# Detection, Estimation modulation theory part 1 - Vantress H.

ISBN 0-471-09517-6

**Download**(direct link)

**:**

**166**> 167 168 169 170 171 172 .. 254 >> Next

Notationally, it is convenient to denote the N messages by a column matrix,

'«lOT

à(ò) —

à2(ò)

(129)

Using this notation, the transmitted signal is

s(t, a(r)) = VlP sin |a>ct + dfc J z(u) </wJ,

where

Ô) = 2 Võ2 gj sin [ tOjU + dfj J fl;(r) dr^.

(130)t

(131)

t The notation s(t, a(r)) is an abbreviation for s(/; a(r), Tt < ò < 0- The second variable emphasizes that the modulation process has memory.

448 5.4 Multidimensional Waveform Estimation

The channel adds noise to the transmitted signal so that the received waveform is

r(f) = s(t, a(r)) + n(t). (132)

Here we want to estimate the N messages simultaneously. Because there are N messages and one received waveform, we refer to this as an N x 1dimensional problem.

FM/FM is typical of many possible multilevel modulation systems such as SSB/FM, AM/FM, and PM/PM. The possible combinations are essentially unlimited. A discussion of schemes currently in use is available in [8].

Case 2. Multiple-Channel Systems. In Section 4.5 we discussed the use of diversity systems for digital communication systems. Similar systems can be used for analog communication. Figure 5.12 in which the message a(t) is frequency-modulated onto a set of carriers at different frequencies is typical. The modulated signals are transmitted over separate channels, each of which attenuates the signal and adds noise. We see that there are M received waveforms,

Ô) = Æ, à(ò)) + è,(0, (/ = 1, 2,..., M),

where

si(t, o(T)) = gi VlPi sin {wct + dfl J a{u)duj.

Once again matrix notation is convenient. We define

'Ji0, a{r))~

sit, a(r))

s2(t, a(r)) _sM(t,a( t)).

(133)

(134)

(135)

ni(t)

Fig. 5.12 Multiple channel system.

Examples of Multidimensional Problems 449

array

o'

V<OV‘

{*&

«tf*

Fig. 5.13 A space-time system.

and

Then

n(0 =

«i(0

n2(0

(136)

-ËÌ(0-

KO = s(r, a(r)) + n(t). (137)

Here there is one message a(t) to estimate and M waveforms are available to perform the estimation. We refer to this as a 1 x M-dimensional problem. The system we have shown is a frequency diversity system. Other obvious forms of diversity are space and polarization diversity.

A physical problem that is essentially a diversity system is discussed in the next case.

Case 3. A Space-Time System. In many sonar and radar problems the receiving system consists of an array of elements (Fig. 5.13). The received signal at the rth element consists of a signal component, st(t, a(r)), an external noise term nEi(t), and a term due to the noise in the receiver element, nm{t). Thus the total received signal at the ith element is

Û0 = sit, a(r)) + nRi(t) + nEi(t).

(138)

450 5.4 Multidimensional Waveform Estimation We define

nit) = nRi(t) + nEi(t). (139)

We see that this is simply a different physical situation in which we have M waveforms available to estimate a single message. Thus once again we have a 1 x M-dimension problem.

Case 4. N x Ì-Dimensional Problems. If we take any of the multilevel modulation schemes of Case 1 and transmit them over a diversity channel, it is clear that we will have an N x M-dimensional estimation problem. In this case the ith received signal, rt(t)9 has a component that depends on N messages, a^t), (j= 1,2,..., JV). Thus

ô) = sfa a(r)) + nt(t), ³ = 1,2,..., M. (140)

In matrix notation

r(/) = s(/, a(r)) + n(/). (141)

These cases serve to illustrate the types of physical situations in which multidimensional estimation problems appear. We now formulate the model in general terms.

5.4.2 Problem Formulation!

Our first assumption is that the messages a{{t), (³ = 1, 2,..., N), are sample functions from continuous, jointly Gaussian random processes. It is convenient to denote this set of processes by a single vector process a(/)- (As before, we use the term vector and column matrix interchangeably.) We assume that the vector process has a zero mean. Thus it is completely characterized by an N x N covariance matrix,

Ê Jit, è) ä ?[(a(0 àã(ì))]

Kai a± (t, u) j Kai ttj2 (´, U)

LKdNdxit, u) j

Kax a^it, li)

. (142)

Thus the ijth element represents the cross-covariance function between the /th and y‘th messages.

The transmitted signal can be represented as a vector s(f, a(r)). This vector signal is deterministic in the sense that if a particular vector sample

t The multidimensional problem for no-memory signaling schemes and additive channels was first done in [9]. (See also [10].)

Derivation of Estimator Equations 451

**166**> 167 168 169 170 171 172 .. 254 >> Next