Theory of Stochastic Signals/Cross-Correlation Function and Cross Power-Spectral Density

1. Theory of Stochastic Signals
2. Random Variables with Statistical Dependence
3. Cross-Correlation Function and Cross Power-Spectral Density

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Definition of the cross-correlation function

In many engineering applications,  one is interested in a quantitative measure to describe the statistical relatedness between different processes or between their pattern signals.  One such measure is the  "cross-correlation function",  which is given here under the assumptions of  "stationarity"'  and  "ergodicity".

$\text{Definition:}$  For the  »cross-correlation function«  $\rm (CCF)$ of two stationary and ergodic processes with the pattern functions  $x(t)$  and  $y(t)$  holds:

$$\varphi_{xy}(\tau)={\rm E} \big[{x(t)\cdot y(t+\tau)}\big]=\lim_{T_{\rm M}\to\infty}\,\frac{1}{T_{\rm M} }\cdot\int^{T_{\rm M}/{\rm 2} }_{-T_{\rm M}/{\rm 2} }x(t)\cdot y(t+\tau)\,\rm d \it t.$$

• The first defining equation characterizes the  expected value formation  ("ensemble averaging"),
• while the second equation describes the  "time averaging"  over an  (as large as possible)  measurement period  $T_{\rm M}$.

A comparison with the  $\text{ACF definition}$  shows many similarities.  

• Setting  $y(t) = x(t)$,  we get  $φ_{xy}(τ) = φ_{xx}(τ)$, i.e., the auto-correlation function,
• for which,  however,  in our tutorial we mostly use the simplified notation  $φ_x(τ)$.

$\text{Example 1:}$  We consider a random signal  $x(t)$  with triangular auto-correlation function  $φ_x(τ)$   ⇒   blue curve.  This ACF shape results e.g. for

Cross-correlation function of a binary signal

• a binary signal with equally probable bipolar amplitude coefficients  $(\pm1)$ 
• and a rectangular basic pulse  $g(t)$.

We consider a second signal  $y(t) = \alpha \cdot x (t - t_{\rm 0})$,  which differs from  $x(t)$  only by an attenuation factor  $(α =0.5)$  and a delay time  $(t_0 = 3 \ \rm ms)$.

This attenuated and shifted signal has the auto-correlation function drawn in red:

$$\varphi_{y}(\tau) = \alpha^2 \cdot \varphi_{x}(\tau) .$$

The shift around  $t_0$  is not seen in this auto-correlation function in contrast to the  (green)  cross-correlation function  $\rm (CCF)$  for which the following relation holds:

$$\varphi_{xy}(\tau) = \alpha \cdot \varphi_{x}(\tau- t_{\rm 0}) .$$

Properties of the cross-correlation function

In the following,  essential properties of the cross-correlation function  $\rm (CCF)$  are composed.  Important differences to the auto-correlation function  $\rm (ACF)$  are:

• The formation of the cross-correlation function is  »not commutative«.  Rather,  there are always two distinct functions,  viz.

$$\varphi_{xy}(\tau)={\rm E} \big[{x(t)\cdot y(t+\tau)}\big]=\lim_{T_{\rm M}\to\infty}\,\frac{1}{T_{\rm M}}\cdot\int^{T_{\rm M}/{\rm 2}}_{-T_{\rm M}/{\rm 2}}x(t)\cdot y(t+\tau)\,\, \rm d \it t,$$

$$\varphi_{yx}(\tau)={\rm E} \big[{y(t)\cdot x(t+\tau)}\big]=\lim_{T_{\rm M}\to\infty}\,\frac{1}{T_{\rm M}}\cdot\int^{T_{\rm M}/{\rm 2}}_{-T_{\rm M}/{\rm 2}}y(t)\cdot x(t+\tau)\,\, \rm d \it t .$$

• There is a relationship between the two functions:   $φ_{yx}(τ) = φ_{xy}(-τ)$.  In  $\text{Example 1}$  of the last section,  the second cross-correlation function  $φ_{yx}(τ)$  would have its maximum at  $τ = -3 \ \rm ms$.
• In general,  the  »maximum CCF«  does not occur at  $τ = 0$  $($exception:   $y = α \cdot x)$  and the CCF value  $φ_{xy}(τ = 0)$  does not have any special,  physically interpretable meaning as in the ACF,  where this value reflects the process power.
• For all  $τ$-values,  the  »CCF magnitude«  is less than or equal to the geometric mean of the two signal powers according to the  $\text{Cauchy-Schwarz inequality}$:

$$\varphi_{xy}( \tau) \le \sqrt {\varphi_{x}( \tau = 0) \cdot \varphi_{y}( \tau = 0)}.$$

In  $\text{Example 1}$  in the last section,  the equal sign applies:  $\varphi_{xy}( \tau = t_{\rm 0}) = \sqrt {\varphi_{x}( \tau = 0) \cdot \varphi_{y}( \tau = 0)} = \alpha \cdot \varphi_{x}( \tau = {\rm 0}) .$

• If  $x(t)$  and  $y(t)$  do not contain a common periodic fraction,  the  »CCF limit«  for  $τ → ∞$  gives the product of both means:

$$\lim_{\tau \rightarrow \infty} \varphi _{xy} ( \tau ) = m_x \cdot m_y .$$

• If two signals  $x(t)$  and  $y(t)$  are  »uncorrelated«,  then  $φ_{xy}(τ) ≡ 0$,  that is,  it is  $φ_{xy}(τ) = 0$  for all values of  $τ$.   For example,  this assumption is justified in most cases when considering a useful signal and a noise signal together.
• However,  it should always be noted,  that the CCF includes only the  »linear statistical bindings«  between the signals  $x(t)$  and  $y(t)$.  Bindings of other types  – such as for the case  $y(t) = x^2(t)$  –  are not taken into account in the CCF formation.

Applications of the cross-correlation function

The applications of the cross-correlation function in Communication systems are many.  Here are some examples:

$\text{Example 2:}$  In  $\text{amplitude modulation}$,  but also in  $\text{BPSK systems}$  ("Binary Phase Shift Keying"),  the so-called  $\text{Synchronous Demodulator}$  is often used for demodulation  (resetting the signal to the original frequency range),  whereby a carrier signal must also be added at the receiver,  and this must be frequency and phase synchronous to the transmitter.

⇒   If one forms the CCF between the received signal  $r(t)$  and the carrier signal  $z_{\rm E}(t)$  on the receiver side,  the phase synchronous position between the two signals can be recognized by means of the CCF peak,  and it can be readjusted in case of drifting apart.

$\text{Example 3:}$  The multiple access method  $\text{CDMA}$  ("Code Division Multiple Access")  is used,  for example,  in the mobile radio standard  $\text{UMTS}$.  It requires strict phase synchronism,  with respect to the added  "pseudonoise sequences"  at the transmitter  ("band spreading")  and at the receiver  ("band compression"). 

⇒   This synchronization problem is also usually solved using the cross-correlation function.

$\text{Example 4:}$  The CCF can be used to determine whether or not a known signal  $s(t)$  is present in a noisy received signal  $r(t) = α - s(t - t_0) + n(t)$  and if so,  at what time  $t_0$  it occurs.

• From the calculated  $t_0$  value,  for example,  a driving speed can be determined  ("radar technique"). 
• This task can also be solved with the "matched filter",  which has many similarities with the CCF and is described in a  $\text{later chapter}$.

$\text{Example 5:}$  In the so-called  $\text{correlation receiver}$,  one uses the CCF for signal detection.   Here one forms the cross-correlation function

• between the received signal  $r(t)$  $($distorted by noise and possibly also by distortions$)$ 
• and all possible transmitted signals  $s_i(t)$,  where for the control index  $i = 1$, ... , $I$  shall hold. 

⇒   Deciding  $N$  binary symbols together,  then  $I = {\rm 2}^N$.  One then decides on the symbol sequence with the largest CCF value,  achieving the minimum error probability according to the  "maximum likelihood decision rule".

Cross power-spectral density

For some applications it can be quite advantageous to describe the correlation between two random signals in the frequency domain.

$\text{Definition:}$ 

The two  »cross power-spectral densities«  ${\it Φ}_{xy}(f)$  and  ${\it Φ}_{yx}(f)$  result from the corresponding cross-correlation functions  $\varphi_{xy}({\it \tau})$,  resp.  $\varphi_{yx}({\it \tau})$  by Fourier transform:

$${\it \Phi}_{xy}(f)=\int^{+\infty}_{-\infty}\varphi_{xy}({\it \tau}) \cdot {\rm e}^{ {\rm -j}\pi f \tau} \rm d \it \tau, $$

$${\it \Phi}_{yx}(f)=\int^{+\infty}_{-\infty}\varphi_{yx}({\it \tau}) \cdot {\rm e}^{ {\rm -j}\pi f \tau} \rm d \it \tau.$$

The same relationship applies here as between

• a deterministic signal  $x(t)$  and its spectrum  $X(f)$, 
• the auto-correlation function  ${\it φ}_x(τ)$  of an ergodic process  $\{x_i(t)\}$  and the corresponding power-spectral density  ${\it Φ}_x(f)$.

Similarly,  the  $\text{inverse Fourier transform}$   ⇒   "Second Fourier integral"  describes here the transition from the frequency domain to the time domain.

$\text{Example 6:}$  We refer to  $\text{Example 1}$ 

For the definition of the cross-correlation function

• with the rectangular random variable  $x(t)$   
• and the attenuated and shifted signal  $y(t) = α - x(t - t_0)$.

⇒   Since the auto-correlation function  ${\it φ}_x(τ)$  is triangular,  the power-spectral density  ${\it Φ}_x(f)$  has a  ${\rm sinc}^2$-shaped profile.

In general, what statements can we derive from this graph for the spectral functions?

1. In  $\text{Example 1}$  we found that the autocorrelation function  ${\it φ}_y(τ)$  differs from  ${\it φ}_x(τ)$  only by the constant factor  $α^2$.
2. It is clear that the power-spectral density  ${\it Φ}_y(f)$  differs from  ${\it \Phi}_x(f)$  also only by this constant factor  $α^2$.  Both spectral functions are real.
3. In contrast,  the cross power-spectral density has a complex functional:

$${\it \Phi}_{xy}(f) ={\it \Phi}^\star_{yx}(f)= \alpha \cdot {\it \Phi}_{x}(f) \hspace{0.05cm}\cdot {\rm e}^{- {\rm j } \hspace{0.02cm}\pi f t_0}.$$

Exercises for the chapter

Exercise 4.14: ACF and CCF for Square Wave Signals

Exercise 4.14Z: Echo Detection