Aufgaben:Exercise 3.12: Trellis Diagram for Two Precursors

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Trellis diagram for two precursors

We assume the basic pulse values   $g_0\ne 0$,  $g_{\rm –1}\ne 0$  and  $g_{\rm –2}\ne 0$: 

• This means that the decision on the symbol  $a_{\rm \nu}$  is also influenced by the subsequent coefficients  $a_{\rm \nu +1}$  and  $a_{\rm \nu +2}$. 

• Thus,  for each time point   $\nu$,  exactly eight  metrics   $\varepsilon_{\rm \nu}$  have to be determined, from which the  minimum accumulated metrics   ${\it \Gamma}_{\rm \nu}(00)$,  ${\it \Gamma}_{\rm \nu}(01)$,  ${\it \Gamma}_{\rm \nu}(10)$  and  ${\it \Gamma}_{\rm \nu}(11)$  can be calculated.

• For example,   ${\it \Gamma}_{\rm \nu}(01)$  provides information about the symbol  $a_{\rm \nu}$  under the assumption that  $a_{\rm \nu +1} = 0$  and  $a_{\rm \nu +2} = 1$  will be.

• Here, the minimum accumulated metric   ${\it \Gamma}_{\rm \nu}(01)$  is the smaller value obtained from the comparison of

$$\big[{\it \Gamma}_{\nu-1}(00) + \varepsilon_{\nu}(001)\big] \hspace{0.15cm}{\rm and}\hspace{0.15cm}\big[{\it \Gamma}_{\nu-1}(10) + \varepsilon_{\nu}(101)\big].$$

To calculate the minimum accumulated metric   ${\it \Gamma}_2(10)$  in subtasks (1) and (2),  assume the following numerical values:

• unipolar amplitude coefficients:  $a_{\rm \nu} ∈ \{0, 1\}$,

• basic pulse values   $g_0 = 0.5$,  $g_{\rm –1} = 0.3$,  $g_{\rm –2} = 0.2$,

• applied noisy detection sample:  $d_2 = 0.2$,

• minimum accumulated metric at time  $\nu = 1$:

$${\it \Gamma}_{1}(00) = 0.0,\hspace{0.2cm}{\it \Gamma}_{1}(01) = 0.2, \hspace{0.2cm} {\it \Gamma}_{1}(10) = 0.6,\hspace{0.2cm}{\it \Gamma}_{1}(11) =1.2\hspace{0.05cm}.$$

The graph shows the simplified trellis diagram for time points   $\nu = 1$  to   $\nu = 8$. 

• Blue branches come from either   ${\it \Gamma}_{\rm \nu –1}(00)$   or   ${\it \Gamma}_{\rm \nu –1}(01)$   and denote a hypothetical  "$0$".

• In contrast,  all red branches – starting from the   ${\it \Gamma}_{\rm \nu –1}(10)$  or   ${\it \Gamma}_{\rm \nu –1}(11)$  states – indicate the symbol  "$1$".

Notes:

• The exercise belongs to the chapter   "Viterbi Receiver".

• All quantities here are to be understood normalized.

• Also, assume unipolar and equal probability amplitude coefficients:   ${\rm Pr} (a_\nu = 0) = {\rm Pr} (a_\nu = 1)= 0.5.$

Questions

1 Calculate the following metrics:

$\varepsilon_2(010) \ = \ $

$\varepsilon_2(011) \ = \ $

$\varepsilon_2(110) \ = \ $

$\varepsilon_2(111) \ = \ $

2 Calculate the following minimum accumulated metrics:

${\it \Gamma}_2(10) \ = \ $

${\it \Gamma}_2(11) \ = \ $

3 What are the symbols output by the Viterbi receiver?

	The first seven symbols are   "$1011010$".
	The first seven symbols are   "$1101101$".
	The last symbol  $a_8 = 1$  is safe.
	No definite statement can be made about the symbol  $a_8$.

Solution

Solution

(1)  The first metric is calculated as follows:

$$\varepsilon_{2}(010) = [d_0 - 0 \cdot g_0 - 1 \cdot g_{-1}- 0 \cdot g_{-2}]^2= [0.2 -0.3]^2\hspace{0.15cm}\underline {=0.01}\hspace{0.05cm}.$$

Correspondingly,  for the other metrics:

$$\varepsilon_{2}(011) \ = \ [0.2 -0.3- 0.2]^2\hspace{0.15cm}\underline {=0.09}\hspace{0.05cm},$$

$$\varepsilon_{2}(110) \ = \ [0.2 -0.5- 0.3]^2\hspace{0.15cm}\underline {=0.36}\hspace{0.05cm},$$

$$\varepsilon_{2}(111) \ = \ [0.2 -0.5- 0.3-0.2]^2\hspace{0.15cm}\underline {=0.64}\hspace{0.05cm}.$$

(2)  The task is to find the minimum value of each of two comparison values:

$${\it \Gamma}_{2}(10) \ = \ {\rm Min}\left[{\it \Gamma}_{1}(01) + \varepsilon_{2}(010),\hspace{0.2cm}{\it \Gamma}_{1}(11) + \varepsilon_{2}(110)\right] = {\rm Min}\left[0.2+ 0.01, 1.2 + 0.36\right]\hspace{0.15cm}\underline {= 0.21}\hspace{0.05cm},$$

$${\it \Gamma}_{2}(11) \ = \ {\rm Min}\left[{\it \Gamma}_{1}(01) + \varepsilon_{2}(011),\hspace{0.2cm}{\it \Gamma}_{1}(11) + \varepsilon_{2}(111)\right] = {\rm Min}\left[0.2+ 0.09, 1.2 + 0.64\right]\hspace{0.15cm}\underline {= 0.29}\hspace{0.05cm}.$$

(3)  The  first and last solutions  are correct:

• The sequence  "$1011010$"  can be recognized from the continuous path:     "red – blue – red – red – blue – red – blue".

• On the other hand,  no final statement can be made about the symbol  $a_8$  at time  $\nu = 8$:

• Only under the hypothesis  $a_9 = 1$  and  $a_{\rm 10} = 1$  one would decide for  $a_8 = 0$,  under other hypotheses for  $a_8 = 1$.