EEE 244-6: Fourier Transform (FFT) and Signal Processing

Signals and frequency ranges Signal Type Frequency Range Speech 200 to 3200 Hz High-fidelity audio 20 to 20,000 Hz AM radio 535 to 1705 kHz (North America) 526.5 to 1606.5 kHz (Europe) FM radio 87.5 to 108.0 MHz 802.11b Wi-Fi networking 2.401 to 2.473 GHz

Applications of Signal Processing Filtering of noise in audio/video signals Multiple sensor data processing for weather, biomedical signals (ECG or EKG, EEG), automobile fuel monitoring Multiplexing and coding of different cellular user signals to send on the same carrier (3G)

Sine Power signal and spectrum

Spectrum of Three phase signal at 60 Hz Amplitude spectrum for all three phases will have one peak at 60 Hz Phase spectrum will have peaks at 0, 120 and 240 degrees for the three phases

Music signal and spectrum https://upload.wikimedia.org/wikipedia/commons/e/ee/Beet5mov1bars1to5.ogg

Example of ECG filtering Courtesy: ScienceProg

Fourier Transform Amplitude spectrum |X(f)|, Volts Phase spectrum /_X(f), Degrees Periodic signals (example: sine signal) will have discrete spectrum Aperiodic signals (example: music signal) will have continuous spectrum

Practical computation of Fourier Transform The time signal x(t) is sampled at interval Dt sec. to give the time vector: x = [x(0), x(1), x(2)…….x(N-1)] The frequency spectrum X(f) is sampled at interval Df Hz. to give the frequency vector: X = [X(0), X(1), X(2)…….X(N-1)] The vectors x and X are related by the Matlab command fft: X = fft(x)

Example: FFT of a power sine wave Time signal x(t) = 110√2 cos(2p x 60t) clear T=1/60 % 1 period of signal N = 32 % Number of points in FFT dt=T/N % Time interval df = 1/(N*dt) % Frequency interval t=0:dt:T-dt; % Time vector x=110*sqrt(2)*cos(2*pi*60*t) % Signal vector X=fft(x) % FFT vector X=round(X) % Rounds spectrum to avoid phase clutter X=fftshift(X) % shifts spectrum to center zero frequency f=-(N/2)*df:df:(N/2-1)*df % Frequency vector subplot(2,1,1) stem(f,abs(X)); % Amplitude plot xlabel(‘Hz’) subplot(2,1,2) stem(f,angle(X)*180/pi) % Phase plot in degrees

Example: FFT of a three phase power wave Time signal x(t) = 110√2 cos(2p x 60t + 120°) clear T=1/60 % 1 period of signal N = 32 % Number of points in FFT dt=T/N % Time interval df = 1/(N*dt) % Frequency interval t=0:dt:T-dt; % Time vector x=110*sqrt(2)*cos(2*pi*60*t+120*pi/180) % Signal vector X=fft(x) % FFT vector X=round(X) % Rounds spectrum to avoid phase clutter X=fftshift(X) % shifts spectrum to center zero frequency f=-(N/2)*df:df:(N/2-1)*df % Frequency vector subplot(2,1,1) stem(f,abs(X)); % Amplitude plot xlabel(‘Hz’) subplot(2,1,2) stem(f,angle(X)*180/pi) % Phase plot in degrees