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Chapter 12 Detection and Parameter In this chapter, we are concerned with optimal receiver structures and near- optimal algorithms for data detection and sync parameter estimation on bot

Chapter 12 Detection and Parameter

In this chapter, we are concerned with optimal receiver structures and near- optimal algorithms for data detection and sync parameter estimation on both flat and frequency-selective fading channels Emphasis is placed on the concept of synchronized detection where the sync parameters are explicitly estimated and then used for detection Based on this mathematical framework, optimal estimator- detector receivers for joint detection and synchronization are derived This chapter focuses on the methodology and fundamental insights rather than on details of implementation; these are addressed in the following chapters on realizable receiver structures and fading channel estimation

12.1 Fading Channel Transmission Models

and Synchronization Parameters

In this section, the transmission models and synchronization parameters of interest are briefly reviewed and put in a mathematical framework suitable for detection and synchronization Based on these models, optimal joint detection and sync parameter estimation strategies and algorithms are systematically derived Even though the optimal algorithms as well as most of their simplified versions admittedly suffer from an extremely high complexity, the ideas, strategies, and valuable insights worked out here provide a universal framework that can be applied to the systematic development of both the data-aided (DA) and non-data- aided (NDA) synchronization algorithms covered in Chapters 14 and 15

As before, we are concerned with linearly modulated, possibly encoded QAM or PSK symbol sequences Although of practical importance, nonlinear modulations such as MSK (Section 3.2) are not discussed here, but the basic principles of deriving optimal detectors and synchronizers remain the same

As detailed in Chapter 11 [eqs (11-l@, (ll-19), (ll-20), and (ll-22),

631

Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing

Heinrich Meyr, Marc Moeneclaey, Stefan A Fechtel Copyright  1998 John Wiley & Sons, Inc Print ISBN 0-471-50275-8 Online ISBN 0-471-20057-3

632 Detection and Parameter Synchronization on Fading Channels

respectively], the following transmission models are of interest:

+ ng’

Flat fading, pulse MF output, imperfect timing: (12-1)

Flat fading, pulse MF output, perfect timing:

with noise autocorrelation functions [eq (ll-21)] l&?(n) = NaS,, (AWGN) and I&?(n) = Nob,, Rm(0),m(l)(n) = Nog(r = [n+0.5]2’)

We consider the transmission of an isolated block of N symbols

starting at reference time index p = 0, where p may be unknown at the receiver The sequence length N may range from a small number (short data packet) up

to near infinity (quasi-continuous transmission) Some of the symbols al, may

be known (training); the unknown (random data) symbols may be uncoded or coded and possibly interleaved Depending on the channel transmission model and the availability of prior knowledge, the sync parameters 8 of interest may encompass the start-of-frame index p, the (relative) frequency offset s2’, the set h

of time-variant selective channel impulse response vectors or the set c of flat fading channel weights, the timing offset E, and carrier phase cp Since the channel is band-limited, the (partial) channel impulse response (CIR) vector h(,‘) as defined in

eq (1 l-59) comprises, in theory, infinitely many samples hz!k The same applies

to the sampled pulse shaping filter response $T,n (i) (E) and the concatenation gg )( e)

of pulse shaping and matched filters

While it is important to represent weak taps (channel pre- and postcursors) in the channel simulator (Section 11.3), the derivation of manageable detection and synchronization algorithms most often requires that the number of sync parameters

to be considered is kept at a minimum In this vein, the CIR model used for the

12.1 Fading Channel Transmission Models and Synchronization Parameters 633 derivation of receiver algorithms must be truncated to some finite length L (symbol intervals) so that the (partial) CIR vector of interest becomes

h(ki) = ( h(d:; h(;:i ha,,)’ , I , (12-3) The T/2-spaced CIR vector hk may be expressed in a short-hand notation as the

“sum” (@) of the two partial CIR vectors:

hk = h(,O) $ h(,‘)

= ( hO;k hl;h h2;k h3;k - haL;lc h&1$ > (12-4)

The set of all CIR vectors hk (Ic = 0, 1, , N- 1, ) may then be collected in a vector h encompassing all channel tap processes that are relevant for the detection

of sequence a Likewise, the (single) weight process trajectory of a flat fading channel can be collected in a vector:

Table 12-l Synchronization Parameters for Flat and Selective Channels

I Selective fading 1 p, i-2, h = (h;f h? hs-, .)T

The parameters p, a’, E, cp are assumed to be invariant over the block length

N Notice that the dimensionality of the sync parameter vector 8 may be very large; for instance, in the case of a selective fading channel, at least (N-1)2(,5+1) channel taps [2( L+l) channel tap processes of duration 2 N - 1 symbol intervals] are to be estimated

A suficient statistic for detection and synchronization is given by the collec- tion of all received samples that are dependent on the data a Since the channel

634 Detection and Parameter Synchronization on Fading Channels

memory is assumed to be finite,, the true sufficient statistic, i.e., the infinite-length received sequence {$k , , r(&}, can also be truncated to a finite-length vector: Receiver input signal: (flat fading: L = 0)

in eq (12-6)] is unknown but bounded within a finite interval -PO 5 ~1 5 1~0 about zero, additional samples of the received signal have to be included into the

(i> (i>

observation vector which then ranges from rBc10 to ~~~+(~+~)-r (receiver input signal) At any rate, the observation may be expressed concisely as a function of the data sequence a and the sync parameters 0, plus additive random noise n or m:

r = &(a, 0) + n

z = f&,e)+m

(receiver input) (pulse MF output) (12-7)

The observation is linearly dependent on the data a but may be nonlinearly dependent on some sync parameters 6

To further illustrate the construction of transmission models, we will now take a closer look at some important special cases of vector-matrix formulations

of the observation

12.1.1 Flat Fading Channel Transmission

As a first example, consider the (relatively simple) case of fkzt fading channel transmission with pulse matched filtering and perfect timing error compensation, with known start-of-frame instant p = 0 but (possibly) unknown frequency Q’ and unknown flat fading process vector c Via eqs (12-1) and (12-6), the (T-spaced) observation z can be expressed in terms of the N-dim data vector a, an N x N- dim diagonal channel matrix C and an N x N-dim diagonal frequency offset

12.1 Fading Channel Transmission Models and Synchronization Parameters

12.1.2 Selective Fading Channel Transmission

Consider now selective fading channel transmission, again with known start- of-frame instant p = 0, (possibly) unknown frequency Q’, and unknown channel impulse response process vector h with (maximum) delay spread L Using eqs (12- 1) and (124, the observation 1: can be expressed in terms of the N-dim data

636 Detection and Parameter Synchronization on Fading Channels

vector a, a (2[N + L] x N)-dim banded channel transmission matrix H and a (2[N+L] x 2[N+L])-d im diagonal frequency offset matrix W(@):

I

rN+L-1

ejn’ (N+L-1)

t W(‘)(W)

ho,m

h , N+L-2 hL,N+m

0, 2, 4 , (1, 3, 5 , ) of the main diagonal of W

In eq (12-lo), the transmission matrix M(B) with sync parameter vector

8 = (0’ hT )’ is, of course, nonlinearly dependent on the frequency shift 0’ but linearly dependent on the channel h (here arranged in matrix H) This linear dependence motivates a second, equivalent transmission model where the observation r is expressed in terms of the B[L+l]N-dim CIR process vector h, an

12.1 Fading Channel Transmission Models and Synchronization Parameters 637 (2[N+L] x 2[L+l]N)-d im data matrix A and (as above) the (2[N+L] x 2[N+L])- dim frequency offset matrix W(Q’):

i)

s,(i)

(12-11) where the (2[N+L] x 2[L+l]N)-d’ im matrix A is constructed from the ([N+ L] x [L+l]N)-d im matrix A(‘) by doubling the size of the ([L+l

diagonal submatrices al, .I&1 (Ic = 0, , N - 1) contained in A (1

x [L+l])-dim

‘1 to dimension (2[L+l] x 2[L+l]) and abutting them just like in A(‘), but with an offset of two rows between abutted submatrices

Of course, eqs (12-10) and (12-11) are just two of many possible ways of representing the observation resulting from transmission through a selective fading channel; vector and matrix arrangements should be selected such that they best suit the application and detection / synchronization problem at hand

12.1.3 Main Points

Vector/matrix models for fading channel transmission have been established which yield vector output processes z (flat fading channel, including pulse matched filtering and perfect timing sync) and r (selective channel) By inspection of these models, the sync parameter vector 8 of interest is easily identified

638 Detection and Parameter Synchronization on Fading Channels

Of particular interest are the flat and selective fading channel trajectories c and h, respectively In this case, the observation (2; or r) is of the form

and thus linearly dependent on the sync parameters c and h, respectively

12.2 Optimal Joint Detection and Synchronization

Up to the present, only relatively few attempts have been made to apply the concepts of joint data detection and parameter synchronization to (static or fading) dispersive channels The basic idea of joint detection and channel estimation is

a - conceptually simple but computationally very complex - exhaustive search for the “overall best fit” between the “model output” (hypothetical data sequence transmitted over its associated hypothetical channel) and the observation (received signal), most often aided by some side information on the channel dynamics Recently, several approaches to joint detection and synchronization, in par- ticular selective channel estimation, have been investigated Most of these assume that the fading is very slow so that the CIR h can be taken as time-invariant within

a sufficiently large time interval Neglecting oscillator frequency offsets, one may attempt to maximize the joint likelihood function of (a, h):

LS = arg min I]r - Ah112

a,h

(12-14)

Fe minimization may be accomplished by first computing the channel estimate h(a) for each possible symbol hypothesis a (within the finite interval where h is stationary) and then select the one which best fits the observation (minimum LS error) [3] In order to restrict the per-symbol computational effort to some upper limit, suboptimal algorithms usually search for the optimum in a recursive manner For instance, the generalized Viterbi algorithm [4] retains a fixed number of “best” data sequences (survivors), together with their associated channel estimates, after progressing from time instant (k-l) to k Generating a distinct channel estimate for every single survivor is sometimes referred to as per-survivor processing (PSP) [S, 6,7] The number of channel estimates may be reduced further by computing only

12.2 Optimal Joint Detection and Synchronization 639 those i(a) that are based on the assumption of binary data a (reduced constellation approach [ 81)

An alternative scheme termed quantized channel approach [9] is based on a fi- nite set of candidate channels h that is selected before the joint detection/estimation algorithm is started In principle, ML or LS detection and estimation is performed

by detecting the optimal data sequence B(h) associated with each candidate h and then selecting the one with the smallest LS error During recursive Viterbi data detection, the candidate channels themselves should be adapted (by driving the residual error to a minimum) so that they converge against the (unquantized) true value

Simulation results [lo, 81 indicate that convergence of such joint detec- tion/estimation algorithms can be remarkably fast, e.g., within 50 to several hun- dred iterations Hence, such algorithms have the potential to tolerate some degree

of fading and therefore remain to be a hot topic for further research

12.2.1 The Bayesian Approach to Joint Detection and Synchronization

Let us now focus on joint detection and estimation in the context of truly fading channels For the following discussion of Bayesian detection and estimation techniques, let r denote the observation in general (either r or the pulse MF output z) Then optimal maximum a posteriori (MAP) detection of the data a contained

in A calls for maximizing the probability of a, conditioned on the observation r,

ii~~p = arg max P(alr)

a sync parameter estimate 6 must be formed and subsequently used for detection

as if it were the true parameter All joint estimation-detection structures of this chapter and, more importantly, virtually all realizable receiver structures are based

on synchronized detection

640 Detection and Parameter Synchronization on Fading Channels

The sync parameters 8 can be brought into play by reformulating the like- lihood p(rla):

P(rla) = I p(r, Wa) &J J

= J drla, 0) - p(ela) de

Following the same high-SNR argument as in Section 4.3, maximizing the inte- grand

yields the joint estimation-detection rule

(h/m, 8) = arg max p(r, 0lo)

(12-18)

(12-19)

In the important case that all random quantities are Gaussian, this high-SNR approximation is actually the optimal estimation-detection rule, regardless of the SNR (see Sections 12.2.2 and 12.2.6)

At this point, it is necessary to take a closer look at the properties of the sync parameters 8 to be estimated In particular, we distinguish between parameters 8s that are essentially static, such as p, E, cp or time-invariant channels h, and parameters 80 that may be termed dynamic in the sense that they are taken from time-variant processes, such as flat or selective fading channel process vectors

c or h, respectively Usually, there is little or no probabilistic information on static parameters other than that they are in a given region Therefore, p(8s) does not exist or can be assumed constant within a finite region In view of the second representation of eq (12- 1 S), joint detection and static parameter estimation thus reduces to maximizing the joint likelihood function p(rla,&) with respect to (a, 0s) (see introduction above) On the other hand, probabilistic side information

on dynamic parameters is usually available and should be made use of Hence, joint detection and dynamic parameter estimation calls for maximizing either the

12.2 Optimal Joint Detection and Synchronization 641 first or second representation of the density function p(r, 80 ]a) [eq (1%18)]: Static sync parameters:

A

( %@S = arg max ) p(r(a, 6~)

a, OS Dynamic sync parameters:

( ii, &J > = arg max p(6D Ir, a) e p(r18)

a, @D

It is immediately recognized that 6~ is an ML estimate and 6~) a MAP estimate,

so that one may speak of ML detection with MAP (dynamic) or ML (static) parameter synchronization

Since there are infinitely many possible realizations of sync parameters 6 whereas the number of possible sequences a is finite, the most natural joint maximization procedure consists in first maximizing the joint likelihood p(rla,&) with respect to 8s (ML) or the conditional pdf p(8D I r, a) with respect to 80 (MAP) for each of the possible a, and then selecting the sequence a with the largest likelihood:

ML estimation of static sync parameters;

joint likelihood for decision:

A es(a) = w3 max p(r 1 a, 63)

es As(a) = p(r 1 a, e,=&(a))

(12-21)

ii = arg max nsca>

a

MAP estimation of dynamic sync parameters;

conditional pdf for decision:

n

@D(a) = arts mm P@D I rj a>

@D AD(a) = p(r I a, eD=bD(a)) ‘p( 80 =bD(a))

(12-22)

ii = arg max AD ca>

Detection and Parameter Synchronization on Fading Channels

The first maximization step yields a conditional sync parameter estimate 6(a) that is subsequently used in the decision likelihood computation as if it were the true parameter Hence, this procedure resembles the concept of synchronized detection Here, however, each candidate sequence a carries its own sync estimate

@(a) conditioned on that sequence

12.2.2 Optimal Linear Estimation of Static and Gaussian

Dynamic Synchronization Parameters

In the previous parts of this book, the estimation of fixed sync parameters 8s has been discussed in great detail Naturally, we shall focus on fading channels with dynamic sync parameters 80 On a number of occasions, however, the quasi-stationarity assumption on the sync parameters holds (e.g., in the context

of blockwise channel acquisition), so that we will also deal with estimating static parameters es

In this section, it is assumed that the observation is linearly dependent on

8 (this applies to both static and dynamic parameters), i.e., the observation is of the form

with an appropriately defined data matrix A and zero-mean complex Gauss- ian noise n with covariance matrix R, Notice that the transmission mod- els of eqs (12-9) and (12-11) comply with this linearity assumption when there is no frequency offset Concerning dynamic sync parameters 80, we shall further assume that these follow a multivariate complex Gaussian distri- bution with known mean vector PD = E[@D] and Hermitian covariance matrix

Ho = E[(@D -pD).(& -pD)H] Under th ese assumptions the likelihood / density functions for ML static / MAP dynamic parameter estimation - as well

as the likelihood function for detection - are also Gaussian

ML Static Parameter Estimation

In the case of static parameters, only the noise is random, and the likelihood function becomes

m(a, 0s) = [r - A@,# + R;‘n[r - A&]

= e,H - [A~R;~A] es - [rHRGIA] es (12-25)

-eg [AHR;‘r] +[rHRilr]

12.2 Optimal Joint Detection and Synchronization 643

By setting the derivative of m(a,Bs)

(12-27)

respectively

MAP Dynamic Parameter Estimation

In the case of dynamic parameters, both the parameters 80 and the additive noise n are Gaussian random variables MAP parameter estimation calls for maximizing the pdf p( or> 1 r, a) [eq (l2-21)], conditioned on the observation

r, with respect t0 80

When two random vectors x and y with means CL,, py, covariances ES, EY and cross covariance EC,, are jointly Gaussian, the conditional pdf p(x 1 y) is also Gaussian, i.e., p(x 1 y) = N (CL+, E+), with mean and covariance [ 1 l]

644 Detection and Parameter Synchronization on Fading Channels

here) and its error covariance matrix become

@D(a) = clz + &,~Jy - py)

Decision Metric for Synchronized Detection

Having obtained a sync parameter estimate &a) and its error covariance matrix E(a) associated with a (hypothetical) data sequence a, the estimate @a) can now be inserted into the decision metric A(a) of eq (12-21) in order to perform synchronized ML detection using the ML or MAP sync parameter estimate Since all random quantities are Gaussian, we have

&la, OS> = N(A@s, Rn)

oc exp {-[r - ABslH Ril e [r - A6s]}

p(rb, 0,) = N(A@D, Rn)

CC exp {-[r - A@D]~ Ril * [r - ABol) (12-32) p(@D) = N(pD, RD)

OC exp -[eD - pDIH - Ril - [eD - PDI}

where the inverse matrices R;l, Rgl are understood to be pseudoinverses in the case that R, and/or RD are singular This is not just an academic subtlety but arises frequently in practice whenever the underlying processes (especially the fading channel processes) are band-limited

Via eqs (12-21) and (12-32), the metrics m(a, 0s) cc - In Ip(r 1 a, OS)] and ??%(a, 8~) Cc - hr b(r 1 a, 8~) p(e~)] for ML detection with ML/MAP

12.2 Optimal Joint Detection and Synchronization 645 synchronization, respectively, can be expressed as

Static sync parameters:

m(a, 0s) = [r - A@~]~Ril[r - A6s]

Dynamic sync parameters:

m(a, 0,) = [r - Atl~]~R,‘[r - A801 + [Oo - pDIHRil[OD - po]

(12-33) Inserting the ML/MAP parameter estimates of eqs (12-27) and (12-3 1) into

eq (12-33) immediately yields the decision metrics m(a) = rn(a, e=e> (to be minimized):

Static sync parameters:

ms(a) = [r - A6s(a)]HR,1 [r - A6S(a)]

mD(a) = [r - Ab(a)]HR;l [r - AiD(

+ [6D(a) - /JD]~R~~ [bD(a) - pD]

These metrics can be reformulated by expanding the estimate &a) For static sync parameters, expanding eq (12-34) (ML) yields

- [@(a) e A~R,‘~] - [rHR;‘n&(a)]

By reformulating the term

rHRi1Ae8S(a) = E?R,~A.[A~R,‘A]-~ AHRzlr

646 Detection and Parameter Synchronization on Fading Channels

one observes that the last three terms of eq (12-35) are equal Furthermore, the first term can be dropped since it is independent of a Hence, the decision metric based on ML sync can be equivalently expressed as

ms(a) = rHR;lAe [A~R;~A]-’ AHRilr

whereby the following identities are established:

ED(a) = [AHR;‘A + RD’1-l = [Es1 + RD’]-’

= RD - RDA~ [ARDA~ + &]-‘ARD first

12.2 Optimal Joint Detection and Synchronization 647 expression

is identified as the MAP estimate 6~ (a) With these identities, the last term of

eq (12-38) can be expressed as

mD(a) = [AHRG1r + R~'/.JD]~ * [AHR;lA + R>l]wla[AHRilr + Ri’pD]

= @(a) - [AHR;h + Rjjl] dD(a) +max

~~'(a>

(12-43)

In summary, the decision metrics can be expressed in terms of an inner product

of the ML or MAP sync parameter estimate O(a), whichever applies

So far, the results on synchronized detection - especially the detection part - have been derived using the high-SNR approximation of eq (12-21) In the linear Gaussian case, however, the ML decision rule of eq (12-43) [which is based

648 Detection and Parameter Synchronization on Fading Channels

ML decision based on ML sync parameter estimation

ML decision based on MAP sync parameter estimation

MAP parameter estimate

decision metric

Figure 12-l Generation of Decision Metrics via ML/MAP Sync Parameter Estimation

on MAP estimation according to eq (12-31)] and, as a consequence, all results

on synchronized detection established in this chapter, are not approximative but indeed optimal at all SNR The proof for this claim is given in Section 12.2.6

To summarize the discussion of joint detection and estimation so far, Figure 12- 1 displays the generation of the ML decision metric based on ML/MAP sync parameter estimation For joint detection and ML estimation, the optimal decision metric rns (a) is obtained by first generating the ML estimate 4s (a) [eq (12-27)] and then forming an inner product according to eq (12-37) Likewise, for *joint detection and MAP estimation, the optimal decision metric ?nD (a) is obtained by first generating the MAP estimate 80 (a) [eq (12-3 l)] and then forming an inner product according to eq (12-43)

Due to its tremendous computational complexity, the full optimal joint detec- tion and estimation procedure involving ML detection based on ML [eqs (12-27) and (12-37)] or MAP [eqs (12-31) and (12-43)] sync parameter estimation will most often not be feasible to implement in practice, especially when dynamic (fading) parameters are involved Nevertheless, it is instructive to notice that there exists a closed-form optimal solution to the joint estimation-detection problem in a fading environment just as for nonfading transmission and reception The optimal receiver for fading channels is derived by applying the same standard elements of estimation and detection theory that have been used for nonfading transmission Furthermore, the expressions of eqs (12-27) and (12-37) [ML] and (12-31) and (12-43) [MAP] can serve as the basis for deriving simplified receivers that can actually be implemented

MAP Dynamic Sync Parameter Estimation via ML Estimation

In the case that the ML estimate of a dynamic parameter 0~ exists (the MAP estimate always exists), it is interesting to establish a link between the ML and MAP estimates as well as their error covariances [eqs (12-27) and (12-31)]

12.2 Optimal Joint Detection and Synchronization 649 From eq (12-40) one notices that the error covariance matrix E:s of ML estimation also appears in the context of MAP dynamic parameter estimation This observation gives rise to a relationship that can be established between MAP and ML linear Gaussian sync parameter estimation, provided that Ils and thus the ML estimate exists As we know, the MAP estimate BD (a) [eq (12-31)] is the optimal dynamic parameter estimate which exploits prior knowledge of some statistical channel parameters, Suppose now that we choose not to consider this prior knowledge for the moment, the dynamic channel parameters are estimated

as if they were static, i.e., one ends up with the ML estimate @s(a) of dynamic parameters 80 The expressions for 6s (a) and Es(a) of eq (12-27) are then valid not only for (optimal) ML estimation of static parameters but likewise for (suboptimal) ML estimation of dynamic sync parameters

The existence of an ML estimate of dynamic parameters is by no means guaranteed In the flat fading case (0~ = c), matrices A, R;i [eq (12-9)] are

N x N square so that AHR; ‘A is also N x N square and, assuming nonzero data symbols ak, full rank and thus invertible Hence, the ML estimate 6s = 6s of the flat fading channel 80 = c exists On the other hand, in the selective fading case

@D = h), matrix A is 2(N+L)x2(L+l)N and R;l 2(N+L)x2(N+L) [q (12- 11)] so that AHF$‘A is 2(L+l)Nx2(L+l)N square but not full rank Hence,

an ML estimate 8s = 6s of the selective fading channel 80 = h does not exist, This is intuitively clear since 2( L + 1) N sync parameters h cannot be estimated from only 2( N +L) (with L > 1) observed samples 1: without side information Case 1: ML Estimate 6s of 80 Exists (Flat Fading)

In this case, dynamic sync parameters can also be viewed as if they were static, i.e., es = eD = 8 Now the second identity of eq (12-40) reveals that the MAP error covariance matrix ED (a) can be obtained from the ML error covariance matrix E:s (a) by a linear matrix operation:

ED(a) = [I - %(a) a [RD + %(a)]-‘\ %(a) (12-44)

6

A similar relationship can be established between the MAP and ML estimates via eqs (12-27) and (12-41):

bD(a) = [AHRilA + Ri’1-l [AHR;‘r + R,‘pD]

= ED(a) [AHR;‘r] + ED(a) [~;lpD]

= N(a) e Es(a) [AHR;‘r] + [xD(a) RE1] pD (12-45)

= N(a) k(a) + M(a) - pD

Therefore, the MAP parameter estimate 6,(a) can be obtained from the ML estimate 6s (a) of the same parameters by premultiplication with a matrix N(a), followed by the addition of the weighted mean From eq (12-44) it is observed

650 Detection and Parameter Synchronization on Fading Channels

that the MAP error covariance matrix E:L> (a) can be obtained from the ML error covariance matrix ES (a) by premultiplication with the same matrix N(a) This relationship may be termed the separation property of MAP dynamic parameter estimation in the sense that MAP estimation can be performed by the two- step procedure of (i) computing the ML estimate from the observation and (ii) computing the MAP estimate from the ML estimate:

observation ML estimate MAP estimate

premultiply by premultiply by N(a),

Es(a) e AHRil add weighted mean M(a) pD

The transition matrix N(a) that links both the ML/MAP estimates and the ML/MAP error covariances, as well as the weighting matrix M(a) are given by

N(a) = I - %(a) [RD + r;ls(a)] -1

= I - [A~R$A]-~ (RD + [A~R;~A]-~)-~

M(a) = ED(a) - Rgl

= N(a) - ES(a) - Ril

(12-47)

= I - RDA* [ARDA~ + R,]-lA

As indicated above, ML estimation does not require any knowledge of channel parameters other than the covariance matrix & of the additive noise In the case

of AWGN, not even the noise power needs to be known since then the matrix

by which the observation must be multiplied collapses to Es(a) e AHRzl = [AHR;~A]-‘.A~R;~ = [ADA]-’ AH On the other hand, matrices N(a) and M(a) needed for MAP estimation incorporate the prior knowledge about the channel, as quantified by the mean ~0 and covariance RD The reduction in the MAP error covariance with respect to the ML error covariance (elements along the main diagonal of I30 and E S, respectively) is also determined by the transition matrix N(a) and thus the channel statistics

Case 2: ML Estimate 6s of 80 Does Not Exist (Selective Fading)

In this case, the entire vector of dynamic sync parameters cannot be viewed

as if it were static However, one may try and find a smaller subset 8s C 80 = 8

of 80 for which an ML estimate 8s exists For example, from the vector h of N selective channel impulse responses hk (Ic = 0, , N-l), one may select a number

N of channel “probes” hkK (K = 0, , N- 1) such that the number r+2( L+ 1)

of unknowns does not exceed the number 2(N + L) of observed samples Then MAP estimation can again be performed using the ML estimate 4s However, in order to make sure that the MAP estimate 6~ so obtained is actually optimal, the subset 8s must be chosen tacitly such that ML estimation does not entail a loss

of information with respect to the observation r In other words, the ML estimate

12.2 Optimal Joint Detection and Synchronization 651

8, must be a sufficient statistic for MAP parameter estimation These and other aspects are discussed in greater detail in Section 12.2.4

12.2.3 Joint Detection and Estimation for Flat Fading Channels

Let us now further explore joint detection and channel estimation for the - seemingly simple - case of flat fading channels Here, zero frequency offset and perfect pulse matched filtering (implying perfect timing compensation) is assumed

We remark that close-to-perfect timing estimation may be accomplished by the following procedure: (i) store the entire received signal (or a sufficiently long section thereof); (ii) in a first processing pass, acquire a timing estimate by applying

a non-data-aided (NDA) timing estimation algorithm just as for nonfading channels (Chapter 5); and (iii) in a second pass, perform timing-compensating matched filtering on the entire stored received signal Thus no data is lost during the timing acquisition phase Then the model of eq (12-9) [with W(Q’) = I] applies:

as the average symbol energy E0 = E{ 1 ak I”}

On mobile channels, the scattered component c&k is often modeled to exhibit the U-shaped Jakes Doppler spectrum S,, ($‘) of eq (11-53) The Jakes spectrum, which is based on the isotropic scattering assumption, accentuates instantaneous Doppler shifts near the cutoff frequency Xb As we shall see later in this section, the actual shape of the Doppler spectrum has no noticeable effect on the estimator performance so that, for the purpose of receiver design, the Jakes Doppler spectrum may as well be replaced by an ideal lowpass spectrum with the same cutoff frequency Xb :

Jakes Doppler spectrum and respective autocorrelation:

otherwise with ad(m) = Jo(27rX/D - m)

652 Detection and Parameter Synchronization on Fading Channels

Ideal lowpass approximation:

if ]$‘I 5 27r&

otherwise with ad(m) = si(2aXb m)

&I = E[(@D - PD) (&I - /-JD)~] = E[Cd * $1

Making use of prior knowledge of the channel statistics pi (thus the IC- factor), RD and the noise power No, the optimal MAP channel estimator, condi- tioned on data sequence a, and its error covariance matrix become [eq (12-31)]

6D(a) = LAHR;‘~+ Ril]-:.[AHR;b + R$pD]

= I - Ro (Ro + lVoP-l(a))-l

Matrices N(a) and M( a are seen to be dependent on the symbol powers pk = )

12.2 Optimal Joint Detection and Synchronization 653

1 ak 1” but not the symbol phases, and, of course, dependent on the channel parameters I<, Xb, NO Therefore, the ML and MAP flat fading Rician channel estimates and covariances can be cast into the form:

ML channel estimate and error covariance:

&(a) = [&(a) - AHR,l] s z = Pml(a)*AH z

Es(a) = [A~R;~A]-~ = NO e P-‘(a)

MAP channel estimate and error covariance:

CI

eD Cal = N(a,K AD’, NO) * &(a) + p(a,K AD’, No)

ED(a) = N(a,K AD’, No) - Wa)

(12-53)

with matrix h(m), weighted mean vector /x( l ) = M( l ) e pr> = CY, M(a) 1 (ps = a; = I</( 1 + K)), and channel covariance matrix RD(K, X(, , No) [eq (12-50) with Pd = l/(1 + I<)] given by

N(a,JQ~,No) = I - NoP-l(a) [R&K&) + NoP-‘(a)]-’

I - RD(IC&) - [R&K&) + Non-‘]-‘] -1

by P-‘(a) compensates for the amplitude variations The error covariance in the ML estimate is the noise power NO weighted by the inverse (hypothetical) symbol powers The MAP channel estimate 80 (a) is obtained from 8s (a) by premultiplication with matrix N(o), being effectively a smoothing operation based

on the knowledge of the channel dynamics and the noise level, followed by the addition of vector p(o) based on the knowledge of the LOS path strength The decision metric m(a) [eq (12-37)] associated with ML flat fading estimation,

A ms(a) = @(a) Ejl(a) es(a) -+max