Lecture 2,3: Signals in communication systemsFourier review

Aliazam Abbasfar

OutlineSignals

Fourier SeriesFourier TransformFourier properties

Linear systemsChannel model

Signals in communication systems

Analog systemsm(t) is a continuous signal

Digital systemsm[n] is a discrete signalm[n] takes limited values

Sourcedecoder

Channel ReceiverSourc

eencoder

message

m(t)m[n]

x(t) y(t)m(t)m[n]

x(t)

x(t)

t

t

T

Signals : Important parametersEnergy, power

Frequency componentsDC levelBandwidthPower spectral density

Energy and Power Signals

x(t) is an energy signal if E is finitex(t) is an power signal if P is finite

Energy signals have zero powerPower signals have infinite energy

dt|x(t)|E 2x

T/2

T/2

2

Tx dt|x(t)|

T

1limP

Tone signals Single 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 ) Power = A2/2

Multi-tone signal Bandwidth

]Re[)2cos()( 020

tfjj eAetfAtx

tfjj

tfjj

eAe

eAe

tfAtx 00 220 22

)2cos()(

Fourier seriesPeriodic signals with period T0

f0 = 1/T0 : fundamental frequency

cn :Line(discrete) spectrum of the signal

Parseval’s theorem :

tnfj

nnp ectx 02)(

dtetxT

c tnfj

T

pn0

0

2

0

)(1

Fourier Transform

Continuous spectrum

Real signals : X(-f) = X*(f)

Even signals : X(f) is realOdd signals : X(f) is imaginary

dtetxfX ftj 2)()(

dfefXtx ftj 2)()(

Rectangular pulseRect(t) : a pulse with unit amplitude and

width

Sinc(f) = sin(f)/(f)

Band-limited and time-limited signals

Fourier Transform PropertiesUseful properties

LinearityTime shiftTime/Freq. scalingModulationConvolution/multiplicationDifferentiation/integration

Duality:

Parseval’s equation :Energy and energy spectral density

Special signalsDC x(t) = 1 X(f) = (f) Impulse x(t) = (t) X(f) = 1Sign x(t) = sgn(t) X(f) = 1/jfStep x(t) = u(t) X(f) = 1/j2f+(f)

Tone x(t) = ej2f0t X(f) = (f-f0)

Periodic signalstnfj

nnp ectx 02)(

)()( 0nffcfXn

np

Fourier examplesImpulse train:

x(t) = (t-nT0) X(f) = 1/T0(f-nf0)

Repetition

y(t) = repT(x) = x(t-nT)

Y(f) = 1/T X(n/T)(f-n/T)

Sampling

y(t) = combT(x) = x(nT)(t-nT)

Y(f) = 1/T X(f-n/T)

Fourier Transform and LTI systemsAn LTI system is defined by its impulse

response, h(t)

H(f) : frequency response of systemx(t) = ej2fot y(t) = H(f0) ej2fot

Eigen-functions and Eigen-values of any LTI system

Bandwidth

Channel modelChannels are often modeled as LTI systems

h(t) : channel impulse responseH(f) : channel frequency response

Noise is added at the receiverAdditive noise

Lowpass and passband channels

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

ReadingCarlson Ch. 2 and 3.1

Proakis 2.1, 2.2