Fuhrmann, D.R. "Complex Random Variables and Stochastic Processes"
`Digit al S ignal P ro ce s s ing Handbo o k
`Ed. Vijay K. Madisetti and Douglas B. Williams
`Boca Raton: CRC Press LLC, 1999
`
`o1999 by CRC Press LLC
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`
`6
`
`Complex Random Variables and
`Stochastic Processes
`
`60.1
`60.2
`
`Introduction
`Complex Envelope Representations of Real Bandpass
`Stochastic Processes
`Representations of Deterministic Signals' Finite-Energy
`Second-Order Stochastic Processes ' Second-Order Com-
`plex Stochastic Processes ' Complex Representations of
`Finite-Energy Second-Order Stochastic Processes' Finite-
`Power Stochastic Processes' Complex Wide-Sense-Stationary
`Processes . Complex Representations of Real Wide-Sense-
`Stationary Signals
`The Multivariate Complex Gaussian Density Function
`Related Distributions
`Complex Chi-Squared Distribution' Complex F Distribution
`. Complex Beta Distribution . Complex Student-, Distribu-
`tion
`60.5 Conclusion
`References
`
`60.3
`60.4
`
`Daniel R. Fuhrmann
`Washington University
`
`60.1 Introduction
`
`Much of modern digital signal processing is concerned with the extraction of information from signals
`which are noisy, orwhich behave randomly while still revealing some attribute or parameter of a system
`or environment under observation. The term in popular use now for this kind of computation is
`statistical signal processing, and much of this Handbook is devoted to this very subject. Statlstical
`signal processing is classical statistical inference applied to problems ofinterest to electrical engineers,
`with the added twist that answers are often required in "real time", perhaps seconds or less. Thus,
`computational algorithms are often studied hand-in-hand with statistics.
`One thing that separates the phenomena electrical engineers study from that of agronomists,
`economists, or biologists, is that the data they process are very often complex, that is, the data
`points come in pairs of the form x + jy, where x is called the real part, y the imaginary part, and
`j : J=. Complex numbers are entirely a human intellectual creation: there are no complex
`physical measurable quantities such as time, voltage, current, money, employment, crop yield, drug
`efficacy, or anything else. However, it is possible to attribute to physical phenomena an underlying
`mathematical model that associates complex causes with real results. Paradoxically, the introduction
`of a complex-number-based theory can often simplify mathematical models.
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`s(r)
`
`LPF
`
`LPF
`
`2coso¡t
`
`-2sinqt
`
`x(r)
`
`v(t)
`
`FIGURE 60.1: Quadrature demodulator
`
`Beyond their use in the development of analytical models, complex numbers often appear as
`actual data in some information processing systems. For representation and computation purposes,
`a complex number is nothing more than an ordered pair of real numbers. One just mentally attaches
`the "j" to one of the two numbers, then carries out the arithmetic or signal processing that this
`interpretation of the data implies.
`One of the most well-known systems in electrical engineering that generates complex data from
`real measurements is the quadrature, or IQ, demodulator, shown in Fig. 60.1. The theory behind this
`system is as follows. A real bandpass signal, with bandwidth small compared to its center frequency,
`has the form
`s(r) : 41¡¡ cos(a;,/ + QQÐ
`(60.1)
`where ø. is the center frequency, and A(¡) and d(¡) are the amplitude and angle modulation,
`respectively. Byviewing A(t)andó(t) togetherasthepolarcoordinatesforacomplexfunctiong(/),
`i.e.,
`
`(60.2)
`
`(60.3)
`
`g(t): ¡çù.iÓ(t) ,
`we imagine that there is an underlying complex modulation driving the generation of s(l), and thus
`s(r) : Re le7¡ej'"tr.
`Again, s (/) is physically measurable, while g (r) is a mathematical creation. However, the introduction
`of g(t) does much to simplify and unify the theory of bandpass communication. It is often the case
`that information to be transmitted via an electronic communication channel can be mapped directly
`into the maghitude and phase, or the real and imaginary parts, of g(/). Likewise, it is possible to
`demodulate s(l), and thus "retrieve" the complex function g(t) and the information it represents.
`This is the purpose of the quadrature demodulator shown in Fig. 60.1. In Section 60.2 we will
`examine in some detail the operation of this demodulator, but lor now note that it has one real input
`and two real outputs, which are rhfe4pretedas the real and imaginary parts of an information-bearing
`complex signal.
`Any application of statistical inference requires the development of a probabilistic model for the
`received or measured data. This means that we imagine the data to be a "realization" of a multivariate
`random variable, or a stochastic process, which is governed by some underlying probability space of
`which we have incomplete knowledge. Thus, the purpose of this section is to give an introduction to
`probabilistic models for complex data. The topics covered are 2nd-order stochastic processes and their
`complex representations, the multivariate complex Gaussian distribution, and related distributions
`which appear in statistical tests. Special attention will be paid to a particular class of random variables,
`called circular complex random variables. Circularity is a type of symmetry in the distributions of
`the real and imaginary parts of complex random variables and stochastic processes, which can be
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`physically motivated in many applications and is almost always assumed in the statistical signal
`processing literature. Complex representations for signals and the assumption of circularity are
`particularþ useful in the processing of data or signals from an array of sensors, such as radar antennas.
`The reader will flnd them used throughout this chapter of the Handbook.
`
`60.2 Complex Envelope Representations of Real Bandpass
`Stochastic Processes
`
`60.2.1 Representations of Deterministic Signals
`The motivation for using complex numbers to represent real phenomena, such as radar or com-
`munication signals, may be best understood by first considering the complex envelope of a real
`deterministic fl nite-energy signal.
`Let s (¿) be a real signal with a well-defined Fourier transform S(ø). We say that s (¿) is bandlimited
`if the support of S(r¿) is finite, that is,
`S(a.¡) 0 ølB
`+ 0 øeB
`where B is the frequency band of tl:'e signal, usually a finite union of intervals on the ¿.¡-axis such as
`B : l-rt¡2, -r..¡rl U frot, øzl.
`The Fourier transform ofsuch a signal is illustrated in Fig. 60.2.
`
`(60.5)
`
`(60.4)
`
`S(co)
`
`úl
`
`úl
`
`ú)
`
`FIGURE 60.2: Fourier transform of a bandpass signal.
`
`Since s(t) is real, the Fourier transform S(ø) exhibits conjugate symmetry, i.e., S(-ø) : S*(ø).
`This implies that knowledge of S(ø), for ø > 0 only, is sufficient to uniquely identify s(t).
`The complex envelope of s (r), which we denote g (r), is a frequency-shifted version of the complex
`signal whose Fourier transform is S(a.¡) for positive a.¡, and 0 for negative a,l. It is found by the
`operation indicated graphically by the diagram in Fig. 60.3, which could be written
`8(/) : LPF{2 s(t)e-ia"t 1.
`cr.r. isthe center frequency of the band B, and "LPF" represents an ideal lowpass filterwhosebandwidth
`is greater than half the bandwidth of s(l), but much less than 2ar. The Fourier transform of g(t) is
`given by
`
`(60.6)
`
`G(ø)
`
`2S(a - a,)
`lal < BW
`0
`otherwise .
`
`(60.7)
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`61999 by CRC Press LLC
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`

`
`s(t)
`
`x(Ð
`
`LPF
`
`2¿jolt
`
`FIGURE 60.3: Quadrature demodulator
`
`G(o¡)
`
`LPFGpoDs
`
`-
`
`FIGURE 60.4: Fourier transform of the complex representation.
`
`(Ð
`
`The Fourier transform of s(r), for s(r) as given in Fig. 60.2, is shown in Fig. 60.4
`The inverse operation which gives s(l) from g(f ) is
`
`s(l) : Ps1t1t)ei'"t¡ '
`
`(60.8)
`
`Our interest in g(f) stems from the information it represents. Real bandpass processes can be
`written in the form
`
`s(r) : A(r) cos(¿dcr + d(r))
`
`(60.e)
`
`where A(t) and d(t) are slowly varying functions relative to the unmodulated carfier cos(ørl), and
`carry information about the signal source. From the complex envelope representation ( 60.3), we
`know that
`
`s(t) : A(t¡siÓ(tt
`
`(60.10)
`
`and hence g(t), in its polar form, is a direct representation of the information-bearing part of the
`signal.
`In what follows we will outline a basic theory of complex representations for real stochastic pro-
`cesses, instead of the deterministic signals discussed above. We will consider representations of
`second-order stochastic processes, those with ñnite variances and correlations and well-deûned spec-
`tral properties. Tho classes of signals will be treated separately: those with finite energy (such as
`radar signals) and those with finite power (such as radio communication signals).
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`60.2.2 Finite-Enerry Second-Order Stochastic Processes
`Let x(l) be a real, second-order stochastic process, with the defining property
`¿{x2(r)}<oo, allt.
`
`Furthermore, let x(f ) be finite-energy, by which we mean
`læ
`J_*E{x'(t)ldt
`The autocorrelation function for x(t) is defined as
`R*(/r, tù: E{x(tùx(tùl,
`and from (60.1 1) and the Cauchy-Schwartz inequality we know that R* is finite for all t1, t2.
`The bi-frequency eneryy spectral density function is
`
`< æ.
`
`(60.11)
`
`(60.12)
`
`(60.13)
`
`s*(a,r, -r: I:f]**frr,tz)e-iottte+i'2î2dttdt2.
`It is assumed that Sn(¿rlr, a.2) exists and is well defrned. In an advanced treatment of stochastic
`processes (e.g., Loeve [1]) it can be shown that S*(ør, a.r2) exists if and only if the Fourier transform
`of x(r) exists with probability l; in this case, the process is said to be harmonizable.
`If x(t) is the input to a linear time-invariant system H, and y(l) is the output process, as shown in
`Fig.60.5,theny(t)isalsoasecond-orderfinite-energystochasticprocess. Thebi-frequencyenergy
`
`(60.14)
`
`x(r)
`
`v(t)
`
`FIGURE 60.5: LII system with stochastic input and output.
`
`spectral density ofy(/) is
`
`S¡,y(alr, øz): H(aùH*(a2)Su(øt,az) .
`
`(60.15)
`
`This last result aids in a natural interpretation of the function ,S*(rr;, ø), which we denote as the
`energy spectral density. For any process, the total energy .E* is given by
`1 /noo
`2tr -/__
`If we pass x(t) through an ideal frlter whose frequency response is I in the band B and 0 elsewhere,
`then the total energy in the output process is
`lr
`Ev : Zo Jrt*(r,<,t)dø
`
`(60.16)
`
`(60.17)
`
`E*:
`
`t*(r, ø)da .
`
`.
`
`This says that the energy in the stochastic process x(t) can be partitioned into different frequency
`bands, and the energy in each band is found by integrating S*(a;, ø) over the band.
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`We can define a bandpass stochastic process, with band B, as one that passes undistorted through
`an ideal filter H whose frequency response is I within the frequency band and 0 elsewhere. More
`precisely, if x(t) is the input to an ideal filter H, and the output process y(t) is equivalent to x(t) in
`the mean-square sense, that is
`
`El(x(t)-y(r))2Ì:o
`then we say that x(t) is a bandpass process with frequency band equal to the passband of H. This is
`equivalent to saying that the integral of S*(øl , a.¡z) outside of the region @r, @z e B is 0.
`
`allt,
`
`(60.1s)
`
`60.2.3 Second-Order Complex Stochastic Processes
`A complexstochastic process z(t) is one given by
`z(t):aç¡¡+jy(t)
`where the real and imaginary parts, x(l) and y(r), respectively, are any two stochastic processes
`defined on a common probabitity space. A finite-energy, second-order complex stochastic process
`is one in which x(t) and y(l) are both finite-energy, second-order processes, and thus have all the
`properties given above. Furthermore, because the two processes have a joint distribution, we can
`define the cross-correlation function
`
`(60. le)
`
`R,(y(rr, tù: E{x(tt)yçùl,
`By far the most widely used class of second-order complex processes in signal processing is the
`class of circular complex processes. A circular complex stochastic process is one with the following
`two defining properties:
`R*(lt, tz): Rw(h,tz)
`
`(60.20)
`
`(60.21)
`
`and
`
`A"'y(rr, t2) : -RyxQt, tz)
`From Eqs. (60.21) and (60.22) we have that
`E{z(tùz* (tùl :2Ru(h, tù i 2i RwQt, tù
`
`all t1, t2 .
`
`$0.22)
`
`(60.23)
`
`and furthermore
`
`Elz(tùz(tù] :0
`(60.24)
`for all /¡, t2. This implies that all of the joint second-order statistics for the complex process z(t) are
`representedinthefunction
`E{z(tr)z*(tz)r
`(60.25)
`^rr(t1,t2):
`which we define unambiguously as the autocorrelation function for z(t). Likewise, the bi-frequency
`spectral density functionfor z(t) is given by
`
`s,,(at ' rr, : I: f o,,rtr, t2)e-ioú e+iaztz¿¡r¿¡r' (60.26)
`
`The functions R"r(^, tz) and S""(ø1 , øz) exhibit Hermitian symmetry, i.e.,
`R,,(t1, t2) : RLQ2, tù
`
`and
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`e1999 by CRC Press LLC
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`S,,(øt,aù: SL@2,øt).
`
`(60.27)
`
`(60.28)
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`However, there is no requirement that Srr(a.r1 , a.r2) exhibit the conjugate symmetry for positive and
`negative frequencies, given in Eq. (60.6), as is the case for real stochastic processes.
`Other properties of real second-order stochastic processes given above carry over to complex
`processes. Namely, if H is a linear time-invariant system with arbitrary complex impulse response
`å(t), frequency response H (a), and complex input z(r), then the complex output w(r) satisfies
`s**(arl ,øz): H(¿¿t)H"(r¡z)Sr,(at,az).
`A bandpass circular complex stochastic process is one with finite spectral support in some arbitrary
`frequency band B.
`Complex stochastic processes undergo a frequency translation when multiplied by a deterministic
`complex exponential. If z(t) is circular, then
`w(r) : ej'úz(t)
`
`(60.30)
`
`(60'29)
`
`is also circular, and has bi-frequency energy spectral density function
`S*(ør, o,z) : Srr(at - a)c, o)z - r¿r) .
`
`(60.31)
`
`60.2.4 Complex Representations of Finite-Enerry Second-Order
`Stochastic Processes
`Let s(l) be a bandpass finite-energy second-order stochastic process, as defined in Section 60.2.2.
`The complex representation of s(/) is found by the same down-conversion and filtering operation
`described for deterministic signals;
`
`g(r) : LPF{2s(ùe-ia"tl.
`
`(60.32)
`
`The lowpass filter in Eq. (60.32) is an ideal filterthat passes the baseband components of the frequency-
`shifted signal, and attenuates the components centered at frequency -2a.r..
`The inverse operation for Eq. (60.32) is given by
`3(r):Re{g(r)ej'"tr.
`
`(60.33)
`
`Because the operation in Eq. (60.32) involves the integral of a stochastic process, which we define
`using mean-square stochastic convergence, we cannot say that s(r) is identically equal to 3(r) in
`the manner that we do for deterministic signals. However, it can be shown that s(/) and 3(r) are
`equivalent in the mean-square sense, that is,
`E{(s(r)-3(r))2}:0
`
`atlt.
`
`(60.34)
`
`With this interpretation, we say that g(/) is the unique complex envelope representation for s(r).
`The assumption of circularity of the complex representation is widespread in many signal process-
`ing applications. There is an equivalent condition which can be placed on the real bandpass signal
`that guarantees its complex representation has this circularity property. This condition can be found
`indirectly by starting with a circular g(r) and looking at the s(r) which results.
`Let g(r) be an arbitrary lowpass circular complex finite-energy second-order stochastic process.
`The frequency-shifted version of this process is
`pçt) : g()e+io,t
`
`(60.35)
`
`and the real part of this is
`
`o1999 by CRC Press LLC
`
`,t¡l:åtptrl+p*(r))
`
`(60.36)
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`By the defrnition of circularity, p(t) and p*(t) are orthogonal processes (E{p(t)(p*(tz))* : 0}) and
`from this we have
`
`(Soo(rr;r, az) I Sp*o*(a1,a2)
`
`(60.37)
`
`(S*(ør - @c, @2 - a") I si*(-ør - (ùc, -@2 - ar))
`
`I 4 1 4
`
`Ss(øt, arz)
`
`Since g(t) is a baseband signal, the frrst term in Eq. (60.37) has spectral support in the first quadrant
`in the (ø1 , ø2) plane, where both øl and @2 are positive, and the second term has spectral support
`only for both frequencies negative. This situation is illustrated in Fig. 60.6.
`
`{or
`
`q
`
`FIGURE 60.6: Spectral support for bandpass process with circular complex representation.
`
`It has been shown that a necessary condition for s(l) to have a circular complex envelope repre-
`sentation is that it have spectral support only in the first and third quadrants of the (ø1 , ø2) plane.
`ifg(t)isnofcircular,thenthes(t)whichresultsfromtheoperation
`Thisconditionisalsosufficient:
`in Eq. (60.33) will have non-zero spectral components in the second and fourth quadrants of the
`(at, ooz) plane, and this contradicts the mean-square equivalence of s(r) and 3(l).
`An interesting class of processes with spectral support only in the first and third quadrants is the
`class of processes whose autocorrelation function is separable in the following way:
`
`R,"(r1,r2): Rr(rr -tùRz(ry).
`
`For these processes, the bi-frequency energy spectral density separates in a like manner:
`
`s,.(ør, az) : st(at - aùsz (t' :"\.
`-'-\
`2 /
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`61999 by CRC Press LLC
`
`(60.38)
`
`(00.39)
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`
`(ù2
`
`ú)l
`
`rN."
`
`/lilntw$tfN
`;lilIlilililillililltN
`,rrfiüHlllllTllllllilìil"
`lllll!llllillllliilllli¡llilr¡'
`\qffiiilflìlülr,,'
`,,, \lll1,,
`
`FIGURE 60.7: Spectral support for bandpass process with separable autocorrelation.
`
`In fact, 51 is the Fourier transform of R2 and vice versa. If 51 is a lowpass function, and 52 is a
`bandpass function, then the resulting product has spectral support illustrated in Fig. 60.7.
`The assumption of circularity in the complex representation can often be physically motivated.
`For example, in a radar system, if the reflected electromagnetic wave undergoes a phase shift, or if the
`reflector position cannot be resolved to less than a wavelength, or if the reflection is due to a sum of
`reflections at slightly different path lengths, then the absolute phase ofthe return signal is considered
`random and uniformly distributed. Usually it is not the absolute phase of the received signal which
`is of interest; rather, it is the relative phase of the signal value at two different points in time, or of
`two different signals at the same instance in time. In many radar systems, particularly those used for
`direction-of-arrival estimation or delay-Doppler imaging, this relative phase is central to the signal
`processing objective.
`
`60.2.5 Finite-Power Stochastic Processes
`The second major class of second-order processes we wish to consider is the class of finite power
`signals. A finite-power signal x(r) as one whose mean-square value exists, as in Eq. (60.4) , but whose
`total energy, as defined in Eq. (60.12), is infinite. Furthermore, we require that the time-averaged
`mean-square value, given by
`
`lrr
`& : r* n J_rR*(t,t)dt,
`
`(60.40)
`
`exist and be finite. P* is called the power of the process x(r).
`The most commonly invoked stochastic process of this type in communications and signal process-
`ing is the wide-sense-stationaryprocess, one whose autocorrelation function Ro(/r, lz) is a function
`of the time difference t1 - t2. only. In this case, the mean-square value is constant and is equal to
`the average power. Such a process is used to model a communication signal that transmits for a long
`period of time, and for which the beginning and end of transmission are considered unimportant.
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`A wide-sense-stationary (w.s.s.) process may be considered to be the limiting case of a partic-
`ular type of finite-energy process, namely a process with separable autocorrelation as described by
`Eqs. (60.38) and (60.39). if in Eq. (60.38) the function -R2 (ltllz) is equal to a constant, then the pro-
`cess is w.s.s. with second-order properties determined by the function l?r (¡r - t). The bi-frequency
`energy spectral density function is
`
`s*"(ør, øz) :2nô(at - øz)sz (t#)
`
`(60.4r)
`
`where
`
`sz(o): [* or(ùe-iardr.
`J_æ
`This last pair of equations motivates us to describe the second-order properties of x(r) with functions
`of one argument instead of two, namely the autocorrelation function ,Ro (r) and its Fourier transform
`S*(ar), known as the power spectral densÍty. From basic Fourier transform properties we have
`I roo
`P" : ,; J_*S*(ro)dr,t
`If w.s.s. x(t) is the input to a linear time-invariant system with frequency response H@;) and
`output v(/), then it is not difûcult to show that
`l. y(l) is wide-sense-stationary, and
`2. S,'(ar) : lH (@)12 su(r.l).
`These last results, combined with Eq. (60.43), lead to a natural interpretation of the power spectral
`densityfunction. Ifx(t)istheinputtoanidealbandpassfilterwithpassbandB,thenthetotalpower
`of the filter output is
`
`(60.42)
`
`(60.43)
`
`.
`
`,r: * l"t*t )0,
`
`(60.44)
`
`This shows how the total power in the process x(t) can be attributed to components in different
`spectral bands.
`
`60.2.6 Complex Wide-Sense-Stationary Processes
`Two real stochastic processes x(r) and y(t), defined on a common probability space, are said to be
`jointly wide-sense-stationary it
`1. Both x(t) and y(t) are w.s.s., and
`2. The cross-correlation R y(/r, tù : E{x(tt)y(r2)} is a function of t1 - t2 only.
`For jointly w.s.s. processes, the cross-correlation function is normally written with a single argu-
`ment, e.9., Àr(z), with z : tt - tz.From the definition we see that
`R*y(z) : R*(-z).
`
`(60.45)
`
`A complex wide-sense-stationary stochastic process z(r) ís one that can be written
`z(t) : ¡1¡¡ + jv(r)
`where x(r) and y(r) are jointly wide-sense stationary. A circular complex w.s.s. process is one in
`which
`R*(r): Ro(z)
`
`(60.46)
`
`(60.47)
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`and
`
`Rrv(z) : -Ryx(t)
`all z '
`The reader is cautioned not to confuse the meanings of Eqs. (60.45) and (60.48)
`For circular complex w.s.s. processes, it is easy to show that
`E{z(tt)z(tùl :0
`
`(60.48)
`
`(60.4e)
`
`for all 11 , t2, and therefore the function
`
`Rrr(h, tz)
`
`E{z(tt)z*(tùl
`2Ru(tt, tù t 2 j Rv"(tt, tz)
`(60.50) arefunctions
`definesallthesecond-orderpropertiesofz(r). AllthequantitiesinvolvedinEq.
`of r : tt - t2 only, and thus the single-argument function Rrr(t) is defined as the autocorrelation
`function for z(t).
`The power spectral density for z(r) is
`
`(60.50)
`
`s72Q,t) :
`
`fæ
`
`J_*n"{t)e-t" dr .
`Rrr(z) exhibits conjugate symmetry (R,r(r) : ni,Gt)); Sr"(ro) is non-negative but otherwise has
`no symmetry constraints.
`If z(l) is the input to a complex linear time-invariant system with frequency response I/ (ar), then
`the output process w(l) is wide-sense-stationarity with power spectral density
`s*(ø) : lH (@)12 szzkù .
`A bandpass w.s.s. process is one with frnite (possible asymmetric) support in frequency.
`If z(¿) is a circular w.s.s. process, then
`
`(60,51)
`
`(60.52)
`
`w(t) : ei'úz(t)
`
`is also circular, and has power spectral density
`S*(a.¡) - S'r(a-ør)
`
`(60.53)
`
`(60.54)
`
`60.2.7 Complex Representations of Real Wide-Sense-Stationary Signals
`Let s(t) be a real bandpass w.s.s. stochastic process. The complex representation for s(¡) is given by
`the now-familiar expression
`
`g(l) : LpF{2 s(t)e-ia,t)
`
`(60.s5)
`
`with inverse relationship
`
`3(r):Re{g(r)ei'."t¡.
`In Eqs. (60.55) and (60.56) , ar, is the center frequency for the passband of s(/), and the lowpass filter
`has bandwidth greater than that of s(t) but much less than2o4. s(r) and 3(t) are equivalent in the
`mean-square sense, implying that g(/) is the unique complex envelope representation for s(r).
`For arbitrary real w.s.s. s(t), the circularity of the complex representation comes without any
`additional conditions like the ones imposed for finite-energy signals. If w.s.s. s(l) is the input to a
`quadrature demodulator, then the output signals x(f) and y(t) are jointly w.s.s., and the complex
`Process
`g(/) : x(/) + jy(t)
`
`(60.56)
`
`(60.5i)
`
`s1999 by CRC Press LLC
`
`TCL Exhibit 1310
`Page 12 of 19
`
`

`
`is circular. There are various ways of showing this, with the simplest probably being a proof by
`contradiction. IfS(t) is a complex process that is nofcircular, then the process Re{g(r)ejo'r} can be
`shown to have an autocorrelation function with nonzero terms which are a function of tt I tz, and
`thus it cannot be w.s.s.
`Communication signals are often modeled as w.s.s. stochastic processes. The stationarity results
`from the fact that the carrier phase, as seen at the receiver, is unknown and considered random,
`due to lack of knowledge about the transmitter and path length. This in turn leads to a circularity
`assumption on the complex modulation.
`In many communication and surveillance systems, the quadrature demodulator is an actual elec-
`tronic subsystem which generates a pair of signals interpreted directly as a complex representation
`of a bandpass signal. Often these signals are sampled, providing complex digital data for further
`digital signal processing. In array signal processing, there are multiple such receivers, one behind
`each sensor or antenna in a multi-sensor system. Data from an array of receivers is then modeled as
`a vector of complex random variables. In the next section, we consider multivariate distributions for
`such complex data.
`
`60.3 The Multivariate Complex Gaussian Density Function
`
`The discussions of Section 60.2 centered on the second-order (correlation) properties of real and
`complex stochastic processes, but to this point nothing has been said aboutjoint probability distri-
`butions for these processes. In this section, we consider the distribution of samples from a complex
`process in which the real and imaginary parts are Gaussian distributed. The key concept of this
`section is that the assumption of circularity on a complex stochastic process (or any collection of
`complex random variables) leads to a compact form of the density function which can be written
`directly as a function of a complex argument z rather than its real and imaginary parts.
`From a data processing point-of-view, a collection of N complex numbers is simply a collection
`of 2N real numbers, with a certain mathematical significance attached to the N numbers we call
`the "real parts" and the other N numbers we call the "imaginary parts". Likewise, a collection of
`N complex random variables is really just a collection of 2N real random variables with some joint
`distribution in lR2N. Because these random variables have an interpretation as real and imaginary
`parts of some complex numbers, and because the 2N-dimensional distribution may have certain
`symmetries such as those resulting from circularity, it is often natural and intuitive to express joint
`densities and distributions using a notation which makes explicit the complex nature of the quantities
`involved. In this section we develop such a density for the case where the random variables have a
`Gaussian distribution and are samples of a circular complex stochastic process.
`Let z¡ , i : 1 .. N be a collection of complex numbers that we wish to model probabilistically. Write
`zi : xi + jyi
`
`(60.58)
`
`and consider the vector of numbers lxl,yr,..,x¡¡,y¡s]r as a set of 2N random variables with a
`distribution over lR.2N. Suppose further that the vector [x1 , yl, .., xN, y¡¿]r is subject to the usual
`multivariate Gaussian distribution with 2N x I mean vector ¡r, and 2N x 2N covariance matrix R.
`For compactness, denote the entire random vector with the symbol x. The density function is
`
`f*(x): (2r)4(detR)i¿-f5! .
`We seek a way of expressing the density function of Eq. (60.59) directly in terms of the complex
`variable z, i.e., a density of the form f"(z). In so doing it is important to keep in mind what such
`a density represents. fr(z) will be a non-negative real-valued function ,f , CN -+ lR.+, with the
`
`(60.5e)
`
`61999 by CRC Press LLC
`
`TCL Exhibit 1310
`Page 13 of 19
`
`

`
`(60.60)
`
`(60.6r)
`
`(60.62)
`
`property that
`
`IJ
`
`o' 'f'{''ta': t '
`The probability that z e A, where A is some subset of CN, is given by
`r
`P(A) :
`Jo.f,k)az.
`
`The differential element de is understood to be
`dz : d x td y tdx2dy2..dx ¡¡ dy ¡¡
`
`(60.59),
`ThemostgeneralformofthecomplexmultivariateGaussiandensityisinfactgivenbyEq.
`and further simplification requires further assumptions. Circularity of the underlying complex
`process is one such key assumption, and it is now imposed. To keep the following development
`simple, it is assumed that the mean vector ¡r is 0. The results for nonzero ¡.r, are not difficult to obtain
`by extension.
`Consider the four real random variables x¡ , y i , xk, yk. If these numbers represent the samples of a
`circular complex stochastic process, then we can express the 4 x 4 covariance as
`
`"{lilr.,,,*,*1¡ :;[î ü ii!
`tL ;; i
`
`'lir:,
`
`¿:' I "é-
`
`- þi*
`dik
`
`0
`dkk
`
`I
`
`II
`
`I I
`
`I II I
`
`I
`
`aN2 -þxz
`o¿Nr - þut
`þut
`þnz dN2
`(yryl
`The key thing to notice about the matrix in Eq. (60.66) is that, because of its special structure, it
`is completely specified by N2 real quantities: one for each of the 2 x 2 diagonal blocks, and two for
`each of the 2 x 2 upper off-diagonal blocks. This is in contrast to the N(2N * I ) free parameters
`one ûnds in an unconstrained 2N x 2N real Hermitian matrix.
`
`o1999 by CRC Press LLC
`
`(60.63)
`
`(60.64)
`
`(60.65)
`
`(60.66)
`
`dlN
`0y
`
`dzN
`ozJ
`
`- þtx
`
`(Y1¡r/
`
`-þzw
`d2N
`
`II
`Il
`
`c,NN
`0
`
`;
`
`c[NN
`
`I II I I
`
`where
`
`a¡*:28{x¡xtl:2Ely¡ytl
`
`and
`
`þit : -ZÍ.,x¡yrl : -l2E{xd¡) -
`Extending this to the full 2N x 2N covariance matrix R, we have
`dtz -þn
`þn
`dzz
`
`dtt
`0øn
`
`0
`
`c,21
`þzt
`
`-þzt
`(I21
`
`Il
`II
`
`o¿n
`
`0
`
`:Y
`
`Il
`II
`
`R:1
`
`2
`
`TCL Exhibit 1310
`Page 14 of 19
`
`

`
`Consider now the complex random variables z¡ and z¡r. We have that
`
`E[z¡zil : E{(x¡ * jy;)(x¡ - jy¡)}
`: øtx!+yf¡:oii
`
`and
`
`E{z¡2f,}
`
`: E{(x¡ * jy;)(xr - jyù}
`: E{x¡x¡ * y¡yr - jx¡y¡ -l jx¡y¡l
`: a*-liþ*.
`
`(60.67)
`
`(60.68)
`
`Similarly
`
`and
`
`(60.69)
`
`Elz¡zil : aik - jþik
`E{z¡2f,}:dkk.
`(60.70)
`Using Eqs. (60.66) through (60.70), it is possible to write the following N x N complex Hermitian
`matrix:
`
`dll
`
`azt -l iþzt
`
`Elzzwl:
`
`ani iþn
`
`d22
`
`ø-tN * iþt¡'t
`az¡¡ * iþz¡'t
`
`(60.71)
`
`ll I Il I
`
`aut I iþut
`avz'l iþxz
`Note that this complex matrix has exactly the same N2 free parameters as did the 2N x 2N real
`matrix R in Eq. (60.66) , and thus it tells us everything there is to know about the joint distribution
`of the real and imaginary components of z. Under the symmetry constraints imposed on R, we can
`define
`C: E{zzH:l
`and call this matrix the covariance matrix for z. In the O-mean Gaussian case, this matrix parameter
`uniquely identifies the multivariate distribution for z.
`The derivation of the density function fr(z) rests on a set of relationships between the 2N x I real
`vector x, and its N x I complex counterpart z. We say that x and z are isomorphic to one another,
`and denote this with the symbol
`zr=x.
`Likewise we say that the 2N x 2N real matrix R, given in Eq. (60.66) , and the N x N complex matrix
`C, given in Eq. (60.71) are isomorphic to one another, or
`
`dNN
`
`(60.72)
`
`(60.73)
`
`C¡vR.
`
`(60.74)
`
`The development of the complex Gaussian density function fr(z) is based on three claims based
`on these isomorphisms.
`Proposition 1. If z x x, and R ¡v C, then
`*r(zR)* : rHcr.
`
`(00.75)
`
`o1999 by CRC Press LLC
`
`TCL Exhibit 1310
`Page 15 of 19
`
`

`
`Proposition 2. If R æ C, then
`
`Proposition 3. If R æ C, then
`
`lR-l -c-l
`
`4
`
`(60.76)
`
`(60 77)
`
`(60.28)
`
`(60.ig)
`
`detR: ldetCr2 /1\"
`(
`\¡)
`The density function fr(z) is found by substituting the results from Propositions 1 through 3
`directly into the density function /*(x). This is possible because the mapping from z to x is one-to-
`one and onto, and the Jacobian is I [see Eq. (60.62)] . We have
`f,k) : lzz¡*{qdetR)+"-¿+
`: lll-t (2ir)-N(det c¡-t r-z'c-tz .
`\2/
`:
`¡r-N(detc)-l ,-zHc-tz.
`At this point it is straightforward to introduce a non-zero mean p, which is the complex vector
`isomorphic to the mean of the real random vector x. The resulting density is
`.
`f,(z):z-NqdetC¡-l "-(z-rùHc-t(z-u)
`The density function in Eq. (60.80) is commonly referred to as the complex Gaussian density
`function, although in truth one could be more general and have an arbitrary 2N-dimension Gaussian
`distribution on the real and imaginary components of z. It is important to recognize that the use
`of Eq. (60.80) implies those symmetries in the real covariance of x implied by circularity of the
`underlying complex process. This symmetry is expressed by some authors in the equation
`Elzzr\ :0
`where the superscript "T" indicates transposition without complex conjugation. This comes directly
`from Eqs. (60.24) and (60.49).
`For many, the functional form of the complex Gaussian density in Eq. (60.80) is actually simpler
`and cleaner than its N-dimensional real counterpart, due to elimination of the various factors of 2
`which complicate it. This density is the starting point for virtually all of the multivariate analysis of
`complex data see