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Correlation of Discrete-Time Signals. Transmitted Signal, x(n). Reflected Signal, y(n) = x(n-D) + w(n). 0. T. Cross-Correlation. Cross-correlation of x(n) and y(n) is a sequence, r xy (l) Reversing the order, r yx (l) => . Similarity to Convolution. No folding (time-reversal)

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Correlation of Discrete-Time Signals

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Correlation of discrete time signals l.jpg

Correlation of Discrete-Time Signals

Transmitted Signal, x(n)

Reflected Signal,

y(n) = x(n-D) + w(n)

0

T


Cross correlation l.jpg

Cross-Correlation

  • Cross-correlation of x(n) and y(n) is a sequence, rxy(l)

  • Reversing the order, ryx(l)

  • =>


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Similarity to Convolution

  • No folding (time-reversal)

  • In Matlab:

    • Conv(x,fliplr(y))


Auto correlation l.jpg

Auto-Correlation

  • Correlation of a signal with itself

  • Used to differentiate the presence of a like-signal, e.g., zero or one

  • Even function


Properties l.jpg

Properties

  • Two sequences, x(n) and y(n), with finite energy

  • Find energy of z(n)

Ex

Ey


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Quadratric in (a/b) and positive, discriminant is non-negative:

For crosscorrelation case:

For autocorrelation case:

Maximum value occurs with zero lag (when signals are perfectly matched)

Often normalized to range [-1,1]:


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Periodic Sequences

  • Power signals crosscorrelation:

  • Define auto and crosscorrelations over one period of the signals

  • If x(n) and y(n) are periodic signals with period N:

  • Correlations are also periodic with period N


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y(n)=x(n)+w(n)


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LTI Systems

  • Convolution, output of LTI system

  • Crosscorrelation between the output and input signal:

  • Similarly, input to output is:


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  • Autocorrelation of output:


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