3.6 Exercises 3.1 (Known zero). Suppose that the process is known to have a zero at z = γ, i.e., ...
2.4 Let (e) be a zero mean white noise process. Suppose that the observed process is Y = e, + 0,-1, where is either 3 or 1/3. (a) Find the autocorrelation function for {Y} both when 0 = 3 and when 0 = 1/3. (b) You should have discovered that the time series is stationary regardless of the value of and that the autocorrelation functions are the same for 0 = 3 and 0 = 1/3. For simplicity, suppose that...
3. Suppose you have X Binom(n, p) where n is known and p is unknown. Typically, people use p = _ to estimate p, where X = Xi is simply a sample of size l. This might represent simultaneously flipping n coins (just once!) and counting the number of heads you see, where each coin has Pheads - p. Now, if both n and p are known, we know the variance V of X is just np(1-p) If p is...
1. Pole-zero placement. We wish to design a stable and causal second-order discrete-time (DT) filter (i.e., having two poles and two zeros, including those at 0 and oo) using pole-zero placement. (a) [5 pts] Where might you place the poles and zeros to achieve the following magnitude frequency response? Sketch the pole-zero plot in the complex z-plane. -Π -Tt/2 0 (b) [3 pts] Give an expression for the transfer function H(z). Justify your answer. (c) [2 pts] Write an expression...
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Exercises with Vectors-II Name [1] Suppose you have two vectors, a and b, that have the same length, so that lal-lb but they point in different directions. Denote the angle between them by . Show that tan(0/2) la-bMa+b Hint: Compute the right-hand side using the fact that lal-bl, and the trig identities 1-cos θ-2sin'(9/2) and 1+cos9-2cos(θ/2) 12] Vectors in 3-dimensions are often parameterized in terms of their length and two angles, as shown in the figure (think of a...
E. If you double the value of kp, what are the new closed-loop pole locations and [5 points] how much overshoot does the step response have? Hint: It is possible to determine the original value for kp. However, with the knowledge at this point, you can compute the pole locations without actually knowing kp (simply double the zero-order term in the denominator polyno- mial). Problem 2 You are confronted with a process that has the unknown transfer function G(s). It...
2. Let (et) be a zero mean white noise process with variance 1. Suppose that the observed process is h ft + Xt where β is an unknown constant, and Xt-et- Explain why {X.) is stationary. Find its mean function μχ and autocorrelation function p for lk0,1,.. a. b. Show that {Yt3 is not stationary. C. Explain why w. = ▽h = h-K-1 is stationary. d. Calculate Var(Yt) Vt and Var(W) Vt . (Recall: Var(X+c)-Var(X) when c is a constant.)...
Please answer both exercises and show work.
Pre-Lab Exercises Exercise 1 Suppose there is a uniform electric field pointing in the positive a-direction with a magnitude of 5.0 V/m. The electric potential is measured to be 50 V at the position x = 10 m, what is the electric potential at other positions? Position [m] -200.00|10 1199 Electric Potential VI Exercise 2 Suppose there is a non-uniform electric field and you measure the electric potential at two different locations along...
Suppose f(z) is a holomorphic function in a domain U, and z0 ∈ U. Prove that f has a zero of order m at z0 if and only if f(z) = g(z)(z − z0)^m, where g(z) is holomorphic in U and g(z0) not equal to 0. Please prove both directions of the if and only if statement and use series expansion to prove. We have not learned calculus of residues yet.
a) List the relative attributes of using digital processing techniques compared to traditional analogue hardware for signal processing. [5 marks] b) Sketch a z-plane diagram including the unit circle. You have four Poles and two Zeros that you can place on the z-plane diagram. Place them in a position which would provide a digital band-stop filter characteristic with the 'notch' at a n/2 Justify your placement of the poles and zeros. 5 marks] c) The z-plane pole-zero plots of two...
Question 3: A problem with analagous mathematical structure to electrostatics (12 marks) In the static (i.e. time-independent) case, the temperature throughout a material can be described by the scalar-valued function T(x, y, z). In addition, the flow of heat can be described by the vector-valued function h(x, y, z), the magnitude of which indicates the amount of heat energy that flows per unit area per unit time and the direction of which is that of the heat flow. (a) (2...