Theory of Stochastic Signals/Creation of Predefined ACF Properties

1. Theory of Stochastic Signals
2. Filtering of Stochastic Signals
3. Creation of Predefined ACF Properties

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ACF at the output of a non-recursive filter

We consider a   non-recursive  $M$–th order delay  filter according to the following diagram.

$M$-th order non-recursive filter

The discrete-time input variable  $〈\hspace{0.05cm}x_ν\hspace{0.05cm}〉$  is

• mean-free  $(m_x = 0)$,
• Gaussian distributed  (with  scatter  $σ_x)$,  and
• without memory ("white noise")   ⇒   statistically independent samples.

This results in the following properties:

• The discrete-time auto-correlation function (ACF) at the input is:

$$\varphi _x ( {k \cdot T_{\rm A} } ) = \left\{ {\begin{array}{*{20}c} {\sigma _x ^2 } & {\rm{f\ddot{u}r}\quad {\it k} = 0,} \\ 0 & {\rm{f\ddot{u}r}\quad {\it k} \ne 0.} \\\end{array}} \right.$$

• The ACF of the discrete-time output sequence  $〈\hspace{0.05cm}y_ν\hspace{0.05cm}〉$  is given as follows:

$$\varphi _y ( {k \cdot T_{\rm A} } ) = \sigma _x ^2 \cdot \sum\limits_{\mu = 0}^{M - k} {a_\mu \cdot a_{\mu + k } } \quad {\rm{f\ddot{u}r}}\quad {\it k} = 0, 1,\,\text{...}\,,\,{\it M}.$$

• All ACF values with  $k > M$  are zero, and all ACF values with  $k < M$  are symmetric about  $k = 0$:

$$\varphi _y ( { - k \cdot T_{\rm A} } ) = \varphi _y ( {k \cdot T_{\rm A} } ).$$

$\text{Example 1:}$  If discrete-time white noise with dispersion  $σ_x = 2$  is present at the input of a first-order non-recursive filter  $($filter coefficients  $a_0 = 0.6$,  $a_1 = 0.8),$  the discrete ACF values of the output signal are (all other ACF values are zero):

ACF at the output of a first order filter

$$\varphi _y (0) = \sigma _x ^2 \cdot ( {a_0 ^2 + a_1 ^2 }) = 4,$$

$$\varphi _y ( { - T_{\rm A} } ) = \varphi _y ( {T_{\rm A} } ) = \sigma _x ^2 \cdot a_0 \cdot a_1 = 1.92.$$

The diagram can be interpreted as follows:

• Because of  $a_0^2 + a_1^2 = 1$,  the output signal  $y(t)$  has exactly the same variance  $σ_y^2 = φ_y(0) = 0.4$  as the input signal:   $σ_x^2 = φ_x(0)$.
• Unlike the input sequence  $〈\hspace{0.05cm}x_ν\hspace{0.05cm}〉$,  the output sequence  $〈\hspace{0.05cm}y_ν\hspace{0.05cm}〉$  has statistical ties between adjacent samples.

Determining the coefficients

Now the following question is to be answered:   How can the coefficients  $a_0$, ... , $a_M$  of a non-recursive filter  $M$–th order can be determined,

• if the desired ACF values  $φ_y(0)$, ... , $φ_y(M · T_{\rm A})$  are given, and
• outside the range from  $-M · T_{\rm A}$  to  $+M · T_{\rm A}$  all ACF values are to be zero.

For  $σ_x = 1$,  the following nonlinear system of equations is obtained, using  $φ_k = φ_y(k · T_{\rm A})$  for simplicity of notation:

$$\begin{align*}\varphi _0 & = \sum\limits_{\mu = 0}^M {a_\mu^2 ,}\\ \varphi _1 & = \sum\limits_{\mu = 0}^{M - 1} {a_\mu \cdot a_{\mu + 1} ,} \\ & ... &\\ \varphi _{M - 1} & = a_0 \cdot a_{M - 1} + a_1 \cdot a_M , \\ \varphi _M & = a_0 \cdot a_M .\end{align*}$$

$\text{Conclusion:}$ 

• Thus, for the  $M + 1$  coefficients, one also obtains  $M + 1$  independent equations.
• By successive elimination of the coefficients  $a_1$, ... , $a_M$,  finally a nonlinear equation of higher order remains for  $a_0$. 

$\text{Example 2:}$  We consider the following constellation:

• a recursive filter of first order   ⇒   $M = 1$,
• a discrete-time input sequence  $〈\hspace{0.05cm}x_ν\hspace{0.05cm}〉$  with mean  $m_x =$ 0   and  dispersion  $σ_x = 1$,
• desired ACF values of the sequence  $〈\hspace{0.05cm}y_ν\hspace{0.05cm}〉$:   $ φ_y(0) = φ_0 =0.58$  and  $φ_y(±T_{\rm A}) = φ_1 = 0.21$.

Thus, the above system of equations is:

$$\varphi _0 = a_0 ^2 + a_1 ^2 = 0.58,$$

$$\varphi _1 = a_0 \cdot a_1 = 0.21.$$

This leads to an equation of degree  $4$, namely

$$a_0 ^2 + \left( { { {0.21} }/{ {a_0 } } } \right)^2 = 0.58\quad \Rightarrow \quad a_0 ^4 - 0.58 \cdot a_0 ^2 + 0.21^2 = 0.$$

A solution represents  $a_0 = 0.7$.    By inserting it into the second equation, we find  $a_1 = 0.3$.

One recognizes from this example that already in the simplest case   ⇒   $M = 1$  for  $a_0$  a nonlinear determination equation of degree  $4$  results.

Ambiguities in the determination of coefficients

As the last example showed, with  $M = 1$  the determination equation for  $a_0$  is of degree  $4$.  At the same time, this means that there are also four sets of coefficients, all leading to the same ACF.

This is obvious for the following reasons:

• The coefficients  $a_0$  and  $a_1$  can change sign simultaneously without changing the system of equations.
• Replacing  $a_0$  by  $a_1$  and vice versa results in the same equation of determination.
• This operation corresponds to a mirroring and shifting of the impulse response.

$\text{Example 3:}$  As shown in the  last section,  the parameter set $a_0 = 0.7$,  $a_1 = 0.3$  is suitable to generate the ACF values $φ_0 = 0.58$  and  $φ_1 = 0.21$.  The desired ACF of the output sequence is then in detailed notation:

Example of ACF calculation

$$\varphi_y(\tau) = 0.58 \cdot \delta(\tau) + 0.21 \cdot \delta(\tau - T_{\rm A}) + 0.21 \cdot \delta(\tau + T_{\rm A}) .$$

The same ACF is also obtained with the coefficients

• $a_0 = - 0.7,\quad a_1 = -0.3,$
• $a_0 = +0.3,\quad a_1 = +0.7,$
• $a_0 = - 0.3,\quad a_1 = -0.7.$

These configurations are obtained by

• simultaneously multiplying all coefficients by  $-1$,  and
• swapping the numerical values of  $a_0$  and  $a_1$.

The diagram shows the respective impulse responses leading to the desired ACF.

Exercises for the chapter

Exercise 5.5: ACF-Equivalent Filters

Exercise 5.5Z: ACF after 1st order filter

Exercise 5.6: Filter Dimensioning

Exercise 5.6Z: Filter Dimensioning again