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Autocorrelation
Definition: If the process {X(t)} is stationary either in the strict sense or in the wide sense, then $E{X(t).X(tτ)}$ is a function of $τ$, denoted by $R_xx (τ) \ or R(τ) \ or \ R_x (τ)$. This function $R(τ)$ is called the autocorrelation function of the process {X(t)}
**Properties:**
1. R(t) is a even function of $τ$
i.e. $R(τ)=R(τ)$
2. R(τ)is maximum at $τ=0$
3. If the autocorrelation function R(t) of a real stationary process {X(t)} is continuous at τ=0, it is continuous at every other point
4. If R(τ) is the autocorrelation function of a stationary process {X(t)} with no periodic component, then $\lim_{τ→∞)}R(τ)=μ_{x}^2$, provided the limit exists
**Proof:**
1. **R(t) is a even function of τ**
i.e. $R(τ)=R(τ)$
Proof: $R(τ)=E(X(t)×X(tτ))$
R(τ)=E(X(t)×X(t+τ))$
put $t+τ=t'$
∴$ t=t'τ$
∴ $R(τ)=E(X(t'τ)×X(t'))==E(X(t')×X(t'τ))$
∴ $R(τ)=R(τ)$
R(τ)is maximum at τ=0
i.e. R(τ)≤R(0)
Proof: Cauchy Schwartz inequality is
$${E(XY)}^2≤E(X^2 )×E(Y^2)$$
$$put X=X(t) \ and \ Y=X(tτ)$$
Then
$$\{\{E(X(t)X(tτ))\}^2 \}≤E(X^2 (t))×E(X^2 (tτ))$$
i.e. $\{R(τ)\}^2≤[E\{X^2 (t)\}]^2$
[Since E{X(t)} and Var {X(t)} are constant for a stationary process]
$$i.e.\{R(τ)\}^2≤\{R(0)\}^2$$
Taking square roots on both sides
R$(τ)$ ≤ R(0) $\space$ [since R(0)=E{X^2 (t)} is positive]
 If R($τ$) is the autocorrelation function of a stationary process {X(t)} with no periodic component, then $\lim_{τ→∞}R(τ)=μ_{x}^2 $, provided the limit exists
Proof:
: $R(τ)=E(X(t)×X(tτ))$
When τis very large, X(t) and $X(tτ)$ are two sample functions (members)of the process {X(t)} observed at a very long interval of time.
Therefore, X(t) and $X(tτ)$ tend to become independent [X(t) and X$(tτ)$ may be dependent, when X(t) contains a periodic component, which is not true]
$\lim_{τ→∞}R(τ)=E(X(t) × X(tτ))=μ_x^2$ $\space$ (since E(X(t)) is constant)
$$∴μ_x=\sqrt{\lim_{τ→∞}R(τ) }$$
Practical Applications:
Use of the autocorrelation function to quantify the effect of noise on a periodic signal
In absence of noise, the autocorrelation function oscillates with a constant amplitude and a maximum of 1. The period of the autocorrelation correspond to the period of the signal.
In presence of noise, the envelope of the autocorrelation function decreases exponentially. More the noise, faster is this decreasing.
This phenomenon is also called "phase diffusion"
Autocorrelation analysis is used heavily in fluorescence correlation spectroscopy.
It is also used in measurement of optical spectra and the measurement of veryshortduration light pulses produced by lasers, both using optical autocorrelators.
In optics, normalized autocorrelations and crosscorrelations give the degree of coherence of an electromagnetic field.
In signal processing, autocorrelation can give information about repeating events like musical beats (for example, to determine tempo) or pulsar frequencies, though it cannot tell the position in time of the beat. It can also be used to estimate the pitch of a musical tone.
In astrophysics, autocorrelation is used to study and characterize the spatial distribution of galaxies in the Universe and in multiwavelength observations of Low Mass Xray Binaries.