HW1.4 Josh Long ECE301Fall2008mboutin - Rhea

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