SCRAMBLING

A scrambler (or randomizer) is an algorithm that converts a source bit-stream into a pseudo-random one of the same length in order to avoid long sequences of bits of the same value [1]. In telecommunications, this is important for two main reasons:

to ease and improve timing recovery on receiver side,
to better disperse signal power spectral density and reduce intermodulation interference.

Focusing on the second aforementioned point, that means trying to make the transmitted spectrum as white as possible. In fact, another popular kind of randomization process (not examined here) is the data whitening.

Here the implementation of the two most common scrambling techniques (additive and multiplicative) in MATLAB (without employing any of its built-in functions) , C/C++ and Python languages is proposed. Both follow an N-cell linear-feedback shift register (LFSR) pattern, as depicted in the examples of Fig.1 and Fig.2.

Fig.1 - Additive scrambler / descrambler scheme

Fig.2 - Multiplicative scrambler (a) and descrambler (b) schemes

The additive (aka synchronous) approach foresees the same scheme for both scrambler (on Tx side) and descrambler (on Rx side) and needs for the content of the LFSR cells to be initialized to a known synchronization word each time a new incoming bit-stream begins. The correct operation of the scrambler / descrambler pair is reported in Fig.3, where the original info bits are recovered with no errors since both employ the same sync word and no bit corruptions occur between the scrambler output and the descrambler input. The main advantage of the additive scrambling technique is that it avoids error multiplication in case of wrong bits arise at the descrambler input, as can be seen from the MATLAB simulation of Fig.4, which makes use of a simple Binary Symmetric Channel (BSC) with an error probability of 1%. However, in case there is a mismatch between the Tx and Rx LFSR initial states a massive error propagation results, as shown in Fig.5.

Fig.3 - Additive scrambler - #1 : Correct operation

Fig.4 - Additive scrambler - #2 : Errors at descrambler input

Fig.5 - Additive scrambler - #3 : Tx/Rx LFSR initial state mismatch

The multiplicative (aka self-synchronizing) approach does not require scrambler and descrambler to be initialized to a common and fixed synchronization word. In fact, after at most N processed bits (needed for self-synchronization) the descrambler is always able to correctly recover the original info stream (an example is reported in Fig.8, where the case of different Tx/Rx LFSR initial states is simulated). However, the disadvantage of the multiplicative scrambling is that even a single wrong bit at the descrambler input can cause a significant error propagation at its output, as shown in Fig.7. Again, in Fig.6 the correct operation of the scrambler / descrambler pair is reported as well, where all the original info bits are recovered with no errors since both employ the same LFSR initial state and no bit corruptions occur between the scrambler output and the descrambler input. Therefore, under ideal conditions, the two techniques exhibit the same error-free performance. Moreover, it is important to notice that, unlike the additive approach, the multiplicative one is characterized by different schemes for scrambler and descrambler, as shown in Fig.2.

Fig.6 - Multiplicative scrambler - #1 : Correct operation

Fig.7 - Multiplicative scrambler - #2 : Errors at descrambler input

Fig.8 - Multiplicative scrambler - #3 : Tx/Rx LFSR initial state mismatch

All the three implementations (MATLAB, C/C++ and Python) give the possibility to customly set the connection vectors (i.e. the transfer function) and the initial states of the scambler / descrambler LFSRs. However, by default these (represented in polynomial form) are set as follows:

where additive and multiplicative polynomials are respectively in accordance with DVB and AX.25 standards.

Finally, with the C/C++ implementation it is also shown the scrambling effective capability of the two algorithms. In fact, for instance starting from an all-zero info byte-stream (INF) the results displayed in Fig.9 and Fig.10 show how both scramblers perform an almost random shuffle bringing about a satisfying alteration between ones and zeros (SCR), before being recovered by the descrambler (DES).

Fig.9 - Scrambling capability for the additive technique

Fig.10 - Scrambling capability for the multiplicative technique

Below there the links through which the aforementioned codes can be free downloaded. For further details and explanations about the practical implementation of the Reed-Solomon encoder / decoder pair take a look at the comments within these files.

Download MATLAB project : Scrambling.m
Download C/C++ project : Scrambling.cpp
Download Python project : Scrambling.py

WARNING#1: The Scrambling.m and Scrambling.py projects have been developed in MATLAB R2017a and Python 2.7.12. Therefore, the correct operation of the script may not be assured with different software versions.

Last update : 18/11/2018

References

[1] Wikipedia - Scrambler (link/a)