Aufgaben:Exercise 4.13: Gaussian ACF and PSD

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Two Gaussian ACF

Let the random process considered here  $\{x_i(t)\}$  be characterized by the autocorrelation function (ACF) outlined above  This random process is mean-free and the equivalent ACF duration is  ${ {\rm \nabla} }\tau_x = 5 \hspace{0.08cm} \rm µ s$:

$$\varphi_x(\tau)=\rm 0.25 V^2\cdot \rm e^{-\pi \hspace{0.03cm}\cdot \hspace{0.03cm} ({\tau}{/ 5 \hspace{0.08cm}{\rm µ}s })^2} .$$

The bottom figure shows the ACF of the process  $\{y_i(t)\}$  This reads with the equivalent ACF duration  ${ {\rm \nabla} }\tau_y = 10 \hspace{0.08cm} \rm µ s$:

$$ \varphi_y(\tau)=\rm 0.16 V^2 + \rm 0.09 V^2\cdot\rm e^{-\pi \hspace{0.03cm}\cdot \hspace{0.03cm} ({\tau}/{\nabla \it \tau_y})^2} .$$

In this exercise, the Power spectral densities of the two processes are sought.

Hints:

• This exercise belongs to the chapter  Power-Spectral Density.
• Reference is also made to the chapter  Auto-correlation function.

• To solve this exercise you can use the following Fourier correspondence:

$$\rm e^{-\pi \hspace{0.03cm}\cdot \hspace{0.03cm} ({\it f}/{\rm \Delta\it f})^2}\ \bullet\!\!-\!\!-\!\!-\!-\!\!\circ\, \ {\rm \Delta \it f} \cdot \rm e^{-\pi \hspace{0.05cm}\cdot \hspace{0.05cm} ({\rm \Delta\it f} \hspace{0.05cm}\cdot \hspace{0.05cm}\it t {\rm )}^{\rm 2}}.$$

Questions

1

What is the äequivalent PSD bandwidth of the process  $\{x_i(t)\}$?

$ {\rm \nabla} \hspace{-0.05cm} f_x \ = \ $ $\ \rm kHz$

2

What is  ${\it \Phi}_x(f)$?  Give the PSD values for  $f= 0$  and  $f = 200 \hspace{0.08cm} \rm kHz$  on.

${\it \Phi}_x(f = 0)\ = \ $ $\ \cdot 10^{-6} \ \rm V^2\hspace{-0.1cm}/Hz$

${\it \Phi}_x(f = 200 \hspace{0.08cm} \rm kHz)\ = \ $ $\ \cdot 10^{-6} \ \rm V^2\hspace{-0.1cm}/Hz$

3

Which statements are valid, if the random process has no periodic parts?  Furthermore, a constant power is assumed.

	The process power is the integral over the PSD.
	If the process is zero mean, the PSD is always continuous.
	The wider the ACF, the wider the PSD.
	A wider ACF results in higher PSD values.

4

Which of the following statements are true regarding the process  $\{y_i(t)\}$?

	The PSD involves a Dirac at frequency $ f = {\rm \nabla} \hspace{-0.05cm} f_y$.
	The PSD involves a Dirac at frequency $f= 0$.
	Dirac weight and continuous PSD have the same unit.

Solution

Solution

(1)  The equivalent PSD bandwidth is the reciprocal of the equivalent ACF duration:

$$\nabla f_x = 1 / \nabla \tau_x \hspace{0.15cm}\underline{= {\rm 200\hspace{0.1cm}kHz}}.$$

(2)  One can adapt the given Fourier correspondence to the task as follows:

$$K\cdot{\rm e}^{-\pi({\tau}/{\nabla\tau_x})^2}\ \circ\!\!-\!\!-\!\!-\!\!-\!-\!\bullet\,\ \frac{\it K}{\nabla \it f_x}\cdot{\rm e}^{-\pi({f}/{\nabla f_x})^2}.$$

• With  $K = 0.25 \hspace{0.05cm}\rm V^2$  and  $ {\rm \nabla} \hspace{-0.05cm} f_x = 200\hspace{0.05cm} \rm kHz$  obtains:

$${\it \Phi_x}(f)=1.25\cdot\rm 10^{-\rm 6}\hspace{0.1cm}\frac{V^2}{Hz}\cdot\rm e^{-\pi({\it f}/{\nabla\it f_x})^2}\hspace{0.3cm} \Rightarrow \hspace{0.3cm}{\it \Phi_x}(f = 0)=\hspace{0.15cm}\underline{\rm 1.25 \cdot 10^{-6} \hspace{0.1cm} V^2\hspace{-0.1cm}/Hz}, \hspace{0.5cm}{\it \Phi_x}(f = 200 \hspace{0.05cm} \rm kHz)=\hspace{0.15cm}\underline{\rm 0.054 \cdot 10^{-6} \hspace{0.1cm} V^2\hspace{-0.1cm}/Hz}.$$

(3)  Correct solutions 1, 2, and 4:

• A mean-free process always results in a continuous PSD.  This is narrower the wider the ACF is (reciprocitylaw).
• The process power is equal to the integral of the PSD.
• Therefore, at constant power, a wider ACF  (narrower PSD)  must be compensated by higher PSD values.
• A DC component or periodic components always result in dirac functions in the PSD;  otherwise, the PSD is always continuous in value.

(4)  Analogous to subtask  (2)  holds with  $ {\rm \nabla} \hspace{-0.05cm} f_y = 100\hspace{0.05cm} \rm kHz$:

$${\it \Phi_y}(f)=\frac{\rm 0.09 V^2}{\nabla\it f_y}\cdot\rm e^{-\pi({\it f}/{\nabla\it f_y})^2}+\it m_y^{\rm 2}\cdot\delta(f).$$

• Because of the DC component, there is a Dirac at frequency $f = 0$ in addition to the continuous PSD component.
• The continuous PSD–part at $f= 0$ is  ${\it \Phi_y}(f = 0)=\hspace{0.15cm}\underline{\rm 0.9 \cdot 10^{-6} \hspace{0.1cm} V^2\hspace{-0.1cm}/Hz}.$
• The fraction at $f = 2 \cdot {\rm \nabla} \hspace{-0.05cm} f_y = 200 \hspace{0.05cm}\rm kHz$  is increased by a factor  ${\rm e}^{-4} \approx 0.0183$  lower   ⇒   ${\it \Phi_y}(f )=\hspace{0.15cm}\underline{\rm 0.0165 \cdot 10^{-6} \hspace{0.1cm} V^2\hspace{-0.1cm}/Hz}.$

(5)  Correct is only the second proposed solution:

• The PSD of a mean-valued process generally involves a Dirac function at $f=0$  with weight  $m_y^2$.
• In the present case, this value is equal to  $0.16 \ \rm V^2$.
• Since  $\delta(f)$  has unit  $\rm 1/Hz = s$ , the units of the continuous and discrete PSD components differ.