Part A - Pre-work: 12 Marks) A.1 Consider the continuous-time signals given in Figure 1. al)...
(b) Consider continuous-time signals xi(t) and x2(t) respectively given as (t +1 -1 <t<o x1(t) = { 2 Ost<2 , I 0 otherwise x2(t) = u(t) – uſt – 2). Find the convolution xı(t) * x2(t). (15 marks)
There are 3 questions on this assignment. The marks awarded for each part are indi- cated in boxes. 1. Consider the function defined by f(x) = 0 and f(x)-f(x +4) 1 (a) Sketch the graph of f(x) on the interval -6,6 (b) Find the Fourier series representation of f(z). You must show how to evaluate any integrals that are needed 2. Consider the function f(x) (a) Sketch the odd and even periodic extension of f(x) for-3< x < 3m (b)...
solve 2.40 a,b,c, e using Fourier series. 2.40 part a,b,c,e 2.40 Consider the continuous-time signals depicted in Fig. P2.40. Evaluate the following convolution integrals: (a) m(t) x(t) y(t) (b) m(t)x(t)z(t) (c) m(t) x(t) ft) (d) m(t) x(t) a(t) (e) m(t)y(t) z(t) (f) m(t) -y(t) w(t) (g) m(t) y(t)g(t) (h) m(t)y(t) c(t) (i) m(t) z(t) f(t) (j) m(t) z(t) g(t) (k) m(t) z(t)b(t) (1) m(t) w(t) g(t) (m) m(t) w(t) a(t) (n) m(t) f(t) g(t (o) m(t) fo) . do) (p)...
Consider a continuous-time LTI system S with impulse response h(t) = 2(u(t + 1)-u(t 1)). Determine the values of the amplitude scaling and the tme shifting that takes place when each of the following input signals is provided to the system S. Don't use the convolution integral, instead use the result about how LTI systems respond to complex exponential signals. (a) x(t) 2 (b) x(t) ej0.5Tt (c) x(t) = e-j0.5πt (d) x(t) = e-jmt (e) x(t) = cos (0.5t) (f)...
Problem 1: Consider the continuous-time signal r(t) as shown in Figure 1. r(t) Figure 1: A continuous-time signal r(t) (a) Determine the fundamental period and the fundamental angular frequency of r(). 5 (b) Write down the equation for z(0) as the Fourier Series in exponential form and identify (c) Sketch the spectrum of this signal indicating the complex amplitudes and the frequen- points the Fourier Series coefficients. (15 points cies. [10 points
(a) Determine the Fourier transform of x(t) 26(t-1)-6(t-3) (b) Compute the convolution sum of the following signals, (6%) [696] (c) The Fourier transform of a continuous-time signal a(t) is given below. Determine the [696] total energy of (t) 4 sin w (d) Determine the DC value and the average power of the following periodic signal. (6%) 0.5 0.5 (e) Determine the Nyquist rate for the following signal. (6%) x(t) = [1-0.78 cos(50nt + π/4)]2. (f) Sketch the frequency spectrum of...
Consider the following continuous-time signal. x(t) = 1 for m < t < m + 1 −1 for m + 1 < t < m + 2 for m = − 4,−2, 0, 2, 4, · · · Sketch the following signals. (a) x(t) (b) y(t) = 2x(2t − 1) + 1
(1) Consider the following continuous-time signal: (1) 2ua(-t+t)ua(t), where its energy is 20 milli Joules (2 x 103Joules). The signal ra(t) is sampled at a rate of 500 samples/sec to yield its discrete-time counter part (n) (a) Find ti, and hence sketch ra(t). (b) From part (a), plot r(n) and finds its energy (c) Derive an expression for the Fourier transform of a(n), namely X(ew). (d) Plot the magnitude spectrum (1X(e)) and phase spectrum 2(X(e). (e) Consider the signal y(n)...
For full credit, you must show all work and box answers 1. If functions f and g are piecewise continuous on the interval [0, oo), then the convolution of f and g is a function defined by the integral The Convolution Theorem (theorem 7.4.2 in your book and formula 6 in your table) states: If j(t) and g) are piecewise continuous on [0, oo) and of exponential order, then We are going to use convolution to solve y"-y,-t-e-,, y(0)-0, y'(0)-0....
For the continuous time signal() shown in the Figure 1, sketch each of the following signals: (a) x(t-3)-3 points(b) x(2-t)-5 points(c) x(2 t+2)-5 points(d) x(2-t/3)-4 points(e) x(t)(σ(t+3/2)-σ(t-3/2))-2 points(f) (x(t)+x(2-t)) u(1-t)-6 points(g) x(t/2-3)+x(t/3-2)-6 points(h) x(t-1) u(t-1)-4 pointsσ(t) is the impulse function and u(t) is the unit step function.