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The similarity of two signals is calculated by cross correlation. But, how to define an indicator which quantitatively measures the degree of similarity between two signals? Thanks.

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  • $\begingroup$ normalize your cross correlation, Stephan. $\endgroup$ – robert bristow-johnson Oct 22 '16 at 3:20
  • $\begingroup$ Can you clarify what it is that you need that is not provided by correlation? $\endgroup$ – MBaz Oct 22 '16 at 3:21
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assuming finite power signals:

$$ \lVert x \rVert^2 \triangleq \lim_{N \to \infty} \ \frac{1}{2N+1} \sum\limits_{n=-N}^{+N} \big|x[n] \big|^2 \ < +\infty $$

this is a Hilbert Space sorta thingie.

define inner product:

$$ \langle x,y \rangle \triangleq \lim_{N \to \infty} \ \frac{1}{2N+1} \sum\limits_{n=-N}^{+N} x[n] \cdot \overline{y}[n] $$

where $\overline{y}[n] $ is the complex conjugate of $y[n]$. so this is true about the norm:

$$ \lVert x \rVert = \sqrt{\langle x, x \rangle} $$

Cross-Correlation:

$$ R_{xy}[k] \triangleq \langle x[n], y[n+k] \rangle $$

Autoorrelation:

$$ R_{xx}[k] \triangleq \langle x[n], x[n+k] \rangle \ \le R_{xx}[0] = \lVert x \rVert^2 $$

Normalized Autocorrelation (sometimes called "autocovariance")

$$ -1 \le \frac{R_{xx}[k]}{R_{xx}[0]} \triangleq \frac{\langle x[n], x[n+k] \rangle}{\langle x[n], x[n] \rangle} \ \le 1 $$

Normalized Crosscorrelation:

$$ -1 \le \frac{R_{xy}[k]}{\lVert x \rVert \lVert y \rVert} \triangleq \frac{\langle x[n], y[n+k] \rangle}{\sqrt{\langle x[n], x[n] \rangle}\sqrt{\langle y[n], y[n] \rangle}} \ \le 1 $$

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  • $\begingroup$ Dear @robertbristow-johnson, thanks for your nice solution. $\endgroup$ – Amin Oct 22 '16 at 4:02
  • $\begingroup$ just a mistake. Excuse me. $\endgroup$ – Amin Oct 22 '16 at 4:03
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The general topic of finding similarities between signals is wide ranging:

  • are the signals of same sampling, length, offset, shift or scale?
  • where do they take their values (discrete, real, complex)?
  • are they stationary? noisy?
  • what do you consider similar (whole signals, chunks, specific features)?
  • which are the invariances looked for?
  • and most important: what is your goal?

The following works provide an overview of some common similarity metrics, which one can normalize into an index:

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    $\begingroup$ This is called a useful answer. $\endgroup$ – msm Oct 22 '16 at 6:44
  • $\begingroup$ Dear @Laurent Duval, thanks for your nice answer. $\endgroup$ – Amin Oct 22 '16 at 12:42
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if you are searching for similarity between two signals in frequency domain, you can go for coherence. Coherence indicates frequency components common to both signals

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