# 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,

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