Discrete Maths

Question

Messages are transmitted through a communication channel using two signals. The transmission of one signal requires $1 $ microsecond, the transmission of other signal requires $2$ microseconds. Assume that each signal in a message is immediately followed by the next signal. The number of different messages consisting of sequences of these two signals that can be sent in $10 \mu s$ is _________ ?

Please explain ...?

Answer 1

Let $T_{n}$ be the number of ways to send the messages in n microseconds.

If there is no message or message of 0 length , we can send it in 1 way (do nothing)

Initially we had a message,to send it  we have two choices like

 1) we can start with a signal of 1 microsecond , this leaves the message can be send in remaining n − 1 microseconds.

2) We can start with a signal of 2 microseconds , this leaves the message can be send in then send the remaining n−2 microsec.

So here the  recurrence relation is $T_{n} = T_{n−1} + T_{n−2}$ for n ≥ 2.

What about base conditions $T_{0}$  and   $T_{1}$

$T_{0}$ = 1 and $T_{1}$=1 as nothing can be sent in 1 way (above) and 1 micro second message can be sent in 1 way.

and 2 micro second  message in 2 ways (sending 1micro second message two times or sendind 2microsecond message at a time).

Compute $T_{10}$  you can find this.

Comments

Here $T_0$ means number of messages that can be sent at $t = 0$.
So, $T_0 = 0$ (or) $T_0 = 1$

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$T_{0}$  means number of messages we can send in 0 micro seconds.

which is = 1 means in 0 micro seconds we can send nothing or empty message (only 1 way)