Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
12.1 Demodulator Structure and Error Probability in Additive
12.1.2 Signal Space Diagram and Optimum Receivers
We now derive the structure of the optimum receivers for digital modulation. In the process, we will also find an additional motivation for using signal space diagrams. For these derivations we make the following assumptions:
1. All transmit symbols are equally likely.
2. The modulation format does not have memory.
3. The channel is an AWGN channel, and both absolute channel gain and phase rotation are completely known. Without loss of generality, we assume henceforth that phase rotation has been compensated completely, so that the channel attenuation is real, so thatα= |α|.
The ideal detector, called theMaximum A Posteriori(MAP) detector, aims to answer the follow- ing question: “If a signalr(t) was received, then which symbolsm(t )was most likely transmitted?”
In other words, which symbol maximizesm?:
Pr[sm(t )|r(t )] (12.6)
Demodulation 223
Bayes’ rules can be used to write this as (in other words, find the symbolmthat achieves) maxm Pr[n(t )=r(t )−αsm(t )]Pr[sm(t )] (12.7) Since we assume that all symbols are equiprobable, the MAP detector becomes identical to the Maximum Likelihood (ML) detector:
maxm Pr[n(t )=r(t )−αsm(t )] (12.8) In Chapter 11, we introduced the signal spaced diagram for the representation of modulated transmit signals. There, we treated the signal space diagram just as a convenient shorthand. Now we will show how a transmit and received signal can be related, and how the signal space diagram can be used to derive optimum receiver structures. In particular, we will derive that the ML detector finds in the signal space diagram the transmit symbol that has the smallest Euclidean distance to the receive signal.
Remember that the transmit signal can be represented in the form:2 sm(t )=
N n=1
sm,nϕn(t ) (12.9)
where
sm,n= TS 0
sm(t )ϕn∗(t ) dt (12.10) Now we find that the received signal can be represented by a similar expansion. Using the same expansion functionsϕn(t )we obtain
r(t )= ∞ n=1
rnϕn(t ) (12.11)
where
rn= TS
0
r(t )ϕn∗(t ) dt (12.12)
Since the received signal contains noise, it seems at first glance that we need more terms in the expansion – infinitely many, to be exact. However, we find it useful to split the series into two parts:
r(t )= N n=1
rnϕn(t )+ ∞ n=N+1
rnϕn(t ) (12.13)
and similarly:
n(t )= N n=1
nnϕn(t )+ ∞ n=N+1
nnϕn(t ) (12.14)
Using these expansions, the expression that the ML receiver aims to maximize can be written as
maxm Pr[n=r−αsm] (12.15)
2Note that these equations are valid for both the baseband and the bandpass representation – it is just a matter of inserting the correct expansion functions.
where the received signal vector r is simply r=(r1, r2, . . .)T, and similarly for n. Since the noise components are independent, the probability density function (pdf) of the received vectorr, assuming thatsmwas transmitted, is given as
p(rLP|αsLP,m)∝exp
− 1
2N0||r−αsm||2
(12.16)
= ∞ n=1
exp
− 1
2N0(rn−αsm,n)2
(12.17) Sincesm,nis nonzero only forn≤N, the ML detector aims to find:
maxm
N n=1
exp
− 1
2N0(rn−αsm,n)2 ∞
n=N+1
exp
− 1 2N0(rn)2
(12.18) A key realization is now that the second product in Eq. (12.18) is independent ofsm, and thus does not influence the decision. This is another way of saying that components of the noise (and thus of the received signal) that do not lie in the signal space of the transmit signal are irrelevant for the decision of the detector (Wozencraft’s irrelevance theorem). Finally, as exp(.)is a monotonic function, we find that it is sufficient to minimize the metric:
μ(sLP,m)= ||rLP−αsLP,m||2 (12.19) Geometrically, this means that the ML receiver decides for symbolmwhich transmit vectorsLP,m has the smallest Euclidean distance to the received vectorrLP. We need to keep in mind that this is an optimum detection method only in memoryless, uncoded systems. Vector r contains “soft”
information – i.e., how sure the receiver is about its decision. This soft information, which is lost in the decision process of finding the nearest neighbor, is irrelevant when one bit does not tell us anything about any other bit. However, for coded systems and systems with memory (Chapters 14 and 16), this information can be very helpful.
The metric can be rewritten as
μ(sLP,m)= ||rLP||2+ ||αsLP,m||2−2αRe{rLPs∗LP,m} (12.20) Since the term||rLP||2 is independent of the consideredsLP,m, minimizing the metric is equivalent to maximizing:
Re{rLPs∗LP,m} −αEm (12.21)
(remember thatEm= ||sLP,m||2/2, see Chapter 11).
One important consequence of this decision rule is that the receiver has to know the value of channel gainα. This can be difficult in wireless systems, especially if channel properties quickly change (see Part II). Thus, modulation and detection methods that do not require this information are preferred. Specifically, the magnitude of channel gain does not need to be known if all transmit signals have equal energy,Em=E. The phase rotation of the channel (argument of alpha) can be ignored if either incoherent detection or differential detection is used.
The beauty of the above derivation is that it is independent of the actual modulation scheme. The transmit signal is represented in the signal space diagram, which gives all therelevantinformation.
The receiver structure of Figure 12.2 is then valid for optimum reception for any modulation alpha- bet represented in a signal space diagram. The only prerequisite is that the conditions mentioned at the beginning of this chapter are fulfilled. This receiver can be interpreted as a correlator, or as a matched filter, matched to the different possible transmit waveforms.
Demodulation 225
sLP,1(t)
sLP,m(t)
sLP,M(t)
∫T0
∫T
0
∫T0
Re{...}
Re{...}
Re{...}
r(t)
aEm
aEM aE1
Select largest
....dt ....dt ....dt
Figure 12.2 Structure of a generic optimum receiver.
In the following, we will find that the Euclidean distance of two points in the signal space diagram is a vital parameter for modulation formats. We find that in the bandpass representation for equal-energy signals, which we assumed in Chapter 11, the Euclidean distance is related to the energy of the signal component as
d122 =2E(1−Re{ρ12}) (12.22)
whereρj k is the correlation coefficient as defined in Chapter 11:
Re{ρk,m} = sksm
|sk||sm| (12.23)