Aufgaben:Exercise 3.1Z: Frequency Response of the Coaxial Cable

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Some coaxial cable types

A so-called standard coaxial cable with core diameter  $2.6 \ \rm mm$, outer diameter  $9.5 \ \rm mm$  and length  $l$  has the following frequency response:

$$H_{\rm K}(f) \ = \ {\rm e}^{- \alpha_0 \hspace{0.05cm} \cdot \hspace{0.05cm} l} \cdot

 {\rm e}^{- \alpha_1  \hspace{0.05cm}\cdot \hspace{0.05cm}l \hspace{0.05cm}\cdot \hspace{0.05cm}f}  \cdot
 {\rm e}^{- \alpha_2  \hspace{0.05cm}\cdot \hspace{0.05cm}l\hspace{0.05cm}\hspace{0.05cm}\cdot
 \sqrt{f}}  \cdot 
 {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_1  \hspace{0.05cm}\cdot \hspace{0.05cm} l \hspace{0.05cm}\cdot \hspace{0.05cm}f}  \cdot
 {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2  \hspace{0.05cm}\cdot \hspace{0.05cm}l\hspace{0.05cm}\hspace{0.05cm}\cdot
 \sqrt{f}}   \hspace{0.05cm}.$$

The attenuation parameters  $\alpha_0$,  $\alpha_1$  and  $\alpha_2$  are to be entered in Neper  $(\rm Np)$, the phase parameters  $\beta_1$  and  $\beta_2$  in Radian  $(\rm rad)$.  The following numerical values apply:

$$\alpha_0 = 0.00162 \hspace{0.15cm}\frac{\rm Np}{\rm km} \hspace{0.05cm},\hspace{0.2cm}

 \alpha_1 = 0.000435 \hspace{0.15cm}\frac{\rm Np}{\rm km\cdot{\rm MHz}} \hspace{0.05cm},
 \hspace{0.2cm}
 \alpha_2 = 0.2722 \hspace{0.15cm}\frac{\rm Np}{\rm km\cdot\sqrt{\rm MHz}} \hspace{0.05cm}.$$

Often, to describe a  linear time-invariant system  (LZI) in terms of system theory, one uses

• the attenuation function $($in  $\rm Np$  or  $\rm dB)$:

$$a_{\rm K}(f) = - {\rm ln} \hspace{0.10cm}|H_{\rm K}(f)|= - 20 \cdot {\rm lg} \hspace{0.10cm}|H_{\rm K}(f)|

   \hspace{0.05cm},$$

• the phase function $($in  $\rm rad$ or $\rm degrees)$:

$$b_{\rm K}(f) = - {\rm arc} \hspace{0.10cm}H_{\rm K}(f)

   \hspace{0.05cm}.$$

In practice one often uses the approximation

$$H_{\rm K}(f) =

 {\rm e}^{- \alpha_2  \hspace{0.05cm}\cdot \hspace{0.05cm}l\hspace{0.05cm}\hspace{0.05cm}\cdot
 \sqrt{f}}  \cdot
 {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2  \hspace{0.05cm}\cdot \hspace{0.05cm}l\hspace{0.05cm}\hspace{0.05cm}\cdot
 \sqrt{f}}\hspace{0.3cm}\Rightarrow \hspace{0.3cm} a_{\rm K}(f) = \alpha_2 \cdot l \cdot
 \sqrt{f}, \hspace{0.2cm}b_{\rm K}(f) = a_{\rm K}(f) \cdot
 \frac{\rm rad}{\rm Np}\hspace{0.05cm}.$$

This is allowed because  $\alpha_2$  and  $\beta_2$  have exactly the same numerical value – just different pseudo units. Using the definition of the  characteristic cable attenuation  (in Neper or decibels)

$$a_{\rm * (Np)} = a_{\rm K}(f = {R_{\rm B}}/{2}) = 0.1151 \cdot a_{\rm * (dB)}$$

digital systems with different bit rate  $R_{\rm B}$  and cable length  $l$  can be treated uniformly.

Notes:

• The exercise belongs to the chapter  Causes and Effects of Intersymbol Interference.
• Reference is made in particular to the section  Signals, Basis Functions and Vector Spaces.

Questions

1 Which terms of  $H_{\rm K}(f)$  do not lead to distortions? The

	$\alpha_0$–term,
	$\alpha_1$–term,
	$\alpha_2$–term,
	$\beta_1$–term,
	$\beta_2$–term.

2 What length  $l_{\rm max}$  could such a cable have to attenuate a DC signal by no more than  $1\%$? 

$l_{\rm max} \ = \ $ $\ {\rm km} $

3 What is the attenuation $($in  $\rm Np)$  at the frequency  $f = 70\,{\rm MHz}$, if the cable length is  $l = 2\,{\rm km}$? 

$a_{\rm K}(f = 70\,{\rm MHz})\ = \ $ $\ {\rm Np} $

4 All other things being equal, what attenuation results when only the  $\alpha_2$–term is considered?

$a_{\rm K}(f = 70\,{\rm MHz})\ = \ $ $\ {\rm Np} $

5 What is the formula for the conversion between  $\rm Np$  and  $\rm dB$? What is the  $\rm dB$ value for the attenuation calculated in (4)?

$a_{\rm K}(f = 70\,{\rm MHz})\ = \ $ $\ {\rm dB} $

6 Which of the statements are true provided that one restricts oneself to the  $\alpha_2$–value with respect to the attenuation function?

	One can also do without the phase term  $\beta_1$
	One can also do without the phase term  $\beta_2$
	$a_* ≈ 40\,{\rm dB}$  is valid for a system with  $R_{\rm B} = 70\,{\rm Mbit/s}$  and  $l = 2\,{\rm km}$.
	$a_* ≈ 40\,{\rm dB}$  is valid for a system with  $R_{\rm B} = 140\,{\rm Mbit/s}$  and  $l = 2\,{\rm km}$.
	$a_* ≈ 40\,{\rm dB}$  is valid for a system with  $R_{\rm B} = 560\,{\rm Mbit/s}$  and  $l = 1\,{\rm km}$.

Solution

Solution

(1)  Solutions 1 and 4 are correct:

• The $\alpha_0$–term causes only frequency-independent attenuation and the $\beta_1$–term (linear phase) causes frequency-independent delay.
• All other terms contribute to the (linear) distortions.

(2)  With $a_0 = \alpha_0 \cdot l$, the following equation must be satisfied:

$${\rm e}^{- a_0 } \ge 0.99

 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}a_0 < {\rm ln}
 \hspace{0.10cm}\frac{1}{0.99}\approx 0.01\,\,{\rm (Np)}
 \hspace{0.05cm}.$$

This gives the following for the maximum cable length

$$l_{\rm max} = \frac{a_0 }{\alpha_0 } = \frac{0.01\,\,{\rm Np}}{0.00162\,\,{\rm Np/km}}\hspace{0.15cm}\underline {\approx 6.173\,\,{\rm km}}

 \hspace{0.05cm}.$$

(3)  For the attenuation curve, considering all terms:

$$a_{\rm K}(f) \ = \ \big[\alpha_0 + \alpha_1 \cdot f + \alpha_2 \cdot

 \sqrt{f}\hspace{0.05cm}\big] \cdot l = \big [0.00162 + 0.000435  \cdot 70 + 0.2722   \cdot \sqrt{70}\hspace{0.05cm}\big]\, \frac{\rm Np}{\rm km} \cdot 2\,{\rm km} = \hspace{0.15cm}\underline {= 4.619\, {\rm Np}}\hspace{0.05cm}.$$

(4)  According to the calculation at point (3), the attenuation value $\underline {4.555\,{\rm Np}}$ is obtained here.

(5)  For any positive quantity $x$ holds:

$$x_{\rm Np} = {\rm ln} \hspace{0.10cm} x = \frac{{\rm lg} \hspace{0.10cm} x}{{\rm lg} \hspace{0.10cm} {\rm e}}

 =   \frac{1}{{20 \cdot \rm lg} \hspace{0.10cm} {\rm e}} \cdot
 (20 \cdot {\rm lg} \hspace{0.10cm} x) = 0.1151 \cdot x_{\rm dB}\hspace{0.3cm} \Rightarrow \hspace{0.3cm} x_{\rm dB} = 8.686 \cdot x_{\rm
Np}\hspace{0.05cm}.$$

Thus, the attenuation value $4.555\,{\rm Np}$ is identical to $\underline{39.57\,{\rm dB} }$.

(6)  Solutions 1, 4 and 5 are correct:

• With the restriction to the attenuation term with $\alpha_2$, the following holds for the frequency response:

$$H_{\rm K}(f) =

 {\rm e}^{- \alpha_2  \hspace{0.05cm}\cdot \hspace{0.05cm}l\hspace{0.05cm}\hspace{0.05cm}\cdot
 \sqrt{f}}  \cdot
 {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_1  \hspace{0.05cm}\cdot \hspace{0.05cm} l \hspace{0.05cm}\cdot f}  \cdot
 {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2  \hspace{0.05cm}\cdot \hspace{0.05cm}l\hspace{0.05cm}\hspace{0.05cm}\cdot
 \sqrt{f}}   \hspace{0.05cm}.$$

• If we omit the $\beta_1$ phase term, nothing changes with respect to the distortions. Only the phase and group delay would become smaller (both equal) by the value $\tau_1 = (\beta_1 \cdot l)/2\pi$.
• On the other hand, if the $\beta_2$–term is omitted, completely different ratios result:

The frequency response $H_{\rm K}(f)$ now no longer satisfies the requirement of a causal system; for such a system $H_{\rm K}(f)$ must be minimum-phase.

The impulse response $h_{\rm K}(t)$ is symmetrical about $t = 0$ with real frequency response, which does not correspond to the conditions.

• Therefore as an approximation for the coaxial cable frequency response is allowed:

$$a_{\rm K}(f) = \alpha_2 \cdot l \cdot

 \sqrt{f}, \hspace{0.2cm}b_{\rm K}(f) = a_{\rm K}(f) \cdot
 \frac{\rm rad}{\rm Np}\hspace{0.05cm}.$$

• That means: $a_{\rm K}(f)$ and $b_{\rm K}(f)$ of a coaxial cable are identical in form and differ only in their units.
• For a digital system with bit rate $R_{\rm B} = 140\,{\rm Mbit/s}$ ⇒ $R_{\rm B}/2 = 70\,{\rm Mbit/s}$ and cable length $l = 2\,{\rm km}$, $a_* ≈ 40\,{\rm dB}$ is indeed valid – see solution to (5).
• A system with four times the bit rate ($R_{\rm B}/2 = 280\,{\rm Mbit/s}$) and half the length ($l = 1\,{\rm km}$) results in the same characteristic cable attenuation.
• In contrast, for a system with $R_{\rm B}/2 = 35\,{\rm Mbit/s}$ and $l = 2\,{\rm km}$ to

$$a_{\rm dB} = 0.2722 \frac{\rm Np}{\rm km\cdot \sqrt{\rm MHz}} \cdot 2 \ \rm km \cdot \sqrt{35 \ \rm MHz} \cdot 8.6859 \frac{\rm dB}{\rm Np} ≈ 28 \ \rm dB.$$