Advanced Communication Systems

Q1. Consider transmission of QPSK over a narrowband Rayleigh fading channel such that,

y(t) = R(t)s(t) + n(t),

where, y(t) is the channel output,

R(t) is the Rayleigh fading,

s(t) is the QPSK signal,

and,

n (t) is the zero mean, white Gaussian noise with power spectral density N_o/2 Watts/Hz.

Q2.

Consider the case where the channel fading over a channel is slow with respect to a symbol period and and the Rayleigh fading can be considered to be constant over the symbol period. Then, during a symbol time,

y (t ) = Rs (t ) + n(t), where, R is a Rayleigh random variable.

In general, has a probability density function given of R by,

f_R(r ) = r/σ²exp(-r²/2σ²)u[r], where, u[r ] = 1,  r ≥ 0,

                                                                     0,  r < 0.

The fade pameter R modifies the envelope of the signal.

Let α = R².  α has the exponential probability density function given by f_α(a) = 1/2σ² exp(-a/2σ²)u[a].

(i) Prove that for σ² = 1/2, then,

E [α] = E [R²] = 1,

f_R (r) = 2r exp(-r²)u[r ],

f_α (a) = exp(-a)u[a],

and, for constant E_b/N_o, E[α.E_b/N_o] = E_b/N_o

(ii) Justify why for a Rayleigh fading channel with average E_b/N_o the average bit error probability P_B is given by P_B = ₀∫^∞Q (√(2E_b/N_o.α)f_α(a)da, where (E_b/N_o) is the average energy per bit to one-sided noise density ratio and (E_b/N_o.α) is the energy per bit to one-sided noise density ratio and α is an exponentially distributed random variable with probability density function given by f_α(a) = exp(-a)u[a].

(iii) Let γ‾_b= E_b/N_o and γ_b = E_b/N_o.α

Using part (ii) verify that P_B = ₀∫^∞Q(√(2γ_b)).f_γb(γ_b)dγ_b'

where, fγ_b(γ_b) = 1/γ‾_b.e^-γb/γb.u[γ_b]

(iv)  It can be shown that,

₀∫^∞Q (√x)e^-x/η/η.dx = 1/2[1 - √(η/(2+n))]

Use this result to show that P_B = ₀∫^∞Q(√2γ_b)f_γb(γ_b)dγ_b= 1/2[1 -√(γ‾_b/(1+γ‾_b)]

(v) Show that for large γ‾_b, P_B ≡ 1/4γ‾_b

Question 2

A Rayleigh random variable has the probability density function given by,

f_R(r) = r/σ².exp(-r²/2σ²).u[r],  where, u[r] = 1,    r≥0,

                                                                     0,    r<0,

Let σ² = 1/2 such that, f_R (r) = 2rexp(-r²)u[r]

An objective for simulation is to generate a sequence of Rayleigh random variables with σ² = 1/2. Using the rand function in Matlab, it is possible to generate a sequence of uniformly distributed random variables taken from the distribution f_U(u) = 1, for 0 ≤ x ≤ 1.

(i) If U is uniformly distributed in [0,1], prove that,

R = √log_e[1/1-U] has the Rayleigh distribution, f_R(r ) = 2rexp(-r²)u[r]

(ii) Use Matlab to generate 1000 values of the Random Variable R derived from uniformly generated values of U which are generated using the rand function. Using a histogram, verify that the values of R have the correct Rayleigh distribution.

(iii) Modify the QPSK for a white Gaussian noise channel simulation in the lecture notes to a Rayleigh fading channel with additive, zero mean white Gaussian noise.

Generate a new fade variable, R = √(log_e[1/1-U]) for every QPSK symbol.

Plot P_B as a function of E_b/N_o (dB), where E_b/(d_B) is the average ratio of energy per bit to noise power spectral density for the following cases:

1. The simulation result for P_B.

2. The analytic result for P_B == 1/2[1 -√(γ‾_b/(1 + γ‾_b))], γ‾_b = E_b/N_o

3. The analytic result for P_B ≡ 1/(4γ‾_b), γ‾_b= E_b/N_o

Plot the results for -10 dB ≤ E_b/N_o (dB)0 ≤ 20 dB

(iii) Discuss the impact upon communications system design created by Rayleigh fading channels in relation to additive white Gaussian noise channels. Discuss the requirements placed upon elements of the wireless communications system including antennas, transmit power, low noise amplifiers, and other equipment considerations. Discuss the limitations imposed upon supported average transmission data rate by Rayleigh fading.

Also, discuss possible methods of maintaining reliable communications with an average data transmission rate on a wireless channel with deep signal fades.