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

2. 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 kHz (Europe)
FM radio
87.5 to MHz
802.11b Wi-Fi networking
2.401 to GHz

3. 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)

4. Sine Power signal and spectrum

5. 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

6. Music signal and spectrum

7. Example of ECG filtering
Courtesy: ScienceProg

8. 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

9. 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)

10. Example: FFT of a power sine wave
Time signal x(t) = 110√2 cos(2p x 60t)
clear
T=1/ % 1 period of signal
N = % 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

11. Example: FFT of a three phase power wave
Time signal x(t) = 110√2 cos(2p x 60t + 120°)
clear
T=1/ % 1 period of signal
N = % 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