Lecture 3,4: Correlation, Spectral densityDistortion

Aliazam Abbasfar

OutlineFourier examples

Signals correlationEnergy/power spectral density

Channel modelSignal distortion

Fourier examples DC : x(t) = 1 X(f) = (f) Impulse : x(t) = (t) X(f) = 1 Sign : x(t) = sgn(t) X(f) = 1/jf Step : x(t) = u(t) X(f) = 1/j2f+(f)

Impulse train:

x(t) = T0(t-nT0) X(f) = (f-nf0) Repetition

y(t) = repT(x) = x(t-nT)

Y(f) = 1/T X(n/T)(f-n/T)

Sampling

y(t) = combT(x) = x(nT)(t-nT)

Y(f) = 1/T X(f-n/T)

Energy and Power Signals

x(t) is an energy signal if E is finitex(t) is an power signal if P is finite

Energy signals have zero powerPower signals have infinite energy

dt|x(t)|E 2x

T/2

T/2

2

Tx dt|x(t)|

T

1limP

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

Correlation of energy signalsCorrelation shows the similarity of 2 signalsCross-correlation of 2 signals

Auto-correlation of a signal

Example : pulse

)(y)x(τ)dt(tx(t)y)(R **xy

yx

2

xy EE)(R

)(x)x(τ)dt(tx(t)x)(R **xx

xxxxxx E)(RE(0)R

Correlation of power signalsCross-correlation of 2 power signals

Auto-correlation of a signal

Example : periodic signals

τ)(tyx(t)τ)dt(tyx(t)lim)(R *T/2

T/2

*

Txy

yx

2

xy PP)(R

xxxxxx P)(RP(0)R

τ)(tx(t)xτ)dt(tx(t)xlim)(R *T/2

T/2

*

Txx

Correlations for LTI systems

Ryx() = h() Rxx()

Rxy() = R*yx(-)= h*(-) Rxx()

Ryy() = h() h*(-) Rxx()

Energy/Power spectral densityEnergy/Power spectral density

ESD :

PSD :

Filtering :

)]([R(f)G xxx F

2

x X(f)(f)G

T

(f)Xlim(f)G

2

T

Tx

(f)GH(f)(f)Gh(t)x(t)y(t) x

2

y

Channel modelChannels are often modeled as LTI systems

h(t) : channel impulse responseH(f) : channel frequency response

Noise is added at the receiverAdditive noise

Lowpass and passband channels

Signal distortionDistortion-less transmission

y(t) = K x(t-td)

Channels distort signalsLinear distortion

Amplitude Phase (delay)

Time delay : td(f) = -(f)/2f Group delay : tg(f) = -1/2 d(f)/df

Non-linear distortion compression

Equalization used to cure linear distortionNoise amplification

ReadingCarlson Ch. 3.2, 3.3, 3.5, and 3.6

Proakis 2.3, 2.4