Continuous-Time Fourier Transform of a Response (matlab)

• Jun 30th 2010, 05:10 PM
1005
Continuous-Time Fourier Transform of a Response (matlab)
A system whose response is y(t) and whose excitation is x(t) is described by a differential equation:

$\displaystyle a_4y'''' + a_3y''' + a_2y'' + a_1y' + a_0y = b_4x'''' + b_3x''' + b_2x'' + b_1x' + b_0x$

The excitation is periodic and found by convolution of an aperiodic function, x_ap(t), with a continuous-time impulse train of period T_0. x_ap(t) is shown below with tri(t) = unit triangle function:

$\displaystyle x_{ap}(t) = c_1 tri( \frac{t - c_2}{c_3} ) + c_4tri( \frac{t - c_5}{c_6} )cos(2 \pi f_o t)$

a0-a4, b0-b4, f_0, T_0, and c1-c6 are all real constants given in my project file. Solutions to the project can be found any way the student wants, and he does not need to show his solutions. Only the answer is graded. The project asks me to graph several things:
x(t)
|CTFT of x(t)| = |X(f)|
angle too.
|frequency response of LTI| = |H(j 2 pi f)|
angle too.
|CTFT of y(t)| = |Y(f)| = |X(f)H(f)|
angle too.
and finally y(t) over one period.

Ok, I've done everything except for the graphing of y(t). I (painfully) found the CTFT of x(t) by hand. To find y(t), I have H(f), Y(f), X(f), and x(t) at my service. I believe finding the inverse FT of Y(f) analytically is impossible, and if not, it is too laborious to be expected. A method of estimation must be possible with matlab. Do you have any ideas how to use my repertoire of functions to find y(t) numerically? EDIT: I'm not asking for you to code it for me. I just need some direction: maybe some matlab functions to look into and some theory that may help.
• Jul 1st 2010, 11:04 AM
1005
I've solved this problem. In case anyone ever finds my post via search and needs the answer, I'll go ahead and detail my solution.

y(t) = sum from k = -inf to inf of c_y[k]exp(j2 pi k t /T0)

c_y[k] = H(k/T0)c_x[k]

c_x[k] = X(k/T0)

where c_y[k] is the harmonic function of y(t), c_x[k] is the harmonic function of x(t), X(k/T0) is the CTFT of x(t) evaluated at K/T0, and H(k/T0) is the frequency response evaluated at k/T0.