Find the analytical expression

Reference no: EM132549963

Let x(s) be the Laplace transform of the signal x(t). The value of x(s) along any contour parallel to the imaginary axis, i.e., s = σO + jω for a constant σO, is equal to the continuous- time Fourier transform of e-a0tx(t). This can be used to compute a numerical approximation to the Laplace transform when you have samples of x(t), but do not have the analytic form of x(t) or x(s). Consider the continuous-time signal

x(t) = e^-tcos(4t)u(t) -1/8 e^-t sin(4t)u(t)

(a) Find the analytical expression for x(s), the Laplace transform of x(t). Generate a pole- zero plot for x(s). Make a sketch of the magnitude of the continuous-time Fourier transform |x(jω)|. Also, sketch |x(s)| along the contour Re(s) = - 1/2.

(b) Notice that the MATLAB function fft could be used to compute an approximation to samples of the continuous-time Fourier transform of a signal x(t) from samples x(nT).

Define x to be 64 samples of x(t) with T = 1/8 for 0 ≤ t < 8. Imagine that you only have these samples, and do not know the analytic form for either x(t) or x(s). Use fft to compute approximate values of x(jω) for ωk = -Π/T + 2Πk/64 T for 0 ≤ K ≤ 63 and plot |x(jω)|. Does this agree with your sketch in problem (a)?

Reference no: EM132549963

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