Bello Measurement of Random Time Variant Linear Channels

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-& NO. 4, JULY 1969 469

Measurement of Random Time-Variant Linear Channels

PHILLIP A. BELLO, SENIOR MEMBER, IEEE

Abstract-This paper concerns the problems of the measurabilityand measurement of random time-variant linear channels. With

regard to measurabilit y, a new, less stringent channel measurabili ty

criteri on is proposed to supersede the BL product int roduced by

Kailath. This criterion involves the area of occupanc y of the Doppler-

delay spread function (or its dual). By using time and bandwidth

constraint s on the input and output of a channel, channel measure-

ment is reduced to the measuremen t of a discrete set of 6nite para-

meters. Optimal measurement techniques are described and their

perfor mances determi ned for both known and unknown channel

correlation functions.

I. INTRODUCTION

'I['

HE MEASUREMENT and characterization of

communicati ons channels have received increasingattention in recent years because of the projected

widespread use of digital communications. Reliable high-

speed digital data transmission requires considerably more

knowledge and equalizati on of channel characteristics

than does conventional highly r edundant analog trans-

mission. This paper is concerned with both the measur-

ability and the measurement of random time-varying

linear channels.

The problem of the measurement of system functions

(input-output relations) of random time-variant channels

differs from the corresponding problem for time-invariant

channels in that even in the absence of noise the random

system function may be unmeasurable. It appears thatthis fact was first pointed out by Kailath [l], who

developed some measurability criteria and introduced

a channel parameter called the “spread factor.” This

parameter is the product of B,,, the maximum rate of

variation of the system in hertz, by L,,,, the maximum

multipath spread of the channel in seconds. For reasons

which will become clear subsequently, we call the product

B,,L,,, the “rectangular” spread factor of the channel

and denote it by S,. Kailath demonstrated that the sys-

tem functions of a linear channel cannot be measured if

S, > 1 and if no further information than B,,, L,, is

known about the channel. Unfortunately the criterion

S, < 1 has been uncritically accepted subsequently asthe channel measurability criterion for random time-

varying linear channels, without paying sufficiently careful

attention to the conditions under which it was derived.

In this paper we shall show that the criterion S, < 1 is

Manuscript received October 22, 1968; revised February 5, 1969.This research was supported by the U. S. Navy Underwater SoundLaboratory, Fort Trumbull, Conn., under Contract N0014C -68-G0051.

The author i s with Signatr on, Inc., Lexington, Mass. 02173.

not the proper channel measurability criterion, and weshall propose a new criterion, S, < 1, where the param-

eter SA is called t he area spread factor of the channel.

Since it will be shown that SA 5 SR, channels which

hitherto were thought unmeasurable are actually mea-

surable.

In defense of Kailath’s result it is worth pointing out

that our new criterion does not necessaril y have to be

regarded as in conflict with the measureability criteria

S, < 1 since we assume somewhat more knowledge of

the fading dispersive characteristics of the channel to be

measured than the gross parameters B,,, and L,,,.

Kailath was very careful to point out that additional

channel knowledge would generally allow exact channelmeasurement even though BmSxLmsx> 1. However, there

is a moot phil osophical point here: if the channel is

wide-sense stationary (WSS), then not only B,,,, L,,,,

but also the additional gross channel information needed

in the present development can be determined without

regard to the size of the product BmBxLmBx. n the other

hand, if the channel is too nonstationary, it is not clear

that B,,, and L,,, can be meaningfull y defined. Thus,

in those cases where B,,, and L,,, can be meaningfully

defined and measured, B,,L,,, < 1 is too stringent a

criterion to use in deciding whether the system functions

of a channel can be measured.

The new channel measurement criterion and optimal

measurement procedures developed here are based’ upon

the input-output representation called the Doppler-delay

spread function [2]. Discrete representations of the channel

are used, corresponding to simultaneous input-time and

output-bandwidth constraints [a]. The Doppler-delay

spread function V(V, E) provides a phenomenological

model of the channel as a continuum of differential scat-

ters subjecting the transmitted signal to a complex gain

V(V, .$) dv do for scatterers providing delays and Doppler

shifts i n the region ([, [ + d[) X (v, v + dv). We define

the region of the v, 2:plane over which V(v, 5) is effectively

nonzero as the delay-Doppler occupancy pattern or simply

the occupancy pattern of the channel. We have sketched

four possible delay-Doppler occupancy patterns i n Fig. 1

with the parameters B,,, and L,, explicitly indicated.

The shaded area of each figure, i.e., the area of the OC-

cupancy pattern, is the new spread factor Sa. Examination

of these figures show that only f or the rectangular shape

of Fig. l(c) is SA = S,. The rectangular spread factor

1 An equivalent development exists for the dual [3] system f unc-tion, the delay-Doppler spread function, U(& Y).



-hl,-

(b)

-----Am-I

(t) Cd)

Fig. 1. Examples of possible delay-Doppler occupancy patterns.

is the area of the smallest rectangle with sides parallel to

the & v axes which just encloses the occupancy pattern.

Fig. 2. Cannonical channel model for input-time output-frequencyconskaint.

Because the channel representation is discrete, the

problem of channel measurement can be reduced to esti-

mation of a set of parameters. We consider two dif ferent

approaches. In the first approach we assume the channel

parameter vector to be an unknown nonrandom vector;

in the second we assume it to be a random vector selected

from an ensemble with known correlation matrix. In the

former case the classical least-squares approach is di-

rectly applicable and has been used first by Levin [4] to

estimate the impulse response of time-invariant filt ers.

In the latter case we enter the domain of estimation theory

originally made popular to engineers by Wiener [5]. For

both cases we determine the structure of the opti mum

estimators and their performance. For simplici ty we

assume the interfering additive noise to be white, but theresults are readily extended to colored noise.

In [2] (Section VI-A-c) it is demonstrated that if

input to a channel is confined by a time gate to the ti

interval 0 < t < T and t’he output spectrum is confinby a bandpass filter to the frequency interval -$W

t < SW,” then in place of the actual channel with Dopple

delay spread function V(V, {) one may use a channel wh

Doppler-delay spread function P(v, .$) has the singu

form

where 6( ) is the unit impulse,

The work reported here attacks a problem area similar

to that studied by Bar-David [6], who considered the

direct estimation of V(v, .$), primarily for radar targets.

In the case of most physical communication channels

V(v, E) has the character of nonstationary white noise in

the E and v variables, and the direct measurement of

V(v, . J does not appear to be useful. However, our use of

input-time and output-bandwidth constraints leads to

discrete channel models whose parameters are finite and

whose measurements are meaningful.

v,, = 11 exp [$rTt,m)] sine [T(v - y)]

,sinc [W({ - k)]V(v, 0 dvdt

and

sin 7rxsine 5 = p.

TX

Examination of (3) shows that { V,, ] are essential ly tw

dimensional sampled versions of the original Dopple

delay spread function, the sampling taking place w

“pulses” of width of the order of l/T in the v direction

l/W in the W direction.

II. CHANNEL REPIZESENTATION

There exist many different system functions for repre-

senting the input-output relations of time varying linear

channels [l], [a], [7]. For the development in the present

paper we need only the input-output relation correspond-

ing to the Doppler-delay spread function [a], i.e.,

w(t) = I/ 4t - 0 exp BWt - t)lVb, t> dv dt (1)

where x(t) is the input and w(t) i s the output.’

The discrete channel model, which consists of a finset of delays and Doppler shifts, is shown in Fig. 2, wh

we have used the notation

rect (x) = 1; /Xl < l/2

LO; 1x1L l/2

If we define the input to the band-limiting filter

w,(t) and the gated channel input of duration T as x

then (2) tells us that the following discrete channel r

tionship applies:

2 In this paper we use complex envelope notation throughout

without explicitl y indicating so in the text. Thus z(t) and w(t) are

3 The analysis is carried out i n this paper with these time

the complex envelopes of the actual real input and output processes.frequency intervals to simplif y the notation. The extension to ointervals is straightforward.

IEEE TRANSACTIONS ON INFORMATION THEORY, JULY 1

SHIFT +ps SHIFT; cps

t

---

vm,n-I I

b-i



BELL0 : MEASUREMENT OF RANDOM TIME-VARIANT LINEAR CHANNELS 471

w(t) = F &‘z,(t - k) exp [jar F (t - $)]vmn. (6)where o( is a number accounting for edge effects and be-

coming negligible compared with TW as T and W become

Since the input signal i s time gated and thus not band-

limited, w,(t) cannot be bandlimit ed in the strict sense of

the word. However, since we assume that both W and the

bandwidth of z(t) greatly exceed l/T, the bandwidth

broadening caused by the time gating has no practical

significance. We then assume that the bandwidth W, of

wl(t) equals the sum of the bandwidth W, of x(t) and the

maximum Doppler spreading of the channel. It is then

convenient to assume W = W, so that the output filter

in Fig. 2 can be removed. The input-output relation (7)

can then be changed to

w(t) = G F rect (“- “‘1: - T’2)z(t - $)

.exp [j27r; (t - &)]Vms. (7) Or

large.

If the multipath spread of the channel is L,,,, and Wis taken as the bandwidth of the received process, the

duration of the output signal is T + L,,,, and the number

of linearly independent complex samples is approximately

W( T + L,,,). Assuming W, 1’ large, we can express the

number of independent observations

N O”t,= WT+P (11)

In the subsequent discussion we shall present some

illustrative examples in which the WSSUS (wide-sense-stationary uncorrelated-scattering) channel is used. For

this channel [2]

v*bJ, E)V(P, 7) = S(E, VI S(?l - 0 6 - VI (8)

where S($, r~) is the scattering function of the channel.

It is shown in [2] that

v;,v,, czi

O;m f r,

n + s (9)

d(n/W, m/T)/TW; m = r, n=s

when the scattering function varies very little for changes

in $ of the order of l/W and changes in v of the order of

l/T. Thus for the WSSUS channel and a sufficientlysmooth scattering function, the gains of the discrete point

LLscatterers” become uncorrelated, and the strength of the

reflection from a particular scatterer becomes proportional

to the amplitude of the scattering function at the same

value of delay and Doppler shift.

III. NECESSITY OF AREA SPREAD-FACTOR CRITERION

In this section we shall demonstrate the necessity of

the area spread-factor criterion. For T >> L,,, and W >>

B,,, the number of V,, coeffici ents significantly different

from zero will be determined by how many rectangles of

dimension l/TW can be fit into the delay-Doppler

occupancy pattern, i.e., into regions of the v, .$plane over

which V(Y, E) is significantly different from zero.4 We have

already illustrated such regions in Fig. 1 and defined the

area of one such region to be S,. The number of V,,

coeffi cients of significant amplitude can be expressed as

Neh = Sa(TW + 4 (10)

.4 The condltlons T >> L,,,, W > B,,, were chosen to allow the

edge effects in two-dimensional sampling to be made negligible.However, the analysis can be extended to smaller values of T, Wif desired.

where ,B s a number accounting for edge effects (including

samples over L,,,) and becoming negligible compared with

TW as T, W becomes large.

Clearly, in order to be able to solve for t he Nch unknowns

s IwT+L1*-WTfa

(13)

for TW >> 1, which demonstrates the necessity of the

area spread-factor measurability criterion.

IV. SUFFICIEXCY OF ARES SPREAD- FACTOR CRITERION

In order to demonstrate the suffici ency of the area

spread-factor criterion, it is necessary to demonstrate a

procedure whereby the set {V,,) can be determined if

As* 5 1.

For convenience, and with no loss in generality, we

assume that the delay-Doppler occupancy pattern of the

channel is bounded by a rectangle whose lower l eft-hand

coordinates are at zero delay and Doppler shift. The maxi-

mum Doppler and multipath spreads can be represented

as

B,,, z M/T (14)

Lax z N/W (15)

where M, N are integers.

In the discussion t o follow we adopt the notation:

wz =w(-;+;) (16)

(17)

PETW (18)

If the output is sampled at the Nyquist rate, the result-

ing set of samples { 20~ provides the minimum number

of parameters needed to represent the received signal.

In terms of the foregoing notation

wl = 2 5 zcen exp [jmS(Z - n)]Vmn.n=O n=o

(20)

If t he input waveform is known, (20) provides a set of

P + N + 1 linear equati ons i n the unknown channel gain



472 IEEE TRANSACTIONS ON INFORMATION THEORY, JULY

in which the integral representation follows from an

plication of the sampling theorem.

One way to make the matrix E nonsingular is to f

the off-diagonal terms to be sufficiently small compa

with the diagonal terms. The diagonal terms are all eq

coefficients ( V,, 1. The number of V,, is approximately

TWX,.

Equation (20) may be expressed in the form

w = zv cm

where the matrix 2, parti tioned into column vectors, has

the form

2 = &o IZlOl . . . IZiwol &l I411 .-- IZMll **-

l&N I &Nl . . . P&v (22)

and the transposes of the column vectors wT, ZE,, and

VT are given by

wT = (wo, w, WA, * * *, WlJ+AJ (23)

n zeros

zf, = (0 0 . . . oxox,ei~B . . . zDefnBp . . zpei”Bpo 0 . . 0) (24)

VT = (v,,v,,~~~v,,v,,v,,~~~v,,~~~v,,v,,~~~v,,).

(25)

If none of the IV,,} (0 g m 5 M, 0 5 n 5 N) arezero, the occupancy pattern has the rectangular shape of

Fig. 1 (c), and the area spread factor is the same as Kai-

lath’s spread factor. When some of the (V,,) are known

to be zero, the matrix 2 can be simplified by omitt ing

columns, the column Z,, being omitted if V,, is zero.

In order to prove the sufficiency of the area spread-

factor criterion, it is necessary to find at least one trans-

mitted signal, i.e., at least one vector {x~, x1, . . . , x,],

for which the set of equations (21) is consistent. These

equations will be consistent if the vectors {Z,,) constitute

a linearly independent set. A necessary and sufficient

condition for a set of vectors to be linearly independentis that the Gramian of these vectors does not vanish.

The Gramian is the determinant of the matrix E given by

E = Z*‘Z. (26)

The matrix E is Hermitian symmetric. It is also non-

negative definite since, if X is a column vector

X*TEX = Z*X*TZX 2 0 . (27)

We now examine the structure of the matrix E. Consider

the typical term in the matrix

c mn,r* = z;,’ z,,. cw

From (24) we note that Z,, differs from Z,, , in a time shift

of n units (n/W seconds) and a frequency shift of m units

(m/T Hz). The inner product (28) is thus found to dif-

fer from the ambiguity function of Z,, only by a phase

factor, i.e.,

c mn.vs= exp [ - j(s - n)r0]x(

+ , r?>

(2%

where

( >1

x W’T = 8 x%~-, exp [jpZ0]

= W LT2*(t)2 (t - +) exp [I?] dt (30)

C =rnR,rnl c (x,12 = W lT liz( dt.

If one could find a time function of effective bandwid

W - c (with E + 0) and time duration T whose ambigu

function could be made sufliciently small at the se

lattice points (n/W, m/T) (excluding (m = 0, n = 0

course), then apart from a constant factor E would

nearly an identity matrix. Although there are undou

edly simpler ways by which to make E nonsingular, th

are sound reasons for constructing the E matrix with s

off -diagonal terms. In the following section it will be sh

that such a choice minimizes the variance of the meas

ment error in the presence of noise and greatly simpli

the structure of the optimum estimator.

It should be recognized that the ambiguity function signal with time duration T and bandwidth W cannot

forced to be identically zero at the complete set of lat

points since

When {x,) is chosen to be a properly truncated m

imal-length shift-register sequence of fl, it is rea

shown from the ambiguity function of this sequence

that

IG, 781m TW-‘/2

C p # r, q # s, TW >> 1.m7t,mn

From (32) we see that this is the smallest level that

ambiguity function can have if all the side lobes are

formly small. If only a small fraction are to be m

uniformly small, as in the present case, and the ot

are unconstrained, it should be possible to make t

considerably smaller than (TW)-I”. Some results of P

and Hofstetter [lo] are suggestive here. Their work

gests that it should be possible to clear away a symme

convex region of low level around the origin of the

biguity function as long as this area is less than four. S

we deal with more general areas, their results would

to be generalized to be made applicable. Equation states that the off-diagonal terms of E can be mad

small as desired relative to the diagonal terms. Un

tunately, since the order of the matrix E is TWSA, w

S, is the area spread factor, the reduction in size of

off-diagonal terms is accompanied by an increase in o

of the matrix E, and the well -known techniques

determining matrix singulari ty by inspecting the elem

of a matrix, such as those dealing with diagonally d

nant matrices [l l], cannot be applied.

There is a simple physical argument for asserting

when S, > 1 there must exist transmitted seque

lx,) for which E is invertible. E will be invertible if positive definite, i.e., if the quadratic form in (27) is



BELL0 : MEASUREMENT OF RANDOM TIME-VARL4NT LINEAR CHANNE LS

tive no matter what vector X is chosen. But this quadratic

form Z*P’ZX is just the total energy at the output of

the channel when the input is Z and the channel gains

are X. Since Z is of length TW and X of length SaTW,

when SA < 1 the dimensionality of X is less than that of

2. Although degenerate forms of Z could be chosen so

that X could be adjusted to force the output to zero for

all time, it is intuitively clear that there must be infinite

sets of suitably chosen Z for which no X can force the

output to zero.

Assuming that E is invertible, we can solve for V with

the aid of E-l by noting that if (21) is pre-multipli ed by

Z** and then by E-’ there results

V = E-lz*TW. (34)

V. MEASUREMENTINTHEPRESENCEOFNOISE:UNKNOWN

CHANNEL CORRELATION FUNCTIONS

We consider now the problem of measuring the coef-

ficients (V,) in the presence of noise. The Zth observed

signal sample becomes

w1 = 2 2 Z1-, exp [jme(Z - n>lVmn + n,,n=O n=O

1 = O,l, 2, *.. , P + N + 1 (35)

or in vector notation

w=ZVfiv (36)

where n, are the noise samples and N is the corresponding

noise vector.

The method of least squares selects as estimates of

{V,,] those parameters { p,,j which make the vector

w = zp (37)

as close as possible t o the observed vector in the sense of

minimizing the energy of the difference vector, 8 - w.

These estimates satisfy the equations

z**zp = z**w (3%

or in terms of the E matrix

P = E-lPTw. (39)

If, by proper selection of the transmitted signal, E

can be made nearly a unit matrix, i.e.,

EwI (40)

where we have normalized so that

,. l%J1217 (41)

then the estimate becomes

P = i?*w. (42)

An examination of the right-hand side of (42) reveals

that when E is an identity matrix, the estimator is real-

izable as a matched-filter bank. Thus, from (42) and (46)

the typical estimate p,, is given by

473

8,, = 225 = C wkzL exp [- jm(k - n) 191

= w 1 w(t)? *f - 5)

.exp[--ij((t-&)I&. (43)

In words, P,,,, is estimated by passing the received wave-

form t hrough a filter matched to a time- and frequency-

shifted version of x(t), where the time shift is n/W seconds

and the frequency shift i s m/T Hz. If E cannot be re-

garded as a unit matrix, the above matched-filter bank

must be followed by the linear operation E-l.

Aside from ease of instrumentation, there is another

reason for trying to make E a unit matrix; for a given

transmitted signal energy and white noi se the estimation

error variance is minimized if E can be made a unit matrix.

The proof is essentially identical to Levin’s proof [4] for

the time-invariant channel. For white noise the covari-

ante matrix of the estimate vector P is given by

cov P = (fV*‘) - (?)(f*‘) = (r,fE-’ (44)

where

Levin [4] shows that if the diagonal terms of a positive

definite matrix A are 1, the diagonal elements of the

inverse matrix A-’ will reach the mini mum values of 1 if

and only if A is a unit matrix. Since the diagonal terms of

cov P are just the error variances, these will be mini-

mized if E can be made into a unit matrix. Thus the mini-

mum estimation error is

e2 = c c ( IG,, - G,,12 > 2 u: TWSA. (46)WI n

For the convenience of the engineer we now compute an

input and output SNR for a WSSUS channel. As our

output SNR we may take the ratio

=(TW)2SAa: ’

(47)

where S(.$, v) is the scattering function and we have

made use of (9).

As the received signal power we take the ratio of re-

ceived signal energy to the time interval T. Then if the

noise power is computed in the observation bandwidth W

we find that the input SNR is given by

(48)



474 IEEE TRANSACTIONS ON INFORMt\TION THEORY, JULY 19

From the general results of [S], the error in our spec

case is

6’ = 2 Tr (&A-’n + El-l (5

where Tr A is the trace of the matrix A (the sum of t

diagonal terms).

It appears intuitively obvious, and it can be prove

that the measurement error will decrease when st,atistic

knowledge is made available. To illustrate this point the simplest situation we consider the case where E m

be taken as the identity matrix. Then

and the ratio of output to input SNR bounded by

1P&s;-X.4

(4%Pin

where S, is the area spread factor. The equality obtains

in (49) when E is an identity matrix.

VI. MEASUREMENT IN THE PRESEXCE OF NOISE: KNOWN

CHAXNEL-CORRELATIOPT FUSCTIONSThe previous section was concerned with optimal

channel measurement for unknown channel-correlation

function. On the basis of experimental evidence and

theoretical considerations, it appears reasonable to postu-

late a quasi-WSSUS channel model for many t ime variant

channels. In this section we shall assume such a channel

model and white Gaussian noise and determine a mini-

mum-mean-squared-error linear channel estimator. The

approach used is called Bayes estimation and can be found

extensively in the literature. A recent reference is Balak-

rishnan [S]. To describe the optimum estimator we must

define the moment matrix of the { V,, 1,A = (vv*‘). (50)

For WT large we may use (9), which makes the matrix

A diagonal with the typical term on the diagonal given by

(1 v nln~) = s(nIW), (m/T).TW

For white noise the optimal estimate is given by

t = AZ*T{ZAZ*’ + u:zj-‘w. (52)

The inversion in (52) can always be performed since

the unit matrix Z is positive definite; and the sum of twononnegative definite matrices is positive definite if either

is positive definite. By factoring matrices (52) can be

expressed in the alternative form when A is nonsingular,

v = [o:A-’ + El-‘Z**w. (53)

A comparison of (53) with (39) reveals that the least-

squares estimate assuming the channel unknown d.ff ers

from the minimum-mean-squared-error estimate assum-

ing the channel Gaussian, only in a replacement of the

matrix E in the former case, by E + aih-‘. The matrix

a:A-’ is diagonal for TW sufficiently large, and from (47)

and (49) the typical term is proportional to the areaspread factor divided by the input SNR. At large input

SNR or very low spread factor, the two estimates (39)

and (53) become identical, while at sufficiently low input

SNR the matrix E can be ignored, and the estimator be-

comes approximately the matched-fi lter bank previously

discussed. From (53) it is readily seen that when E may

be regarded as diagonal, the receiver is a matched-fi lter

receiver at all SNR, with weightings dependent upon

the strength of each path.

The estimator (52) and (53) minimizes the error term

E2 = c c (lPmn - KJ). (54)

(5

which by comparison with (46) is seen to be less than t

error when the channel scatteri ng function is unknown

The estimator discussed in this section differs fro

that of the previous section in that it distorts, i.e., t

“signal” component of the estimator output differs fro

the channel complex-gain vector by a linear transforma

tion. As a consequence t here are actually two types

errors, linear distort ion and additive noise. The measquared error t2 in (55) is infl uenced by both types

errors. From (53) and (36) it is readily seen that the sign

component N of the estimator output is given by

H = Gv = [n:A-’ + El-‘EV (5

where & denotes an average over the additive noise onl

It is clear that the additive-noise component M of t

estimator output is given by

M = (o;A-’ + E}-‘Z*‘N. (5

An output signal-to-noise ratio based upon the ratio

the strength of H to the strength of M can le: d to phy

ically meaningless results because H and L’ are related

Hm = %,1/-,, + c C’ Prn?wSVIS (5T s

where CL,, and Prnnvs are constants and the prime

CT c: denotes exclusion of the mn term. Ideally (

would equal 1 and the pm,,, = 0. In practice, howeve

the amplitude factor o(,, < 1 and the interference ter

c 2, PmnrSVrSwill be present. When the gain coef

cients V,, are uncorrelated, the second term in (59) c

safely be regarded as “noise” and the output sign

strength identified as

When the double sum in (59) is correlated with the fir

term, the signal term must be identif ied as the sum of t

first term and that portion of the double sum correlate

with the first term.

When E is the identity matrix and the V,, are u

correlated,

H = [u;A-’ + I]-‘V (6

where the matrix &fA-’ is diagonal. In particular,

Hnm ($“+)-’ Vm (6



BELL0 : MEASUREMENT OF RANDOM TIME-VARIANT LINEAR CHANNELS 475

so that in this simple case there is no uncorrelated part

and H is “all” signal. The signal output and noise powers

are then

Since the operation Q-‘[ .]Q is a similarity transformation,

it does not change the trace of the matrix. Thus

(H*TH)= c c ( /Vmnj2 $$+]-’ (63)mn

(M* TM) = Tr[(MM*T)]

= Tr[ { crib-’ + I] -“gz]

(64)

When all gain coefficients have the same strength,

wJ> = WI”>> (65)

it is readily found that

( 1vr > _ $2Pout = -z -

un A

which is the same as when t he scattering function is un-

known and E = Z [see (49)].

Although in some cases knowledge of the scattering

function may provide little benefit when E is “close” to

an identity matrix, one can show that when E is nearly

singular, the measurement error for unknown channel

statistics becomes very much larger than when the sta-

tistics are known. In particular, suppose that E is singular

so that the measurement technique of the previous sec-

tion fails. We now determine the performance of the

present technique. To simplify the computation we shall

confine our attention to the computation of the mean

squared error E’ at larger input SNR.

Let us suppose that all paths are of equal strength, so

that

A = (pq”)Z. (67)

The matrix E, being a nonnegative definite Hermitian

symmetric matrix, can be factored in the form

E = Q*%Q (68)

where Q is a unitary matrix

Q*’ = Q-’(69)

and 3, is a nonnegative diagonal matrix whose entries are

the eigenvalues of E.

Using (67))(69) in (55)

c2 = Tr (Q-‘[(IV\“)-‘Z + r,“k]Q]-‘. (70)

where X, is the lcth eigenvalue of E. As pi,/Sa becomes

large

lim c2 + ( /G/’ ).O as ” + mA

(72)

where 0 is the number of zero eigenvalues. Since the

percentage measurement error is given by

?J 2/c c ( IlJmn12= & ( PI2> (73)

we note that for E singular and uncorrelated V,,

11= O/TWX, (74)

which will be small if the number of zero eigenvalues i s

small compared with the length of the vector I’.

ACKNOWLEDGMENT

The author would like to thank A. Ellinthorpe of the

U. S. Navy Underwater Sound Laboratory for pointing

out the need for clarification of the channel measure-

ability problem.

Ill

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REFEREXCES

T. Kailath, “Sampli ng model s for linear time-vari ant filters,”M. I. T. Research Lab. of Electronics, Cambridge, Mass., Rept.;52$;Y ?,5, 19ng.

Characterizati on of randomly time-variant linearchannels: ” IEEE ‘I’rans. Communications Systems, vol. CS-11,pp. 360-393, D ecember 1963.

-7 “Time-frequency duality,” IEEE Trans. InformationTheory, vol. IT-IO, pp. 18-33, January 1964.RI. J. Levin, “Optimum estimation of impulse response inthe presence of noise,” IRE Trans. Circuit Theory, vol. CT-7,pp. 50-56, March 1960.N. Wiener, Extrapolation, Interpolation, and Smoothing ofStationary Time Series. New York: Wiley, 1949.I. Bar-David, “Esti mation of linear weighting functions inGaiissi an noise,” IEEE Trans. Informati on Theory, pp. 395-407, May 1968.A. Gersho, “Characterization of time-varying linear systems,”Proc. IEEE (Correspondence), vol. 51, p. 238, January 1963.A. V. Balakri shnan, “Filter mg and prediction t,heory,” inCommunication Theory, A. T’. Balakrishnan, Ed. New York:McGraw-Hill , 1968, pp. 88-159.R. M. Lerner, ‘Si gnals with uniform ambi guity functions,”1958 IRE Natl. Conv. Rec., pt. 4, pp. 27-36.R. Price and E. M. Hofstetter,. “Bounds on the volume andheight distribxit ions of ambiguity functions,” IEEE Trans.

Informati on Theor? , vol. IT-11, pp. 207-214, Apri l 1965.R. S. Varga, Matrzx Iterative Anulysis. Englewood Cliffs, N. J.:Prentice-Hall, 1962, p. 23.

Bello Measurement of Random Time Variant Linear Channels

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