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Electrical Communications Systems ECE.09.331 Spring 2007

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### Electrical Communications SystemsECE.09.331Spring 2007

Lecture 2aJanuary 23, 2007

Shreekanth Mandayam

ECE Department

Rowan University

http://engineering.rowan.edu/~shreek/spring07/ecomms/

Plan

- Recall:
- Intoduction to Information Theory
- 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

Measures of Information

- Definitions
- Probability
- Information
- Entropy
- Source Rate
- Recall: Shannon’s Theorem
- If R < C = B log2(1 + S/N), then we can have error-free transmission in the presence of noise

MATLAB DEMO:

entropy.m

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

Communications SystemsSignal Properties: Terminology

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

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

Power and Energy Signals- Although “real” signals are energy signals, we analyze them pretending they are power signals!

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

Fourier Series

Infinite sum of

sines and cosines

at different frequencies

Any periodic

power signal

Fourier Series

Fourier Series Applet:

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

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

f

Fourier SeriesExponential Representation

Periodic Waveform

w(t)

t

T0

2-Sided Amplitude Spectrum

f0 = 1/T0; T0 = period

Fourier 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 with

T0 ----> infinity

Then, f0 -----> 0

The spectrum is continuous!

Continuous Fourier Transform

Aperiodic Waveform

- 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!

w(t)

t

T0 Infinity

|W(f)|

f

f0 0

Continuous Fourier Transform (CFT)

Frequency, [Hz]

Phase

Spectrum

Amplitude

Spectrum

Inverse Fourier Transform (IFT)

DefinitionsSee p. 45

Dirichlet Conditions

Properties 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. 50

FT Theorems

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