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Energy and Power Correlation in Communication Systems

This lecture focuses on the correlation of energy and power in signals within communication systems. We will explore the principles of signal correlation, including cross-correlation and auto-correlation, as well as their properties such as linearity and orthogonality. The lecture will also cover energy and power spectral density, providing a fundamental understanding of these concepts essential for analyzing signal performance in linear time-invariant systems. Through examples, we will illustrate the application of correlation functions in determining relationships between signals.

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Energy and Power Correlation in Communication Systems

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  1. EE354 : Communications System I Lecture 4,5: Energy and power Correlation Spectral density Aliazam Abbasfar

  2. Outline • Energy/power • Signals correlation • Energy/power spectral density

  3. Correlation • Correlation shows the similarity of 2 signals • Energy signal • Power signal • Properties • Linearity • Corr( x(t+t), y(t+t) ) = corr( x(t), y(t) ) • Corr( y(t), x(t) ) = Corr( x(t), y(t) )* • Corr( x(t), x(t) ) = Ex or Px • Orthogonal signals • Corr( x(t), y(t) ) = 0

  4. Correlation of energy signals • Correlation functions: • Cross-correlation of 2 signals • Auto-correlation of a signal • Example : pulse

  5. Correlation of power signals • Correlation function • Cross-correlation of 2 power signals • Auto-correlation of a signal • Example : periodic signals

  6. Properties • Rx(-t) = Rx*(t) • Ryx(t) = Rxy*(-t) • Correlations for LTI systems • Ryx(t) = h(t)  Rx(t) • Rxy(t) = Ryx*(-t)= h*(-t)  Rx(t) • Ry(t) = h(t)  h*(-t)  Rx(t)

  7. Energy/Power spectral density • Energy/Power spectral density • ESD : • PSD : • Filtering :

  8. Examples • z(t) = x(t) + y(t) • Rz(t) = Rx(t) + Ry(t) + Rxy(t) + Ryx(t) • x(t) , y(t) orthogonal for all t • Rz(t) = Rx(t) + Ry(t) • Gz(f) = Gx(f) + Gy(f) • y(t) = repT( x(t)) • Y(f) = 1/T comb1/T( X(f)) • Gy(f) = 1/T2 comb1/T( |X(f)|2) • y(t) = x(t) ejWot • Y(f) = X(f-f0) • Gy(f) = Gx(f-f0)

  9. Reading • Carlson Ch. 3.2, 3.3, 3.5, and 3.6 • Proakis 2.3, 2.4

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