Exercise 2.4: 2D Transfer Function

2D impulse response $|h(\tau, \hspace{0.05cm}t)|$

The graph shows the two-dimensional impulse response $h(\tau, \hspace{0.05cm}t)$ of a mobile radio system in magnitude representation.

It can be seen that the 2D impulse response only has components at delays $\tau = 0$ and $\tau = 1 \ \rm µ s$ .
At these times:

$$h(\tau = 0\,{\rm µ s},\hspace{0.05cm}t) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac{1}{ \sqrt{2}} = {\rm const.}$$

$$h(\tau = 1\,{\rm µ s},\hspace{0.05cm}t) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \cos(2\pi \cdot {t}/{ T_0})\hspace{0.05cm}.$$

For all other values of $\tau$, we have $h(\tau, \hspace{0.05cm}t) \equiv 0$.

We want to obtain the two-dimensional transfer function $H(f, \hspace{0.05cm} t)$ as the Fourier transform of $h(\tau, t)$ with respect to the delay $\tau$:

$$H(f,\hspace{0.05cm} t) \hspace{0.2cm} \stackrel {f,\hspace{0.05cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} h(\tau,\hspace{0.05cm}t) \hspace{0.05cm}.$$

Notes:

This task belongs to chapter Multi–Path Reception in Mobile Communications.
A similar problem is treated in Exercize 2.5 but with a different nomenclature.

Questionnaire

$T_0 \ = \ $

$\ \ \rm ms$

$t_1 \ = \ $

$\ \ \rm ms$

$t_2 \ = \ $

$\ \ \rm ms$

	We have $H_0(f) = H_0(f + i \cdot 1 \ {\rm MHz}), \ i = ±1, ±2, \ \ \text{...}$
	We have approximately $0.293 ≤ \|H_0(f)\| ≤ 1.707$.
	$\|H_0(f)\|$ has a maximum at $f = 0$ .

	We have $H_{10}(f) = H_{10}(f + i \cdot 1 \ {\rm MHz}),\ i = ±1, ±2, \ \ \text{...}$
	We have approximately $0.293 ≤ H_{10}(f) ≤ 1.707$.
	$\|H_{10}(f)\|$ has a maximum at $f = 0$ .

Solution

(1) The period can be read from the given graph. If the magnitude representation is taken into account, the result is $T_0 \ \underline {= 20 \ \ \rm ms}$.

(2) At time $t_1 \ \underline {= 5 \ \ \rm ms}$ we have $h(\tau = 1 \ {\rm µ s}, t_1) = 0$. Accordingly, the following applies

$$h(\tau = 1\,{\rm µ s},\hspace{0.05cm}t_1) = \frac{1}{ \sqrt{2}} \cdot \delta(\tau)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} H(f,\hspace{0.05cm}t_1) = \frac{1}{ \sqrt{2}} = {\rm const.}$$

The same applies to $t_2 \ \underline {= 15 \ \ \rm ms}$:

$$h(\tau = 1\,{\rm µ s},\hspace{0.05cm}t_2) = \frac{1}{ \sqrt{2}} \cdot \delta(\tau)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} H(f,\hspace{0.05cm}t_2) = \frac{1}{ \sqrt{2}} = {\rm const.}$$

(3) At time $t = 0$ the impulse response with $\tau_1 = 1 \ \ \rm µ s$ is

$$h(\tau,\hspace{0.05cm}t = 0) = \frac{1}{ \sqrt{2}} \cdot \delta(\tau)+ \delta(\tau - \tau_1)\hspace{0.05cm}.$$

Its Fourier transform is

$$H_0(f) = H(f,\hspace{0.05cm}t = 0) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac{1}{ \sqrt{2}} + 1 \cdot {\rm e}^{- {\rm j}\cdot 2 \pi f \tau_1}=\frac{1}{ \sqrt{2}} + \cos( 2 \pi f \tau_1)- {\rm j}\cdot \sin( 2 \pi f \tau_1)$$

$$\Rightarrow \hspace{0.3cm} |H_0(f)| \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sqrt { \left [ {1}/{ \sqrt{2}} + \cos( 2 \pi f \tau_1) \right ]^2 + \left [\sin( 2 \pi f \tau_1)\right ]^2}= \sqrt { 0.5 + 1 + {2}/{ \sqrt{2}} \cdot \cos( 2 \pi f \tau_1)} = \sqrt { 1.5 + { \sqrt{2}} \cdot \cos( 2 \pi f \tau_1)}\hspace{0.05cm}.$$

Therefore,

$H_0(f)$ is periodic with $1/\tau_1 = 1 \ \rm MHz$.
For the maximum and minimum values, we have

$${\rm Max}\, \left [ \, |H_0(f)|\, \right ] \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sqrt { 1.5 + { \sqrt{2}} } \approx 1.707 \hspace{0.05cm},\hspace{0.5cm}{\rm Min}\, \left [ \, |H_0(f)|\, \right ] \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sqrt { 1.5 - { \sqrt{2}} } \approx 0.293 \hspace{0.05cm}. $$

At $f = 0$, $|H_0(f)|$ has a maximum.

Therefore, all three solution suggestions are correct.

(4) For the time $t = 10 \ \rm ms$ the following equations apply:

$$h(\tau,\hspace{0.05cm}t = 10\,{\rm ms}) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac{1}{ \sqrt{2}} \cdot \delta(\tau)- \delta(\tau - \tau_1)\hspace{0.05cm},$$

$$H_{10}(f) = H(f,\hspace{0.05cm}t = 10\,{\rm ms}) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac{1}{ \sqrt{2}} - \cos( 2 \pi f \tau_1)+ {\rm j}\cdot \sin( 2 \pi f \tau_1)\hspace{0.05cm},$$

$$ |H_{10}(f)| \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sqrt { 1.5 - { \sqrt{2}} \cdot \cos( 2 \pi f \tau_1)}\hspace{0.05cm}.$$

2D impulse response $|h(\tau, \hspace{0.05cm}t)|$ and 2D transfer function $|H(f, \hspace{0.05cm}t)|$

Solutions 1 and 2 are correct:

The frequency period does not change from $t = 0$.
The maximum value is still $1,707$ and the minimum value $0,293$ does not change compared to the subtask (3).
For $f = 0$ there is now a minimum and not a maximum.

The graph on the right shows the magnitude $|H(f, t)|$ of the 2D transfer function.