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Cross Correlation Function and its Properties
Cross Correlation Function
The cross correlation function between two different signals is defined as the measure of similarity or coherence between one signal and the time delayed version of another signal.
The cross correlation function is defined separately for energy (or aperiodic) signals and power or periodic signals.
Cross Correlation of Energy Signals
Consider two energy signals $\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t})}$ and $\mathit{x_{\mathrm{2}}}\mathrm{(\mathit{t})}$. The cross correlation of these two energy signals is defined as −
$$\mathit{R_{\mathrm{12}}}\mathrm{(\tau)}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t})}x_{\mathrm{2}}^{*}\mathrm{(\mathit{t-\tau})}\mathit{dt} \:\mathrm{=}\: \int_{-\infty}^{\infty}\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t+\tau})}\mathit{x_\mathrm{2}^*}\mathrm{(\mathit{t})}\mathit{dt}$$
Where, the variable $\tau$ is called the delay parameter or scanning parameter or searching parameter.
The cross correlation of two energy signals is defined in another form as −
$$\mathit{R_{\mathrm{12}}}\mathrm{(\mathit{\tau})} \:\mathrm{=}\: \int_{-\infty}^{\infty}\mathit{x_\mathrm{2}}\mathrm{(t)}\mathit{x_\mathrm{1}^*}\mathrm{(t-\tau)}\:\mathit{dt}$$
Properties of Cross Correlation Function for Energy Signals
The properties of cross correlation function for energy signals are given as follows −
Property 1
The cross correlation functions of energy signals exhibit conjugate symmetry property, that is,
$$\mathit{R_\mathrm{12}}\mathrm{(\mathit{\tau})}\:\mathrm{=}\:\mathit{R_\mathrm{21}^*}\mathrm{(-\tau)}$$
Property 2
The cross correlation functions of energy signals are not in general commutative, i.e.,
$$\mathit{R_\mathrm{12}}\mathrm{(\mathit{\tau})}\:\mathrm{
eq}\:\mathit{R_\mathrm{21}}\mathrm{(-\tau)}$$
Property 3
If,
$$\mathit{R_\mathrm{12}}\mathrm{(0)}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x_\mathrm{1}}\mathrm{(\mathit{t})}\mathit{x_\mathrm{2}^*}\mathrm{(\mathit{t})}\mathit{dt}\:\mathrm{=}\:\mathrm{0}$$
Then, the two energy signals $\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t})}$ and $\mathit{x_{\mathrm{2}}}\mathrm{(\mathit{t})}$ are said to be orthogonal signals over the entire time interval. The cross correlation of orthogonal signals is zero.
Property 4
The cross correlation of two energy signals is equivalent to the product of the Fourier transform of one signal and the complex conjugate of Fourier transform of another signal, i.e.,
$$\mathit{R_\mathrm{12}}\mathrm{(\tau)}\:\leftrightarrow\:\mathit{X_\mathrm{1}}\mathrm(\omega).\mathit{X_\mathrm{2}^*}\mathrm{(\mathit{\omega})}$$
This property of cross correlation is known as correlation theorem.
Cross Correlation of Power Signals
Consider two power (or periodic) signals $\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t})}$ and $\mathit{x_{\mathrm{2}}}\mathrm{(\mathit{t})}$ having the same time period (say T), then the cross correlation of these two power signals is defined as,
$$\mathit{R_\mathrm{21}}\mathrm{(\mathit{\tau})}\:\mathrm{=}\:\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathrm{(\mathit{T}\diagup2)}}^{\mathrm{(\mathit{T}\diagup2)}}\mathit{x_\mathrm{1}}\mathrm{(\mathit{t})}\mathit{x_\mathrm{2}^*}\mathrm{(\mathit{t-\tau})}\mathit{dt}$$
The cross correlation of two periodic functions is defined in another form as −
$$\mathit{R_\mathrm{21}}\mathrm{(\tau)}\:\mathrm{=}\:\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathrm{(\mathit{T}/\mathrm{2})}}^{\mathrm{(\mathit{T/\mathrm{2}})}}\mathit{x_\mathrm{2}}\mathrm{(\mathit{t})}\:\mathit{x_\mathrm{1}^*}\mathrm{(\mathit{t-\tau})}\:\mathit{dt}$$
Where, the variable $\tau$ is called the delay parameter.
Properties of Cross Correlation Function for Power Signals
The properties of cross correlation for power signals are given as follows −
Property 1
The cross correlation of two power signals exhibits complex conjugate symmetry, i.e.,
$$\mathit{R_\mathrm{12}}\mathrm{(\mathit{\tau})}\:\mathrm{=}\:\mathit{R_\mathrm{21}^*}\mathrm{(-\tau)}$$
Property 2
The cross correlation of two power signals is not commutative, that is,
$$\mathit{R_\mathrm{12}}\mathrm{(\mathit{\tau})}\:\mathrm{
eq}\:\mathit{R_\mathrm{21}}\mathrm{(-\tau)}$$
Property 3
The cross correlation function of two power signals is equivalent to the multiplication of Fourier transform of one signal and the complex conjugate of Fourier transform of the other signal, i.e.,
$$\mathit{R_\mathrm{12}}\mathrm{(\tau)}\:\leftrightarrow\:\mathit{X_\mathrm{1}}\mathrm{(\omega)}.\mathit{X_\mathrm{2}^*}\mathrm{(\omega)}$$
Property 4
If,
$$\mathit{R_\mathrm{12}}\mathrm{(0)}\:\mathrm{=}\:\lim_{T \rightarrow \infty}\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathrm{(T/\mathrm{2})}}^{\mathrm{(T/\mathrm{2})}}\mathit{x_\mathrm{1}}\mathrm{(t)}\:\mathit{x_\mathrm{2}^*}\mathrm{(\mathit{t})}\mathit{dt}\:\mathrm{=}\:\mathrm{0}$$
Then, the two power signals $\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t})}$ and $\mathit{x_{\mathrm{2}}}\mathrm{(\mathit{t})}$ are called the orthogonal signals over the entire time interval.
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