Solution:

A scrambler is a telecommunication device that can be used to make information coming out of a transmitter unintelligible.

In order to be able to interpret the information the receiver has to have a descrambler.

A scrambler is normally applied to analog systems while an encryptor, a device with the same objectives, is applied to digital systems.

A scrambler can also be applied to a telecommunication system for a different purpose; to randomize input data. When a bit stream consists of a long sequence of 1s or 0s, the clock in the transmitter fails to track the receiver clock and the system lose synchronization.

A scrambler can be deployed to change a long string of 1s or 0s to a more random sequence and hence aid in synchronization. Scramblers can be classified into two classes; additive scramblers and multiplicative scramblers.

Additive scramblers use modulo-two addition to transform the input data stream and in order to achieve synchronization between the transmitter and the receiver a sync word is used.

The sync word has a unique pattern which is placed at the beginning of each frame and is known at the receiver.

The receiver searches for it and determines the beginning of the incoming frame.

Multiplicative scramblers have been named so because in order to determine the scrambled the output they perform a multiplication between the input signal and the scrambler transfer function in the z-domain.

They do not need a sync word for frame synchronization and that is why they are referred to as self-synchronizing scramblers.

The figure shows an ITU-T standard multiplicative scrambler and descrambler defined by the polynomial,

$ 1+x^{18}+x^{23}\\ $

In general, the scrambler output is given by

$ y(n)=\sum_{k=k}^N h(k) x(n-k)\\ $

and the descrambler output is given by

$ x(n)=\sum_{i=1}^N h(i) y(n-i) .\\ $