Chapter 2 Signal Processing at Receivers: Detection Theory

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1 Chapter Signal Processing at Receivers: Detection Theory As an application of the statistical hypothesis testing, signal detection plays a key role in signal processing at receivers of wireless communication systems. To accept or reject a hypothesis based on observations, the hypotheses are possible statistical descriptions of observations using statistical hypothesis testing tools. As realizations of a certain random variable, observations can be characterized by a set of candidate probability distributions of the random variable. In this chapter, based on the statistical hypothesis testing, we introduce the theory of signal detection and key techniques for performance analysis. We focus on the fundamentals of signal detection in this chapter, while the signal detection over multiple-antenna systems will be considered in the following parts of the book..1 Principles of Hypothesis Testing Three key elements are carried out in the statistical hypothesis testing, including 1 Observations. Set of hypotheses. 3 Prior information. The decision process or hypothesis testing is illustrated in Fig..1. InFig..1, is shown that observations and prior information are taken into account to obtain the final decision. However, considering the cases that no prior information is available or prior information could be useless, the hypothesis test can also be developed with observations only. Under the assumption that there exist M hypotheses, we can have an M-ary hypothesis testing in which we need to choose one of the M hypotheses that explains observations and prior information best. In order to choose a hypothesis, different criteria can be considered. According to these criteria, different hypothesis tests are. Bai et al., ow Complexity MIMO Receivers, DOI: 1.17/ _, 5 Springer International Publishing Switzerland 14

2 6 Signal Processing at Receivers: Detection Theory Fig..1 Block diagram for hypothesis testing Prior information Observations Hypothesis Testing Decision outcome available. Based on the likelihood ratio R 1 hypothesis test; three well-known hypothesis tests are given as follows: 1 Maximum a posteriori probability MAP hypothesis test. Baysian hypothesis test. 3 Maximum likelihood M hypothesis test. In the following section, the hypothesis tests in the above are illustrated respectively.. Maximum a Posteriori Probability Hypothesis Test et us first introduce the MAP hypothesis test or MAP decision rule. Consider that there are different balls contained in two boxes A and B, where a certain number is marked on each ball. Under the assumption that the distribution of the numbers on balls is different for each box, as a ball is drawn from one of the boxes, we want to determine the box where the ball is drawn from based on the number of the ball. Accordingly, the following two hypotheses can be founded: H : the ball is drawn from box A; H 1 : the ball is drawn from box B. For example, suppose that 1 balls are drawn from each box as shown in Fig... Based on the empirical distribution results in Fig.., conditional distributions of the number on balls are given by and Pr 1 H = 4 1 ; Pr H = 3 1 ; Pr 3 H = 3 1 ; 1 Note that in Chaps. and 3, we use R to denote the term likelihood ratio, while in the later chapters of the book, the R is used to represent lattice reduction.

3 . Maximum a Posteriori Probability Hypothesis Test 7 Fig.. Balls drawn from two boxes A 3,1,,1,1,3,1,,3} B 3,1,3,4,1,,4,,3,3} Observation Knowledge C 1 D A new observation Which box is this ball drawn from? Pr 1 H 1 = 1 ; Pr H 1 = 1 ; Pr 3 H 1 = 4 1 ; Pr 4 H 1 = 1. In addition, the probability that A H orbh 1 box is chosen is assumed to be the same, i.e., Pr H = Pr H 1 = 1..1 Then, we can easily have Pr 1 = Pr H Pr 1 H + Pr H 1 Pr 1 H 1 = 6 ; Pr = Pr H Pr H + Pr H 1 Pr H 1 = 5 ; Pr 3 = Pr H Pr 3 H + Pr H 1 Pr 3 H 1 = 7 ; Pr 4 = Pr H Pr 4 H + Pr H 1 Pr 4 H 1 =,

4 8 Signal Processing at Receivers: Detection Theory where Pr n denotes the probability that the ball with number n is drawn. Taking PrH k as the a priori probability APRP of H k, the a posteriori probability APP of H k is shown as follows: Pr H 1 = 3 ; Pr H = 3 5 ; Pr H 3 = 3 7 ; Pr H 4 =, and Pr H 1 1 = 1 3 ; Pr H 1 = 5 ; Pr H 1 3 = 4 7 ; Pr H 1 4 = 1. Here, Pr H k n is formed as the conditional probability that the hypothesis H k is true under the condition that the number on the drawn ball is n. For example, if the number of the ball is n = 1, since Pr H 1 = 3 is greater than Pr H 1 1 = 1 3,we can decide that the ball is drawn from box A, where the hypothesis H is accepted. The corresponding decision rule is named as the MAP hypothesis testing, since we choose the hypothesis that maximizes the APP. Generally, in the binary hypothesis testing, H and H 1 are referred to as the null hypothesis and the alternative hypothesis, respectively. Under the assumption that the APRPs PrH and PrH 1 are known and the conditional probability, PrY H k, is given, where Y denotes the random variable for an observation, the MAP decision rule for binary hypothesis testing is given by H : Pr H Y = y > Pr H 1 Y = y ; H 1 : Pr H Y = y < Pr H 1 Y = y,. where y denotes the realization of Y. Note that H is chosen if Pr H Y = y > Pr H 1 Y = y and vice versa. Here, we do not consider the case of Pr H Y = y = Pr H 1 Y = y in., where a decision can be made arbitrarily. Thus, the decision outcome in.3 can be considered as a function of y. Using Bayes rule, we can also show that H : Pr Y = y H Pr Y = y H 1 > Pr H 1 Pr H ; H 1 : Pr Y = y H Pr Y = y H 1 < Pr H.3 1 Pr H.

5 . Maximum a Posteriori Probability Hypothesis Test 9 f y =,σ f y = s,σ 1 s y Fig..3 The pdf of the hypothesis pair Notice that as Y is a continuous random variable, Pr Y = y H k is replaced by f Y = y H k, where f Y = y H k represents the conditional probability density function pdf of Y given H k. Example.1. Define by N μ, σ the pdf of a Gaussian random variable i.e., x with mean μ and variance σ, where N μ, σ = exp x μ πσ σ..4 et the noise n be a Gaussian random variable with mean zero and variance σ, while s be a positive constant. Consider the case that a constant signal, s, is transmitted, while a received signal, y, may be corrupted by the noise, n, as shown in Fig..3. Then, we can have the following hypothesis pair to decide whether or not s is present when y is corrupted by n: H : y = n;.5 H 1 : y = s + n. Then, as shown in Fig..3, wehave f y H = N, σ ; f y H 1 = N s, σ,.6 and when s >. etting f Y = y H f Y = y H 1 = exp sy s σ,.7 ρ = PrH PrH 1, the MAP decision rule is simplified as follows:

6 1 Signal Processing at Receivers: Detection Theory Table.1 MAP decision rule Accept H H 1 H is true Correct Type I false alarm H 1 is true Type II miss Correct detection Table. The probabilities of type I and II errors Error type Case Error probability Type I Accept H 1 when H is true P A Type II Accept H when H 1 is true P B H : y < s + σ ln ρ ; s H 1 : y > s + σ ln ρ. s.8 Since the decision rule is a function of y, we can express the decision rule as follows: ry = : y A ;.9 ry = 1 : y A 1, where A and A 1 represent the decision regions of H and H 1, respectively. Therefore, for the MAP decision rule in.8, the corresponding decision regions are given by A = A 1 = y y < s } σ ln ρ s y y > s σ ln ρ s ; },.1 where A and A 1 are regarded as the acceptance region and the rejection/critical region, respectively, in the binary hypothesis testing. Table.1 shows four possible cases of decision, where type I and II errors are usually carried out to analyze the performance. Note that since the null hypothesis, H, normally represents the case that no signal is present, while the other hypothesis, H 1, represents the case that a signal is present, the probabilities of type I and II errors are regarded as the false alarm and miss probabilities, respectively. The two types of decision errors are summarized in Table.. Using the decision rule ry, P A and P B are given by and P A = Pr Y A 1 H.11 = ry f y H dy.1 = E [ry H ].13

7 . Maximum a Posteriori Probability Hypothesis Test 11 P B = Pr Y A H 1.14 = 1 ry f y H 1 dy.15 = E [1 ry H 1 ],.16 respectively. Then, the probability of detection becomes P D = 1 P B.17 = E [ry H 1 ] Baysian Hypothesis Test In order to minimize the cost associated with the decision, the Baysian decision rule is carried out. Denote by D k the decision that accepts H k, while by G ik the associated cost of D i when the hypothesis H k is true. Assuming the cost of erroneous decision to be higher than that of correct decision, we have G 1 > G and G 1 > G 11.The average cost E [G ik ] is given by Ḡ = E [G ik ].19 = G ik Pr D i, H k. i k = G ik P D i H k Pr H k..1 i k et A c denote the complementary set of the decision region, A, and assume that A 1 = A c for convenience. Since the average cost in.1 is rewritten as Pr D 1 H k = 1 D H k,. Ḡ = G 1 PrH + G 11 PrH 1 + g 1 y g ydy, A.3 where g y = PrH G 1 G f y H ; g 1 y = PrH 1 G 1 G 11 f y H 1..4 Then, it is possible to minimize the average cost Ḡ by properly defining the acceptance regions, while the problem is formulated as

8 1 Signal Processing at Receivers: Detection Theory Since.3 follows min A,A 1 Ḡ..5 Ḡ = Constant + g 1 y g ydy,.6 A we can show that } min Ḡ min g 1 y g ydy,.7 A A A while the optimal regions that minimize the cost are given by A = y g 1 y g y} ; A 1 = y g 1 y >g y}..8 Hence, we can conclude the Baysian decision rule that minimizes the cost as follows: H : g y >g 1 y; H 1 : g y <g 1 y,.9 or H : H 1 : f y H f y H 1 > PrH G 1 G 11 ; PrH 1 G 1 G f y H f y H 1 < PrH G 1 G 11, PrH 1 G 1 G.3 where G 1 G and C 1 C 11 are positive. More importantly, for binary hypothesis testing, we can find out that the ratio of the cost differences, G 1 G 11 G 1 G, is able to characterize the Baysian decision rule rather than the values of individual costs, G ik s. Specifically, as G 1 G 11 G 1 G = 1, the Baysian hypothesis test becomes the MAP hypothesis test..4 Maximum ikelihood Hypothesis Test The MAP decision rule can be employed under the condition that the APRP is available. Considering the case that the APRP is not available, another decision rule based on likelihood functions can be developed. For a given value of observation, y, the likelihood function is defined by f y = f y H ; f 1 y = f y H 1..31

9 .4 Maximum ikelihood Hypothesis Test 13 Notice that the likelihood function is not a function of y since y is given, but a function of the hypothesis. With respect to the M function, the M decision rule is to choose the hypothesis as follows: where the ratio, f y H : H : f y > f 1 y; H 1 : f y < f 1 y, f 1 y > 1; f y H 1 : f 1 y < 1, f f 1 y, is regarded as the R. For convenience, given by.3 Ry = log f y f 1 y.33 the e-based log-likelihood ratio R, the M decision rule can be rewritten as H : Ry >; H 1 : Ry <..34 Note that the M decision rule can be considered as a special case of the MAP decision rule when the APRPs are the same, i.e., PrH = PrH 1. In this case, the MAP decision rule is reduced to the M decision rule..5 ikelihood Ratio-Based Hypothesis Test et the R-based decision rule be H : H 1 : f y f 1 y > ρ; f y f 1 y < ρ,.35 where ρ denotes a predetermined threshold. Consider the following hypothesis pair of received signals: H : y = μ + n;.36 H 1 : y = μ 1 + n, where μ 1 > μ and n N, σ. Then, it follows that f y = N μ, σ ; f 1 y = N μ 1, σ,.37

10 14 Signal Processing at Receivers: Detection Theory Fig..4 The relationship of different decision rules R-based decision rule Bayesian decision rule MAP decision rule M decision rule while the R becomes Rx = μ 1 μ y μ +μ 1 σ..38 The corresponding R-based decision rule is given by H : y μ + μ 1 H 1 : y μ + μ 1 < σ ln ρ μ 1 μ ; > σ ln ρ μ 1 μ..39 etting ρ = σ ln ρ + μ + μ 1,.4 μ 1 μ.39 can be rewritten as H : y < ρ; H 1 : y > ρ..41 Specifically, if we let ρ = PrH G 1 G PrH 1 G 1 G 11,.4 then the R-based decision rule becomes the Baysian decision rule. In summary, The R-based decision rule can be regarded as a generalization of the MAP, Baysian, and M decision rules, where the relationship of various decision rules for binary hypothesis testing is shown in Fig..4.

11 .6 Neyman Pearson emma 15.6 Neyman Pearson emma It is possible to define the Correct detection and type I error false alarm probabilities for each decision rule. On the contrary, for a given target detection probability or error probability, we may be able to derive an optimal decision rule. et us consider the following optimization problem: max d P D d subject to P A d σ,.43 where d and σ represent a decision rule and the maximum false alarm probabilities, respectively. In order to find the decision rule, d, with the maximum false alarm probability constraint, the Neyman Pearson emma is presented as follows. emma.1. et the decision rule of d s be 1, if f 1 s >η f s; d 1, with probability p; s = γs = if f, with probability 1 p, 1 s = η f s;, if f 1 s <η f s,.44 where η > and p is decided such that P A = σ. The decision rule in.44 is named as the Neyman Pearson NP rule and becomes the solution of the problem in.43. Proof. Under the condition of P A σ, we can assume that the decision rule becomes ˆd. In order to show the optimality of the problem in.43, we need to verify that for any ˆd, wehavep D t P D ˆt. Denote by S the observation set, where s S. Using the definition of d, for any s S, it shows that d s ˆds f 1 s η f s..45 Then, it is derived that and s S s S d s ˆds f 1 s η f s ds.46 d s f 1 sds ˆds f 1 sds η d s f sds ˆds f sds, s S s S s S.47 which can further show that P D d P D ˆd η P A d P A ˆd..48

12 16 Signal Processing at Receivers: Detection Theory Since.48 shows that P D d P D ˆd, the proof is completed. From.43, we can show that the NP decision rule is the same as the R-based decision rule with the threshold ρ = 1 η except the randomized rule, i.e., f 1s = η f s. Example.. et us consider the following hypothesis pair: H : y = n; H 1 : y = s + n,.49 where s > and n N, σ.using.41, the type I error probability is shown as = = = ρ ρ = Q dy f ydy f ydy 1 exp y πσ σ ρ, σ where Qx denotes the Q-function and is defined by Qx = x dy 1 π e t / dt..5 Note that the function Qx, x, is the tail of the normalized Gaussian pdf i.e., N, 1 from x to. Then, it follows that = Q ρ σ.51 or ρ = σq 1..5 Thus, the detection probability becomes P D = = ρ ρ f 1 ydy 1 πn y s exp dy N

13 .6 Neyman Pearson emma P D s = s = 1 s = Δ Fig..5 Receiver operating characteristics ROCs of the NP decision rule. ρ s = Q σ = Q Q 1 s,.53 σ which shows that the detection probability is a function of the type I error probability. In Fig..5, the receiver operating characteristics ROCs of the NP decision rule are shown for different values of s. Note that the ROCs are regarded as the relationship between the detection and type I error probabilities..7 Detection of Symmetric Signals Symmetric signals are widely considered in digital communications. In this section, we focus on the detection problem of symmetric signals, namely the symmetric signal detection. Again, let us consider the hypotheses of interest as follows: H : y = s + n; H 1 : y = s + n,.54

14 18 Signal Processing at Receivers: Detection Theory where y is the received signal, s >, and n N, σ. According to the symmetry of transmitted signals s and s, the probabilities of type I and II errors become the same and given by For a given Y = y, the R function is written as P Error = PrAccept H H 1.55 = PrAccept H 1 H..56 f Y RY = log f 1 Y = 1 σ Y s Y + s = s Y..57 σ In addition, based on the M decision rule, we have H : RY >; H 1 : RY <, which can be further simplified as H : Y > ; H 1 : Y <..58 From this, we can show that the M detection is simply a hard-decision of the observation Y = y..7.1 Error Probability With the symmetry of transmitted signals s and s in.54, the error probability is found as which can be derived as P Error = PrRy > H 1.59 = PrRy < H,

15 .7 Detection of Symmetric Signals 19 P Error = PrRy > H 1 = Pry > H 1 1 = exp πσ 1 Y + s σ dy.6 and P Error = = Q s σ 1 π exp Y dy s..61 σ etting the signal-to-noise ratio SNR be SNR = s σ, then we can have P Error = Q SNR. Note that the error probability decreases with the SNR, since Q is a decreasing function..7. Bound Analysis In order to show the characteristics of the error probability, let us derive the bounds on the error probability. Define the error function as erfy = π y and the complementary error function as exp y dy.6 erfcy = 1 erfcy.63 = exp t dt, for y >, π y where the relationship between the Q-function and the complementary error function is given by erfcy = Q y ; Qy = 1 y erfc..64 Then, for a given Y = y, we can show that the complementary error function has the following bounds:

16 Signal Processing at Receivers: Detection Theory 1 1 exp Y Y < erfcy < exp Y,.65 πy πy where the lower bound is valid if Y > 1/. Accordingly, the Q-function QY is bounded as follows: 1 1 exp Y / Y < QY < exp Y /,.66 πy πy where the lower bound is valid if x > 1. Thus, the upper bound of error probability is given by P Error = Q exp SNR/ SNR <..67 π SNR In order to obtain an upper bound on the probability of an event that happens rarely, the Chernoff bound is widely considered, which can be used for any background noise. et py denote the step function, where py = 1, if y, and py =, if y <. Using the step function, the probability for the event that Y y, which can be also regarded as the tail probability, is given by PrY y = = y f Y ρdρ pρ y f Y ρdρ = E[pY y],.68 where f Y ρ represents the pdf of Y.FromFig..6, we can show that py expy..69 Thus, we can have or PrY y E[expY y].7 PrY y E [ exp λy y ].71 for λ. By minimizing the right-hand side with respect to λ in.71, the tightest upper bound can be obtained which is regarded as the Chernoff bound and given by PrY y min λ exp λy E [ exp λy ]..7 In.7, E [ exp λy ] is called the moment generating function MGF.

17 .7 Detection of Symmetric Signals 1 Fig..6 An illustration of step function expy py y et Y be a Gaussian random variable with mean μ and variance σ.themgfof Y is shown as E[exp λy ] =exp λμ + 1 λ σ,.73 while the corresponding Chernoff bound is given by PrY y min λ = min λ exp exp λy exp λμ + 1 λ σ λμ y + 1 λ σ..74 The solution of the minimization is found as λ = arg min λμ exp y + 1 λ λ σ = max, y μ } σ..75 Under the condition that λ >, the Chernoff bound is derived as y μ PrY y exp σ..76 With respect to the error probability in.6, the Chernoff bound is given by P Error exp s σ = exp SNR..77 In summary, the Chernoff bound can be given by

18 Signal Processing at Receivers: Detection Theory Qy exp y which is actually a special case of the Chernoff bound in.7.,.78.8 Binary Signal Detection In general, signals are transmitted by waveforms rather than discrete signals over wireless channels. Using binary signaling, for t < T, the received signal can be written as Y t = St + Nt, t < T,.79 where T and Nt denote the signal duration and a white Gaussian random process, respectively. Note that we have E[Nt] = and E[NtNρ] = N δt ρ, where δt represents the Dirac delta function. The channel in.79 is called the additive white Gaussian noise AWGN channel, while St is a binary waveform that is given by under hypothesis H : St = s t;.8 under hypothesis H : St = s 1 t. Note that the transmission rate is T 1 bits per second for the signaling in.79. et a heuristic approach be carried out to deal with waveform signal detection problem. At the receiver, the decision is made with Y t, t < T. Denote by yt a realization or observation of Y t, where samples are taken from yt. etting st and nt be a realization of St and Nt, respectively, we can have y l = s m,l = n l = lt l 1T lt l 1T lt l 1T ytdt, s m tdt, ntdt,.81 where H : y l = s,l + n l ; H 1 : y l = s 1,l + n l..8 In addition, since Nt is a white process, the n l s are independent, while the mean of n l is zero and the variance can be derived as

19 .8 Binary Signal Detection 3 σ = E[nl ] lt = E l 1T Ntdt = lt l 1T lt l 1T E[NtNρ]dtdρ = lt l 1T lt l 1T N δt ρdtdρ = N T..83 etting y =[y 1 y y ] T, the R becomes Ry = l=1 f y l H l=1 f y l H 1.84 = log f y f 1 y, which follows that Ry = = l=1 l=1 log f y l f 1 y l [ log exp 1N y l s,l y l s 1,l ] = 1 y l s 1,l y l s,l N l=1 = 1 N l=1 y l s,l s 1,l + s1,l s,l..85 In addition, letting s m =[s m,1 s m,... s m, ] T,.85 can be rewritten as Ry = 1 y T s s 1 s T N s s1 T s l=1 From the R in.86, the MAP decision rule is given by

20 4 Signal Processing at Receivers: Detection Theory H : y T s s 1 >σ PrH1 log PrH H 1 : y T s s 1 <σ PrH1 log PrH + 1 st s s T 1 s 1; + 1 st s s T 1 s For the R-based decision rule, we can replace PrH PrH 1 by the threshold ρ, which is H : y T s s 1 >σ log ρ + 1 st s s1 Ts 1; H 1 : y T s s 1 <σ log ρ st s s1 Ts 1. As the number of samples during T seconds is small, there could be signal information loss due to sampling operation using the approach in.81. To avoid any signal information loss, suppose that is sufficiently large to approximate as y T s i 1 T T yts m tdt..89 Accordingly, the R-based decision rule can be written as H : T yts t s 1 tdt > σ log ρ + 1 T s t s1 t dt; H 1 : T yts t s 1 tdt < σ log ρ + 1 T s t s1 t dt,.9 or H : T yts t s 1 tdt > W T ; H 1 : T yts t s 1 tdt < W T,.91 if we let W T = σ log ρ + 1 T s t s 1 t dt..9 Note that the decision rule is named as the correlator detector, which can be implemented as in Fig..7. et us then analyze its performance. For the M decision rule, we can have ρ = 1 in the R-based decision rule, which leads to W T = 1 X = T yt s t s 1 t dt W T, we can show that T s t s 1 t dt. etting PrD H 1 = PrX > H 1 ; PrD 1 H = PrX < H..93

21 .8 Binary Signal Detection 5 Fig..7 Correlator detector using binary signalling In order to derive the error probabilities, the random variable X has to be characterized. Note that X is a Gaussian random variable, since Nt isassumedtobea Gaussian process. On the other hand, if H m is true, the statistical properties of G depend on H m as Y t = s m t+ Nt. Then, letting H m be true, to fully characterize X, the Gaussian mean and variance of X are given by E[X H m ]= = T T E[Y t H m ] s t s 1 t dt W T s m t s t s 1 t dt W T.94 and σ m = E[X E[X H m] H m ],.95 respectively. Denote by E s the average energy of the signals, s m t, m = 1,, which are equally likely transmitted, we can show that E s = 1 T s t + s1 t dt.inthe meanwhile, letting τ = E 1 T s s ts 1 tdt,wehaveσ = σ = σ 1 = N E s 1 τ and E[X H ]= E s 1 τ;.96 E[X H 1 ]= E s 1 τ. Up to this point, the pdfs of X under H and H 1 become exp g E s1 τ N E s 1 τ f g = ; πn E s 1 τ exp g + E s1 τ N E s 1 τ f 1 g =. πn E s 1 τ.97

22 6 Signal Processing at Receivers: Detection Theory By following the same approach in Sect..7.1, the error probability can be derived as P Error = Q E s 1 τ..98 N For a fixed signal energy, E s, since the error probability can be minimized when τ = 1, the corresponding minimum error probability can be written as P Error = Q E s..99 N Note that the resulting signals that minimize the error probability are regarded as antipodal signals, which can be easily shown as s t = s 1 t. In addition, for an orthogonal signal set where ρ =, we have the corresponding error probability to be P Error = Q E s N..1 Therefore, it can be shown that there is a 3 db gap in SNR between the antipodal signal set in.99 and orthogonal signal set in.1..9 Detection of M-ary Signals After introducing the binary signal detection for M =, let us consider a set of M waveforms, s 1 t, s t,...,s M t}, t < T,forM-ary communications, where the transmission rate is given by R = log M T.11 bits per seconds. In.11, it can be observed that the transmission rate increases with M, while a large M would be preferable. However, the detection performance becomes worse as M increases in general. et the received signal be yt = s m t + nt.1 for t < T under the mth hypothesis. The likelihood with samples is shown as

23 .9 Detection of M-ary Signals 7 f m y = = f m y l l=1 l=1 exp y l s m,l πn N..13 Taking the logarithm on.13, we can rewrite the log-likelihood function as follows: 1 log f m y = log + πn 1 = log πn which can be further simplified as log exp y l s m,l l=1 N y l s m,l N,.14 log f m y = 1 N y l s m,l 1 s m,l l=1.15 by canceling the common terms for all the hypotheses. In addition,.15 can be rewritten as log f m yt = 1 T yts m tdt 1 T sm N tdt,.16 when goes infinity. Note that E m = T s m tdt in.16 represents the signal energy, while T yts mtdt denotes the correlation between yt and s m t. Based on the log-likelihood functions, the M decision rule can be found by If log f m yt log f m yt accept H m.17 or If log f myt f m yt accept H m,.18 for m 1,,...,M}\m}, where \ denotes the set minus i.e., A \ B =x x A, x / B}. In Fig..8, the implementation of M decision rule using a bank of the correlator modules is carried out, while the MAP decision rule can also be derived by taking into account the APRP on the M decision rule.

24 8 Signal Processing at Receivers: Detection Theory Fig..8 A bank of correlator modules using M-ary signals.1 Concluding Remarks In this chapter, different decision rules have been introduced with their applications to signal detection. Since multiple signals are transmitted or received via multiple channels simultaneously, in multiple-input multiple-output MIMO systems, it is preferable to describe signals in vector forms. In the next chapter, we will focus on signal detection in a vector space and the idea of MIMO detection.

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