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EE354 : Communications System I

EE354 : Communications System I. Lecture 4,5: Energy and power Correlation Spectral density Aliazam Abbasfar. Outline. Energy/power Signals correlation Energy/power spectral density. Correlation. Correlation shows the similarity of 2 signals Energy signal Power signal

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EE354 : Communications System I

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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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