Exercise 3.11: Viterbi Receiver and Trellis Diagram

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Trellis diagram
for one precursor

The Viterbi receiver allows a low-effort realization of the maximum likelihood decision rule.  It contains the system components listed below:

• A matched filter adapted to the basic transmission pulse with frequency response  $H_{\rm MF}(f)$  and output signal   $m(t)$,

• a sampler spaced at the symbol duration  $T$,  which converts the continuous-time signal   $m(t)$  into the discrete-time sequence   $〈m_{\rm \nu}〉$, 

• a decorrelation filter with frequency response   $H_{\rm DF}(f)$  for removing statistical ties between noise components of the sequence   $〈d_{\rm \nu}〉$,

• the Viterbi decision,  which uses a trellis-based algorithm to obtain the sink symbol sequence  $〈v_{\rm \nu}〉$. 

The graph shows the simplified trellis diagram of the two states  "$0$"  and  "$1$"  for time points   $\nu ≤ 5$.  This diagram is obtained as a result of evaluating the two minimum accumulated metrics   ${\it \Gamma}_{\rm \nu}(0)$  and  ${\it \Gamma}_{\rm \nu}(1)$  corresponding to   Exercise 3.11Z.

Notes:

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

• Reference is also made to the section  "Maximum-a-posteriori and Maximum–Likelihood decision rule".

• 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

Which of the following statements are true?

	The matched filter  $H_{\rm MF}(f)$  is mainly used for noise power limitation.
	The decorrelation filter removes ties between samples.
	The noise power is only affected by  $H_{\rm MF}(f)$,  not by  $H_{\rm DF}(f)$.

2

At what times  $\nu$  can we finally decide the current symbol  $a_{\rm \nu}$? 

	$\nu = 1,$
	$\nu = 2,$
	$\nu = 3,$
	$\nu = 4,$
	$\nu = 5.$

3

What is the total sequence decided by the Viterbi receiver?

$a_1 \ = \ $

$a_2 \ = \ $

$a_3 \ = \ $

$a_4 \ = \ $

$a_5 \ = \ $

4

Which of the following statements are true?

	It is certain that the detected sequence was also sent.
	A MAP receiver would have the same error probability.
	Threshold decision is the same as this maximum likelihood receiver.

Solution

Solution

(1)  The  first two solutions  are correct:

• The signal  $m(t)$  after the matched filter  $H_{\rm MF}(f)$  has the largest possible signal-to-interference power ratio  $\rm (SNR)$.
• However,  the noise components of the sequence  $〈m_{\rm \nu}〉$  are (strongly) correlated due to the spectral shaping.
• The task of the discrete-time decorrelation filter with the frequency response  $H_{\rm DF}(f)$  is to dissolve these bindings,  which is why the name  "whitening filter"  is also used for  $H_{\rm DF}(f)$.
• However,  this is possible only at the cost of increased noise power   ⇒   consequently,  the last proposed solution does not apply.

(2)  The two arrows arriving at   $\underline {\nu = 1}$  are each drawn in blue and indicate the symbol  $a_1 = 0$.  Thus,  the initial symbol  $a_1$  is already fixed at this point.  Similarly,  the symbols  $a_3 = 1$  and  $a_5 = 0$  are already fixed at times  $\underline {\nu = 3}$  and  $\underline {\nu = 5}$,  respectively.

In contrast,  at time $\nu = 2$,  a decision regarding symbol  $a_2$  is not possible.

• Under the hypothesis that the following symbol  $a_3 = 0$  would result in symbol  $a_2 = 1$  $($at  "$0$"  a red path arrives,  thus coming from  "$1$"$)$.
• In contrast,  the hypothesis  $a_3 = 1$  leads to the result  $a_2 = 0$  $($the path arriving at  "$1$"  is blue$)$.

The situation is similar at time  $\nu = 4$.

(3)  From the continuous paths at  $\nu = 5$  it can be seen:

$$a_{1}\hspace{0.15cm}\underline {=0} \hspace{0.05cm},\hspace{0.2cm} a_{2}\hspace{0.15cm}\underline { =0} \hspace{0.05cm},\hspace{0.2cm}a_{3}\hspace{0.15cm}\underline {=1} \hspace{0.05cm},\hspace{0.2cm} a_{4}\hspace{0.15cm}\underline {=0} \hspace{0.05cm},\hspace{0.2cm} a_{5}\hspace{0.15cm}\underline {=0} \hspace{0.05cm}.$$

(4)  Only the  second statement  is correct:

• Since the source symbols  "$0$"  and  "$1$"  were assumed to be equally probable,  the ML receiver  (Viterbi)  is identical to the MAP receiver.
• A threshold decision  $($which makes a symbol-by-symbol decision at each clock$)$  has the same BER as the Viterbi receiver only if there is no intersymbol interference.
• This is obviously not the case here,  otherwise it should be possible to make a final decision at every time  $\nu$.
• The first statement is also false.  Indeed,  this would mean that the Viterbi receiver can have error probability  $0$  in the presence of AWGN noise. 
  This is not possible for information-theoretic reasons.