321 1.825 9.85E-03 2.825 6.47E-05 3.825 6.32E-08 0.85 0.229332 1.85 8.89E-03 2.85 5.57E-05 3.85 5.19E-08 0.875 0.215925 1.875 8.01E-03 2.875 4.79E-05 3.875 4.25E-08 0.9 0.203092 1.9 7.21E-03 2.9 4.11E-05 3.9 3.48E-08 0.925 0.190823 1.925 6.48E-03 2.925 3.53E-05 3.925 2.84E-08 0.95 0.179109 1.95 5.82E-03 2.95 3.02E-05 3.95 2.32E-08 0.975 0.167938 1.975 5.22E-03 2.975 2.58E-05 3.975 1.89E-08 1 0.157299 2 4.68E-03 3 2.21E-05 4 1.54E-08

A.3 Derivatives

d dw dv du (uvw) = uv + uw + vw (A.25) dx dx dx dx % & d u 1 du dv = v − u (A.26) dx v v2 dx dx d du dv (uv) = vuv−1 + (ln u)uv (A.27) dx dx dx d du dv (uv) = vuv−1 + (ln u)uv (A.28) dx dx dx d 1 du (log u) = (log e) (A.29) dx a a u dx − dn n dnu n dv dn 1u n dnv (uv) = v + +···+ u (A.30) n 0 n 1 n−1 n n dx  dx dx dx dx d x f (t)dt = f (x) (A.31) dx c  d g(x) g(x) ∂h(x, t) h(x, t)dt = g(x)h(x, g(x)) − f (x)h(x, f (x)) + dt dx f (x) f (x) ∂x (A.32) f (x) f (x) lim = lim . (A.33) x→a g(x) x→a g(x)

A.4 Definite Integrals

 ∞ √ 1√ xe−x dx = π (A.34) 0 2  ∞ 2 1√ e−x dx = π (A.35) 0 2 A Mathematical Tables and Algorithms 845

 ∞ x z−1 e−x dx = Γ(z) (A.36) 0 ∞ a e−ax cos bxdx = (A.37) 2 + 2 0 a b ∞ a e−ax sin bxdx = (A.38) a2 + b2 0 ∞ e−ax sin bx b dx = tan−1 (A.39) x a 0 ∞ e−ax − e−bx b dx = ln (A.40) 0 x  a  ∞ 2 1 π e−ax dx = (A.41) 0 2 a   ∞ 2 1 π 2 e−ax cos bxdx = e−b /4a (A.42) 0 2 a ∞ Γ(n + 1) xne−axdx = , a > 0, n > −1 an+1 0 (A.43) n! = , a > 0, n = 0, 1, 2,... an+1  ∞ [1 − erf(βx)] exp −μx2 xdx 0 ' ( (A.44) 1 β = 1 −  , Reμ>−Reβ2, Reμ>0 2μ μ + β2  2π x cos θ e dθ = 2π I0(x) (A.45) 0   ∞ π 2 −(ax2+bx+c) b −4ac e dx = e 4a (A.46) −∞ a  √ ∞ νπΓ(ν/2) (1 + x2/ν)−(ν+1)/2dx = . (A.47) −∞ Γ((ν + 1)/2))

Interesting definite integrals:

 n b ∞ 2−1   f (x)dx = (b − a) (−1)m+12−n f a + m(b − a)2−n (A.48) a = =  n 1 m 1 1 [ln(1/x)]p dx = p! (A.49) 0 846 A Mathematical Tables and Algorithms  1 ∞ x−x dx = n−n = 1.29128599706266 ... (A.50) 0 n=1  1 ∞ x x dx = −(−1)nn−n = 0.78340510712 .... (A.51) 0 n=1

A.5 Indefinite Integrals   udv = uv − vdu (A.52)   1 f (ax)dx = f (u)du (A.53) a  sin udu =−cos u (A.54)  cos udu = sin u (A.55)  tan udu = ln sec u =−ln cos u (A.56)  u sin 2u 1 sin2 udu = − = (u − sin u cos u) (A.57) 2 4 2  u sin 2u 1 cos2 udu = + = (u + sin u cos u) (A.58) 2 4 2  eax eaxdx = (A.59) a  eax 1 xeaxdx = x − (A.60) a a  eax 2x 2 x2eaxdx = x2 − + (A.61) a a a2  1 dx = ln |x| (A.62) x  ax ax dx = (A.63) ln(a)  ln(x)dx = x ln(x) − x (A.64)  = − logb(x)dx x logb(x) x logb(e) (A.65) A Mathematical Tables and Algorithms 847  du 1 u = tan−1 (A.66) u2 + a2 a a  du u √ = sin−1 . (A.67) a2 − u2 a

A.6 Linear Matrix Transformations of Two-Dimensional Spaces

The transformations given below are particularly useful in the context of Chap. 5 to transform two-dimensional signal constellations according to the linear mapping rule: the original coordinates are mapped into the new coordinates according to [xy]T ← A × [xy]T, where [xy] is a row vector and A is given below, depending on the desired mapping:

1. Rotation by θ degrees counterclockwise:   cos(θ) − sin(θ) A = . (A.68) sin(θ) cos(θ)

2. Reflection against the horizontal axis. A horizontal reflection is a map that trans- forms a matrix into its horizontal mirror image:   −10 A = . (A.69) 01

3. Scaling by n in all directions:   n 0 A = . (A.70) 0 n

4. Horizontal shear mapping. A horizontal shear mapping keeps the vertical coordi- nate and shifts the horizontal coordinate by a distance m times the distance from the horizontal axis:   1 m A = . (A.71) 01

5. Squeezing. A squeeze mapping preserves Euclidean area of regions in the Carte- sian plane:   k 0 A = . (A.72) 01/k 848 A Mathematical Tables and Algorithms

6. Projection onto the vertical axis:   00 A = . (A.73) 01

A.7 Mathcad Routine for Converting Decimal to Binary Numbers

The routine (A.74) below converts a decimal number x into its binary form. The result is presented as a transposed vector (row matrix).   bin(x):=  A ← 0   log(x + 1)  ←  n ceil  log(2)   B − ← 0  n 1  for y ∈ (n − 1) .. 0   (A.74)   y  A ← A + 2    ← ≤  Bn−y−1 1 if A x    A ← A − 2y otherwise   BT

A.8 Mathcad Routine for a Binary Counter

The routine (A.75) below generates all ordered binary b-bit words. The result is presented as a matrix.

=  Counter(b):  ←  Bb−1,2b−1 0   ∈ .. b −  for x 0 2 1    A ← 0      for y ∈ b − 1 .. 0    (A.75)    ← + y   A A 2      Bb−y−1,x ← 1 if A≤ x       ← − y  A A 2 otherwise  B A Mathematical Tables and Algorithms 849

A.9 Mathcad Routine for a Gray Counter

The routine (A.76) below generates all ordered binary b-bit words using the Gray code. The result is presented as a matrix.    Bb−1,2b−1 ← 0 =  CounterG(b):  b  for x ∈ 0 .. 2 − 1     ←   A 0     for y ∈ b − 1 .. 0       ← + y    A A 2       B − − , ← 1 if A ≤ x (A.76)    b y 1 x    ← − y   A A 2 otherwise     G , ← B ,   0 x 0 x     for y ∈ 1 .. b − 1 if b > 1    G , ← B , ⊕ B − ,  y x y x y 1 x  G

A.10 Mathcad Routine for Generating Walsh-Hadamard Sequences

The routine (A.77) below generates all Walsh-Hadamard sequences of length n.The result is presented as an n × n matrix H(n).  =  ← H(n):  X 1   log(n)  for i ∈ 0.. − 1   log(2) (A.77)   X ← augment(stack(X, X), stack(X, −X))  X

A.11 Mathcad Routine for Generating the IOWEM of a Block Code

The routine (A.78) below finds the input-output weight enumerating matrix (IOWEM) of a block code. The limitation of the parameters (n, k) is governed by the lim- itations of Mathcad when working with matrices. The input to this routine is the values of n and k, the matrix m with all message words and the matrix C with all 850 A Mathematical Tables and Algorithms codewords. This routine becomes attractive for long codes with a large number of codewords.

 =  ∈ .. k − B :  for i 0 2 1     k−1     wmi ← mi, j     j=0     n−1     wc ← C ,   i i j  j=0   ← (A.78)  Countk,n 0   for x ∈ .. k −  0 2 1   for i ∈ 0..k   for j ∈ ..n  0  ← + = ∧ =  Counti, j Counti, j [(wmx i) (wcx j)]  Count

A.12 Mathcad Routine for Computing the Euler Function

The routine (A.79) below can be used to compute the Euler function value of the argument t. It has a Mathcad built-in greatest common divisor (gcd) function that can be implemented by the GCD(a, b) function also given by (A.80).

 φ =  = (t):  1 if t 1   otherwise     ←   x 0   (A.79)   for i ∈ 1..t − 1     ← + , =   x x 1 if GCD(i t) 1   x

  GCD(a, b):=  while b = 0     ←   r b     b ← mod(a, b) (A.80)     ←  a r  r A Mathematical Tables and Algorithms 851

References

1. Abramowitz, M. and Stegun, I. (Ed.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C, 1972. 2. Bronshtein, I. N., Semendyayev, K. A., Musiol, G. and Muehlig, H. Handbook of Mathematics, 5th Ed. Springer-Verlag: Berlin-Heidelberg, 2007. 3. Gradshteyn, I. S. and Ryzhik, I. M. Table of Integrals, Series, and Products. 7th Ed. Academic Press/Elsevier: USA/UK, 2007. 4. Polyanin, A. D. and Manzhirov, A. V. Handbook of Mathematics for Engineers and Scientists. Boca Raton, Fl, USA: Chapman & Hall/CRC, 2007. 5. Poularikas, A. D. The Handbook of Formulas and Tables for Signal Processing. USA. CRC Press LLC, 1999. 6. NIST: National Institute of Standards and Technology. NIST Digital Library of Mathematical Functions. Available at: http://dlmf.nist.gov/(last access on July 2009). 7. Zwillinger, D. Standard Mathematical Tables and Formulae. 31st Ed. London – New York: Chapman & Hall/CRC Press, 2003. Abbreviations

A/D analog-to-digital AC alternate current ACPR adjacent channel power ratio AFC automatic frequency control AGC automatic gain control AM amplitude modulation AMI alternate mark inversion AoA angle-of-arrival APK amplitude-phase keying AR autoregressive ARQ automatic repeat request ASK amplitude shift keying AWGN additive white Gaussian noise BCH Bose-Chaudhuri-Hocquenghem BCM block-coded modulation BER bit error rate BFSK binary frequency-shift keying BIBO bounded-input, bounded-output BICM bit-interleaved coded modulation BICM-ID BICM with iterative decoding BMA Berlekamp-Massey’s algorithm BPF band-pass filter BPL broadband over power-line BPSK binary phase-shift keying BRGC binary reflected Gray code BS base-station BSC binary symmetric channel BTC block turbo code CAP carrierless amplitude-phase CD compact disc CDF cumulative distribution function CDM code-division multiplexing

853 854 Abbreviations

CDMA code-division multiple-access CE controlled equalization CF characteristic functions CIR channel impulse response CLT central limit theorem CMI code mark inversion CPFSK continuous-phase frequency-shift keying CPM continuous-phase modulation CRB Cramer-Rao´ bound CRC cyclic redundancy check CSI channel state information CTC convolutional turbo code CW continuous wave D/A digital-to-analog DBPSK binary differential phase-shift keying DC direct current DC down-conversion DCDJ duty-cycle dependent jitter DC-DS-CDMA double spreading code direct-sequence CDMA DC-MC-CDMA double spreading code multi-carrier CDMA DDJ data-dependent jitter DEBPSK differentially encoded binary phase-shift keying DEM digital elevation map DEMPSK differentially encoded M-ary phase-shift keying DEQPSK differentially encoded quaternary phase-shift keying DFT discrete Fourier transform DJ deterministic jitter DLL delay-locked loop DLMF digital library of mathematical functions DMC discrete memoryless channel DMS discrete memoryless source DoA direction-of-arrival DoD direction-of-departure DS-CDMA direct-sequence CDMA DSP digital signal processing DS-SS direct-sequence spread-spectrum DS-UWB direct-sequence ultra wideband DVB digital video broadcasting EA Euclidian algorithm EGC equal-gain combining EHF extremely high frequency EMI electromagnetic interference ERP effective radiated power ESD energy spectral density EXIT extrinsic information transfer Abbreviations 855

FCC Federal Communications Commission (USA) FEC forward error correction FFT fast Fourier transform FH-SS frequency-hopping spread-spectrum FIR finite impulse response FM frequency modulation FSC frequency-spread coding FSK frequency shift keying FSM finite-state machine FSO free-space optical GCD greatest common divisor GF Galois field GFSK Gaussian-filtered frequency-shift keying GMSK Gaussian minimum-shift keying GPR ground penetrating radar GPS global positioning system GSM global system for mobile communication HDBn high density bipolar n HDSL high-rate digital subscriber line I&D integrate and dump I&Q in-phase and quadrature i.i.d. independent and identically distributed ICI inter-carrier interference IDFT inverse discrete Fourier transform IFFT Inverse fast Fourier transform IIR infinite impulse response IOWEF input-output weight enumerating function IOWEM input-output weight enumerating matrix ISI intersymbol interference ISM industrial, scientific and medical LAN local area network LBW loop bandwidth LDPC low-density parity check LED light-emitting diode LFSR linear feedback shift-register LLR log-likelihood ratio LMS least mean square LoS line-of-sight LPF low-pass filter LPI low probability of interception LSB least significant bit LTI linear, time-invariant LTV linear, time-variant LUT look-up table MAC medium access control 856 Abbreviations

MAI multiple access interference M-AM M-ary amplitude modulation MAP maximum a posteriori M-ASK M-ary amplitude shift keying MC multi-carrier MC-CDMA multi-carrier CDMA MC-DS-CDMA multi-carrier direct-sequence CDMA MCM multi-carrier modulation M-DPSK M-ary differential phase-shift keying MDS maximum-distance-separable M-FSK M-ary frequency shift keying MGF moment-generating function MIMO multiple-input, multiple-output ML maximum likelihood MLCM multilevel coded modulation MLSD maximum likelihood sequence detection MLSE maximum likelihood sequence estimation MMSE minimum mean square error MMSEC minimum mean square error combining M-PAM M-ary pulse amplitude modulation MPI multipath interference M-QAM M-ary quadrature amplitude modulation MRC maximum ratio combining MSB most significant bit MSE mean square error MSK minimum-shift keying MT mobile terminal MT-CDMA multi-tone CDMA MUD multiuser detection NRZ non-return-to-zero NRZ-b bipolar non-return-to-zero NRZ-u unipolar non-return-to-zero OFDM orthogonal frequency-division multiplexing OOK on-off keying OQPSK offset quaternary phase-shift keying (see SQPSK) ORC orthogonality restoring combining P/S parallel-to-serial PAM pulse amplitude modulation PAPR peak-to-average power ratio PCB printed circuit board PCBC parallel concatenated block codes PCCC parallel concatenated convolutional codes PCL power-line communications PCM pulse-coded modulation PDF probability density function Abbreviations 857

PDP power delay profile PG processing gain PJ periodic jitter PLL phase-locked loop PM phase modulation PMF probability mass function PN pseudo-noise (or pseudo-random) PPM pulse-position modulation PSD power spectral density PSF pulse-shaping filter PSK phase shift keying PSM pulse-shaping modulation QAM quadrature amplitude modulation QPSK quaternary phase-shift keying RASE rapid acquisition by sequential estimation RF Radiofrequency RFID radiofrequency identification RJ random jitter RM Reed-Muller rms root-mean-square ROC region of convergence RRC root-raised cosine RS Reed-Solomon RSC recursive systematic convolutional RSS received signal strength RT real time RZ-b bipolar return-to-zero RZ-u unipolar return-to-zero S&H sample and hold S/P serial-to-parallel S/STP screened shielded twisted pair S/UTP screened unshielded twisted pair S/V scalar-to-vector SC selection combining SC single carrier SCBC serially concatenated block codes SCCC serially concatenated convolutional codes SCR silicon-controlled rectifier SDR software-defined radio SER symbol error rate SIHO soft-input, hard-output SISO soft-input, soft-output SNR signal-to-noise ratio 858 Abbreviations

SNRQ signal-to-quantization noise ratio SOVA soft-output Viterbi algorithm SPC single-parity check SPC-PC single-parity check product code SPC-TPC single-parity check turbo product code SQ scalar quantization SQPSK staggered quaternary phase-shift keying (see OQPSK) SS spread-spectrum SSA serial-search acquisition SSMA spread-spectrum multiple access SSP sound speed profile STC space-time coding SV Saleh-Valenzuela SVD singular value decomposition TCM trellis-coded modulation TDL tapped delay line TDL tau-dither loop TH-BPSK time-hopping binary phase-shift keying TH-OOK time-hopping on-off keying TH-PAM time-hopping pulse amplitude modulation TH-PPM time-hopping pulse-position modulation TH-PSM time-hopping pulse-shaping modulation TH-UWB time-hopping ultra wideband TJ total jitter TPC turbo product code TTCM turbo trellis-coded modulation UC up-conversion UEP unequal error protection UI unit interval US uncorrelated scattering UTP unshielded twisted pair UWA underwater acoustic UWB ultra-wideband V/S vector-to-scalar VA Viterbi algorithm VCC voltage-controlled clock VCO voltage-controlled oscillator VQ vector quantization VT variance transfer WDM wavelength division multiplexed WEF weight enumerator function WLAN wireless local area network WPAN wireless personal area network WSS wide-sense stationary WSSUS wide-sense stationary, uncorrelated scattering Abbreviations 859

WUSN wireless underground sensor network XOR exclusive-or ZF zero forcing π/4-DQPSK π/4-shifted differential quaternary phase-shift keying π/4-QPSK π/4-shifted quaternary phase-shift keying Index

A Autocovariance of a random process, 64 Abelian group, 734 Automatic frequency control (AFC), 215 Absolute zero temperature, 175 Automatic gain control (AGC), 231, 441 Absorption, 253 Automatic repeat request (ARQ), 732, 831 Acceptance/rejection method, 57 Autoregressive random process (AR(4)), Adaptive equalization, 231 156–157 Additive white Gaussian noise (AWGN), 80 Average fade duration, 225 Adjacent channel power ratio (ACPR), 511 Average power of a signal, 89 A-law, 126 AWGN channel, 175 Algebraic decoding, 734 Axiomatic definition, 4 Aliasing, 116 Alternate mark inversion (AMI), 269 B Amplitude modulation (AM), 412 Bandwidth, 162 Amplitude-phase keying (APK), 412 absolute, 162 Amplitude-shift keying (ASK), 412 -3 dB or half power, 163 Analog-to-digital conversion, 122 equivalent noise, 163 Analytic signal, 146 null-to-null or main lobe, 164 Angle-of-arrival (AoA), 251 root mean square, 163 Angular spread, 252 of a signal, 162 Anomalous propagation, 243 spectral mask, 165 Anti-aliasing filter, 118 Baseband and passband signals, 89 Anti-jamming communication, 586 Baseband signaling, 265 Antipodal signaling, 415 Baseband transmission, 265 Aperiodic cross-correlation parameter, 602 Base-function, 364 Aperiodic signal, 88 Base vectors, 730 Aperture-medium coupling loss, 671 Bathtub diagram, 323, 331 a priori information, 822 Baud rate, 159 Area-mean, 200 Bayes rule, 1, 7–9, 16 Area-mean power, 233 BCH bound, 764 Array codes, 810 BCJR, 817 Array processing, 250 Berlekamp-Massey’s algorithm (BMA), 769 Array response vector, 252 Bernoulli random variable, 23 Asymptotic coding gain, 800 Big O, 111 Atmospheric stratification, 244 Bilinear transformation, 144 Attenuation constant, 187 Binary BCH codes, 763 Augmenting, 736 Binary reflected Gray code (BRGC), 397 Autocorrelation function of a random Binary reflection, 397 process, 68 Binary symmetric channel (BSC), 9, 53, Autocovariance-ergodic random processes, 64 173, 704

861 862 Index

Binomial coefficient, 7 Channel state information, 727, 832 Binomial random variable, 24 Channel transition matrix, 181 Binomial theorem, 400 Characteristic function, 21 Bipolar code, 266 Characteristic impedance, 187 Bipolar non-return to zero (NRZ-b), 269 Characteristic polynomial, 615 Bipolar return to zero (RZ-b), 269 Chebyshev inequality, 50 Bit error probability, 362 Chien search, 769 Bit error rate, 53 Chi-square PDF, 43 Bit-interleaved coded modulation (BICM), Class A amplifier, 509 796, 832 Classic definition, 4–5 Bit-interleaved coded-modulation with Class I duobinary signaling, 336 iterative decoding (BICM-ID), 829 Class IV duobinary signaling, 338 Block code, 730 Clipping, 123 Block-coded modulation (BCM), 831 Cluster arrival rate, 238 Block error probability, 772 Cluttering, 671 Block interleaver, 735 Coaxial cable, 187 Block turbo codes, 826 Code acquisition, 621 Bluetooth, 585 Code diversity, 604, 731, 829 Blu-ray, 505 Code-division multiple access (CDMA), 601 Boltzmann constant, 79 Code-division multiplexing (CDM), 644 Bounded distance decoder, 730, 773 Code mark inversion (CMI), 271 Bounded-input, bounded-output system Code polynomial, 753 (BIBO), 136 Code rate, 701, 730 Box-Muller method, 58 Code search, 637 Branch metric, 781 Codeword, 730 Burst errors, 183 Coding gain, 718 Coherence bandwidth, 226–227 C Coherence time, 227 Capacity-achieving codes, 710 Coherent detection, 415, 427 Capacity of fading channels, 715 receiver, 384 Capacity of the MIMO channel, 723 Colored noise channel Cartesian product, 437 capacity of, 719 Catastrophic collision, 668 Combinations, 4, 6 Causal linear system, 136 Combinatorial formulas, 5 Centralized random variable, 19 Combining techniques, 648 Central Limit theorem, 35, 37 Communication channel, 171 Central moment, 19 Companding, 126 Channel capacity, 723 Compatibility constraint, 715 Channel capacity theorem, 729 Complementary error function, 30 Channel coder, 172 Complex envelope, 147 Channel coding, 730 Complex Gaussian random process, 78 theorem, 701 Complex Gaussian random variable, 31 Channel correlation function, 219 Complex random processes, 61 Channel covariance matrix, 656 Complex representation, 146 Channel matrix, 701 of signals, 146 Channel modeling, 174 Concatenated codes, 812 Channel models Concatenation, 735 empirical, deterministic and stochastic, 175 Conditional entropy, 700 narrowband, 213 Conditional expected value, 23 Rummler, 245, 249 Conditional probability, 7, 16–17, 28 wideband, 214 Confidence coefficient, 50 wireless spatial, 256 Confidence interval, 50 Channel sounding, 174, 596 Congruential methods, 56 Index 863

Constellation, 177, 370 Cyclic extension, 653 Constraint length, 731, 794 Cyclic prefix, 651 Continuous memoryless random variable, 28 Cyclic redundancy check (CRC), 731 Continuous-phase FSK (CPFSK), 454 Cyclic suffix, 651 Continuous-phase modulation (CPM), 455 Cyclostationary random processes, 73 Continuous random variable, 12, 16, 18, 20, 22 Cyclotomic cosets, 732 Continuous-valued random process, 61 Controlled equalization (CE), 660 D Convolutional code, 731, 808 Data compaction, 690 decoding, 801 Data compression, 690 encoding, 795 1 dB Compression point, 510 free distance, 803 Decimation, 122 performance function, 802 split-state diagram, 789 Decision depth, 785, 795 transfer function, 789 Delay-locked loop (DLL), 638 Convolutional encoder Delay modulation, 271 catastrophic, 780 Delay spread, 220, 227 feedback or recursive, 780 typical values, 236 feed-forward, 780 Delta-sigma converter, 130 systematic and non-systematic, 780 Design distance, 765 Convolutional turbo codes, 808 Detection, 177 Convolution integral, 133 Deterministic event, 2 Convolution sum, 133 Deterministic and random signals, 89 Cooley-Tukey, 111 Deviation, 19 Cophasing, 606, 608 Difference equation, 91, 138 Copy-type multi-carrier, 645 Differential encoder, 563 Correction filter, 119 Differential phase-shift keying (DBPSK), 534 Correlation, 21 Diffraction, 206 coefficient, 21 Diffraction loss, 241 receiver, 292 Digital elevation map (DEM), 206 Correlative coding, 333 Digital filtering, 166 Correlator, 292 Digital filters, 122, 145–146 Coset, 744 Digital modulation, 411 Cosine filter, 341 Dimensionality theorem, 366 Costas loop, 552 Dirac delta function, 91 Cost function, 351 Direction-of-departure (DoD), 252 Counting methods, 1, 5 Direct-sequence UWB (DS-UWB), 668 Covariance, 21 Dirichlet conditions, 95, 103 Coverage prediction, 206 Discrete channel, 173 Craig’s method, 445 Discrete channels with memory, 183 Cramer-Rao´ bound (CRB), 550 Discrete to continuous frequency mapping, 109 Crest factor, 124 Discrete Fourier transform (DFT), 108 Critical distance, 202 Discrete memoryless channel (DMC), 180, 699 Cross constellation, 442 Discrete memoryless source, 691 Cross-correlation function of a random Discrete random variable, 12–13, 18–19, process, 70 23, 26 Cross-covariance of a random process, 70 Discrete-time, continuous-valued random Crossover probability, 181 process, 61 Cross-spectral density, 75 Discrete-time, discrete-valued random Cumulative distribution function, 10–13, process, 61 37, 56 Discrete-time filter, 311 CW jammer, 593 Discrete-time Fourier transform Cyclic code, 731 convergence, 107 864 Index

periodicity, 106 training mode, 349 properties and pairs, 108 training sequence, 349 Distortion-free channel, 307 zero forcing, 344 Distortion-free linear system, 133 ZF, 344 Distributed primary parameters, 185 Equally likely symbols, 362 Diversity, 231, 572, 606 Equivalent codes, 732 Divide and conquer, 731 Equivalent low-pass, 148 Doppler shift, 215–216 Equivalent noise bandwidth, 81 Doppler spectrum, 216 Equivalent noise temperature, 80, 176 Doppler spread, 215, 222 Ergodic, 63 Down-sampling, 122 Ergodic capacity, 726 Dual code, 732 Error correcting codes, 231 Dual spaces of vector spaces, 732 Error detection and error correction, 732 Ducting, 243 Errored-second, 244 Duobinary conversion filter, 336 Error floor, 603 Duobinary signaling, 336 Error locator polynomial, 769 Duty-cycle, 96 Error pattern, 736 Error probability analysis E antipodal signaling, 289 Early-late synchronizer, 565 Error syndrome, 737 E-carrier hierarchy, 270 Error-trapping, 759 Eigenfunction, 136 Error value, 769 Eigenvalue, 137 Euclidian algorithm, 769 Electromagnetic interference (EMI), 185 Euclidian distance, 365 Elementary row operations, 737 Even and odd signals, 89 Energy and power signals, 89 EXIT chart, 831 Energy of a signal, 104 Expectation, 17 Energy signal, 362 Expected value, 17 Energy spectral density, 104, 107, 153, 155, Exponential random variable, 28 161 Expurgating, 736 Ensemble averages, 62 Extended source, 693 Entropy, 690 Extending, 736 Entropy function, 691 Extension field, 733 Envelope, 148 Extremely high frequency (EHF), 189 Envelope delay, 134 Extrinsic information, 816, 821 Equal-gain combining (EGC), 604, 648 Extrinsic information transfer, 831 Equalization, 320 Eye diagram, 307, 319 Equalizer, 343 adaptive, 343 F blind, 354 Fading decision-directed mode, 349 frequency flat and fast, 224 fixed, 343 frequency flat and slow, 224 fractionally-spaced, 345 frequency selective and fast, 224 gradient algorithm, 351 frequency selective and slow, 224 LMS, 344, 349 Failure, 23 MLSE, 344 False alarm, 622 MMSE design criterion, 344 Fast Fourier Transform, 109 noise enhancement, 348 Fast FSK, 476 peak distortion, 349 Fast and slow FH-SS, 632 steepest descent algorithm, 352 Fibonacci LFSR, 680 stochastic gradient algorithm, 352 Finite impulse response (FIR), 144 symbol-spaced, 345 Finite-state machine (FSM), 778, 796 synchronous, 345 Finite trace-back Viterbi algorithm, 783 Index 865

Fire codes, 759 Gaussian random variable, 29 Fixed-length source code, 697 Generalized ML receiver, 385 Flat fading, 211 Generalized Nyquist criterion, 316, 366 Follower jammer, 630 Generator coefficient, 777 Forney’s algorithm, 769 Generator matrix, 732 Forward error correction (FEC), 732 Generator polynomial, 733, 764 Fourier analysis, 94 Generator sequence, 792 Fourier series, 94 Geometric random variable, 24 complex exponential, 94 Geometric representation of signals, 363 convergence, 95 Gibbs phenomenon, 97 trigonometric, 94 Gilbert-Elliott, 184 truncated, 96 Global positioning system (GPS), 585 Fourier transform (FFT), 109, 111 Global System for Mobile Communication continuous-time, 103–104 (GSM), 505 convergence of the, 103 Gold sequences, 622 discrete-time, 106 Goodness-of-fit test, 55 for periodic signals, 103 Gram-Schmidt, 367 properties and pairs, 104 Gray code, 397 via VisSim/Comm, 111 Gray labeling, 380, 397 Fourier transform pair, 105 Gray mapping, 380, 397 Fractional bandwidth, 665 Green machine, 744 Free distance, 733, 785 Gross bit rate, 692 Free-space, 201 Ground penetrating radar (GPR), 666 Free-space optical (FSO) channel, 252, 254 Group, 733 Frequency diversity, 604 Group delay, 134 Frequency-domain spreading, 642 Guard interval, 653 Frequency-hopping spread-spectrum H (FH-SS), 630 Hadamard matrix, 626 Frequency interleaving, 645 Hadamard transform decoding, 744 Frequency modulation (FM), 412 Hamming bound, 734 Frequency offset, 653 Hamming codes, 739 Frequency offset correction, 655 Hamming distance, 734 Frequency selective fading, 214, 222 Hanning window, 158 Frequency-shift keying (FSK), 412 Hard-decision decoding, 734 Frequency warping, 144 Harmonic, 511 Fresnel ellipsoids, 240 Heaviside step function, 91 Fresnel integral, 241 High density bipolar n (HDBn), 271 Fresnel zones, 240 Hilbert transform, 146 Full-period cross-correlation, 602 Histogram, 14 Fully-interleaved channel, 774 5 Huffman code, 694–695 Functional relation, 76 Huffman decoding tree, 698 Fundamental angular frequency, 94 Huffman tree, 696 Huygens-Fresnel principle, 241 G Huygen’s secondary source, 207 Galois field, 729 Galois LFSR, 680 I Gamma random variable, 44 Impulse noise, 79 Gaussian-filtered frequency-shift keying Impulse radio, 665 (GFSK), 521 Impulse response, 131 Gaussian-filtered MSK (GMSK), 476 Impulsive noise, 79 Gaussianization, 718, 728, 828 In-band distortion, 511 Gaussian noise, 79 Incomplete product codes, 813 Gaussian random process, 76 Independent events, 8, 18 866 Index

Independent random variables, 18–19 Laser, 192 Index of refraction, 191 Last mile, 185 Indoor channel, 232 Laurent’s expansion, 577 large-scale propagation, 233 μ-law, 126 small-scale propagation, 234 Law of diminishing returns, 605 Industrial, scientific and medical, 588 Law of large numbers, 54 Infinite bandwidth, 162 Leaky cable, 188 Infinite impulse response (IIR), 144 Lempel-Ziv code, 699 Information, 710 Lengthening, 736 Information bit rate, 709 Level codes, 266 In-phase component, 147 Level crossing rate, 225 Input-output weight enumerating function L’Hopital’sˆ rule, 93 (IOWEF), 739 Light emitting diode (LED), 192, 255 Integrator and dump, 291 Likelihood function, 374 Inter-carrier interference (ICI), 655 Linear code, 735 Interleaving, 231, 735 Linear differential equations, 138 Intermodulation product, 509 Linear feedback shift register (LFSR), 612 Interpolating function, 116 Linear span, 612 Interpolation, 122 Linear system, 130 Intersymbol interference (ISI), 191, 307, 312, properties, 135 651 scaling, 130 Inverse CDF method, 56 superposition, 130 Inverse discrete Fourier transform (IDFT), 109 Linear time-variant system (LTV), 135 Inverse fast Fourier transform (IFFT), 109, 111 Line codes, 266 IP2, 510 types of, 266 I&Q imbalance, 578 Line repeaters, 266 I&Q modulator, 429 Line-of-sight (LoS), 239 Irreducible polynomial, 613, 735 optical, 239 ISM bands, 588 radio, 239 Iterative decoding, 735 Link margin, 243, 250 LMS algorithm, 352 J Local-mean, 203 Jacobian of the transformation, 39 power, 209 Jamming margin, 633 Log-distance model, 201 Jitter, 324, 327 Log-distance path loss, 202 Johnson-Nyquist noise, 79 Log-likelihood algebra, 818 Joint central moment, 20 Log-likelihood function, 382, 551 Joint moment, 20 Log-likelihood ratio (LLR), 816 Joint probability density function, 18 Look-up table (LUP), 371 Joint probability mass function, 18 Loop filter, 552 K Low-density parity check (LDPC), 806 Karnaugh map, 398 Low probability of interception (LPI), 588 Key equations, 769 Knife-edge, 241 M Kraft-McMillan inequality, 695 MacWilliams identity, 766 Kronecker delta function, 91 Magnitude response, 120 Kronecker product, 625 Majority-logic decoding, 744 Manchester code, 270 L MAP decision rule, 305 Lagrange multipliers, 720 Marcum Q-function, 225, 519 Landau-Pollak theorem, 366 Marginal probability, 8 Laplace distribution, 32 Mark, 96 Laplace transform, 113 Markov chains, 162, 166, 184, 198 Index 867

M-ary differentially-coherent PSK (M-DPSK), MSK-type receivers, 525 576 M-th power loop, 561 Matched filter, 284 Multichannel reception, 605 non-coherent, 538 Multi-h CPM, 455, 505 Matched filter vs. correlator Multilevel coded modulation (MLCM), 831 equivalence between, 293 Multimodal, 16 Maximum-distance-separable (MDS), 767 Multipath delay profile, 220 Maximum-length sequences, 612 Multipath fading, 208 Maximum likelihood, 16 Multipath intensity profile, 220 Maximum likelihood sequence detection Multipath interference, 620 (MLSD), 333 Multipath propagation, 207 Maximum likelihood sequence estimation Multiple access interference, 619 (MLSE), 344, 525, 673 Multiple symbol differential detection, 546 Maximum a posteriori, 16 Multiplexing gain, 723 Maximum ratio combining (MRC), 604, 648 Multiplicative congruential method, 56 Mean ergodic random processes, 64 Multiplicative fading, 213 Mean of a random process, 64 Multistage decoding, 831 Mean square error (MSE), 203 Multiuser detection (MUD), 603, 648, 657 Mean-square value, 19 Multivariate distributions, 34 Meggitt decoder, 758 Multivariate Gaussian random variables, 34 Memoryless channel, 180 Muting, 732 Memoryless LTI systems, 137 Mutually exclusive events, 3, 8 Mersenne prime, 56 N Message-passing, 737 Narrow-sense BCH, 764 Microstrip, 189 Near-far problem, 603 Microwave link Noise Figure, 510 clearance, 241 Noise, 79 Mie effect, 255 -vector, 177 Miller code, 271 Nominal carrier frequency, 451 Millimetre-wave, 234 Non-binary BCH, 769 MIMO capacity, 727 Non-coherent code synchronization, 635 Minimal encoder, 779 Non-coherent detection, 413 Minimal polynomial, 732 Nonlinear amplification, 509 Minimum energy constellation, 388 Nonlinear distortion, 513 Minimum Hamming distance, 732 Nonlinear system, 131 Minimum mean square error, 349 Non-minimum phase, 246 Minimum mean square error combining Non-periodic signal, 88 (MMSEC), 648 Non-square QAM, 442 Mixed random variables, 12 Non-uniform quantization, 123 ML decision rule, 304 Normalized histogram, 14 MMSE combining, 643 Normalized power, 90, 99 Modified duobinary signaling, 335 Normalized standard error, 51 Modulation, 151 Normal random variable, 30 Moment, 19 Norm of a vector, 365 -generating function, 21 Nyquist criterion for ISI-free, 312, 315 theorem, 22 Nyquist pulse, 317 Monocycle, 668 Nyquist rate, 116 Monte Carlo, 51, 55 Nyquist sampling criterion, 116 Moving average filter, 140 Nyquist sampling theorem, 122 Moving median filter, 209 M-PAM signaling, 276 O M-sequences, 612 Odd symmetry, 313 MSK modulator, 502 One-tap equalization, 648 868 Index

Only discrete memoryless random variable, 25 Personal area network, 232 On-off keying (OOK), 463, 671 Phase ambiguity, 579 Optical fiber, 172, 190 Phase conjugation, 673 acceptance cone, 191 Phase constant, 187 chromatic dispersion, 192 Phase discriminator, 554 intermodal dispersion, 192 Phase-locked loop (PLL), 269, 552 mono-mode, 191 lock points, 579 multimode, 191 Phase modulation (PM), 412 refractive index, 191 Phase noise, 554, 655 single-mode, 191 Phase response, 135 total reflection, 191 Phase-shift keying (PSK), 415 Optimum receiver, 278 PN sequences, 627 Order, 736 Poisson random variable, 25 Ordered, 5–7 Polarization diversity, 604 Order of the joint moment, 20 Polybinary coding, 335 Orthogonal Polybipolar, 335 random variables, 21 coding, 335 Orthogonal frequency-division multiplexing, Post-detection combining, 681 640 Power amplifier, 510 Orthogonality restoring combining (ORC), 648 Power delay profile, 230 Orthogonalization, 367 Power efficiency, 425 Orthogonal multiplexing, 428 Power-line channel, 194 Orthogonal random processes, 76 Power-line communications (PLC), 199 Orthonormal base functions, 177 BPL, 194 Orthonormal basis, 363 broadband over power-line, 194 Outage capacity, 726 channel Outage probability, 244 asynchronous impulsive noise, 197 background noise, 197 P frequency response, 195 Pair-wise error probability, 392 generalized background noise, 197 Parameter estimation, 50, 565 impulse noise, 198 Parity-check matrix, 755 narrowband noise, 196 Parity-check polynomial, 755 periodic impulsive noise, 196 Parity equations, 811 indoor PLC channel, 195 Parseval’s relation, 110 in-home PLC, 194 Parseval’s theorem, 104, 107, 155, 284 Power overdrive, 578 Partial cross-correlation, 623, 661 Power of a signal, 105 Partial response signaling, 341, 356, 523 Power spectral density, 81, 162 Passband signaling, 415 Power spectral density of a random process, 69 Path diversity, 608–609 Pre-detection combining, 680–681 Path loss exponent, 201–204, 208, 233–234, Pre-envelope, 146 236 Preferred m–sequences, 624 Path metric, 785 Prefix codes, 695 PDF of the sum of random variables, 36 Primary distributed parameters, 188 Peak-to-average power ratio (PAPR), 437, 531, Prime decomposition, 615 534 Prime factorization, 615 Peak-to-rms ratio, 124 Primitive BCH, 765 Perfect code, 774 Primitive element, 768 Periodically stationary random processes, 64 Primitive polynomial, 618, 768 Periodic cross-correlation, 626 Principle of orthogonality, 351 Periodic signal, 94 Printed circuit board (PCB), 189 Periodogram, 156, 158 Probability density function, 12–13, 15–16, 18, Permutations, 6 22, 27–29, 31, 33–36, 38, 40–44, 76 Index 869

Probability generating function, 22 Received signal strength (RSS), 183 Probability mass function, 12–13, 17–18, Received signal-vector, 177 23–25, 37, 54 Reciprocal polynomial, 615 Processing gain, 634 Reconstruction filter, 116 Product codes, 825 Rectangular function, 92 Projection theorem, 351 Rect function, 92 Propagation prediction, 201 Recursive systematic convolutional (RSC) Properties of the variance, 19 code, 780 PSD estimation, 166 Reed algorithm, 741 bias, 158 Reed-Muller codes, 739 frequency resolution, 158 Reed Solomon codes, 766 leakage, 156 Reflection, 191 via Markov chains, 166 Region of convergence, 114 periodogram, 155 Relative frequency, 3, 14, 33, 51 smearing, 158 Residual mean square error, 353 smoothing, 156 Resolvability condition, 599 window type, 158 Resonant cavity, 511 Pseudo-noise, 612 Reversed polynomial, 616 Pulse Amplitude Modulation (PAM), 159, 273 Ricean fading, 213 Pulse jammer, 591 Rice factor, 226 Pulse signal-to-noise ratio, 296 Rich scattering, 250 Puncturing, 736 Rolloff factor, 317 Root-cosine filter, 342 Q Root-mean-square, 19 Quadrature amplitude modulation (QAM), 412 Root-raised cosine, 319 Quadrature component, 148 Root of unity, 108 Quantization error, 123 Rotation and translation invariance, 387 Quasi-distortion-free transmission, 319 Rummler, 240, 247 R Radiofrequency identification (RFID), 585 S Raised cosine, 317 Sa function, 93 RAKE receiver, 605–606 Saleh-Valenzuela, 232, 235, 237–239 Random binary wave, 152 Sample average, 46 power spectral density, 152 Sample-functions, 61 Random-coding exponent, 702 Sample and hold, 118 Random event, 2 Sample Mean, 47 Random FM noise, 216 Sample space, 2, 4–5, 7–8, 61, 73 Random jammer, 591 Sampling Random process, 46 of deterministic signals, 115 Random processes through linear systems, 73 of passband signals, 122 Random variables, 1, 10 of stochastic processes, 117 Rapid acquisition by sequential estimation Sampling oscilloscopes, 122 (RASE), 636 Scalar quantization, 695 Rate-distortion theory, 695 Scale parameter, 32 Rate parameter, 28 Scattering, 207 Ray arrival rate, 236 Scattering function, 219 Raylaunching, 201 Screened shielded (S/STP), 185 Rayleigh effect, 255 Screened unshielded (S/UTP), 185 Rayleigh fading, 213 S-curve, 554 Rayleigh loss coefficient, 193 Search dwell time, 636 Rayleigh random variable, 31 Secondary parameters, 188 Rayleigh’s energy theorem, 155, 284 Second central moment, 19 Ray-tracing, 201, 233 Seed, 56 870 Index

Selection combining, 604 S/P-type multi-carrier, 645 Serial-search acquisition (SSA), 621, 637 Square QAM, 434 Set partitioning, 797 Stable linear system, 136 Shadowing Standard array, 744 lognormal, 203 Standard error, 51 Shannon limit, 707 Standard refraction, 243 Shared time early-late, 639 State transition diagram, 778 Shift-invariant system, 135 State transition matrix, 184 Shortening, 736 Stationary process, 63 Sifting property, 91, 296 Stirling’s formula, 6 Sigma-delta converter, 130 Stochastic convergence, 55 Signaling rate, 159 Stop criterion, 819 Signal-to-quantization noise ratio, 124 Strict-sense stationary, 63 Signals Stripline, 189 continuous-time, continuous-valued, 87 Strong law of large numbers, 55 continuous-time, discrete-valued, 87 Subrefraction, 243 discrete-time, continuous-valued, 87 Substrate, 189 discrete-time, discrete-valued, 87 Success, 23 Signal-space, 177 Sum-product algorithm, 808 representation, 363–364 Sum of random variables, 35–36 Signal-vectors, 177, 367 Sunde’s BFSK, 479 Sign function, 126, 146 Sunde’s BFSK modulation, 459 Silicon-controlled rectifiers (SCR), 198 Superrefraction, 243 Simulation Surface resistance, 186 of communication channels, 217 Survivor path, 788 Sinc function, 93 Symbol, 176 properties, 93 vs. bit error probability, 396 Sine cardinal, 93 error probability, 362 Single-bit A/D converter, 130 rate, 159 Single-parity check product code (SPC-PC), -timing 810 Single-parity check turbo product codes early-late gate, 565 (SPC-TPC), 817 Synchronization Singular value decomposition (SVD), 724 acquisition, 547 Skin depth, 186 acquisition time, 550 Skin effect, 188 algorithmic approach, 578 Smart antenna, 250 blind, 546 Soft-decision decoding, 737 classical approach, 578 Soft-input, hard-output (SIHO) decoder, 737 discrete-time approach, 578 Soft-input, soft-output (SISO) decoder, 737 independent estimation, 551 Soft-output Viterbi algorithm (SOVA), 807 joint estimation, 551 Software-defined radio (SDR), 122 for other modulations, 564 Sound speed profile (SSP), 253 pilot-aided, 546 Source coding, 172 self-synchronization, 546 theorem, 694 of SS signals Space diversity, 604 acquisition, 621 Space-time coding, 723, 830 acquisition and tracking, 634 Spatial channel models, 251 false alarm, 622 Spectral efficiency, 413 tracking, 621 Spectral regrowth, 511 symbol-timing, 555 Sphere packing bound, 734 tracking, 547 Spreading factor, 633 trained, 546 Spread-spectrum multiple access (SSMA), 601 Syndrome decoding, 744 Index 871

Syndrome vector, 737 Uniform error property, 773 Systematic encoding, 737 Uniform quantization, 123 Uniform random variable, 27 T Union bound, 4, 391, 393 Tanner graph, 808 Unipolar code, 266 Tapped delay line (TDL), 195, 214, 639 Unipolar non-return-to-zero (NRZ-u), 267 Tau-dither loop, 639 Unipolar return-to-zero (RZ-u), 268 Taylor polynomial, 509 Uniquely decodable codes, 695 T-carrier hierarchy, 270 Unit impulse, 91 Temperature inversion, 244 properties, 92 Terrestrial microwave channel, 244 Unit interval, 329 Theorem of irrelevance, 376 Unit-step function, 90 Thermal noise, 79 Unshielded twisted pair (UTP), 185 Thevenin´ circuit, 79 Up-sampling, 122 Time averages, 62 Time diversity, 604 V Time-frequency duality, 681 Variable-length source code, 698 Time hopping, 668 Variance, 19 Timehopping multiple-access, 668 Variance of a random process, 64 Time-hopping ultra wideband (TH-UWB), 668 Variance transfer, 831 Time reversal, 677–678 Vector AWGN channel model, 176 Timing jitter, 323, 327 Vector channel, 178 Total expectance, 67 Vector quantization, 695 Total expectation law, 23 Vector space, 732 Total probability theorem, 9, 16–17 Vector subspace, 732 Training sequence, 635 Venn diagram, 2 Transformations of random variables, 38 Vestigial symmetry, 315 Transform-based code, 695 VisSim/Comm Transition codes, 266 documentation, viii Transition probabilities, 699 instalation, xxv Transmission line, 187, 189 license and support, xxv Transmit diversity, 604, 723 manufacturers, xxv Transmitted reference, 673 Viterbi algorithm, 525, 738, 795 Transversal filter, 143 Trellis-coded modulation (TCM), 795 W Trellis diagram, 738, 788 Wagner decoding, 831 Trellis truncation length, 794–795 Walsh–Hadamard sequences, 624 Trivial codes, 749 Water-filling, 641, 722 Turbo code, 806 Water-pouring, 641, 722 Turbo processing, 807 Waveform channel, 173 Twisted pair, 184 Waveguide, 189–190 Two-dimensional Gaussian random variable, Wavelength division multiplexing (WDM), 34 191 Two-ray model, 201 Weak law of large numbers, 55 Weight, 738 U Weight distribution, 739 Ultra-wideband (UWB), 235, 678 Weight enumerator function (WEF), 737 Unbiased estimator, 50 White noise, 80 Uncertainty, 690 Wide-sense stationary, 63 Uncertainty principle, 366 Wide-sense stationary, uncorrelated scattering Uncorrelated random processes, 66, 70 (WSSUS), 219, 607 Underground radio channel, 255 Wiener-Khintchine relations, 68 Underwater acoustic channel (UWA), 252 WiFi, 585 Unequal error protection, 832 Windowing, 653 872 Index

Wireless channel, 199 Wireless local area network (WLAN), 232 correlation functions, 219 Wireless personal area networks (WPAN), 232 Doppler spread, 215 Wireless underground sensor network flat fading, 211 (WUSN), 256 frequency selective, 214 Wireline channels, 184 large scale propagation, 200 multipath propagation, 207 Z outdoor, 200 Zeroth-order hold, 118, 290 small scale propagation, 207 Ziv-Lempel code, 694 tapped delay line, 214 Z-transform, 113