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Unformatted text preview: l 2012 September 7, 2012 115 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example 4 8 6 4 2 x(τ) h(t−τ) 0 −2 x(τ) h(t−τ) −4 conv(x,h) −6 −8 −2 −1 0 1 2 3 4 5 6 t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 116 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example 5 8 6 4 2 x(τ) h(t−τ) 0 −2 x(τ) h(t−τ) −4 conv(x,h) −6 −8 −2 −1 0 1 2 3 4 5 6 t EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 117 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example 6 Input Filter 100 Output 600 100 500 50 400 0 300 y(t) h(t) n(t) 50 200 0 100 −50 −50 0 −100 0 10 −100 20 0 time (s) 0.05 0.1 −100 10 20 time (s) 30 −20 20 10 −40 0 −10 −20 |Y(f)| (dB) 0 20 |H(f)| (dB) 30 |n(f)| (dB) 0 time (s) −60 −80 −100 −30 −140 −500 0 frequency(Hz) EECS 455 (Univ. of Michigan) 500 0 −10 −20 −120 −40 −500 10 0 frequency(Hz) Fall 2012 500 −30 −500 0 500 frequency(Hz) September 7, 2012 118 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example 7 Input Filter 100 Output 600 100 500 50 400 0 300 y(t) h(t) n(t) 50 200 0 100 −50 −50 0 −100 0 10 −100 20 0 time (s) 0.05 0.1 −100 30 10 20 time (s) 0 30 20 20 0 −10 −20 |Y(f)| (dB) −50 10 |H(f)| (dB) |n(f)| (dB) 0 time (s) −100 10 0 −10 −150 −20 −30 −40 −500 0 frequency(Hz) EECS 455 (Univ. of Michigan) 500 −200 −500 0 frequency(Hz) Fall 2012 500 −30 −500 0 500 frequency(Hz) September 7, 2012 119 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example 8 Consider noise multiplied by a sinusoid into an integrator. That is, T n(t )A cos(2π fc t )dt η= 0 where n(t ) is white Gaussian noise with two sided power spectral density N0 /2. Assume that fc T ≫ 1. Because we are performing a linear operation on Gaussian noise, the resulting random variable η is Gaussian. Since the mean on n(t ) is zero, the mean of η is zero. The variance is determined as follows. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 120 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example 8 Var[η ] = E [η 2 ] T = E[ T n(t )A cos(2π fc t )dt 0 T = T n(t )n(s)A2 cos(2π fc t ) cos(2π fc s)dtds] E[ 0 0 T T E [n(t )n(s)]A2 cos(2π fc t ) cos(2π fc s)dtds = 0 0 T T = 0 0 T = 0 = EECS 455 (Univ. of Michigan) n(s)A cos(2π fc s)ds] 0 N0 δ (t − s)A2 cos(2π fc t ) cos(2π fc s)dtds 2 N0 2 A cos2 (2π fc t )dt 2 A2 N0 4 T (1 + cos(2π 2fc t ))dt = 0 Fall 2012 A2 TN0 4 September 7, 2012 121 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example 9 Consider zero mean white Gaussian noise with power spectral density Sx (f ) = N0 /2 or autocorrelation Rx (τ ) = N0 δ(τ ). If the noise is 2 multiplied by a waveform x (t ) and integrated then the variance can be calucated as follows. η= n(t )x (t )dt Var[η ] = E [η 2 ] = E[ n(t )x (t )dt = E [n(t )n(s)]x (t )x (s)dtds = = EECS 455 (Univ. of Michigan) n(s)x (s)ds] N0 δ(t − s)x (t )x (s)dtds 2 N0 2 N0 x (t )dt = x 2 (t )dt 2 2 Fall 2012 September 7, 2012 122 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Summary: White Gaussian noise Let n(t ) be white Gaussian noise. Then η1 = η2 = ∞ −∞ n(t )h(T η3 = T 0 ⇒ Var[η1 ] = n(t )x (t )dt − t )dt n(t ) cos(2π fc t )dt EECS 455 (Univ. of Michigan) N0 2 ⇒ Var[η2 ] = N0 2 ⇒ Var[η3 ] = N0 T 4. Fall 2012 x 2 (t )dt ∞ 2 −∞ h (t )dt September 7, 2012 123 / 174 Lecture Notes 2 m-sequences Maximal length sequences (m-sequences) Maximal length shift register sequences are used in many applications including spread-spectrum systems. They are sometimes called pseudo-noise (PN) sequences because they seem to have noise like properties. They are sometimes called linear feedback shift register sequences (LFSR) because they are generated using linear feedback (mod 2). They are usually not used as error control codes but as signaling waveforms. The usefulness stems from the nice autocorrelation property m-sequences posses. m-sequences are similar to Fibonacci sequences except for m-sequences the arithmetic is done mod(2). Because of the limited number of m-sequences another class of sequences (Gold sequences) is often used. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 124 / 174 Lecture Notes 2 m-sequences Fibanacci sequences F n = F n −1 + F n −2 , F0 = 0, F1 = 1 MATLAB CODE clear all F(1)=0; F(2)=1; for n=3:51 F(n)=F(n-1)+F(n-2); end f(51) EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 125 / 174 Lecture Notes 2 m-sequences n Fn 0 0 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 n Fn 10 55 11 89 12 144 13 233 14 377 15 610 16 987 17 1597 18 2584 19 4181 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 126 / 174 Lecture Notes 2 m-sequences Suppose the the n-th term was of the form Fn = Asn . Then F n = F n −1 + F n −2 Asn = Asn−1 + Asn−2 s2 = s1 + 1 s2 − s1 − 1 = 0 1 ± (1 + 4) s= √ √2 1− 5 1+ 5 , s1 = s0 = 2 2 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 127 / 174 Lecture Notes 2 m-sequences n n Fn = As0 + Bs1 ; Initial Conditions F0 = 0, F1 = 1; n = 0, n = 1, F0 s1 − F1 A= , s1 − s0 EECS 455 (Univ. of Michigan) F0 = A + B ; F1 = As0 + Bs1 F1 − F0 s0 B= , s1 − s...
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This note was uploaded on 02/12/2014 for the course EECS 455 taught by Professor Stark during the Fall '08 term at University of Michigan.

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