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Introduction to Information Theory and Signal Properties in Communication Systems

Explore the fundamentals of information theory, signal properties, and Fourier analysis in communication systems. Understand measures of information, signal types, power and energy signals, Fourier series, and continuous Fourier transform.

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Introduction to Information Theory and Signal Properties in Communication Systems

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  1. Electrical Communication SystemsECE.09.331Spring 2008 Lecture 2aJanuary 29, 2008 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring08/ecomms/

  2. Plan • 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

  3. ECOMMS: Topics

  4. 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

  5. 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

  6. ECOMMS: Topics

  7. Signal Properties: Terminology • Waveform • Time-average operator • Periodicity • DC value • Power • RMS Value • Normalized Power • Normalized Energy

  8. 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!

  9. 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

  10. ECOMMS: Topics

  11. 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/

  12. |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

  13. 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!

  14. 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

  15. Continuous Fourier Transform (CFT) Frequency, [Hz] Phase Spectrum Amplitude Spectrum Inverse Fourier Transform (IFT) Definitions See p. 45 Dirichlet Conditions

  16. 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

  17. Summary

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