3. Write the following signal as a combination (sums or products) of scaled and shifted unit...
Linear Systems
(3) Write an expression for the following functions using shifted and scaled versions of the unit step function u(t)
1. Consider the signal represented in Figure 1. a) Write an analytical formula to represent the signal x(t). b) Consider the signal x(t). For each of the following combinations, draw the signal and indicate its duration, extension and area: Xa(t) = x(t – 3); xo(t) = x(2t); Ic(t) = x(-3(t + 1)); Id(t) = 2x(2t) – x(t) – 2x(-(t + 4)); Le(t) = xd(t) + rect(476). c) Compute the Fourier transform of r(t). AX(t) 2 1 2 5 7 t...
(b) The signal f(t) is shown in the figure below 3 2 f(t) _ 0 I 1 -4 -3 -2 -1 0 1 2 3 4 5 6 7 t and is given by 21 (1) + 3A (132), where A is the triangle function defined as 10-{ It a It <a It > a 0 Write the Fourier transform F [A(t)] (s) of f(t) exploiting the fact that FA(t)](s) = sinc-(s) where sin(TTS) sinc(s) ITS and the theorem for...
3. (45 pts) On signal energy and power. From the following signals, identify energy signals and power signals. For energy signals, calculate their energy. For power signals, calculate their average signal power. (g) x(t)= rect(t)) (h) x(t) =Loo rect(A) (i) 2(t)=e(-1-j80%(t) (k) x(t) = e-M/2 (l) x[n] = e-jm/2
4. Riemann sums - ten rectangles same area A) with ten rectangles in Repeat problem 3 (same function f(x) = place of five: a) Draw ten rectangles to visualize a Riemann sum of your choice for the area A. b) Give an estimate of the area A using the Riemann sum (the sum of the areas of the ten rectangles). 3. Riemann sums-five rectangles a) Sketch the graph of the function f(x) = b) Sketch the area A bounded by...
(VII). Compute the following sums and products expressions. (Note: write only one equation in each line.) (1) ?(2i +3) -1 (2) II (2i+3) in 1 (3) ? ?(2i +3j) (2i +3j) 2500 (5) ? (2i +3) ?2000
Matrix Algebra
3. indicated sums and products are defined. Mark each statement True of False. Justify each The following questions concern arbitrary matrices A, B, and C for which the answer (a) If A and B are 2 x 2 matrices with columns ai, a2 and bi, b2, respectively, thern 101 a2b2] (b) Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A. (c) AB + AC-A(B +C (d)...
5. Consider the signal a(t) shown below. x(t) (a) Write an equation for (t) in terms of time-shifted unit impulse functions. Note that in the plot the area of each impulse is indicated by its amplitude. This is a standard convention for plotting impulses. (b) What is the value of Lor(t)dt? (c) Make a plot of f r(1)dt ANSWERS: For part (b), the integral equals 1. For part (c), the plot has value 2 between t = 1 and t...
..3-1 In Fig. P1.3-1, the signal fi(t) = f(-). Express signals fa(t), f(t), fu(t), and f(t) in terms of signals f(t), fi(t), and their time-shifted, time-scaled or time-inverted versions. For instance f2(t) = f(t-T) + fit - T). f (1) f (1) 1,(0) f(t) 0 - 0 1 Fig. P1.3-1
2.22" Convert each of the following expressions into sum of products and product of sums: (a) (AB +C)(B + C'D) (b) x +x(x y)(+z')