Exercise 2.6Z: PN Generator of Length 3

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PN generator with  $L = 3$

The adjacent sketch shows a PN generator of length  $L = 3$  with generator polynomial

$$G( D) = D^{\rm 3} + D^{\rm 2} + \rm 1$$

and thus the octal identifier  $(g_3 \ g_2 \ g_1 \ g_0)$ = $(1 \ 1 \ 0 \ 1)_{\rm bin} = (15)_{\rm oct}$.

The corresponding reciprocal polynomial

$$G_{\rm R}(D) = D^{\rm 3}\cdot ( D^{\rm -3} + D^{\rm -2} + 1) = D^{\rm 3} + D^{\rm 1} + \rm 1$$

has the octal identifier  $(1 \ 0 \ 1 \ 1)_{\rm bin} = (13)_{\rm oct}$.

• At start time,  let the three memory cells be preallocated with the binary values  $1$,  $0$  and  $1$ .
• Both arrangements generate an  "M-sequence".

Hints:

• The exercise belongs to the chapter  Generation of Discrete Random Variables.

• The topic of this chapter is illustrated with examples in the  (German language)  learning video:
      "Erläuterung der PN-Generatoren an einem Beispiel"   $\Rightarrow$   "Explanation of PN generators using an example".

Questions

1

How long is the period length of the configuration  $(15)$?

$P \ = \ $

2

Determine the output sequence  $〈z_ν\rangle$  for the time points  $1$, ... , $P$.  What are the first  $15$  binary values of the output sequence?
Hint: From left to right,  label the cells with  $S_1$,  $S_2$  and  $S_3$.  Output the value  $z_ν$  that is currently  (at time  $\nu$)  entered into the memory cell  $S_1$.

	$1\ 0 \ 0 \ 1 \ 1 \ 0 \ 1 \ 0 \ 1 \ 1 \ 1 \ 1 \ 0 \ 0 \ 0$ . . .
	$1 \ 0 \ 0 \ 1 \ 1 \ 0 \ 0 \ 1 \ 1 \ 0 \ 0 \ 1 \ 1 \ 0 \ 0$ . . .
	$1 \ 1 \ 0 \ 0 \ 1 \ 0 \ 1 \ 1 \ 1 \ 0 \ 0 \ 1 \ 0 \ 1 \ 1$ . . .
	$0 \ 0 \ 1 \ 1 \ 1 \ 0 \ 1 \ 0 \ 0 \ 1 \ 1 \ 1 \ 0 \ 1 \ 0$. . .

3

Which of the following statements are true for each M-sequence?

	The number of  "zeros"  and  "ones"  is equal.
	In each period there is one more  "one"  than  "zeros".
	The maximum number of consecutive  "ones"  is  $L$.
	The sequence  $1 \ 0 \ 1 \ 0 \ 1 \ 0$ ...   is not possible.

4

Now consider the reciprocal order  $(13)$.  What are the first  $15$  binary values of the output sequence with the same initial assignment here?

	$0 \ 0 \ 0 \ 1 \ 1 \ 1 \ 1 \ 0 \ 1 \ 0 \ 1 \ 1 \ 0 \ 0 \ 1$ . . .
	$0 \ 0 \ 1 \ 1 \ 1 \ 0 \ 1 \ 0 \ 0 \ 1 \ 1 \ 1 \ 0 \ 1 \ 0$ . . .
	$0 \ 0 \ 1 \ 0 \ 1 \ 1 \ 1 \ 0 \ 0 \ 1 \ 0 \ 1 \ 1 \ 1 \ 0$ . . .

Solution

Solution

PN generator with octal identifier  $15$

(1)  It is an M-sequence with  $L= 3$.  It follows that $P= 2^L - 1 \hspace{0.15cm}\underline{= 7}$.

(2)  We denote the cells from left to right by  $S_1$,  $S_2$  and  $S_3$.  Then holds:

• $S_2(\nu) = S_1(\nu - 1)$,
• $S_3(\nu) = S_2(\nu - 1)$,
• $S_1(\nu) = S_2(\nu - 1) \ {\rm mod } \ S_3(\nu - 1)$.

The result is entered in the first row of the above table  (marked in red):

• At the clock time  $\nu = 7$  results in the same memory usage as at the time  $\nu = 0$.
• From this follows  ${P = 7}$  and the sequence is from  $\nu = 1$  corresponding to  solution 3:

$$\langle z_\nu \rangle = 1 \ 1 \ 0 \ 0 \ 1 \ 1 \ 0 \ 0 \ 1 \ 0 \ 1 \ 1 \text{...}$$

• In contrast,  proposal 1 describes the M sequence of the PN generator with length  $L=4$  and identifier  $(31)$   ⇒   period length is  $P= 15$.
• In proposal 2,  the period length  $P= 4$  is too short.

• Finally,  the last proposal would have the desired period length  $P= 7$,  but from the modulo 2 addition of  $S_2= 0$  and  $S_3= 1$  $($for  $\nu = 0)$  it necessarily follows at the next time  $(\nu = 1)$:   $S_1= 1$.   This property is not exhibited by sequence 4.

(3)  Correct are solutions 2, 3, and 4:

• The maximum number of consecutive  "ones"  is  $L$  (namely if there is a  "one"  in all  $L$  memory cells).
• On the other hand,  it is not possible that all memory cells are filled with  "zeros".  Therefore,  there is always one more  "one"  than  "zeros".
• The period length of the last sequence is  $P = 2$.  On the other hand,  for an M-sequence  $P= 2^L - 1.$  For no value of  $L$:    $P = 2$  is possible.

PN generator with octal identifier  $13$

(4)  In the adjacent table the emergence of the PN sequence at the reciprocal polynomial  $G_{\rm R}(D)$  is entered.  It can be seen that the  proposed solution 2 applies:

• Also for the reciprocal arrangement,  the period length  $P = 7$  must hold,  so that proposition 1  $($with  $P = 15)$  is eliminated.
• Proposal 3 is just a version of the output sequence of  $(15)$ shifted by two clocks.
• In contrast,  in the (correct) second proposal,  the inverse of ... $1 \ 1 \ 0 \ 0 \ 1 \ 0 \ 1$ ... – thus the sequence ... $1 \ 0 \ 1 \ 0 \ 0 \ 1 \ 1$ ... – are included,  albeit with a phase shift.