Time Shifting and Scaling of Functions

We’ll begin with a square function, f(t), that has a an amplitude of 1, a start time of 2 seconds and an end time of 4 seconds.

square-wave

Next, a time shift is demonstrated. Here our function is changed from f(t) to f(t-2). Notice that subtracting 2 from t in the function results in a positive shift of the graph.

positive-shifted-square-wave

In the next two graphs t will be scaled. Scaling t is not quite as intuitive as we may have expected. When we multiply t by 2, corresponding points of the function now occur at 1/2 the time they previously had. When we divide t by 2, each corresponding time on the graph occurs at a t that is now multiplied by 2. Notice that each of these factors directly affects the duration of the signal.

time-shifted-square-wave
time-shifted-square-wave2

Scaling the amplitude has more intuitive results. If we multiply f(t) by 2, the amplitude of 1 is changed to 2. Multiplying f(t) by 1/2 results in an amplitude of 1/2.

taller-square-wave
truncated-square-wave

Finally, multiplying t by -1 mirrors our function over the y-axis. Each time now occurs at its negative.

mirrored-square-wave


Example:

Here we will attempt to convert f(t) into 2*f(.5t+3). The graph of f(t) is shown below.

triangle-wave

The easiest way to handle this type of problem without error is to manipulate the function one step at a time. First, I have converted f(t) into 2*f(t). Only the peaks are changed here (by a factor of 2).

triangle-wave2

Next, I convert 2*f(t) into 2*f( \frac{1}{2} t). Notice how the \frac{1}{2} actually expands our graph duration by a factor of 2 (from a 6 sec duration to a 12 sec duration).

shifted-triangle-wave

Finally, we move from 2*f( \frac{1}{2} t) to 2*f( \frac{1}{2} t + 3). As shown in the discussion above, this is a time shift. Time shifts can be a little confusing because adding results in a negative shift of our graph. Try to think of it as our signal occurring 3 seconds earlier than before, reading from left to right on the graph. The easiest way to do this part is shift each x-intercept by 3 seconds (to the left, of course).

shifted-triangle-wave2

One Reply to “Time Shifting and Scaling of Functions”

  1. You are correct. I performed the time scaling and time shifting in opposite order. The signal should have been shifted by 6 seconds (3/(.5)) to the left and then scaled appropriately for the 1/2 factor. Updates on the way..

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