00:01
We were asked to answer a question about signal transmission.
00:06
We are asked how many different message can be transmitted in n microseconds using the two signals described in a previous exercise.
00:15
As i recall from that previous exercise, we had that the number of signals in n microseconds, a n -microseconds, a number of signals in n -1 microseconds plus the number of signals in n -minus -2 microseconds.
00:38
And we had that initial conditions are at zero microseconds we have the empty signal, so it's one, and at one microsecond, we can only send one signal, so a1 equals 1.
00:57
And so we have a homogenous linear occurrence relation with characteristic equation r squared minus r minus 1 equals 0.
01:11
And using the quadratic formula, we have that r is equal to 1 plus or minus the square root of 1 plus 4, which is 5 over 2.
01:26
And we have that these roots are distinct.
01:29
So the general solution is alpha 1, and this is 1 plus root 5 over 2, plus alpha 2 times 1 minus root 5 over 2, both of these to the 8.
01:47
And using our initial conditions we have that 1 which is a 0 is equal to alpha 1 plus alpha 2 and we have that 1 which is a 1 is equal to alpha 1 times 1 plus root 5 over 2 plus alpha 2 times 1 minus root 5 over 2 and writing this we can multiply the top equation by 1 minus root 5, and we can subtract the top equation from the bottom equation...
