Electrical Communications Systems ECE.09.331 Spring 2007

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Electrical Communications Systems ECE.09.331 Spring 2007. Lecture 2a January 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

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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

Analog

Continuous set of messages (signals)

Legacy

Predominant

Inexpensive

Communications Systems
Signal 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/

|W(n)|

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

f

Fourier Series

Exponential 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)

Definitions

See 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