Periodic Signals and Non-Periodic Signals
Most of the signals from Homework 1 were boring (including mine) so I thought I'd broaden the periodic signal pool. I chose the CT signal: $ x(t) = |2*cos(.5*t)| $ . A graph of this signal in continuous time is shown below.
The continuous-time signal $ x(t) = |2*cos(.5*t)| $ is periodic.
As can be seen below, when the function mentioned earlier is sampled at the inappropriate frequency, the signal becomes non periodic. This function can be represented by $ x[n] = |2*cos(.5*n)| $ and is graphed below at a sampling rate of $ 1/7\pi $.
The discrete-time signal $ x[n] = |2*cos(.5*n)| $ is not periodic.
In addition to this observation, however, it is important to note that if several different time-shifted portions of this equation were laid on top of each other infinitely many times, it would become a periodic signal again. This is illustrated below.
The discrete-time signal $ x[n] = |2*cos(.5*n)| $ becomes periodic when superimposed with several time-shifted iterations of itself.
