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EE354 : Communications System I

EE354 : Communications System I. Lecture 3,4: Correlation, Spectral density Distortion Aliazam Abbasfar. Outline. Fourier examples Signals correlation Energy/power spectral density Channel model Signal distortion. Fourier examples. DC : x(t) = 1  X(f) = d (f)

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EE354 : Communications System I

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  1. EE354 : Communications System I Lecture 3,4: Correlation, Spectral density Distortion Aliazam Abbasfar

  2. Outline • Fourier examples • Signals correlation • Energy/power spectral density • Channel model • Signal distortion

  3. Fourier examples • DC : x(t) = 1  X(f) = d(f) • Impulse : x(t) = d(t)  X(f) = 1 • Sign : x(t) = sgn(t)  X(f) = 1/jpf • Step : x(t) = u(t)  X(f) = 1/j2pf+ 1/2d(f) • Impulse train: x(t) = T0Sd (t-nT0)  X(f) = Sd(f-nf0) • Repetition y(t) = repT(x) = Sx(t-nT)  Y(f) = 1/T SX(n/T) d (f-n/T) • Sampling y(t) = combT(x) = Sx(nT) d (t-nT)  Y(f) = 1/T SX(f-n/T)

  4. Energy and Power Signals • x(t) is an energy signal if E is finite • x(t) is an power signal if P is finite • Energy signals have zero power • Power signals have infinite energy

  5. Power measurement • PdBW = 10 log10(P/1 W) • PdBm = 10 log10(P/1 mW) = PdBW + 30 • Power gain • g = Pout/Pin • gdB = 10 log10( Pout/Pin) • Power loss • L = 1/g = Pin/Pout • LdB = 10 log10( Pin/Pout) • Transmission gain • Pout = g1g2g3g4 Pin= g2g4 /L1L3 Pin • in dB : Pout = g1 + g2 + g3 +g4 + Pin= g2 + g4 - L1 – L3 + Pin

  6. Correlation of energy signals • Correlation shows the similarity of 2 signals • Cross-correlation of 2 signals • Auto-correlation of a signal • Example : pulse

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

  8. Correlations for LTI systems • Ryx(t) = h(t)  Rxx(t) • Rxy(t) = R*yx(-t)= h*(-t)  Rxx(t) • Ryy(t) = h(t)  h*(-t)  Rxx(t)

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

  10. Channel model • Channels are often modeled as LTI systems • h(t) : channel impulse response • H(f) : channel frequency response • Noise is added at the receiver • Additive noise • Lowpass and passband channels

  11. Signal distortion • Distortion-less transmission • y(t) = K x(t-td) • Channels distort signals • Linear distortion • Amplitude • Phase (delay) • Time delay : td(f) = -(f)/2pf • Group delay : tg(f) = -1/2p d(f)/df • Non-linear distortion • compression • Equalization used to cure linear distortion • Noise amplification

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

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