Download 1. By consecutive tossing of coin twice, explain the meaning of

1. By consecutive tossing of coin twice, explain the meaning of probability space, random
variable, distribution function, and probability density function. (15%)
Probability measure
Define P as a function mapping P: £→R, and P satisfy the following axioms.
(a) P(A)≧0, where A is an event and P(A) is called the probability of the event.
(b) P(Ω)=1.
(c) P(A∪ B)=P(A)+P(B) provided that A,B∈£ and A∩B= Φ,Φ is called the impossible event.
Random variables
In a probability space (Ω，£，P), X:Ω → Rn is a random variable if and only if X is measurable
w.r.t the field £.
Distribution function
The function F(x) = P{ω|X(ω) ≤ x} is defined as the distribution function of X(ω).
Probability Density Function
p(x) is called the probability density function of x(ω) if
x
F(x) = ∫ p(y)dy
−∞
If F(x) is differentiable w.r.t. x then
p(x) = dF(x)/dx
2. Find and sketch the autocorrelation function of a square wave,
1 T2
Rx ( )   T x(t )  x(t   )dt , T : period
T 2
(10)%
<ans> The autocorrelation function in time domain average is
1 T2
Rx ( )   T x(t )  x(t   )dt
T -2
where T is the period of the square wave and τis the time shift. And we let x(t) & x(t+τ)
be
X(t+τ)
X(t)
a
X(t)˙X(t+τ)
-a
T/2
a
a2
t
t
-τ
T
T/2
-a
t
T/2
T
T
-a2
Then we can find that x(t)˙x(t+τ) will be
Therefore, the autocorrelation function of the square wave would be
where a is the amplitude of x(t). When T/2≦τ≦T, that would be
Rx ( ) 
1
T
a2
τ
[2  ( + )  a 2 + 2    a 2 ]  (T +4 )
T
2
T
(-T/2≦τ≦0 )
Rx ( ) 
1
T
a2
[2  (   )  a 2  2    a 2 ] 
(T  4 )
T
2
T
( 0≦τ≦T/2 )
Rx(τ)
a2
T/2
-a2
T/4
3. For a Rayleigh distribution given as
p ( x)  N  e
 ( x-x
2 x 2
2
)
Find the multiplication of N. (10%)
<ans>




0
0
p( x)dr  1
 (x- x )2
e
2 x 2
dx  1/N
令 y  x- x ,
1
2 x 2
c
τ
T


0
 (x- x )2
e
N
2 x 2
c

dx  e cy dy 
2


c
1
2 x 2
4. Give the definition of Gaussian white process. What are the role and use of
Gaussian white process? (10％)
<ans>
Time domain:
A Gaussian process v (t ) define on {Ω，£，P} is a white process if its mean and covariance
functions are given by
(1) Zero mean value: E[v(t )]  0
(2) Covariance function E[v(t )  v( s )]  Q (t  s )
Frequency domain:
Power spectrum is Fourier transform of autocorrelation function.
constant
£s
Sxx ()w
Rxx ( )
£n
£s
Role of Gaussian white process
(a) Mathematics: a model of ideal random signal source
(b) Physics: a model of physical noise
(c) Engineering: signal for dynamic testing
Physics
Math
Eng.
Phenomenon
Ideal Analytical
Model Testing signal
5.What is an aliasing problem in signal processing? Propose a method considering practical
situation to avoid aliasing?(15%)
<ans>取樣頻率不足造成的錯誤，取樣頻率需要大於 2w，其中 w 為 Nyquist 取樣頻率，可
加上 anti- aliasing。
6. Two independent random variables x and y have Gaussian probability density function
of N(0, 1) and N(1, 2) ,respectively. Determine the mean and variance of the random
variable u  x 4 ( y  1)2 . (15%)
z = y − 1=>z ,N(0,2)
μ = E[u] = E[x 4 (y − 1)2 ] = E[x 4 z 2 ] = E[x 4 ]E[z 2 ]
u
= 3σ4 (μx 2 + σz 2 ) = 3 ∗ 1 ∗ (0 + 2) = 6
E[u2 ] = E[x 8 (z)4 ] = E[x 8 ]E[z 4 ] = (1 ∗ 3 ∗ 5 ∗ 7 ∗ σx 8 ) ∗ (3σz 4 ) = 105 ∗ 18 ∗ 3 ∗ 22 = 1260
2
2
2
2
u
u
σ = E [(u − μ ) ] = E [u2 − 2uμ + μ ] = E[u2 ] − μ = 1260 − 62 = 1224
u
u
u
7. Describe an algorithm for generating uniform random number with range from 1to 3.
(10%)
<ans>參照作業二
8. Give an example and explain the specifications of an instrument system. (10%)
Fundamental structure：
Static and Dynamic characteristics：
(1) Static
(2) Dynamic