s o (t) = S(f)H(f; t)e j2πft df,

Transcription

1 Sample Problems for Midterm. The sample problems for the fourth and fifth quizzes as well as Example on Slide 8-37 and Example on Slides ) will also be a key part of the second midterm.. For a causal) time-varying multipath fading channel, the input-output relation can be described as s o t) = hτ; t)st τ)dτ, ) where st) is the input waveform, s o t) is the output waveform and hτ; t) is the impulse response of the time-varying multipath fading channel. a) Can we measure the impulse response hτ; t) by transmitting a single impulse st) = δt) at the input? Justify your answer. b) Based on ), show that s o t) = Sf)Hf; t)e jπft df, where Sf) = st)e jπft dt and Hf; t) = hτ; t)e jπfτ dτ. c) Give st) =Re{ st)e jπfct },s o t) =Re{ s o t)e jπfct } and hτ; t) =Re{ hτ; t)e jπfcτ }. ) Prove that provided that wherefrom),wehave s o t) = hτ; t) st τ)dτ, Sf f c ) H f f c ; t) = S f f c ) Hf f c ; t) =0, Sf) = [ Sf f c )+ S f f c )] and Hf; t) = [ Hf f c ; t)+ H f f c ; t)]. 3) d) If we mistakenly put hτ; t) =Re{ hτ; t)e jπfct } in c), then show that 3) should be replaced by Gτ; ν) = [ Gτ; ν f c )+ G τ; ν f c )], where Gτ; ν) = hτ; t)e jπνt dt and Gτ; ν) = hτ; t)e jπνt dt.

2 3. For a time-flat frequency-flat fading channel, the input signal suffers a multiplicative factor α before it reaches the receiver. The signal-to-noise ratio of the system becomes γ = α E b N 0. Assume γ is Rayleigh distributed with probability density function f γ γ) = γ 0 e γ/γ 0 for γ 0, where γ 0 = E[γ]. a) Show that the average bit error rate BER) for coherent binary FSK is γ0 +γ 0 ), where its BER without fading is equal to Φ ) γ). Show that γ0 +γ 0 is approximately /γ 0 )asγ 0 large. b) Show that the average bit error rate BER) for binary DPSK is,whereitsber +γ 0 ) without fading is equal to e γ. c) Show that the average bit error rate BER) for non-coherent BFSK is +γ 0, where its BER without fading is equal to e γ/. d) Show that subject to γ 0 > 0, we have ) γ0 < +γ 0 + γ 0 ) and lim γ 0 ) γ0 +γ 0 =. γ 0 4. a) Prove, based the below figure, that the Doppler shift ν doppler is equal to v λ cosα), where v is the velocity of the car and λ is the wavelength. Hint: ν doppler = lim t 0 f = lim t 0 π ) φ t = lim t 0 λ b) Derive ν doppler if v = 00 km/hour, α =0andf c =.08 GHz. Note: The light speed if m/sec. 5. Let hτ; t) =Re{ hτ; t)e jπfcτ } denote the impulse response of a time-varying multipath fading channel. a) Suppose hτ; t) is a stationary, zero-mean, complex-valued Gaussian random process in t). Is hτ; t) stationary in t? Doeshτ; t) have zero-mean? Justify your answer. ) L t

3 b) Now suppose hτ; t) = gτ; t)+µτ), where gτ; t) is a stationary, zero-mean, complexvalued Gaussian random process in t) andµτ) is a deterministic function of τ in other words, hτ; t) is not necessarily a zero-mean process for each τ). If gτ; t) satisfies uncorrelated scattering condition, i.e., R g τ,τ ; t) =E [ g τ ; t) gτ ; t + t)] = r g τ ; )δτ τ ). Determine R hτ,τ ; t) =E [ h τ ; t) hτ ; t + t) ]. c) Continue from b). Does R Hf,f ; t) =E[ H f ; t) Hf ; t + t)] depend only on time difference and frequency difference? Justify your answer. Hint: Denote Mf) = µτ)e jπfτ dτ as the Fourier transform of µτ) and use it in your derivation. 6. For a time-flat frequency-flat fading channel, the input-output relation with L diversity can be derived as x l = α l s + w l for l L, where { w l } L l= are zero-mean with E[ w l w k ]= { N0 k = l; 0, k l, and s {± E b } is a BPSK signal. Suppose {α l } L l= receiver. can be accurately estimated at the a) Find the optimal linear combining with real-valued) coefficients {β l } L l= such that L β l x l = l= L β l α l s + l= L ) β l w l maximizes the average signal-to-noise ratio for given {α l } L l=. [ Hint: SNR given {α l } L l= = E L l= β lα l s) ] [ L l= β lw l) ] E l= E b + E b 0 4) b) Further assume that { w l } L l= are Gaussian distributed. Does the linear combining in 4) give the optimal decision rule in the sense of minimizing the error probability under equal prior probability? Hint: Under equal prior probability, the optimal decision rule is the maximumlikelihood decision rule. 7. For the derivation of the rake receiver, Price and Green considered an equivalent system model as follows, where Ĥf) is random and bandlimited and ŵt) is bandlimited white noise. 3

4 We now extend it to a random, bandlimited and also time-varying Ĥf; t). a) Noting that ĥτ; t) = Ĥf; t)ejπfτ df, we obtain from the sampling theorem that ) [ l ĥτ; t) = ĥ W ; t sinc W τ l )] [ = ĥ l t) W sinc W τ l )], W W l= l= where ĥlt) ĥ l ; t) is the lth time-varying channel tap. Represent Ĥf; t) in W W terms of ĥlt). b) Set W = 00 KHz and maximum) delay spread σ τ = 500 ns. Suppose ĥlt) =0for l<0andl/w σ τ. Then, how many channel taps are required to estimate at the receiver? c) Set W =.5 MHz and maximum) delay spread σ τ = 500 ns. Suppose ĥlt) =0for l<0andl/w σ τ. Then, how many channel taps are required to estimate at the receiver? d) Let L σ τ W and assume BPSK transmission i.e., either + st) or st) is transmitted). For the rake receiver to function properly, it is required that the signal waveform satisfies s t l ), s t k ) {, 0 l = k L; W = and Sf) =0for f > W W 0, otherwise, 5) where the inner product between at) andbt) is defined as at)b t)dt See Slide 8-87). Does the simple design { Sf) = W, f <W/; 0, otherwise 4

5 satisfy 5)? Justify your answer. Hint: at)b t)dt = Af)B f)df. 8. An antenna array receiver can be schematically sketched as follows. a) Suppose there are N = users and M = 4 antennas. Let 0 C M N = Find a vector w that can eliminate the interference from the second user via the linear manipulation below) when the receiver only wishes to recover the message from the first user. 5

6 Hint: w should be orthogonal to the second column of C, andyetw cannot be orthogonal to the first column of C. b) Among all w s that satisfy the requirement in a), give one that maximizes the output SNR, provided that the additive noise v is zero-mean uncorrelated with variance N 0 /. You should justify why it maximizes the output SNR. c) Can the antenna array receiver eliminate the interference from the second user, if M =? Justify your answer. 6