What is Correlation in Signals and Systems?

What is Correlation?

The correlation of two functions or signals or waveforms is defined as the measure of similarity between those signals. There are two types of correlations −

• Cross-correlation

• Autocorrelation

Cross-correlation

The cross-correlation between two different signals or functions or waveforms is defined as the measure of similarity or coherence between one signal and the time-delayed version of another signal. The cross-correlation between two different signals indicates the degree of relatedness between one signal and the time-delayed version of another signal.

The cross-correlation of energy (or aperiodic) signals and power (or periodic) signals is defined separately.

Cross-correlation of Energy Signals

Consider two complex signals $\mathit{x_{\mathrm{1}}\mathrm{\left ( \mathit{t} \right )}}$ and $\mathit{x_{\mathrm{2}}\mathrm{\left ( \mathit{t} \right )}}$ of finite energy. Then, the cross-correlation of these two energy signals is defined as

$$\mathrm{\mathit{R_{\mathrm{12}}\mathrm{\left(\tau\right )}\mathrm{=}\int_{-\infty }^{\infty }x_{\mathrm{1}}\mathrm{\left ( \mathit{t}\right)}x_{\mathrm{2}}^{*}\mathrm{\left( \mathit{t-\tau }\right )}dt}\:\mathrm{=}\:\int_{-\infty }^{\infty }\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{t+\tau }\right )}\mathit{x}_{\mathrm{2}}^{*}\mathrm{\left(\mathit{t}\right)}\:\mathit{dt}}$$

If the two signals $\mathit{\mathit{x}_{\mathrm{1}}\mathrm{\left ( \mathit{t} \right )}}$ and $\mathit{\mathit{x}_{\mathrm{2}}\mathrm{\left ( \mathit{t} \right )}}$ are real, then the cross-correlation between them is,

$$\mathrm{\mathit{R_{\mathrm{12}}\mathrm{\left(\tau\right )}\mathrm{=}\int_{-\infty }^{\infty }x_{\mathrm{1}}\mathrm{\left ( \mathit{t}\right)}x_{\mathrm{2}}\mathrm{\left( \mathit{t-\tau }\right )}dt}\:\mathrm{=}\:\int_{-\infty }^{\infty }\mathit{\mathit{x}}_{\mathrm{1}}\mathrm{\left(\mathit{t+\tau} \right )}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{t}\right)}\:\mathit{dt}}$$

If the energy signals $\mathit{x_{\mathrm{1}}\mathrm{\left ( \mathit{t} \right )}}$ and $\mathit{x_{\mathrm{2}}\mathrm{\left ( \mathit{t} \right )}}$ have some similarity. Then, the cross-correlation $\mathit{R_{\mathrm{12}}\mathrm{\left(\tau\right )}}$ between them will have some finite value over the range $\tau$. The variable $\tau$ is called the delay parameter or searching parameter or scanning parameter. The time-delay parameter ($\tau$) is the time delay or time shift of one of the two signals. This delay parameter $\tau$ determines the correlation between two signals.

Cross-correlation of Power Signals

The cross-correlation $\mathit{R_{\mathrm{12}}\mathrm{\left(\tau\right )}}$ for two power (or periodic) signals $\mathit{x_{\mathrm{1}}\mathrm{\left ( \mathit{t} \right )}}$ and $\mathit{x_{\mathrm{2}}\mathrm{\left ( \mathit{t} \right )}}$ may be defined using the average form of correlation. If two power signals $\mathit{x_{\mathrm{1}}\mathrm{\left ( \mathit{t} \right )}}$ and $\mathit{x_{\mathrm{2}}\mathrm{\left ( \mathit{t} \right )}}$ have the same time period (say T), then the cross-correlation between them is defined as follows −

$$\mathrm{\mathit{R_{\mathrm{12}}\mathrm{\left(\tau\right )}\mathrm{=}\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathrm{\left (\mathit{T/\mathrm{2}}\right )}}^{\mathrm{\left(\mathit{T/\mathrm{2}}\right)}}x_{\mathrm{1}}\mathrm{\left ( \mathit{t}\right)}x_{\mathrm{2}}^{*}\mathrm{\left( \mathit{t-\tau }\right )}\:dt}}$$

Autocorrelation

The autocorrelation function is defined as the measure of similarity or coherence between a signal and its time delayed version. Therefore, the autocorrelation is the correlation of a signal with itself.

Like cross-correlation, autocorrelation is also defined separately for energy (or aperiodic) signals and power (periodic) signals.

Autocorrelation of Energy Signals

The autocorrelation of an energy or aperiodic signal $\mathit{x\mathrm(\mathit{t})}$ is defined as −

$$\mathrm{\mathit{R_{\mathrm{11}}\mathrm{\left(\tau\right )}\:\mathrm{=}\:R\mathrm{\left ( \tau \right )}\:\mathrm{=}\int_{-\infty }^{\infty }\:x\mathrm{\left ( \mathit{t}\right)}x^{\mathrm{*}}\mathrm{\left( \mathit{t-\tau }\right )}\:dt}}$$

Where, the variable $\tau$ is called the delay parameter and here, the signal $\mathit{x\mathrm(\mathit{t})}$ is time shifted by $\tau$ units in the positive direction.

If the signal $\mathit{x\mathrm(\mathit{t})}$ is shifted by $\tau$ units in negative direction, then the autocorrelation of the signal is defined as,

$$\mathrm{\mathit{R_{\mathrm{11}}\mathrm{\left(\tau\right )}\:\mathrm{=}\:R\mathrm{\left ( \tau \right )}\:\mathrm{=}\int_{-\infty }^{\infty }\:x\mathrm{\left ( \mathit{t+\tau }\right)}x^{\mathrm{*}}\mathrm{\left( \mathit{t}\right )}\:dt}}$$

Autocorrelation of Power Signals

The autocorrelation of a power signal or periodic signal $\mathit{x\mathrm(\mathit{t})}$ having a time period T is defined as,

$$\mathrm{\mathit{R\mathrm{\left(\tau\right )}\mathrm{=}\displaystyle \lim_{\mathrm{T} \to \infty }\frac{\mathrm{1}}{\mathit{T}}\int_{-\mathrm{\left (\mathit{T/\mathrm{2}}\right )}}^{\mathrm{\left(\mathit{T/\mathrm{2}}\right)}}x\mathrm{\left ( \mathit{t}\right)}x^{*}\mathrm{\left( \mathit{t-\tau }\right )}\:dt}}$$