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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