EE602
Assignment 1
August 8, 2015
1. Consider the experimental setup shown in the figure. The problem is to observe x and estimate
s. Redraw this diagram in the form of Figure 1.2 (Scharf) to show that it fits our structure
of statistical reasoning. Can you describe an experiment where this diagram applies?
2. Consider the Estimator discussed in class (Section 1.4 of Scharf). Assume that θ is an unknown constant and the noises are drawn from a sequence of i.i.d. N [0, σ 2 ] random variables.
(a) Show that the estimator θ̂N is distributed as follows
θ̂N : N (θ,
σ2
)
N
(1)
σ2
)
N
(2)
(b) Show that the estimator error is distributed as
N : N (0,
(c) Show that the probability that |N | exceeds (> 0) is
Z
P [|N | > ]
=
2
√
−( σ ) N
−∞
where
Z
1
2
√ e−x /2 dx
2π
η
Φ (η) =
−∞
=
√ 2Φ −
N
σ
1
2
√ e−x /2 dx
2π
(3)
(4)
3. Figure shows p radiating sources ri with i = 1, 2, ...p transmitting plane waves that are sensed
by N sensors labeled S0 , S1 , ...SN −1 . Sensor S0 is placed at the center of the coordinate system
and the coordinate of sensor Sn is zn .
The source transmits a propagating wave whose complex representation is
T
rm (t, z) = Am ej (wm t−km z )
(5)
The scalar frequency wm , is the radian frequency of the source, and the vector km is the wave
number for the source. This wave number may be written as km = (2π/λm )dm , where dm is
the vector of direction cosines.
The scalar waveform sensed by sensor Sn is the sum of all signals rm (t, z) read at z = zn :
xn (t) =
p
X
m=1
1
rm (t, z)
(6)
EE602
Assignment 1
August 8, 2015
Express the received waveform vector X as




A1 ejw1 t
x0 (t)
A2 ejw2 t 
 x1 (t) 




 ...  = H  .. 




 .. 
 ... 
Ap ejwp t
xN −1 (t)
(7)
where column hm m = 1, 2, ....p of matrix H characterizes the phase delays of source rm to
each of the sensors Sn n = 0, 1, 2....N − 1
If samples are taken in time intervals of T at each sensor, then show that


x0 (0)
.. .. ..
x0 ((M − 1)T )


..
.. .. ..
..



 = HAT
..
.. .. ..
..




..
.. .. ..
..
xN −1 (0) .. .. .. xN −1 ((M − 1)T )
(8)
Where A is the diagonal matrix containing the source amplitudes Am and T is a row Vandermonde matrix determined by the source frequencies wm .
4. Let {xk (ξ)}4k=1 be four IID random variables with exponential distribution (P.1, Manolakis,
Chapter 3) with a = 1. Let
yk (ξ) =
k
X
xl (ξ)
l=1
(a) Determine and plot the pdf of y2 (ξ)
(b) Determine and plot the pdf of y3 (ξ)
(c) Determine and plot the pdf of y4 (ξ)
2
1≤k≤4
(9)
EE602
Assignment 1
August 8, 2015
(d) Compute the pdf of y4 (ξ) with that of gaussian density
5. The Cauchy distribution with mean µ is given by
fx (x) =
1
1
π 1 + (x − µ)2
−∞<x<∞
(10)
Let {xk (ζ)}N
i=k be N IID random variables with the above distribution. Consider the mean
estimator based on {xk (ζ)}N
i=k
N
1 X
µ̂(ζ) =
xk (ζ)
(11)
N
k=1
Determine whether µ̂(ζ) is a consistent estimator of µ.
6. The mean and the covariance of a Gaussian random vector x are given by, respectively,
1
1 21
(12)
µx =
and Γx = 1
2
2 1
Plot the 1σ, 2σ and 3σ concentration ellipses representing the contours of the density function
in the (x1 , x2 ) plane. Hints: The radius of an ellipse with major axis a (along x1 ) and minor
axis b < a (along x2 ) is given by
r2 =
a2 b2
a2 sin2 θ + b2 cos2 θ
(13)
√
√
where 0 ≤ θ ≤ 2π. Compute the 1σ ellipse specified by a = λ1 and b = λ2 and then rotate
h
i
(i)
(i) T
and translate each point x(i) = x1
x2
using the transformation w(i) = Qx x(i) + µx .
7. A causal LTI system, which is described by the difference equation
1
1
y(n) = y(n − 1) + x(n) + x(n − 1)
2
3
(14)
is driven by a zero-mean WSS process with auto-correlation rx (l) = 0.5|l| .
(a) Determine the PSD and auto-correlation of the output sequence y(n).
(b) Determine the cross-calibration rxy (l) and cross-PSD Rxy (ejω ) between the input and
the output signals.
(Illustrate Results in MATLAB)
8. Show that
n X
n
E[(aX + b) ] =
an−i bi E(X n−i )
i
n
(15)
i=0
9. Consider a normal random vector x (ξ) with components that are mutually uncorrelated, that
is, ρij = 0. Show that (a) the covariance matrix Γx is diagonal and (b) the components of
x (ξ) are mutually independent.
3
EE602
Assignment 1
August 8, 2015
10. Determine whether the following matrices are valid correlations matrces:
(a)
R1 =
(c) R3 =
1
1
1
1+j
1
1

1
(b) R2 = 
1
2
1
4
1-j
1
(d)
1
4
1
2
1
2
1
1
2

1
R4 = 
1
2
1


1
1
2
2
1
1

1
2

1
11. A WSS process with PSD Rx (ejω ) = 1/(1.64 + 1.6 cosω) is applied to a causal system
described by the following difference equation
y(n) = 0.6y(n − 1) + x(n) + 1.25x(n − 1)
Compute (a) the PSD of the output and (b) the cross-PSD Rxy (ejω ) between input and
output.
12. For each of the following, determine whether the random process is (1) WSS or (2) m.s.ergodic
in the mean.
(a) X(t) = A, where A is a random variable uniformly distributed between 0 and 1.
(b) Xn = A cosω0 n, where A is a Gaussian random variable with mean 0 and variance 1.
(c) A Bernoulli process with Pr[Xn = 1]=p and Pr[Xn = −1]=1 − p
4