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  (CCF), which is given here under the assumptions of  stationarity  and  ergodicity  .

$\text{Definition:}$  For the  cross-correlation function  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 a (as large as possible) measurement period  $T_{\rm M}$ 

.

A comparison with the  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  $φ_x(τ)$  .

Cross-correlation function of a binary signal

$\text{Example 1:}$  We consider a random signal  $x(t)$  with triangular ACF  $φ_x(τ)$   ⇒   blue curve.  This ACF shape results, for example.

• for a binary signal with equal probability bipolar amplitude coefficients  $(+1$  resp.  $-1)$  and
• for rectangular fundamental momentum.

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 transit time  $(t_0 = 3 \ \rm ms)$  ;

This attenuated and shifted signal has the ACF drawn in red.

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

The shift around  $t_0$  is not seen in the ACF in contrast to the cross correlation function (CCF) (shown in green) 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 are compiled and important differences to the ACF are elaborated.

• 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,  $φ_{yx}(τ)$  would have its maximum at  $τ = -3 \ \rm ms$.
• In general, the  maximum CCF  does not occur at $τ = 0$  $($exception:   $y = α - x)$  and the KKF value  $φ_{xy}(τ = 0)$  does not have any special, physically interpretable meaning as in the ACF, where this value reflects the process power.
• The magnitude of the KKF is less than or equal to the geometric mean of the two signal powers according to the  Cauchy-Schwarz inequality  for all  $τ$-values:

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

• In  $\text{Example 1}$  on the last page, 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 limit of the CCF 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 an interfering signal together.
• It should always be noted, however, 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  $y(t) = x^2(t)$  - on the other hand, are not taken into account in the CCF formation.

Applications of the cross-correlation function

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

$\text{Example 2:}$  In  Amplitude Modulation, but also in  BPSK systems  (Binary Phase Shift Keying), the so-called synchronous demodulator is very 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 receive signal and the receive carrier signal, the phase synchronous position between the two signals can be recognized by means of the peak of the KKF, and it can be readjusted in case of drifting apart

.

$\text{Example 3:}$  The multiple access method  CDMA  (Code Division Multiple Access)  is used, for example, in the mobile radio standard  UMTS  . It requires strict phase synchronism, with respect to the added  pseudonoise sequences  at the transmitter  (bandspread)  and at the receiver  (bandspread) (Bitte um bessere Übersetzung).  This synchronization problem is also usually solved using the cross-correlation function.

$\text{Example 4:}$  The cross-correlation function 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 value for  $t_0$  then, for example, a driving speed can be determined  (radar technique).  This task can also be solved with the so-called matched filter, which is still described in a  later chapter  and which has many similarities with the cross-correlation function.

$\text{Example 5:}$  In the so-called  Correlation receiver  one uses the CCF for signal detection.   Here one forms the cross-correlation between the received signal distorted by noise and possibly also by distortions  $r(t)$  and all possible transmitted signals  $s_i(t)$, where for the running 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 density  ${\it Φ}_{xy}(f)$  and  ${\it Φ}_{yx}(f)$  result from the corresponding cross-correlation functions  $\varphi_{xy}({\it \tau})$  respectively.   $\varphi_{yx}({\it \tau})$  by the 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  $X(f)$  respectively.
• between 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  Inverse Fourier transform describes here   ⇒   "Second Fourier integral" the transition from the spectral domain to the time domain.

For the definition of the cross-correlation function

$\text{Example 6:}$  We refer to  $\text{Example 1}$  with the two "rectangular random variables"  $x(t)$   and  $y(t) = α - x(t - t_0)$.

Since the ACF  ${\it φ}_x(τ)$  is triangular, as described in the chapter  power spectral density  the PSD  ${\it Φ}_x(f)$  has a  ${\rm si}^2$-shaped profile.
In general, what statements can we derive from this graph for spectral functions?

• In the quoted  $\text{Example 1}$  we found that the autocorrelation function  ${\it φ}_y(τ)$  differs from  ${\it φ}_x(τ)$  only by the constant factor  $α^2$  .
• 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.
• 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