HW2-A Ben Laskowski ECE301Fall2008mboutin - Rhea

Part 1

Changing a Periodic Continuous Time Signal to a Non-Periodic Discrete Time Signal

One can take a signal that would be periodic in continuous time and turn it into a signal that is not periodic in discrete time. Consider the continuous time signal $ x(t)=sin(t) $. Plotting this signal yields a smooth waveform that repeats itself with period $ T=2\pi $.

The continuous-time signal $ x(t)=sin(t) $ is periodic.

Sampling this signal at every integer time yields something altogether different.

Sampling the continuous-time signal $ x(t)=sin(t) $ at integer times yields something like this. Note that the new discrete-time function $ x[n]=sin(n) $ is not periodic. Here we have shown five cycles of the formerly-periodic continuous time function.

The new discrete time function looks like this on its own.

The non-periodic discrete-time function $ x[n]=sin(n) $.

For the signal to be periodic, there must exist an integer N such that $ x[n]=x[n+N] $. For the signal defined as it is here, no such integer N exists.

Changing a Periodic Continuous Time Signal to a Periodic Discrete Time Signal

Suppose our sampling frequency, instead of being 1, was $ \frac{\pi}{8} $. Then the newly sampled function overlaid with the continuous function would look something like

The periodic discrete-time function $ x[n]=sin(\frac{\pi}{8}n) $ overlaid with its continuous time equivalent.

The periodic discrete-time function $ x[n]=sin(\frac{\pi}{8}n) $.

Part 2

Consider the non-periodic function $ f(t)=e^{-0.2t}*sin(10t) $.

Non-periodic function $ f(t)=e^{-0.2t}sin(10t) $.

If we run the following MATLAB code, we can shift it in several small increments and add it to itself to produce a function that approximates something periodic. Since we cannot add an infinite number of functions shifted an infinitely small amount, this approximation will work for our purposes.