# lecture05 - Lecture Notes 5 Goals Lecture 5 Goals Be able...

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Lecture Notes 5 Goals Lecture 5 Goals: Be able to determine bandwidth of digital signals. Be able to convert a signal from baseband to passband and back. Be able to convert a set of signals to a set of vectors and a set of orthogonal signals. EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 1 / 130

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Lecture Notes 5 Bandwidth Bandwidth of Digital Data Signals In many digital communication systems the transmitted signal W ( t ) is an infinite sequence of amplitude modulated pulses or waveforms i.e. W ( t ) = summationdisplay = -∞ b x ( t T ) . This signal is a random process because the data sequence b are random. The random process W ( t ) , however, is not wide sense stationary. That is, the autocorrelation R W ( t ) = E [ W ( t ) W ( t + τ )] typically depends on both t and τ . This means we can not take the Fourier transform of the autocorrelation function to get the power spectral density (because the autocorrelation is a function of both t and τ ). EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 2 / 130
Lecture Notes 5 Bandwidth Bandwidth of Digital Data Signals A digital data signal is modeled as a random process Y ( t ) which is a stationary (wide sense) version of a process W ( t ) ; Y ( t ) = W ( t U ) where U is a random variable needed inorder to make Y ( t ) wide sense stationary. In this case if U is uniformly distributed between 0 and T then Y ( t ) is a wide snese stationary random process. We desire then to compute the auto correlation of Y ( t ) and also the spectrum of Y ( t ) . Assume that { b } = -∞ is a sequences of i.i.d. random variables with zero mean and variance σ 2 (e.g. P { b = + 1 } = 1 / 2 P { b = 1 } = 1 / 2 ). Also assume U and b are independent. EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 3 / 130

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Lecture Notes 5 Bandwidth Bandwidth of Digital Data Signals Claim: R Y ( τ ) = σ 2 T integraldisplay -∞ x ( t ) x ( t + τ ) dt S Y ( f ) = σ 2 T | X ( f ) | 2 where X ( f ) = F{ x ( t ) } EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 4 / 130
Lecture Notes 5 Bandwidth Bandwidth of Digital Data Signals Derivation: For any t and τ E [ Y ( t ) Y ( t + τ )] = E 2 4 X = −∞ b x ( t - T - U ) X m = −∞ b m x ( t + τ - mT - U ) 3 5 = X = −∞ X m = −∞ E { b b m } | {z } δ m = σ 2 , ℓ = m 0 , ℓ negationslash = m E [ x ( t - T - U ) x ( t + τ - mT - U )] = X = −∞ σ 2 E [ x ( t - T - U ) x ( t + τ - T - U )] = X = −∞ σ 2 1 T Z T u = 0 x ( t - T - u ) x ( t + τ - T - u ) du = 1 T X = −∞ σ 2 Z ( + 1 ) T T x ( t - v ) x ( t + τ - v ) dv ( v = T + u , dv = du ) = σ 2 T Z −∞ x ( t - v ) x ( t + τ - v ) dv ( w = t - v ) = σ 2 T Z −∞ x ( w ) x ( w + τ ) dw = σ 2 T Z −∞ x ( t ) x ( t + τ ) dt . EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 5 / 130

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Lecture Notes 5 Bandwidth Bandwidth of Digital Data Signals Derivation: (cont.) Thus Y ( t ) is wide sense stationary with autocorrelation R Y ( τ ) = σ 2 T Z −∞ x ( t ) x ( t + τ ) dt . Now let g 1 ( t ) = x ( t ) and g 2 ( t ) = x ( - t ) then g 1 * g 2 ( τ ) = Z −∞ g 1 ( t ) g 2 ( τ - t ) dt = Z −∞ x ( t ) x ( t - τ ) dt = Z −∞ x ( t + τ ) x ( t ) dt .
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