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(a) Let the correlation be defined as r (t) x(T) y (tT) dT T Express R...
3. Let x, and yn be the Fourier series coefficients of (t) and y(t), respectively Assuming the period of r(t) is To. express y, in terms of (a) y(t)-x(at), where a 0. dt 3. Let x, and yn be the Fourier series coefficients of (t) and y(t), respectively Assuming the period of r(t) is To. express y, in terms of (a) y(t)-x(at), where a 0. dt
(b) Let X(ju) denote the Fourier transform of the signal r(t) shown in the figure x(t) 2 -2 1 2 Using the properties of the Fourier transform (and without explicitly evaluating X(jw)), ii. (5 pts) Find2X(jw)dw. Hint: Apply the definition of the inverse Fourier transform formula, and you can also recall the time shift property for Fourier Transform. (c) (5 pts) Fourier Series. Consider the periodic signal r(t) below: 1 x(t) 1 -2 ·1/4 Transform r(t) into its Fourier Series...
Consider a signal x(t) = e-tu(t), and the signal y(t) below: dx(t) y(t) = 3e-33+ z(t – 5) + 5* dt Va) What is X(jw), the Fourier transform of æ(t)? b) Find the phase of the complex number X(j1). c) Find Y(jw), the Fourier transform of y(t). d) Find the magnitude of the complex number Y(j1).
Given that the Fourier transform of x(t) is 3e-jw x(jw) = (1 +ju) find the Fourier transform of the following signals in terms of X (jw). a. y(t) = e'*x(t – 2) b. y(t) = x(-3) c. y(t) = x(t)dt
(a) The MATLAB command trapz(x,y) computes the integral of the function y with respect to the 'variable of integration' r, i.e J ydr. Use MATLAB help to understand how trapz works. (b) Consider r(t) 1 for -1 t1, and r(t) 0 otherwise. Use the trapz command to compute and plot the Fourier transform of r (t). Denote this by X (jw). compute and plot the inverse Fourier transform of 2πX(jw). in part b)? Why, or why not? This is called...
Recall that the one-sided Laplace transform of x(t)is defined as x(s)-J x()e "atfor any complex numl 1-0 0C A special case of this is X(ia) x(t)e-ω'dt, which is called the one-sided Fourier transform (FT) of x( 1-0 transforms the time domain into the frequency domain; a domain often preferred by engineers, as it decom its various frequency components. Consider the following approximation of the unit impulse δφ : x(t)-[u, (t)-4(t-A)] / Δ , where Δ is the pulse width. (a)...
6. (a) The signal y(t) is defined as follows: ' y(t) = r(r)dT Suppose that (t) tu(t). First sketch r(t). Then determine and sketch y(t). (b) Consider the signal p(t) Σ δ(t-2n). First sketch p(t). Then calculate the following integral: 27 p(T)dT -3
Problem 2 In each step to follow the signals h(t) r (t) and y(t) denote respectively the impulse response. input, and output of a continuous-time LTI system. Accordingly, H(), X (w) and Y (w) denote their Fourier transforms. Hint. Carefully consider for each step whether to work in the time-domain or frequency domain c) Provide a clearly labeled sketch of y(t) for a given x(t)-: cos(mt) δ(t-n) and H(w)-sine(w/2)e-jw Answer: y(t) Σ (-1)"rect(t-1-n) Problem 2 In each step to follow...
(e) Consider an LTI system with impulse response h(t) = π8ǐnc(2(t-1). i. (5 pts) Find the frequency response H(jw). Hint: Use the FT properties and pairs tables. ii. (5 pts) Find the output y(t) when the input is (tsin(t) by using the Fourier Transform method. 3. Fourier Transforms: LTI Systems Described by LCCDE (35 pts) (a) Consider a causal (meaning zero initial conditions) LTI system represented by its input-output relationship in the form of a differential equation:-p +3讐+ 2y(t)--r(t). i....
plz 1. Let and y be the Fourier series coefficients of r(t) and y(t), respectively. Assuming the period of r(t) is T, express y in terms of d (b) y(t) (10 pts) () (a) y(t) (at), where a 0. (10 pts) dt 1. Let and y be the Fourier series coefficients of r(t) and y(t), respectively. Assuming the period of r(t) is T, express y in terms of d (b) y(t) (10 pts) () (a) y(t) (at), where a 0....