INTRODUCTION TO DIGITAL COMMUNICATIONS 13 can also be expressed in the frequency domain. The Fourier transformation of φ HH (t, τ ) with respect to t yields the scattering function  HH (f d ,τ)= F { φ HH (t, τ ) } . (1.17) The Doppler frequency f d originates from the relative motions between the transmitter and the receiver. Integrating over τ leads to the Doppler power spectrum  HH (f d ) = ∞  0  HH (f d ,τ)dτ, (1.18) describing the power distribution with respect to f d . The range over which  HH (f d ) is almost nonzero is called Doppler bandwidth B d . It represents a measure for the time variance of the channel and its reciprocal t c = 1 B d (1.19) denotes the coherence time. For t c  T s , the channel is slowly fading, for t c  T s , it changes remarkably during the symbol duration T s . In the latter case, it is called time-selective and time diversity (cf. Section 1.4) can be gained when channel coding is applied. Integrating  HH (f d ,τ) versus f d instead of τ delivers the power delay profile  HH (τ ) = f dmax  −f dmax  HH (f d ,τ)df d (1.20) that describes the power distribution with respect to τ . The coherence bandwidth defined by B c = 1 τ max (1.21) represents the bandwidth over which the channel is nearly constant. For frequency-selective channels, B  B c holds, that is, the signal bandwidth B is much larger than the coherence bandwidth and the channel behaves differently in different parts of the signal’s spectrum. In this case, the maximum delay τ max is larger than T s so that successive symbols overlap, resulting in linear channel distortions called intersymbol interference (ISI). If the coefficients h[k, κ] in the time domain are statistically independent, frequency diversity is obtained (cf. Section 1.4). For B  B c , the channel is frequency-nonselective, that is, its spectral density is constant within the considered bandwidth (flat fading). Examples for different power delay profiles can be found in Appendix A.2. Modeling Mobile Radio Channels Typically, frequency-selective channels are modeled with time-discrete finite impulse response (FIR) filters following the wide sense stationary uncorrelated scattering (WSSUS) approach (H ¨ oher 1992; Schulze 1989). According to (1.11), the signal is passed through a tapped-delay-line and weighted at each tap with complex channel coefficients h[k, κ]as shown in Figure 1.12. 14 INTRODUCTION TO DIGITAL COMMUNICATIONS + x[k] h[k, 0] h[k, 1] h[k, 2] h[k, L t − 1] y[k] n[k] T s T s T s Figure 1.12 Tapped-delay-line model of frequency-selective channel with L t taps Although the coefficients are comprised of transmit and receive filters, as well as the channel impulse response h(t, τ ) and the prewhitening filter g W [k], (as stated in Section 1.2.1), they are assumed to be statistically independent (uncorrelated scattering). The length L t =τ max /T s  of the filter depends on the ratio of maximum channel delay τ max and symbol duration T s . Thus, the delay axis is divided into equidistant intervals and for example, the n κ propagation paths falling into the κ-th interval compose the coefficient h[k, κ] = n κ −1  i=0 e j2πf d,i kT s +jϕ i (1.22) with ϕ i as the initial phase of the i-th component. The power distribution among the taps according to the power delay profiles described in Appendix A.2 (Tables A.1 and A.3) can be modeled with the distribution of the delays κ. The more delays that fall into a certain interval, the higher the power associated with this interval. Alternatively, a constant number of n propagation paths for each tap can be assumed. In this case, h[k, κ] = ρ κ · n−1  i=0 e j2πf d,i kT s +jϕ i (1.23) holds and the power distribution is taken into account by adjusting the parameters ρ κ .The Doppler frequencies f d,i in (1.22) and (1.23) depend on the relative velocity v between the transmitter and the receiver, the speed of light c 0 and the carrier frequency f 0 f d = v c 0 · f 0 · cos α. (1.24) In (1.24), α represents the angle between the direction of arrival of the examined propagation path and the receiver’s movement. Therefore, its distribution also determines that of f d leading to Table A.2. Maximum and minimum Doppler frequencies occur for α = 0and α = 180, respectively, and determine the Doppler bandwidth B d = 2f dmax . The classical Jakes distribution depicted in Figure 1.13  HH (f d ) =  A √ 1−(f d /f dmax ) 2 |f d |≤f dmax 0else, (1.25) INTRODUCTION TO DIGITAL COMMUNICATIONS 15 −1.5 −1 −0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 f d /f dmax → p f d (f d /f dmax ) → Figure 1.13 Distribution of Doppler frequencies for isotropic radiations (Jakes spectrum) is obtained for isotropic radiations without line-of-sight (LoS) connection. For referred directions of arrival, Gaussian distributions with appropriate means and variances are often assumed (cf. Table A.2 for τ>0.5 µs). Unless otherwise stated, nondissipative channels assume meaning, so that E H {  κ |h[k, κ]| 2 }=1 holds. In the following part, the focus is on the statistics of a single channel coefficient and, therefore, drop the indices k and κ. For a large number of propagation paths per tap, real and imaginary parts of H are statistically independent and Gaussian distributed stochastic processes and the whole magnitude |H|= √ H 2 + H 2 is Rayleigh distributed p |H| (ξ) =  2ξ/σ 2 H · exp(−ξ 2 /σ 2 H )ξ≥ 0 0else (1.26) with mean E H {|h|} =  πσ 2 H /2. In (1.26), σ 2 H denotes the average power of H. The instan- taneous power which is chi-squared distributed with two degrees of freedom p |H| 2 (ξ) =  1/σ 2 H · exp(−ξ/σ 2 H )ξ≥ 0 0else (1.27) while the phase is uniformly distributed in [−π, π]. If a LoS connection exists between the transmitter and the receiver, the total power P of the channel coefficient h is shared among a constant LoS and a Rayleigh fading component with a variance of σ 2 H . The power ratio between both parts is called Rice factor K = σ 2 LoS /σ 2 H . Hence, the LoS component has a power of σ 2 LoS = Kσ 2 H and the channel coefficient becomes h =  σ 2 H · K +α (1.28) with total power P = (1 + K)σ 2 H . The fading process α consists of real and imaginary parts that are statistically independent zero-mean Gaussian processes each with variance σ 2 H /2. 16 INTRODUCTION TO DIGITAL COMMUNICATIONS 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 0 1 2 3 0 0.5 1 1.5 2 ξ →ξ → p |H| (ξ) → p |H| (ξ) → K = 0K = 0 K = 1K = 1 K = 3 K = 5 K = 6 K = 11 a) σ 2 H = 1 b) P = 1 Figure 1.14 Rice distributions for different Rice factors K As shown in (Proakis 2001), the magnitude of H is Ricean distributed p |H| (ξ) =  2ξ/σ 2 H · exp  − ξ 2 /σ 2 H − K  · I 0  2ξ  K/σ 2 H  ξ ≥ 0 0else. (1.29) In (1.29), I 0 (·) denotes the zeroth-order modified Bessel function of first kind (Benedetto and Biglieri 1999). With reference to the squared magnitude, we obtain the density p |H| 2 (ξ) =  1/σ 2 H · exp  − ξ/σ 2 H − K  · I 0  2  ξK/σ 2 H  ξ ≥ 0 0else. (1.30) The phase is no longer uniformly distributed. Figure 1.14a shows some Rice distributions for a constant fading variance σ 2 H = 1and varying Rice factor. For K = 0, the direct component vanishes and pure Rayleigh fading is obtained. In Figure 1.14b, the total average power is fixed to P = 1andσ 2 H = P/(K + 1) is adjusted with respect to K. For a growing Rice factor, the probability density function becomes more narrow and reduces to a Dirac impulse for K →∞. This extreme case corresponds to the AWGN channel without any fading. The reason for especially discussing the above channels is that they somehow repre- sent extreme propagation conditions. The AWGN channel represents the best case because noise contributions can never be avoided perfectly. The frequency-nonselective Rayleigh fading channel describes the worst-case scenario. Finally, Rice fading can be interpreted as a combination of both, where the Rice factor K adjusts the ratio between AWGN and fading parts. 1.2.4 Systems with Multiple Inputs and Outputs So far, this section has only described systems with a single input and a single output. Now, the scenario is extended to MIMO systems that have already been introduced in INTRODUCTION TO DIGITAL COMMUNICATIONS 17 x 1 [k] x 2 [k] x N I [k] y 1 [k] y 2 [k] y N O [k] h 1,1 [k, κ] h 2,1 [k, κ] h 2,N I [k, κ] h N O ,N I [k, κ] n 1 [k] n 2 [k] n N O [k] Figure 1.15 General structure of frequency-selective MIMO channel Subsection 1.1.1. However, at this point we are restricted to a general description. Specific communication systems are treated in Chapters 4 to 6. According to Figure 1.1, the MIMO system consists of N I inputs and N O outputs. Based on (1.11), the output of a frequency-selective SISO channel can be described by y[k] = L t −1  κ=0 h[k, κ] · x[k −κ] +n[k]. This relationship now has to be extended for MIMO systems. As a consequence, N I signals x µ [k], 1 ≤ µ ≤ N I , form the input of our system at each time instant k and we obtain N O output signals y ν [k], 1 ≤ ν ≤ N O . Each pair (µ, ν) of inputs and outputs is connected by a channel impulse response h ν,µ [k, κ] as depicted in Figure 1.15. Therefore, the ν-th output at time instant k can be expressed as y ν [k] = N I  µ=1 L t −1  κ=0 h ν,µ [k, κ] · x µ [k −κ] + n ν [k] (1.31) where L t denotes the largest number of taps among all the contributing channels. Exploiting vector notations by comprising all the output signals y ν [k] into a column vector y[k]and all the input signals x µ [k] into a column vector x[k], (1.31) becomes y[k] = L t −1  κ=0 H[k, κ] · x[k − κ] +n[k]. (1.32) In (1.32), the channel matrix has the form H[k, κ] =    h 1,1 [k, κ] ··· h 1,N I [k, κ] . . . . . . . . . h N O ,1 [k, κ] ··· h N O ,N I [k, κ]    . (1.33) Finally, we can combine the L t channel matrices H[k, κ] to obtain a single matrix H[k] = [H[k, 0] ···H[k, L t − 1]]. With the new input vector x L t [k] = [x[k] T ···x[k −L t − 1] T ] T we obtain y[k] = H[k] · x L t [k] + n[k]. (1.34) 18 INTRODUCTION TO DIGITAL COMMUNICATIONS 1.3 Signal Detection 1.3.1 Optimal Decision Criteria This section briefly introduces some basic principles of signal detection. Specific algorithms for special systems are described in the corresponding chapters. Assuming a frame-wise transmission, that is, a sequence x consisting of L x discrete, independent, identically dis- tributed (i.i.d.) symbols x[k] is transmitted over a SISO channel as discussed in the last section. Moreover, we are restricted to an uncoded transmission, while the detection of coded sequences is subject to Chapter 3. The received sequence is denoted by y and com- prises L y symbols y[k]. Sequence Detection For frequency-selective channels, y suffers from ISI and has to be equalized at the receiver. The optimum decision rule for general channels with respect to the frame error probability P f looks for the sequence ˜ x that maximizes the a posteriori probability Pr{X = ˜ x | y}, that is, the probability that ˜ x was transmitted under the constraint that y was received. Applying Bayes’ rule Pr{X = ˜ x | Y = y}=p Y| ˜ x (y) · Pr{X = ˜ x} p Y (y) , (1.35) we obtain the maximum a posteriori (MAP) sequence detector ˆ x = argmax ˜ x∈ X L x Pr{ ˜ x | y}=argmax ˜ x∈ X L x  p Y| ˜ x (y) · Pr{ ˜ x}  (1.36) where X L x denotes the set of sequences with length L x and symbols x[k] ∈ X . 6 It illustrates that the sequence MAP detector takes into account the channel influence by p Y| ˜ x (y) as well as a priori probabilities Pr{ ˜ x} of possible sequences. It has to be emphasized that p Y| ˜ x (y) is a probability density function since y is distributed continuously. On the contrary, Pr{ ˜ x | y} represents a probability because ˜ x serves as a hypothesis taken from a finite alphabet X L x and y represents a fixed constraint. If either Pr{ ˜ x} is not known, a priori to the receiver or all sequences are uniformly distributed resulting in a constant Pr{ ˜ x}, we obtain the maximum likelihood (ML) sequence detector ˆ x = argmax ˜ x∈ X L x p Y| ˜ x (y). (1.37) Under these assumptions, it represents the optimal detector minimizing P f . Since the sym- bols x[k]in ˜ x are elements of a discrete set X (cf. Section 1.4), the detectors in (1.36) and (1.37) solve a combinatorial problem that cannot be fixed by gradient methods. An exhaustive search within the set of all possible sequences ˜ x ∈ X L x requires a computational effort that grows exponentially with |X| and L x and is prohibitive for most practical cases. An efficient algorithm for an equivalent problem – the decoding of convolutional codes (cf. Section 3.4) – was found by Viterbi in 1967 (Viterbi 1967). Forney showed in (Forney 1972) that the Viterbi algorithm is optimal for detecting sequences in the presence of ISI. 6 For notational simplicity, p Y|X=˜x (y) is simplified to p Y|˜x (y) and equivalently Pr{X = ˜ x} to Pr{ ˜ x}.Theterm p Y (y) can be neglected because it does not depend on ˜ x. INTRODUCTION TO DIGITAL COMMUNICATIONS 19 Orthogonal Frequency Division Multiplexing (OFDM) and CDMA systems offer different solutions for sequence detection in ISI environments. They are described in Chapter 4. Symbol-by-Symbol Detection While the Viterbi algorithm minimizes the error probability when detecting sequences, the optimal symbol-by-symbol MAP detector ˆx[k] = argmax X µ ∈X Pr{X [k] = X µ | y}=argmax X µ ∈X  ˜ x∈ X L x ˜x[k]=X µ Pr{X = ˜ x | y} = argmax X µ ∈X  ˜ x∈ X L x ˜x[k]=X µ p Y| ˜ x (y) · Pr{ ˜ x} (1.38) minimizes the symbol error probability P s . Obviously, the difference compared to (1.36) is the fact that all sequences ˜ x with ˜x[k] = X µ contribute to the decision, and not only to the most probable one. Both approaches need not deliver the same decisions as the following example demonstrates. Consider a sequence x = [x[0],x[1]] of length L x = 2 with binary symbols x[k] ∈{X 0 ,X 1 }. The conditional probabilities Pr{ ˜ x | y}=Pr{˜x[0], ˜x[1] | y} are exemplarily summarized in Table 1.1. While the MAP sequence detector delivers the sequence ˆ x = [X 0 ,X 1 ] with the highest a posteriori probability Pr{ ˜ x | y}=0.27, the symbol-by-symbol detector decides in favor to Pr{X [0] = X µ | y}=  X ν ∈X Pr{X [0] = X µ , X [1] = X ν | y} (and an equivalent expression for x[1]) resulting in the decisions ˆx[0] =ˆx[1] = X 0 . How- ever, the difference between both approaches is only visible at low SNRs and vanishes at low error rates. Again, for unknown a priori probability or uniformly distributed sequences, the corre- sponding symbol-by-symbol ML detector is obtained by ˆx[k] = argmax X µ ∈X p Y|X [k]=X µ (y) = argmax X µ ∈X  ˜ x∈ X L x ˜x[k]=X µ p Y| ˜ x (y). (1.39) Table 1.1 Illustration of sequence and symbol-by-symbol MAP detection Pr{˜x[0], ˜x[1] | y}˜x[1] = X 0 ˜x[1] = X 1 Pr{˜x[0] | y} ˜x[0] = X 0 0.26 0.27 0.53 ˜x[0] = X 1 0.25 0.22 0.47 Pr{˜x[1] | y} 0.51 0.49 20 INTRODUCTION TO DIGITAL COMMUNICATIONS Memoryless channels For memoryless channels like AWGN and flat fading channels and i.i.d. symbols x[k], the a posteriori probability Pr{ ˜ x | y} can be factorized into  k Pr{˜x[k] | y[k]}. Hence, the detector no longer needs to consider the whole sequence, but can instead decide symbol by symbol. In this case, the time index k can be dropped and (1.38) becomes ˆx = argmax X µ ∈X Pr{X = X µ | y}. (1.40) Equivalently, the ML detector in (1.39) reduces to ˆx = argmax X µ ∈X p Y|X µ (y). (1.41) 1.3.2 Error Probability for AWGN Channel This section shall describe the general way by which to determine the probabilities of decision errors. The derivations are restricted to memoryless channels but can be extended to channels with memory or trellis-coded systems. In these cases, vectors instead of symbols have to be considered. For a simple AWGN channel, y = x + n holds and the probability density function p Y|X µ (y) in (1.41) has the form p Y|X µ (y) = 1 πσ 2 N · e −|y−X µ | 2 /σ 2 N (1.42) (cf. (1.12)). With (1.42), a geometrical interpretation of the ML detector in (1.41) shows that the symbol X µ out of X that minimizes the squared Euclidean distance |y − X µ | 2 is determined. Let us now define the decision region D µ =  y ||y − X µ | 2 < |y − X ν | 2 ∀ X ν = X µ  (1.43) for symbol X µ comprising all symbols y ∈ C whose Euclidean distance to X µ is smaller than to any other symbol X ν = X µ . The complementary set is denoted by D µ . Assuming that X µ was transmitted, a detection error occurs for y/∈ D µ or equivalently y ∈ D µ .The complementary set can be expressed by the union D µ =  ν=µ D µ,ν of the sets D µ,ν =  y ||y − X µ | 2 > |y − X ν | 2  (1.44) containing all symbols y whose Euclidean distance to a specific X ν is smaller than to X µ . This does not mean that X ν has the smallest distance of all symbols to y ∈ D µ,ν . The symbol error probability can now be approximated by the well-known union bound (Proakis 2001) P s (X µ ) = Pr  y ∈ D µ  = Pr    y ∈  ν=µ D µ,ν    ≤  ν=µ Pr  y ∈ D µ,ν  . (1.45) INTRODUCTION TO DIGITAL COMMUNICATIONS 21 The equality in (1.45) holds if and only if the sets D µ,ν are disjointed. The upper (union) bound simplifies the calculation remarkably because in many cases it is much easier to deter- mine the pairwise sets D µ,ν than to exactly describe the decision region D µ . Substituting y = X µ + n in (1.44) yields Pr{Y ∈ D µ,ν }=Pr  |Y −X µ | 2 > |Y −X ν | 2  = Pr  |N | 2 > |X µ − X ν + N | 2  = Pr  Re  (X µ − X ν ) · N ∗     η < − 1 2 |X µ − X ν | 2    ξ  (1.46) In (1.46), η is new a zero-mean Gaussian distributed real random variable with variance σ 2 η =|X µ − X ν | 2 σ 2 N /2andξ a negative constant. This leads to the integral Pr{Y ∈ D µ,ν }= 1  π|X µ − X ν | 2 σ 2 N  −|X µ −X ν | 2 /2 −∞ e − η 2 |X µ −X ν | 2 σ 2 N dη (1.47) that is not solvable in closed form. With the complementary error function (Benedetto and Biglieri 1999; Bronstein et al. 2000) erfc(x) = 2 √ π  ∞ x e −ξ 2 dξ = 1 − 2 √ π  x 0 e −ξ 2 dξ = 1 − erf(x). (1.48) and the substitution ξ = η/(|X µ − X ν |σ N ) we obtain the pairwise error probability between symbols X µ and X ν Pr{Y ∈ D µ,ν }= 1 2 · erfc  |X µ − X ν | 2 4σ 2 N  . (1.49) Next, we normalize the squared Euclidean distance |X µ − X ν | 2 by the average symbol power σ 2 X  2 µ,ν = |X µ − X ν | 2 σ 2 X = |X µ − X ν | 2 E s /T s (1.50) so that the average error probability can be calculated with (1.14) to P s = E  P s (X µ )  =  X µ P s (X µ ) · Pr{X µ } ≤ 1 2  X µ Pr{X µ }·  X ν =X µ erfc      µ,ν 2  2 · E s N 0   . (1.51) Equation (1.51) shows that the symbol error rate solely depends on the squared Euclidean distance between competing symbols and the SNR E s /N 0 . Examples are presented for various linear modulation schemes in Section 1.4. 22 INTRODUCTION TO DIGITAL COMMUNICATIONS 1.3.3 Error and Outage Probability for Flat Fading Channels Ergodic Error Probability For nondispersive channels, the transmitted symbol x is weighted with a complex-valued channel coefficient h and y = hx +n holds. Assuming perfect channel state information (CSI) at the receiver, that is, h is perfectly known, we obtain Pr{Y ∈ D µ,ν | h}=Pr  |Y −hX µ | 2 > |Y −hX ν=µ | 2  = 1 2 · erfc      µ,ν 2  2 ·|h| 2 E s N 0   . (1.52) Therefore, the symbol error probability is itself a random variable depending on the instan- taneous channel energy |h| 2 . The ergodic symbol error rate can be obtained by calculating the expectation of (1.52) with respect to |h| 2 . A convenient way exploits the relationship 1 2 · erfc(x) = 1 π ·  π/2 0 exp  − x 2 sin 2 θ  dθ for x>0 (1.53) which can be derived by changing from Cartesian to polar coordinates (Simon and Alouini 2000). Inserting (1.53) into (1.52), reversing the order of integration and performing the substitution s(θ) =−( µ,ν /2) 2 E s /N 0 / sin 2 (θ) we obtain E H  Pr{Y ∈ D µ,ν | h}  =  ∞ 0 1 π  π/2 0 p |H| 2 (ξ) ×exp  −ξ ( µ,ν /2) 2 E s /N 0 sin 2 (θ)  dξ dθ = 1 π  π/2 0  ∞ 0 exp(ξs(θ)) ·p |H| 2 (ξ) dξ dθ. (1.54) The inner integral in (1.54) describes the moment generating function (MGF) M |H| 2 (s) = ∞  0 p |H| 2 (ξ) ·e sξ dξ (1.55) of the random process |H| 2 (Papoulis 1965; Simon and Alouini 2000). Using the MGF is a very general concept that will be used again in Section 1.5 when dealing with diversity. For the Rayleigh fading channel, the squared magnitude is chi-squared distributed with two degrees of freedom so that M |H| 2 (s) has the form M |H| 2 (s) = ∞  0 1 σ 2 H e −ξ/σ 2 H · e sξ dξ = 1 1 − sσ 2 H . (1.56) [...]...INTRODUCTION TO DIGITAL COMMUNICATIONS 23 Replacing s(θ ) by −( µ,ν /2) 2 Es /N0 / sin2 (θ ) again, the subsequent integration with respect 2 to θ can be solved in closed-form It yields for a nondissipative channel with σH = 1 EH Pr{Y ∈ Dµ,ν | h} = = 1 π π /2 0 sin2 (θ ) sin2 (θ ) + ( 1 1− 2 µ,ν /2) 2E s /N0 ( µ,ν /2) 2 Es /N0 1 + ( µ,ν /2) 2 Es /N0 dθ (1.57) Contrary to (1.51), the... different channels can be expressed by a correlation matrix   2 σ H1 σH1 σH2 ρ1 ,2 · · · σH1 σHD ρ1,D 2  σH1 σH2 2, 1 σH2 · · · σH2 σHD 2, D    (1. 124 )  HH =    2 σHD σH1 σHD ρD,1 σH2 σHD ρD ,2 · · · whose elements are µ,ν = EH {hµ h∗ } = σHµ σHν ρµ,ν The coefficient ρµ,ν describes the ν correlation between the channel coefficients hµ and hν normalized on σHµ σHν Due to 2 ρµ,µ = 1,... |h |2 = D resulting in an overall average SNR D γ = ¯ γ = ¯ =1 Es · DN0 D 2 σH = =1 Es N0 (1. 121 ) 2 2 after MRC.10 For a linear increase of σH , we obtain σH = δ · (D + 1) The average SNRs of the contributing channels become γ = 2 Es Es /N0 2 · · σH = DN0 D D+1 with with the slope δ = 2/ 1≤ ≤ D (1. 122 ) For an exponential growth, the SNRs are determined by γ =C· Es · exp( δ/D) DN0 with 1 ≤ ≤D (1. 123 )... Hence, m = m /2 bits are mapped onto both real and imaginary symbol parts, according to a real-valued M -ASK with M = 2m The combination of both parts results in a square arrangement, for example, a 16-QAM with m = 2 (see Figure 1.18) Adapting the condition in (1.76) to QAM yields e2 M M −1 M −1 M −1 (2 + 1 − M )2 + (2 + 1 − M )2 = 2e2 µ=0 ν=0 ! (2 + 1 − M )2 = µ=0 Es Ts Due to M = M 2 , the parameter... 10 2 10 =1 =2 =4 = 10 = 20 = 100 2 −3 = 10−1 = 10 2 = 10−3 = 10−4 = 10−5 −4 10 −6 10 −4 0 Pt Pt Pt Pt Pt 10 10 10 b) 10 log10 (Eb /N0 ) = 12 dB 0 10 Pout → 0 41 −8 10 20 30 Eb /N0 in dB → 40 10 0 1 10 2 10 10 3 10 D→ Figure 1 .27 Outage probabilities for BPSK and i.i.d Rayleigh fading channels a) a target error rate of Pt = 10−3 and b) 10 log10 (Eb /N0 ) = 12 dB numerically For BPSK, Figure 1 .27 shows... EH 2Ps M -ASK √ M -ASK (h) − Ps (h) 2 (1.87) The expectation of the linear term is already known from M-ASK Properly replacing M √ by M and Es by Es /2 in (1. 82) delivers √ √ M −1 3Es /N0 M -ASK · (1 − α) with α = (h) = 2 √ 2 EH Ps 2( M − 1) + 3Es /N0 M 32 INTRODUCTION TO DIGITAL COMMUNICATIONS 10 10 Ps → 10 10 10 10 10 0 M =4 M = 16 M = 64 −1 2 −3 −4 −5 −6 0 10 20 30 Eb /N0 in dB → 40 Figure 1 .20 ... =1 2 |h [k] |2 · σN = D h[k] 2 2 · σN (1.106) 2 With EX {|x |2 } = σX the SNR can be expressed by γ [k] = 8 The 2 σX 1 · h[k] · 2 σN D 2 = Es 1 · · N0 D D D |h [k] |2 = =1 γ [k] (1.107) =1 optimum scaling derived here is necessary for multiamplitude modulation in order to apply the correct √ decision thresholds For BPSK, the scalar weighting with D/ h[k] 2 can be neglected 40 INTRODUCTION TO DIGITAL COMMUNICATIONS. .. the determinant in (1. 125 )  a b b  b a b    b b has the form  ··· b ··· b     ··· b a whose determinant is (a − b)D−1 [a + b(D − 1)] (Simon and Alouini 20 00) Substituting 2 2 a = σH − λ and b = σH ρ yields the expression 2 2 2 2 (σH − λ − σH ρ)D−1 [σH − λ + σH ρ(D − 1)] = 0 2 2 The solutions are the (D − 1)-fold zero λ1 = σH (1 − ρ) and the single zero 2 = σH (1 + 2 ρ(D − 1)) Inserting... 10 10 10 0 M M M M −1 2 =2 =4 =8 = 16 −3 −4 −5 −6 0 10 20 30 Eb /N0 in dB → 40 Figure 1 .23 Symbol error probabilities for M-PSK and transmission over an AWGN channel (—) and a frequency-nonselective Rayleigh fading channel (- - -) 36 INTRODUCTION TO DIGITAL COMMUNICATIONS 10 Pout → 10 10 10 10 0 M M M M M −1 =2 =4 =8 = 16 = 32 2 −3 −4 0 10 20 30 Eb /N0 in dB → 40 Figure 1 .24 Outage probability Pout... determined such that the energy constraint in (1.76) is fulfilled leading to M−1 Ts Es 3 2 ! · (1.79) (2 + 1 − M)e = Es ⇒ e = M M 2 − 1 Ts µ=0 for equally likely symbols The minimum normalized squared Euclidean distance amounts to 2 0 = (2e )2 12 = 2 Es /Ts M −1 (1.80) INTRODUCTION TO DIGITAL COMMUNICATIONS 4-ASK 29 16-QAM Im 32- QAM Im Im 5e 3e 3e e -3e -e e 3e -3e Re -e -e e e 3e Re -5e -3e e -e 3e 5e Re . whole magnitude |H|= √ H 2 + H  2 is Rayleigh distributed p |H| (ξ) =  2 /σ 2 H · exp(−ξ 2 /σ 2 H )ξ≥ 0 0else (1 .26 ) with mean E H {|h|} =  πσ 2 H /2. In (1 .26 ), σ 2 H denotes the average power. variance σ 2 η =|X µ − X ν | 2 σ 2 N /2andξ a negative constant. This leads to the integral Pr{Y ∈ D µ,ν }= 1  π|X µ − X ν | 2 σ 2 N  −|X µ −X ν | 2 /2 −∞ e − η 2 |X µ −X ν | 2 σ 2 N dη (1.47) that. nondissipative channel with σ 2 H = 1 E H  Pr{Y ∈ D µ,ν | h}  = 1 π  π /2 0 sin 2 (θ) sin 2 (θ) + ( µ,ν /2) 2 E s /N 0 dθ = 1 2  1 −  ( µ,ν /2) 2 E s /N 0 1 + ( µ,ν /2) 2 E s /N 0  . (1.57) Contrary