# Ece 6604 Final Exam

**Topics:**Probability theory, Probability density function, Random variable

**Pages:**5 (663 words)

**Published:**December 1, 2012

School of Electrical and Computer Engineering

ECE6604 Personal & Mobile Communications

Final Exam

Spring 2010

Tuesday May 6, 11:30am - 2:20pm

• Attempt all questions.

• All questions are of equal value.

• Open book, open notes, exam.

1a) 5 marks: The LCR at the normalized threshold ρ for a 2-D isotropic scattering channel can be expressed as

√

2

LR = 2πfm ρe−ρ ,

where

ρ=

Δ

and Rrms =

R

Ωp

=

R

Rrms

E[α2 ] is the rms envelope level.

i) Find the normalized threshold level ρo at which the LCR reaches its maximum value. ii) Explain why the LCR at ρ decreases as ρ deviates from ρo . 1b) 5 marks: Consider a cellular system with a carrier frequency of 2 GHz. Suppose that the user is in a vehicle travelling at 60 km/h. Assuming that the channel is characterized by 2D isotropic scattering, ﬁnd

i) the LCR at the normalized level ρ = −3 dB.

ii) the AFD at the normalized level ρ = −3 dB.

2) The power delay proﬁle for a WSSUS channel is given by φgg (τ ) =

0.5[1 + cos(2πτ /T )] ,

0,

0 ≤ τ ≤ T /2

otherwise

a) 3 marks: Find the channel frequency correlation function. b) 4 marks: Calculate the mean delay and rms delay spread.

c) 3 marks: If T = 0.1 ms, determine whether the channel exhibits frequencyselective fading to the GSM system.

3) Cellular CDMA systems use soft handoﬀ, where the transmissions to/from multiple base stations are combined to give a macro-diversity. Here we consider the eﬀects of path loss and shadowing and ignore multipathfading. Suppose that the received signal power corresponding to the link with the ith base-station, Ωpi , has the probability density function pΩpi

(x) = √

(dBm)

(x − μΩpi (dBm) )

1

exp −

2

2σΩ

2πσΩ

2

.

where

μΩ pi

(dBm)

= E[Ωpi

(dBm) ]

The Ωpi are assumed to be statistically independent.

a) 5 marks: The reverse link uses selection combining such that the best basestation is always selected. In this case, Ωs

p

(dBm)

An outage occurs if Ωs

p

= max Ωp1

(dBm)

≤ Ωth

(dBm) , . . . , ΩpL (dBm)

(dBm) .

What is the probability of outage?

b) 5 marks: The forward link uses coherent combining such that Ωmr(dBm) = Ωp1

p

(dBm)

+ . . . + ΩpL

(dBm)

Again, an outage occurs if Ωmr(dBm) ≤ Ωth (dBm) . What is the probability of p

outage if

μΩp1 (dBm) = μΩp2 (dBm) = · · · = μΩpL (dBm) ?

4) Consider the reception of a signal in the presence of a single co-channel interferer and neglect the eﬀect of AWGN. The received signal power, C , and interference power, I , due to Rayleigh fading have the exponential distributions 1 −x/C

¯

¯e

C

1

¯

pI (y ) = ¯e−x/I

I

pC (x) =

¯

¯

where C and I are the average received signal power and interference power, respectively.

a) 5 marks: Assuming that C and I are independent random variables, ﬁnd the probability density function for the carrier-to-interference ratio λ=

C

.

I

Hint: If X and Y are independent random variables, then the probability density function of U = X/Y is

pU (u) =

pXY (v, v/u)|v/u2 |dv .

b) 5 marks: Now suppose that the system uses 2-branch selection diversity. The branches are independent and balanced (i.e., the distribution pU (u) is the same for each branch. What is the probability density function of λ at the output of the selective combiner?

5) Suppose that a system uses selection diversity. The branches experience independent Rayleigh fading. However, the average received bit energy-to-noise ratio on each diversity branch is diﬀerent, such that

γi = 2−i γo

¯

i = 1, . . . , L

a) 5 marks: Find the probability density function of the bit energy-to-noise ratio at s

the output of the selective combiner, denoted by γb .

b) 5 marks: If DPSK modulation is used, write down an expression for the probability of bit error. Obtain a closed-form expression if possible; otherwise leave your expression in integral form....

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