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Electrical Communications Systems ECE.09.331 Spring 2007 PowerPoint PPT Presentation


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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 Systems ECE.09.331 Spring 2007

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Electrical communications systems ece 09 331 spring 2007 l.jpg

Electrical Communications SystemsECE.09.331Spring 2007

Lecture 2aJanuary 23, 2007

Shreekanth Mandayam

ECE Department

Rowan University

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


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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


  • Ecomms topics l.jpg

    ECOMMS: Topics


    Measures of information l.jpg

    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


    Communications systems l.jpg

    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 Systems


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    ECOMMS: Topics


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    Signal Properties: Terminology

    • Waveform

    • Time-average operator

    • Periodicity

    • DC value

    • Power

    • RMS Value

    • Normalized Power

    • Normalized Energy


    Power and energy signals l.jpg

    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!


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    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


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    ECOMMS: Topics


    Fourier series l.jpg

    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/


    Fourier series12 l.jpg

    |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 l.jpg

    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 l.jpg

    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


    Definitions l.jpg

    Continuous Fourier Transform (CFT)

    Frequency, [Hz]

    Phase

    Spectrum

    Amplitude

    Spectrum

    Inverse Fourier Transform (IFT)

    Definitions

    See p. 45

    Dirichlet Conditions


    Properties of ft s l.jpg

    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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    Summary


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