S. Mandayam/ ECOMMS/ECE Dept./Rowan University

Electrical Electrical Communication SystemsCommunication Systems

ECE.09.331ECE.09.331 Spring 2010Spring 2010

Shreekanth MandayamECE Department

Rowan University

http://engineering.rowan.edu/~shreek/spring10/ecomms/

Lecture 2aLecture 2aJanuary 27, 2010January 27, 2010

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

PlanPlan• Digital and Analog Communications

Systems• Properties of Signals and Noise

• Terminology• Power and Energy Signals

• Recall: Fourier Analysis• Fourier Series of Periodic Signals

• Continuous Fourier Transform (CFT) and Inverse Fourier Transform (IFT)

• Amplitude and Phase Spectrum• Properties of Fourier Transforms

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

ECOMMS: TopicsECOMMS: Topics

Probability

Inform ation

Entropy

Channel Capacity

Discrete

Pow er & Energy S ignals

Continuous Fourier T ransform

Discrete Fourier T ransform

Baseband and Bandpass S ignals

Com plex Envelope

G aussian Noise & SNR

Random VariablesNoise Calculations

Continuous

Signals

AMSw itching M odulator

Envelop Detector

DSB-SCProduct M odulatorCoherent Detector

Costas Loop

SSBW eaver's M ethodPhasing M ethod

Frequency M ethod

Frequency & Phase M odulationNarrow band/W idebandVCO & S lope Detector

PLL

Analog

Source EncodingHuffm an codes

Error-control EncodingHam m ing Codes

Sam plingPAM

QuantizationPCM

Line Encoding

T im e Division M uxT1 (DS1) Standards

Packet Sw itchingEthernet

ISO 7-Layer Protocol

BasebandCODEC

ASKPSKFSK

BPSK

QPSK

M -ary PSK

QAM

BandpassM ODEM

DigitalDigital Com m Transceiver

System s

Electrical Com m unication System s

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

Communications SystemsCommunications Systems• Digital

• Finite set of messages (signals)

• inexpensive/expensive• privacy & security• data fusion• error detection and

correction• More bandwidth• More overhead (hw/sw)

• Analog• Continuous set of

messages (signals)• Legacy• Predominant• Inexpensive

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

Signal Properties: Signal Properties: TerminologyTerminology

• Waveform• Time-average operator• Periodicity• DC value• Power• RMS Value• Normalized Power• Normalized Energy

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

Power and Energy SignalsPower and Energy Signals

• Power Signal• Infinite duration• Normalized power is

finite and non-zero• Normalized energy

averaged over infinite time is infinite

• Mathematically tractable

• Energy Signal• Finite duration• Normalized energy is

finite and non-zero• Normalized power

averaged over infinite time is zero

• Physically realizable

• Although “real” signals are energy signals, we analyze them pretending they are power signals!

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

The Decibel (dB)The Decibel (dB)

• Measure of power transfer

• 1 dB = 10 log10 (Pout / Pin)

• 1 dBm = 10 log10 (P / 10-3) where P is in Watts

• 1 dBmV = 20 log10 (V / 10-3) where V is in Volts

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

ECOMMS: TopicsECOMMS: Topics

Probability

Inform ation

Entropy

Channel Capacity

Discrete

Pow er & Energy Signals

Continuous Fourier Transform

Discrete Fourier Transform

Baseband and Bandpass Signals

Com plex Envelope

G aussian Noise & SNR

Random VariablesNoise Calculations

Continuous

Signals

AMSw itching M odulator

Envelop Detector

DSB-S CProduct M odulatorCoherent Detector

Costas Loop

SSBW eaver's MethodPhasing M ethod

Frequency M ethod

Frequency & Phase M odulationNarrow band/WidebandVCO & S lope Detector

PLL

Analog

Source EncodingHuffm an codes

Error-control EncodingHam m ing Codes

Sam plingPAM

Q uantizationPCM

Line Encoding

Tim e Division M uxT1 (DS1) Standards

Packet Sw itchingEthernet

ISO 7-Layer Protocol

BasebandCODEC

ASKPSKFSK

BPSK

Q PSK

M -ary PSK

Q AM

BandpassM O DEM

DigitalDigital Com m Transceiver

Systems

Electrical Comm unication Systems

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

Fourier SeriesFourier Series

Fourier Series Applet:

http://www.gac.edu/~huber/fourier/

Any periodicpower signal

Infinite sum ofsines and cosinesat different frequencies

Fourier Series

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

Fourier SeriesFourier Series

dte )t(wT1

W

where

e W)t(w

2/T

2/T

Tnt2j

0n

Tnt2j

nn

0

0

0

0

Exponential Representation Periodic Waveform

w(t)

t

|W(n)|

f-3f0 -2f0 -f0 f0 2f0 3f0

2-Sided Amplitude Spectrum

f0 = 1/T0; T0 = period

T0

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

Fourier TransformFourier Transform

• Fourier Series of periodic signals• finite amplitudes• spectral components separated by discrete frequency

intervals of f0 = 1/T0

• We want a spectral representation for aperiodic signals

• Model an aperiodic signal as a periodic signal withT0 ----> infinity

Then, f0 -----> 0

The spectrum is continuous!

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

Continuous Fourier TransformContinuous Fourier Transform• We want a spectral

representation for aperiodic signals

• Model an aperiodic signal as a periodic signal with

T0 ----> infinity

Then, f0 -----> 0

The spectrum is continuous!

t

T0 Infinity

w(t)

Aperiodic Waveform

|W(f)|

ff0 0

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

DefinitionsDefinitions

)f(j

ft2j

e )f(W)f(W

)f(Y j)f(X)f(W

dte )t(w)t(w)f(W

F

Continuous Fourier Transform (CFT)

Frequency, [Hz]

AmplitudeSpectrum

PhaseSpectrum

dfe )f(W)f(W)t(w ft2j1-

F

Inverse Fourier Transform (IFT)

See p. 45Dirichlet Conditions

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

Properties of FT’sProperties of FT’s• If w(t) is real, then W(f) = W*(f)

• If W(f) is real, then w(t) is even

• If W(f) is imaginary, then w(t) is odd

• Linearity

• Time delay

• Scaling

• Duality

See p. 50FT Theorems

S. Mandayam/ ECOMMS/ECE Dept./Rowan University

SummarySummary