Signals and Spectra (A)
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Energy and Power Instantaneous Power p(t) = x 2 (t) real signal Energy dissipated during ( /2, + /2) Average power dissipated during ( /2, + /2) + / 2 E x = /2 x 2 (t) dt P x = + / 2 /2 x 2 (t) dt Affects the performance of a communication system he rate at which energy is dissipated Determines the voltage Signals & Spectra (A) 3
Energy and Power Signals () Energy dissipated during + / 2 E x = /2 ( /2, + /2) x 2 (t) dt Average power dissipated during P x = + / 2 /2 ( /2, + /2) x 2 (t) dt Energy Signal Nonzero but finite energy Power Signal Nonzero but finite power 0 < E x < 0 < P x < for all time for all time E x + /2 = lim /2 x 2 (t) dt P x = lim + /2 /2 x 2 (t) dt = x 2 (t) dt < < Signals & Spectra (A) 4
Energy and Power Signals (2) Energy Signal Nonzero but finite energy Power Signal Nonzero but finite power 0 < E x < 0 < P x < for all time E x + /2 = lim /2 x 2 (t) dt P x = lim for all time + /2 /2 x 2 (t) dt = x 2 (t) dt < < P x = lim + /2 /2 x 2 (t) dt E x + /2 = lim /2 x 2 (t) dt = lim B 0 = lim B Non-periodic signals Deterministic signals Periodic signals Random signals Signals & Spectra (A) 5
Energy and Power Spectral Densities () otal Energy, Non-periodic Average power, Periodic E x = x 2 (t) dt P x = + /2 /2 x 2 (t) dt Parseval's heorem, Non-periodic Parseval's heorem, Periodic = n= X ( f ) 2 d f = c n 2 = Ψ( f ) d f = Gx ( f ) d f = 2 0 Ψ ( f ) d f = 2 0 Gx ( f ) d f Energy Spectral Density Power Spectral Density Ψ( f ) = X ( f ) 2 G x ( f ) = c n 2 δ( f n f 0 ) n= Signals & Spectra (A) 6
Energy and Power Spectral Densities (2) Energy Spectral Density Power Spectral Density Ψ( f ) = X ( f ) 2 G x ( f ) = c n 2 δ( f n f 0 ) n= otal Energy, Non-periodic Average power, Periodic E x = x 2 (t) dt P x = + /2 /2 x 2 (t) dt = Ψ( f ) d f = Gx ( f ) d f Parseval's heorem, Non-periodic Parseval's heorem, Periodic Non-periodic power signal (having infinite energy)? Signals & Spectra (A) 7
Energy and Power Spectral Densities (3) Power Spectral Density G x ( f ) = lim X ( f ) 2 Non-periodic power signal (having infinite energy)? No Fourier Series Power Spectral Density G x ( f ) = Average power, Periodic c n 2 δ( f n f 0 ) n= truncate x(t) x (t) ( 2 t + 2 ) Fourier ransform X ( f ) P x = + /2 /2 = x 2 (t) dt Gx ( f ) d f P x = lim + /2 /2 x 2 (t) d t Parseval's heorem, Periodic = lim X( f ) 2 d f Signals & Spectra (A) 8
Autocorrelation of Energy and Power Signals Autocorrelation of an Energy Signal Autocorrelation of a Power Signal R x (τ) = x(t)x(t + τ) dt R x (τ) = lim ( τ ) + /2 /2 x(t)x(t + τ) dt ( τ ) Autocorrelation of a Periodic Signal R x (τ) = 0 0 /2 ( τ ) + 0 /2 x(t)x(t + τ) dt R x ( τ) = R x ( τ) R x ( τ) = R x ( τ) R x ( τ) R x (0) R x ( τ) R x (0) R x ( τ) Ψ( f ) R x ( τ) G x ( f ) R x (0) = x 2 (t) dt R x (0) = 0 0 /2 + 0 /2 x 2 (t) dt Signals & Spectra (A) 9
Ensemble Average Random Variable Random Process m x = E {X } m x (t k ) = E {X (t k )} = x p X (x) d x = x p Xk (x) d x for a given time t k E {X 2 } = σ x 2 + m x 2 R x (t, t 2 ) = E {X (t ) X (t 2 )} = x 2 p X (x) d x = x x 2 p X (x, X 2, x 2 ) d x d x 2 Signals & Spectra (A) 0
WSS (Wide Sense Stationary) Random Process WSS Process m x (t k ) = E {X (t k )} = x p Xk (x) d x m x (t k ) = E {X (t k )} = m x for a given time t k R x (t, t 2 ) = E {X (t ) X (t 2 )} = x x 2 p X (x, X 2, x 2 ) d x d x 2 R x (t, t 2 ) = E {X (t ) X (t 2 )} = R x (t t 2 ) Signals & Spectra (A)
Ergodicity and ime Averaging Random Process m x (t k ) = E {X (t k )} = x p Xk (x) d x R x (t, t 2 ) = E {X (t ) X (t 2 )} = x x 2 p X (x, X 2, x 2 ) d x d x 2 for a given time WSS Process by ensemble average m x (t k ) = E {X (t k )} R x (t, t 2 ) = E {X (t ) X (t 2 )} = m x = R x (t t 2 ) = R x (τ) Ergodic Process by time average m x (t k ) = E {X (t k )} = R x (t, t 2 ) = E {X (t ) X (t 2 )} = m x = lim + /2 / 2 X (t) dt = lim + + /2 /2 X (t) X (t+ τ) dt Signals & Spectra (A) 2
Autocorrelation of Power Signals Autocorrelation of a Random Signal R x ( τ) = E {X (t) X (t + τ)} Autocorrelation of a Power Signal R x (τ) = lim + /2 /2 x(t)x(t + τ) dt ( τ ) Autocorrelation of a Periodic Signal R x (τ) = 0 0 /2 ( τ ) + 0 /2 x(t)x(t + τ) dt R x ( τ) = R x ( τ) R x ( τ) = R x ( τ) R x ( τ) R x (0) R x ( τ) R x (0) R x ( τ) G x ( f ) R x (0) = E {X 2 (t)} R x ( τ) G x ( f ) R x (0) = 0 0 /2 + 0 /2 x 2 (t) dt Signals & Spectra (A) 3
Autocorrelation of Random Signals Autocorrelation of a Random Signal Power Spectral Density of a Random Signal R x ( τ) = E {X (t) X (t + τ)} G x ( f ) = lim + X ( f ) 2 = lim + + /2 /2 X (t) X (t+ τ) dt if ergodic in the autocorrelation function R x ( τ) = R x ( τ) G x ( f ) = G x ( f ) R x ( τ) R x (0) G x ( f ) 0 R x ( τ) G x ( f ) G x ( f ) R x ( τ) R x (0) = E {X 2 (t)} P x (0) = G X ( f ) d f Signals & Spectra (A) 4
Ergodic Random Process m X = E {X (t)} DC level m X 2 normalized power in the dc component E {X 2 (t)} total average normalized power (mean square value) E {X 2 (t)} σ X 2 rms value of voltage or current average normalized power in the ac component σ X m x = m X 2 = 0 σ X 2 = E {X 2 } σ X rms value of the ac component var = total average normalized power = mean square value (rms^2) m X = 0 rms value of the signal Signals & Spectra (A) 5
Single one Input y(t) = h(t) x(t ) + = h(τ) x(t τ) d τ x (t) = A e j Φ e j ω t amplitude = A phase = Φ frequency = ω h(t) Fourier ransform y(t) y(t) = h(τ) A e j Φ e j ω(t τ) d τ = h(τ) A e j Φ e j ω t e j ω τ d τ = A e j Φ e j ω t h(τ) e j ω τ d τ = A e j Φ e j ω t H ( j ω) complex number given ω and t = A e j Φ e j ω t ρe j θ changes the amplitude and the phase of the single tone input but not the frequency Signals & Spectra (A) 6
Impulse Response & Frequency Response y(t) = h(t) x(t ) + = h(τ) x(t τ) d τ x(t ) h(t) y(t) Fourier ransform X ( j ω) Y ( j ω) H ( j ω) Y ( j ω) = H ( j ω) X ( j ω) + Y ( j ω) = y (τ) e j ω τ d τ + H ( j ω) = h( τ) e j ω τ d τ + X ( j ω) = x (τ) e j ω τ d τ Signals & Spectra (A) 7
Linear System y(t) = h(t) x(t ) + = h(τ) x(t τ) d τ x(t ) h(t) y(t) δ(t ) h(t) h(t) A e j Φ e jwt h(t) H ( j ω) A e j Φ e j ωt single frequency component : ω Frequency Response + H ( j ω) = h(τ) e j ω τ d τ Signals & Spectra (A) 8
ransfer Function & Frequency Response y(t) = h(t) x(t ) + = h(τ) x(t τ) d τ x(t ) h(t) y(t) ransfer Function Laplace ransform H (s) = H (σ+ j ω) H (s) = h(τ)e s τ d τ initial state differential equation σ = 0 Frequency Response Fourier ransform H (ω) = H ( j ω) H ( j ω) = h(τ)e j ω τ d τ steady state frequency response Signals & Spectra (A) 9
Linear System & Random Variables () R.V R.V X (t ) h(t) Y (t ) mean E [ X (t)] h(t) E [Y (t)] = 0 t E[ X (τ)]h(t τ) d τ correlation E [ X (t ) X (t 2 )] E [Y (t )Y (t 2 )] h(t ) h ( t) Fourier ransform WSS S xx (ω) H (ω)h (ω) S YY (ω) = H (ω) 2 S XX (ω) = + Rxx (τ)e j ω τ d τ = + RYY (τ)e j ω τ d τ Signals & Spectra (A) 20
Linear System & Random Variables (2) R.V R.V X (t ) h(t) Y (t ) ρ(t) = h(t ) h ( t) H (ω) 2 = H (ω) H (ω) R xy (τ) = R xx (τ) h ( τ) R yy (τ) = R xy (τ) h(τ) R yy (τ) = R xx (τ) ρ(τ) S xy (ω) = S xx (ω)h (ω) S yy (ω) = S xy (ω)h (ω) S yy (ω) = S xx (ω) H (ω) 2 R xx (τ) h(t ) h ( t) R yy (τ) S xx (ω) H (ω)h (ω) S yy (ω) Signals & Spectra (A) 2
Summary () Non-periodic signals Periodic signals Random signals Energy Signal Power Signal Power Signal + /2 E x = /2 x 2 (t) dt P x = + /2 /2 x 2 (t) dt P x = + /2 /2 x 2 (t) dt Energy Spectral Density Power Spectral Density Power Spectral Density Ψ( f ) = X ( f ) 2 G x ( f ) = c n 2 δ(f n f 0 ) n= G x ( f ) = lim otal Energy Average Power Average Power X ( f ) 2 Ψ( f ) d f Gx ( f ) d f Gx ( f ) d f Signals & Spectra (A) 22
Summary (2) Energy Signal Autocorrelation Power Signal Autocorrelation Random Signal Autocorrelation R x ( τ) = R x ( τ) = x(t) x(t + τ) dt lim + + /2 /2 x (t)x (t + τ) dt E {X (t) X (t + τ)} R x ( τ) = Non-periodic signals Periodic signals Random signals for a Periodic Signal for a Ergodic Signal R x ( τ) = R x ( τ) = R x ( τ) = x(t) x(t + τ) dt + 0 /2 0 /2 x (t) x(t + τ) dt 0 lim + + /2 /2 X (t) X (t+ τ) dt Signals & Spectra (A) 23
References [] http://en.wikipedia.org/ [2] http://planetmath.org/ [3] B. Sklar, Digital Communications: Fundamentals and Applications