Design Project, ECE583

Exercise 1: Information conveyed by a discrete Gaussian channel [40%]

Consider a discrete-time memoryless channel

yk =

√

SNRxk + nk, k = 1, 2, . . .

where n ∼ CN (0, 1) and E|X|2 = 1 so that SNR denotes the average signal-to-noise ratio at the

receiver. According to Shannon, if the input X is drawn according to a Gaussian distribution, then

the capacity per real dimension is

C =

1

2

log2(1 + SNR) [bits/real dimension].

In practice, however, we cannot use a Gaussian input. In this exercise, we are going to study the

effect of having finite-alphabet constellations on the amount of information conveyed by this channel.

a) What is the conditional PDF of the channel, i.e., pY |X(y|x) for 1) a real constellation 2) a

complex constellation?

b) For a general constellation of size N , derive a computable formula for mutual information

I(X;Y ) [1], i.e.,

I(X;Y ) = EX,Y

(

log2

pX,Y (x, y)

pX(x)pY (y)

)

.

[Hint: The expression should be in terms of SNR, N , the constellation points and pX(x), i.e.,

the input distribution.]

c) Let ΩX denote the set of constellation points. Assume that the input is uniformly distributed,

i.e.,

pX(x) =

1

N

∀x ∈ ΩX .

Plot the average mutual information I(X;Y ) (in bits per real dimension) versus the SNR (from

−10 dB to 40 dB with a 2-dB step) for a 2-PAM, 8-PAM, 16-PAM, 64-PAM, 16-QAM, and 64-

QAM constellations in a single figure. Also, in the same figure, plot the capacity and compare

it with other mutual information curves that you obtained.

d) Consider a 16-QAM constellation shown in Fig. 1. Let p, q, and r be the probability of white,

black and shaded dots, respectively. Find the relation between them. Also, for SNR = 15 dB find

optimal values (p∗, q∗, r∗) such that I(X;Y ) is maximized. Compare this mutual information

with the case p = q = r = 116 .

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ECE583: Digital Communications

Figure 1: Exercise 1 part d

Exercise 2: ISI and Detection [60%]

Consider a system, shown in Fig. 2, employing a 16-QAM modulation over an AWGN channel of the

form

yk =

√

SNRxk + nk, k = 1, 2, . . . ,M

where n ∼ CN (0, 1), E|X|2 = 1 such that SNR denotes the average signal-to-noise ratio at the receiver.

In Fig. 2, g(t) denotes the pulse shape, b(t) represents the channel impulse response and f(t) is the

filter at the receiver. Moreover, we have the transmitted signal of the form

s(t) =

∑

k

xkg(t− kT ).

• a) Non-ISI channel + symbol detection

In this part, you are asked to implement the system shown in Fig. 2 where b(t) = δ(t), i.e., a non-ISI

channel, and the detector is a minimum distance (symbol) detector.

1. Let the symbol time T = 1 and generate a random sequence of M = 104 16-QAM symbols.

2. Use a root-raised-cosine pulse g(t) with a 35% excess bandwidth over the time period [−20,+20]

with 10 samples per symbol to generate the transmitted signal s(t).

3. Pass the signal through the channel and the matched filter (matched with the pulse shape and

channel).

4. Sample the resulting signal at the appropriate time instances in order to generate the decision

variables {uk}.

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ECE583: Digital Communications

mapper ft

T

xhkxk yk

nt

st rtd

detbtgt

Figure 2: Exercise 2 part a and b

5. Now, use the minimum distance detector and plot the average BER versus SNR (0 dB to 26 dB

with a 2-dB step), i.e.,

Pe(SNR) =

1

M log2N

M∑

k=1

w(xˆk, xk)

where w(xˆk, xk) denotes the number of bits in error at time k and (in this example) N = 16.

6. If necessary increase the number of symbols M in order to see a better curve.

• b) ISI channel + symbol detection

In the model of the previous part, consider an ISI channel with

b(t) = δ(t) + βδ(t− T ), |β| < 1.

1. Let the symbol time T = 1 and generate a random sequence of M = 104 16-QAM symbols.

2. Use a root-raised-cosine pulse g(t) with a 35% excess bandwidth over the time period [−30,+30]

with 10 samples per symbol to generate the transmitted signal s(t).

3. Pass the signal through an ISI channel with β = 0.1 and the matched filter (matched with the

pulse shape and channel).

4. Sample the resulting signal at the appropriate time instances in order to generate the decision

variables {uk}.

5. For β = 0, 0.01, 0.1, 0.2, use a symbol detector and plot the average BER versus SNR (0 dB to

26 dB with a 2-dB step).

6. Explain why the detector in this part is not optimal.

7. Explain the behavior of the curves for different values of β.

References

[1] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory, vol. 28,

no. 1, pp. 55–67, Jan. 1982.

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