1. EE354 : Communications System I
Lecture 2:
Fourier review
Aliazam Abbasfar

2. Outline Signals Fourier Series Fourier Transform Fourier properties
Linear systems

3. Signals in communication systems
m(t)
m[n]
m(t)
m[n]
x(t)
y(t)
message
Source
encoder
Transmitter
Source
decoder
Channel
Receiver
Analog systems
m(t) is a continuous function
Digital systems
m[n] is a discrete function
m[n] takes limited values
DC level, energy, power
x(t) t
x(t) T t

4. Fourier series Periodic signals with period T0
f0 = 1/T0 : fundamental frequency
cn :Line(discrete) spectrum of the signal
Parseval’s theorem :

5. Fourier Transform Continuous spectrum Real signals : X(-f) = X*(f)
Even signals : X(f) is real
Odd signals : X(f) is imaginary

6. Rectangular pulse Rect(t) : a pulse with unit amplitude and width
Sinc(f) = sin(pf)/(pf)
Band-limited and time-limited signals

7. Fourier Transform Properties
Useful properties
Linearity
Time shift
Time/Freq. scaling
Modulation
Convolution/multiplication
Differentiation/integration
Duality: 
Parseval’s equation :
Energy and energy spectral density

8. Special signals DC x(t) = 1  X(f) = d(f)
Impulse x(t) = d(t)  X(f) = 1
Sign x(t) = sgn(t)  X(f) = 1/jpf
Step x(t) = s(t)  X(f) = 1/j2pf+ 1/2d(f)
Sinusoids x(t) = ej2pf0t  X(f) = d(f-f0)
Periodic signals .

9. Fourier Transform and LTI systems
An LTI system is defined by its impulse response, h(t)
H(f) : frequency response of system
x(t) = ej2pf0t  y(t) = H(f0) ej2pf0t
Eigen-functions and Eigen-values of any LTI system

10. Reading
Carlson Ch. 2 and 3.1
Proakis 2.1, 2.2