1. EE354 : Communications System I
Lecture 2,3:
Signals in communication systems
Fourier review
Aliazam Abbasfar

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

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 signal
Digital systems
m[n] is a discrete signal
m[n] takes limited values
x(t) t
x(t) T t

4. Signals : Important parameters
Energy, power
Frequency components
DC level
Bandwidth
Power spectral density

5. Energy and Power Signals
x(t) is an energy signal if E is finite
x(t) is an power signal if P is finite
Energy signals have zero power
Power signals have infinite energy

6. Tone signals Single tone signal Multi-tone signal
Periodic with period T0
Frequency content only at f0
Amplitude and phase = phasor
One-sided/Two-sided spectrum
We show the spectrum with respect to f ( NOT w)
Power = A2/2
Multi-tone signal
Bandwidth

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

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

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

10. 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

11. 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) = u(t)  X(f) = 1/j2pf+ 1/2d(f)
Tone x(t) = ej2pf0t  X(f) = d(f-f0)
Periodic signals

12. Fourier examples Impulse train:
x(t) = Sd (t-nT0)  X(f) = 1/T0Sd (f-nf0)
Repetition
y(t) = repT(x) = S x(t-nT) 
Y(f) = 1/T S X(n/T) d (f-n/T)
Sampling
y(t) = combT(x) = S x(nT) d (t-nT) 
Y(f) = 1/T S X(f-n/T)

13. Fourier Transform and LTI systems
An LTI system is defined by its impulse response, h(t)
H(f) : frequency response of system
x(t) = ej2pfot  y(t) = H(f0) ej2pfot
Eigen-functions and Eigen-values of any LTI system
Bandwidth

14. Channel model Channels are often modeled as LTI systems
h(t) : channel impulse response
H(f) : channel frequency response
Noise is added at the receiver
Additive noise
Lowpass and passband channels
Channel bandwidth

15. Power measurement PdBW = 10 log10(P/1 W)
PdBm = 10 log10(P/1 mW) = PdBW + 30
Power gain
g = Pout/Pin
gdB = 10 log10( Pout/Pin)
Power loss
L = 1/g = Pin/Pout
LdB = 10 log10( Pin/Pout)
Transmission gain
Pout = g1g2g3g4 Pin= g2g4 /L1L3 Pin
in dB : Pout = g1 + g2 + g3 +g4 + Pin= g2 + g4 - L1 – L3 + Pin

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