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ECE 459 Fall 2000 Handout # 6 November 7, 2000
HOMEWORK ASSIGNMENT 4
Reading: Lecture notes (lectures 16-18), papers/books referenced in lecture notes.
Due Date: Thursday, November 16, 2000 (in class)
- Optimality of maximal-ratio combining scheme for coherent detection with diversity: Consider BPSK signaling on an L-th order diversity channel. Each channel introduces a fixed attenuation and phase shift so that the received signal at the output of the `-th channel is:
y(t) = ±αejφ`
E g(t) + w(t)
where the processes w`(t) are independent complex WGN processes with PSD N 0. The receiver uses the decision statistic
y =
∑^ L
`=
β〈y(t), g`(t)〉
where the {β`} are complex weighting factors to be determined. A decision in favor of +1 (“bit 1”) is made if yI > 0 and −1 (“bit 0”) otherwise. (i) Determine the p.d.f. of yI when +1 is transmitted. (ii) Show that the probability of bit error Pb is given by:
Pb = Q
2 E
N 0
∑L
=1 √ Re{βαejφ^ } ∑L =1 |β|
2
(iii) Determine the values of {β`} that minimize Pb. Hint: Use the Cauchy-Schwarz inequality.
- Equal gain combining: Consider the same diversity channel model as in problem 1. Suppose the receiver is able to obtain perfect estimates of the phases {φ
} but does not have estimates of {α}. A useful test statistic is formed by equal gain combining that yields:
y =
∑^ L
`=
e−jφ^ 〈y(t), g`(t)〉
and a decision in favor of +1 (“bit 1”) is made if yI > 0 and −1 (“bit 0”) otherwise.
(i) Determine the p.d.f. of yI when +1 is transmitted, for fixed values of {α}. (ii) Find the probability of bit error Pb, for fixed values of {α}.
Note: In class we analyzed the average bit error probability for maximum ratio combining under the assumption that the {α} are i.i.d. Ricean (or Rayleigh) random variables. It would be nice if we could extend this analysis to equal gain combining. Unfortunately, it is not possible to simplify the expression for Pb in this case, and we are left with a multi-dimensional integral over the joint pdf of the {α} (why?). Of course, it is possible (but not necessary for this homework!) to obtain performance results by computing the multidimensional integral either directly or via Monte-Carlo techniques.
- Diversity combining with Non-coherent detection. Consider a binary FSK signal with Rayleigh fading. Let the received signal at the output of the
-th channel be y(t) = αejφ
E gm,(t) + w(t), = 1, 2 ,... , L, m = 0, 1 , where g 0 ,(t) and g 1 ,(t) are orthogonal The statistics ym, = 〈gm,(t), y(t)〉 for = 1, 2 ,... , L, m = 0, 1 are sufficient. We do not assume knowledge of {α} or {φ} at the receiver. Now, if we model the {φ} as i.i.d. Uniform[0, 2 π], then it is possible to show that MPE receiver (for uniform priors on m) can be shown to be constructed as follows. First, we compute the statistics
Vm = (^) N^1 0
∑^ L
`=
|ym,`|^2 m = 0, 1 ,
and then decide ‘0’ if V 0 > V 1 , and ‘1’ otherwise. (The 1/N 0 factor is just for normalization.)
(i) Assuming that the fading is independent (Rayleigh) across the channels, show that, conditioned on ‘0’ being sent, the pdf’s of V 0 and V 1 are given by
pV 0 (x) = x
L− 1 (L − 1)!(1 + γ)L^
exp
− x γ + 1
(^11) {x≥ 0 }, and pV 1 (x) = x
L− 1 (L − 1)!
e−x (^11) {x≥ 0 },
where γ = E/N 0. (ii) It is easy to see that
Pb = P ({V 1 > V 0 } | {‘0’ sent}) = 1 −
0
FV 1 (x)pV 0 (x)dx
where FV 1 (·) is the cdf of V 1. Based on problem 6(ii) of HW#4, you should be able to write down an equation for FV 1 (x). Now show that Pb =
2 + γ
)L L∑− 1
`=
L − 1 + `
`
1 + γ 2 + γ
)`
(iii) Using the fact that γ = γb/L, plot Pb vs. γb for γb ranging from 5 to 40 dB, and for L = 2, 3 , 4.
- BPSK with perfect interleaving and hard decision decoding on a Rayleigh fading channel.
(i) Show that
Pb ≤ Pc e ≤
∑^ n
q=t+
n q
2 −^
γc 1 + γc
)q ( 1 2 +
γc 1 + γc
)n−q ,
where t = b dmin 2 −^1 c (dmin is the minimum distance of the code).
