Project - Optimal Receiver and Symbol-by-Symbol Detection for Bandlimitted Channels

1. Computer Assignment

Consider the transmission of a sequence of binary signals x[k] ∈ {-1, 1} through a bandlimited channel as illustrated in the figure below.

The pulse shaping filter p(t) is a square-root raised-cosine filter with rolloff factor α. The channel c(t) is given by

c(t) = β₀δ(t) + β₁δ(t - T ) + β₂δ(t - 2T),

where β_i are real constants, and ∑_iβ_i² = 1 (i.e., the channel has unit energy). The noise N(t) is assumed to be AWGN with power spectrum density N₀/2.

1. The Pulse Shaping Filter p(t) and the Composite Channel g(t)

(a) Use Matlab to plot the impulse response and the frequency response of the square-root raised-cosine pulse shaping filter p(t) for (i) α = 0 and (ii) α = ½.

(b) Suppose that the rolloff factor α = 0, i.e., the pulse shaping filter is an ideal lowpass filter. Let β₀ = β₂ = ½, β₁ = 1/√2. Plot h(t) =^? p(t) ∗ c(t) and the magnitude of its frequency response |H(f)|=^?|F(h(t))|.

(c) For parameters given in (a)-ii, plot g(t) =^? h(t) ∗ h(-t) and the magnitude of its frequency response |G(f)|=^?|F(g(t))|.

2. Symbol-by-Symbol Detection

When the channel is ideal (c(t) = δ(0)), the composite channel response g(t) is a raised-cosine pulse. Thus, the output of the matched filter r[k] contains no ISI; we return to an AWGN channel where the optimal detector can detect the sequence of transmitted symbols one by one. In this symbol-by-symbol detection, the output of the matched filter is compared with the signal points in the constellation using the minimum-distance criterion. This, as will be shown in the class, is not optimal when the channel is not ideal (thus the output of the matched filter contains ISI).

(a) Let g[i] = g(iT). Show that the output of the matched filter r[k] can be written as

r[k] = _i=-2∑²g[i]x[k - i] + w[k],                      (1)

i.e., g[i] = 0 for |i| > 2. Express g[i] (|i| ≤ 2) as functions of β_i. Show that

w[k] = β₀n[k] + β₁n[k - 1] + β₂n[k - 2]     (2)

where n[k] is a zero mean Gaussian i.i.d. sequence with variance N₀/2. Is w[k] an i.i.d. sequence?

(b) For the parameters given in 1.(a)-ii, generate a sequence of r[k] of 1000 samples based on (1) and (2). Plot r[k] for N₀ = 0 (no noise) and N₀ = 1. If there were no ISI nor noise, we should have two distinct points at -1 and 1. Observe the effect of ISI.

(c) Given g[0], the symbol-by-symbol detector that ignores ISI is given by

xˆ[k] = arg minx_[k]∈{-1,1}|(r[k]/g[0]) - x[k]|²

For the parameters given in 1.(a)-ii, and SNR = E_b/N₀ in the range of 0:2:10dB, provide two curves in a single plot: (I) the analytical symbol error rate of BPSK vs. SNR when the channel is ideal AWGN, and (II) the simulated symbol error rate of the above detector vs. Eb/N0 (dB). Use log scale for the symbol error rate. What is the performance loss due to ISI?

3. Optimal Receiver: MLSD

Consider the same binary communication system given above. Here we implement the Viterbi algorithm for the detection of the entire sequence.

(a) Define the state variables and draw one stage of the trellis diagram.

(b) For SNR=0:2:10dB, simulate the maximum likelihood sequence detector (MLSD) using Viterbi algorithm by detecting blocks of 100 symbols at a time with 500 blocks. Provide three curves in a single plot:

i. the analytical symbol error rate of BPSK vs. SNR when the channel is ideal AWGN;

ii. the simulated symbol error rate of the symbol-by-symbol detector vs. SNR (dB) (results you obtained in Project 3);

iii. the symbol error rate of the maximum likelihood sequence detector (MLSD).

Use log scale for the symbol error rate.