# Of michigan 1 t 1 t fall 2012 2 t f september 7 2012

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Unformatted text preview: perature? PN = = 2.483×106 2.4×106 2.483×106 2.4×106 N0 df + 2 −2.4×106 −2.483×106 N0 df 2 N0 df = 83 × 106 × 4 × 10−21 = 3.32 × 10−13 Watts EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 101 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Correlated Noise RN (τ ) = Λ(τ /T ) = SN (f ) = T sinc2 (fT ) =T EECS 455 (Univ. of Michigan) τ 1 − |T | 0 |τ | ≤ T , |τ | &gt; T sin2 (π fT ) (π fT )2 Fall 2012 September 7, 2012 102 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Correlated Noise RN (τ ) −T τ T SN (f ) 2 −T EECS 455 (Univ. of Michigan) 1 −T 1 T Fall 2012 2 T f September 7, 2012 103 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Noise into linear systems Now consider noise at the input to the receiver. X (t ) - Y (t ) H (f ) - The power spectral density of the output of the ﬁlter is determined from the power spectral density at the input to the ﬁlter and the transfer function of the ﬁlter. SY (f ) = |H (f )|2 SX (f ) EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 104 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Autocorrelation The autocorrelation is given by RY (τ ) = E [Y (t )Y (t + τ )] = E[ ∞ X (t − α)h(α)d α −∞ ∞ ∞ = = −∞ X (t + τ − β )h(β )d β ] E [X (t − α)X (t + τ − β )]h(α)h(β )d αd β −∞ ∞ −∞ ∞ −∞ ∞ −∞ ∞ −∞ = ∞ −∞ ˜ RX (τ − γ − β )h(γ )h(β )d γ d β ˜ RX ((τ − β ) − γ )h(γ )d γ h(β )d β ˜ RY (τ ) = RX (τ ) ∗ h ∗ h, ˜ where h(t ) = h(−t ). EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 105 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise At any particular time the output due to noise alone is a random variable with a certain density function. The mean of the output is the convolution of the mean of the input signal with the impulse response of the system. The variance of the output is σ 2 = Var[Y (t )] = RY (0) = = ∞ −∞ ∞ −∞ EECS 455 (Univ. of Michigan) Fall 2012 RX (β − γ )h(γ )h(β )d γ d β |H (f )|2 SX (f )df September 7, 2012 106 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise White Noise into a Filter For the case when the noise is white with power spectral density N0 /2 the variance of the output is Variance at output of ﬁlter due to WGN σ 2 = Var[Y (t )] N0 ∞ 2 = h (γ )d γ 2 −∞ N0 ∞ = |H (f )|2 df . 2 −∞ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 107 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example: Ideal Brickwall Filter H (f ) A6 N0 /2 W - f σ 2 = A2 WN0 (A ﬁlter for which the noise variance is σ 2 but does not have the brickwall shape is said to have noise bandwidth σ 2 /(A2 N0 ) where A is the peak output). EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 108 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Filtering of Gaussian Noise -4 x 10 Time Domain Frequency Domain 0 |X(f)|^2 x(t) 2 0 -200 -2 -2 0 2 -400 -10 4 -5 0 5 10 -5 0 5 10 -5 0 frequency 5 10 0 H(f) h(t) 1 0.5 -40 0 0 2 -60 -10 4 0 2 |Y(f)|^2 y(t)=x(t)*h(t) -0.5 -2 -4 x 10 -20 0 -200 -2 -2 0 2 4 -400 -10 time EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 109 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Gaussian Density σ3 2 σ1 2 2 &gt; σ2 &gt; σ3 2 2 2 σ1 2 0 EECS 455 (Univ. of Michigan) Fall 2012 X(iT) September 7, 2012 110 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Gaussian Density If η is a Gaussian distributed random variable with mean µ and variance σ 2 then we can calculate various probabilities involving η . In particular x P {η &lt; x } = √ 1 2πσ −∞ (x −µ)/σ 2 /2σ 2 e−(w −µ) dw 1 2 √ e−w /2 dw 2π −∞ x −µ x −µ ) = Q (− ) = Φ( σ σ = where α Φ(α) = Q (α) = EECS 455 (Univ. of Michigan) 1 2 √ e−w /2 dw 2π −∞ ∞ √ 1 −w 2/2 1 √e dw = erfc(α/ 2) 2 2π α Fall 2012 September 7, 2012 111 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise If you put the following Matlab code in the appropriate place then you can use the q function in your other codes. function % y=q(x) % % % % q(x) = q = qfunction(x); is the probabilty that a zero mean, unit variance Gaussian random variable is above x. 1/(sqrt(2 pi)) int from x to infinity exp(-tˆ2/2) dt q = 0.5*erfc(x/sqrt(2)); return EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 112 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example 1 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 113 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example 2 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 114 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example 3 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) Fal...
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