Description
1. (a) The random variable Z is a decision variable for the binary detection in a receiver,
and is given by the following uniform probability density function (pdf) fZ(z) as
shown below.
𝑓(𝑧) = ൜
𝑘 𝑎ଵ ≤ 𝑧 ≤ 𝑎ଶ
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
where k, a1 and a2 are fixed parameters.
(i) Determine the value of k for this pdf.
(ii) Let the value of a1 = ‒1 and a2 = 2, then calculate the value of P( |𝑍 |≤ ½).
(iii) Find the mean value and variance of Z.
(8 Marks)
(b) A Wide Sense Stationary (WSS) random process X(t) has a Power Spectral
Density (PSD) SX(f). It is applied to a differentiator to produce the output
derivative random process Y(t) as shown in Figure 1.
Figure 1
(i) The autocorrelation function RX(τ) = E[X(t)X(t+τ)]. Show the
autocorrelation function RY(τ) of the output Y(t) is related as shown
𝑑
𝑑𝑡 Y(t) X(t)
2
below. Determine the expression for the output power spectral density
SY(f) of Y(t).
RY(τ) = −
ௗ
మோೣ(ఛ)
ௗఛమ
.
(ii) If the autocorrelation function of X(t) is given by RX(τ) = 16 sinc(4τ),
determine the PSD of Y(t) and calculate its average power. Explain the
significance of your results for higher frequency component of the output
Y(t).
(12 Marks)
2. (a) If the Fourier transform of an energy signal x t( ) is denoted by X f ( ), then
2 2
| ( ) | | ( ) | x t dt X f df
.
State the physical meaning of the above relation. Provide the details of the
derivation.
(8 Marks)
(b) Let
2 2 1 ( ) sinc sinc b
b b b b
t t T
x t
T T T T
,
where Tb
is a constant, is the convolution operator and
sin( ) sinc( ) t
t
t
.
Find the spectrum X f ( ) of x t( ) and determine its energy Ex
.
(6 Marks)
(c) The signal x t( ) is sampled by an ideal sampling function
( ) ( )
s
n
x t t nT
,
where Ts
is the sampling period and ( )t is the unit impulse function. Design a
low-pass filter (LPF) to recover the desired signal x t( ) without distortion. What
are the bandwidth and the gain of the LPF? Draw the transfer function H f ( ) of
the LPF for illustration,
(6 Marks)
3. (a) The random variable X is normally distributed with probability density
function (PDF)
2 1 ( 1) ( ) exp
2 2
X
x
p x
.
=
3
Obtain the mean and variance of the random variable X by observation. Suppose
X is applied to an electronic limiter with output Y characterized by
3 / 2, 2,
3, 2.
x x
Y
x
Plot the graph of Y versus X. Determine the PDF of the output Y of the limiter.
(10 Marks)
(b) The random variable V has a cumulative distribution function (CDF)
2
( ) Pr (1 ) ( ) v F v V v e u v V
,
where u(v) is the unit step function. Plot the curve of ( ) F v V
versus v and find the
PDF of V.
Define a new random variable W in terms of V by
2
( ) (1 ) ( ) V W F V e u V V
.
Determine the CDF ( ) F w W
of W for w 0 , 0 1 w and w 1. Finally, plot
the CDF ( ) F w W
and obtain the PDF of W.
(10 Marks)
4. Consider a binary signal detector with the input
r a n ,
where the signal component may be a or a with equal probability. The noise
component n is described by the probability density function (PDF)
1
( ) exp 2 | |
2
n p n n .
(a) Plot the conditional PDFs ( | )
r p r a and ( | )
r p r a together. Determine the
optimum threshold 0 of the binary signal detector and label the position of 0
clearly on the same graph.
(5 Marks)
(b) Given that the signal component a is received at the detector, derive the
probability of bit error, i.e.,
Pr error | Pr | 0 a r a .
(6 Marks)
(c) Compute the overall probability of bit error of the binary signal detector.
(6 Marks)
(d) Suppose that the noise PDF is replaced by a Gaussian PDF with the same mean
and variance. Without going through the detailed calculation, will the overall
probability of bit error of the binary detector be bigger or smaller? Justify your
answer.
(3 Marks)
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