Solve and select the correct answer: 1. The distribution function for random variable X with p(x) = 2e-2x for x>0 is- (a) 1-e-2x (b) e+2x (c) e-2x (d) 1 + e −2x 2. The system function for discrete-time causal system which described by the difference equation y (n) = 2x(n) – 3y(n - 1) is z (a) H(z): (z+1)² 2z (b) H(z) = = (z-1)² (c) H(z): = 2z z+3 2z (d) H(z) = (z+3)² 3. The Z-Transform of the sequence x(n) = {ne" u(n) } and its ROC is ze-1 (a) X(z): = (2-1)² ' |z|<1 (b)X(z) = = ze-1 (ze-1-1)² , z> e¹ (c) X(z) = = ze¹ ze¹-1 ·,|z|> e¯¹. (d) X(z) =,|z|< 1 4. The unit step response for LTI system which has impulse response h(t) = e¹ u(t) is. (a) (1-e) u(t) (b) (1+ e² ) u(t) (c) u(t) (d) e-tu(t) 5. The Eigen values for the matrix [3 10 A= are L2 4 (a) λ₁ = 8, λ₂ = 1 (b) λ₁ = -8, λ₂ = 1 (c) λ₁ = 8, λ₂ = -] (d) λ₁ = 8, λ₂ = -2

Solve d)

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1.37. An important concept in many communications applications is the correlation be- tween two signals. In the problems at the end of Chapter 2, we will have more to say about this topic and will provide some indication of how it is used in practice. For now, we content ourselves with a brief introduction to correlation functions and some of their properties. Let x(t) and y(1) be two signals; then the correlation function is defined as - L. b1,(1) = x(1 + T)y(7)dT. The function ,u(1) is usually referred to as the autocorrelation function of the signal x(t), while oy(t) is often called a cross-correlation function. (a) What is the relationship between 4,y(1) and øy»(1)? (b) Compute the odd part of øx:(1). (c) Suppose that y(t) = x(t + T). Express ,,(1) and Þyy(1) in terms of p«(1).

This question is split into parts from A to E please help with all parts since it's still one question

Figure in the below presents the direct form II realization of a difference equation. Assume that the resulting system is linear and time-invariant. v[n] x[n] yln] z1 1/3 -2 a) Find the direct form I realization. b) Find the difference equation described by the direct form I realization. c) Find the relation between v[n] and y[n]. Find the relation between v[n] and x[n]. Using these relations, verify that the relation between y[n] and x[n] in the direct form II realiza- tion is the same as your answer to part (b).

Sensor and systems