Question

A series of 16 samples starting at index n = −8 with sampling period ∆t = 0.25 µs is observed to be

[ −6.5000 2.3910 −11.1569 −6.5307 1.5000 −8.0615 −8.3284 −1.3045 −2.5000 −12.3910 0.1569 −3.4693 −10.5000 −1.9385 −2.6716 −8.6955 ]

where the 9th sample represents the time origin (t = 0). Use MATLAB to find the FFT (magnitude and phase) of this sampled time vector. Explain how you keep track of the origin of the frequency domain vector so that in a final plot of the spectrum, the origin (f = 0 Hz) is the 9th element of the frequency domain vector. Use the uncertainly principle to make the numbers corresponding to the horizontal axis of your frequency plots correspond to the correct values in Hz. Don’t forget that the spectrum is likely complex valued. The MATLAB command fftshift might be useful to explore for this problem.

Answer 1

MATLAB code to obtain the FFT sequence

x = [-6.5000 2.3910 -11.1569 -6.5307 1.5000 -8.0615 -8.3284 -1.3045 -2.5000...
-12.3910 0.1569 -3.4693 -10.5000 -1.9385 -2.6716 -8.6955]; % Signal x(t)

abs(fftshift(fft(x))) % magnitude of the spectrum

Output:

ans =

Columns 1 through 8

0 16.0001 0 47.9999 4.0000 0.0001 0 0.0001

Columns 9 through 16

80.0000 0.0001 0 0.0001 4.0000 47.9999 0 16.0001

angle(fftshift(fft(x)))*57.3 % phase of the spectrum

Output:

ans =

Columns 1 through 8

0 -180.0132 0 90.0063 0 -104.6037 0 -142.5500

Columns 9 through 16

180.0133 142.5500 0 104.6037 0 -90.0063 0 180.0132