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I am very new to Matlab, and I'm supposed to use the built-in FFT function to do discrete Fourier Transform for f(x) = sin x + 4 cos(5x) + (sin(6x))^2 on the interval [-pi, pi] with a uniform partition for the interval with n = 9. Then I have to

(a) Plot the magnitudes of the Fourier coefficients and

(b) Compute the first-order derivates at the grid points via FFT and compare them with f'(x).

Here's what I have for part (a):

x = -pi:0.25*pi:pi;

y = sin(x)+4*cos(5*x) + sin(6*x).*sin(6*x);

V=fft(y,9);

plot(abs(V));

I'm a little confused with what the function fft returns. Does it return the Fourier coefficients of f(x) in my program?

I got

V =

-0.0000

-5.9965 + 2.1842i

-4.5019 - 4.8898i

-8.3033 -15.3964i

0.8017 + 2.1116i

0.8017 - 2.1116i

-8.3033 +15.3964i

-4.5019 + 4.8898i

-5.9965 - 2.1842i

I also don't know how to find the first-order derivates at the grid points via FFT for part (b). What function do I use?

Thank you very much!

Regards,

Rayne