# Exercise 1.1Z: Non-redundant Binary Source

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Dirac-shaped source signal

Any digital source can be completely described by its source symbol sequence

$$\langle q_\nu \rangle = \langle \hspace{0.05cm}q_0 \hspace{0.05cm}, q_1 \hspace{0.05cm}, q_2 \hspace{0.05cm}, ... \hspace{0.05cm} \rangle.$$

Contrary to the theory part,  here the control variable  $\nu$  starts with zero.  If each individual symbol  $q_\nu$  originates from the symbol set  $\{\rm L, \ H\}$,  it is called a binary source.

Using the symbol spacing  $T$,  one can also characterize the source symbol sequence  $\langle q_\nu \rangle$  in an equivalent way by the Dirac-shaped source signal

$$q(t) = \sum_{(\nu)} a_\nu \cdot {\rm \delta} ( t - \nu \cdot T),$$

which rather corresponds to a system-theoretic approach.  Here, we denote  $a_\nu$  as the amplitude coefficients.

• In the case of a binary unipolar digital signal transmission holds:
$$a_\nu = \left\{ \begin{array}{c} 1 \\ 0 \\ \end{array} \right.\quad \begin{array}{*{1}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} q_\nu = \mathbf{H} \hspace{0.05cm}, \\ q_\nu = \mathbf{L} \hspace{0.05cm}. \\ \end{array}$$
• Correspondingly,  in the case of a bipolar system:
$$a_\nu = \left\{ \begin{array}{c} +1 \\ -1 \\ \end{array} \right.\quad \begin{array}{*{1}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} q_\nu = \mathbf{H} \hspace{0.05cm}, \\ q_\nu = \mathbf{L} \hspace{0.05cm}. \\ \end{array}$$

The diagram shows the Dirac-shaped source signal  $q(t)$  of a binary source.  It is known from this source that it is redundancy-free.  This statement is quite relevant for solving the problems.

Notes:

• The exercise belongs to the chapter  "System Components of a Baseband Transmission System".
• Reference is made in particular to the section  "Descriptive variables of the digital source".
• In the literature,  the two possible binary symbols are usually designated as  $\rm L$  and  $\rm 0$.
• To avoid the somewhat confusing mapping  $a_\nu = 1$  for  $q_\nu =\rm 0$  and  $a_\nu = 0$  for  $q_\nu =\rm L$,  we use the symbols  $\rm L$  ("Low") and  $\rm H$  ("High") in our learning tutorial.

### Questions

1

What is the symbol distance  $T$?

 $T \ = \$ $\ \rm µ s$

2

What is the bit rate  $R$ output by the source?

 $R \ = \$ $\ \rm kbit/s$

3

Is this representation unipolar or bipolar?

 The symbol sequence is unipolar. The symbol sequence is bipolar.

4

What is the source symbol  $q_2$?

 $q_2 = \rm L$, $q_2 = \rm H$.

5

What is the symbol probability  $p_{\rm H} = {\rm Pr}(q_\nu = \rm H$)?

 $p_{\rm H} \ = \$

### Solution

#### Solution

(1)  According to the diagram,  the distance between two symbols is  $\underline{T = 2\ \rm µ s}$.

(2)  With this redundancy-free binary source – and only with such a source – the bit rate is $R = 1/T\hspace{0.15cm}\underline{=500 \ \rm kbit/s}$.

(3)  The possible amplitude coefficients are $\pm 1$.  Therefore,  the given symbol sequence is  bipolar.

(4)  The amplitude coefficient  $a_2$  can be read at  $2T = 4 \ \rm µ s$.  With bipolar mapping,  it follows that  $a_2 = -1$  for symbol  $q_2 \hspace{0.15cm}\underline {=\rm L}$.

(5)  Even if the diagram suggests otherwise for the short time interval shown here:  For a redundancy-free binary source,  in addition to the statistical independence of the symbols,  $p_{\rm H} = p_{\rm L}\hspace{0.15cm}\underline{ = 0.5}$  (equally probable symbols)  must also hold.