Periodic Signal Definition
- For a Continuous-time signal
There exists a positive value of T for which
$ x(t) = x(t - T) $
for all values of t.
- For a Discrete-time signal
There exists a positive integer N for which
$ x[n] = x[n + N] $
for all values of n.
Note: N is the period of the signal.
Problems
- Periodic Continuous-Time Signal
$ x(t) = 3\cos(4t + \frac{\pi}{3}) $As you can see in graph the signal is being repeated and a value T can be used to get the same value during other times of the signal.
- Non-Periodic Discrete-Time Signal
$ x[n] = \cos(\frac{\pi}{8}n^2) $As you can see the graph is non-periodic due to the fact that here is no value of N that could be added so that $ x[n] = x[n+N $
- Bonus Question
$ x(t) = e^{j(\pi t-1)} $
Is periodic, when graphed it produces a straigh line.
Since it is a line, at any time the value of the signal will be equal to any other time.
Credit: Problems were taken from Signals & Systems 2nd ed. (Oppenheim) Page 61
