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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
Correlation of Discrete-Time Signals

Transmitted Signal, x(n)

Reflected Signal,

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

0

T

cross correlation
Cross-Correlation
  • Cross-correlation of x(n) and y(n) is a sequence, rxy(l)
  • Reversing the order, ryx(l)
  • =>
similarity to convolution
Similarity to Convolution
  • No folding (time-reversal)
  • In Matlab:
    • Conv(x,fliplr(y))
auto correlation
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
Properties
  • Two sequences, x(n) and y(n), with finite energy
  • Find energy of z(n)

Ex

Ey

slide6

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]:

periodic sequences
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
lti systems
LTI Systems
  • Convolution, output of LTI system
  • Crosscorrelation between the output and input signal:
  • Similarly, input to output is:
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