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Autocorrelation of a sine wave

ECE 3800

Western Michigan University

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X(t) is given by the

sine functionwhere q is auniform randomdistributionbetween 0 and 2 p .

We have two pathsto solve for theautocorrelation

= 5 2 +

q

f q (q )

0 2p

= 1 = 2 = 12

= + = 5 2 + 5 2( +

= 5 2 + 5 2

= + Autocorrelation = + Time Autocorrelation

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Statistical Autocorrelation

The Statisticalautocorrelation isperformed byintegrating therandom variables.

We will need touse a trigmanipulation toseparate the sinefunctions.

= 5 2 + 5 2 +2 12

= +

= +

+ = 2 +

= 25

2

2(2 +

) + 2 1

2

= 2 + = 2

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We are going tosplit theintegration intoparts.

= 25 2 2(2 + ) + 2 12

= 254 2 254 2(2 + ) + 2

= 254 ( 2 2(2 + ) + 2

254 4 + 2 +2 254 2

= 4 + 2 = 2

252

254 2

254 2

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The 4 p pushes the sine 4 cycles ahead ofthe other, but doesnt change value.

252 25

4 2 252sin 4 + 2 + 2 | 25

2[sin 4 + 4 +2 sin

0, 2 p , 4 p

252[sin 4 + 2 sin 4

254 2 (2 0) 252 2

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Statistical Autocorrelation

The statisticalautocorrelation isstationary and isonly a function ofthe lag time.

= 2 0 = 2

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Time Autocorrelation

Next we will try thetime autocorrelation.The sine function will beintegrated over a singleperiod. For a sine wavethe single period isT=2p.

Again we use a trigmanipulation toseparate the two sinefunctions

= 1 +

= 1 5 2 + 5 2 + 2

+ = 2 + = 2 + = 2

= 252 2 2(2 + )+ 2

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The final steps of the time autocorrelationplays out very similar to the statistical

autocorrelation.

= 25 2 2(2 + ) + 2

= 25 2 4 + 2 + 2

= 252 2 252 4 + 2 + 2

= 254 2 252 sin 4 + 2 +

= 252 2

258 [sin 4 + 2 + 2 sin 4

= 2

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