Signal Transformations

 Objective: Help student gain a visual understanding of several basic
 signal transformations.

 Commands: array operations, plot, labels, figure.

 Background Information: In MATLAB and other programming environments
 continuous signals cannot be represented by vectors or arrays, since
 these are inherently discrete. However, it is possible to approximate a
 continuous signal by taking samples of the signal, where the time
 interval between samples is small enough that the signal exhibits only
 small changes in amplitude between adjacent samples.  For example,
 consider the signal x(t) = exp(-0.5*t)*u(t).  This signal "turns on"
 at t = 0, and then exponentially decays towards 0 with an initial slope
 of -0.5 (at t = 0).  Given this initial slope, the time interval between
 samples should be significantly less than 0.5.  A nice choice might be
 100 times smaller, although this will clearly depend on the application.

 Exercise: Consider the signal x(t) = sin(2*pi/4*t).

 This is a continuous signal, "turning on" at time t = negative infinity and 
 continuing towards t = positive infinity.  Clearly we can't plot this signal 
 for all time in Matlab.  In this exercise, you should limit
 your time axis from -6 s to 6 s, and use a sampling period (interval
 between time points) of 0.01 s.

 First, set up a vector t holding your time values, and create the signal
 x(t) = sin(2*pi/4*t).  This can be done with the following code:

   t_start = -6;
   t_end = 6;
   t_interval = 0.01;

   t = t_start:t_interval:t_end;

   x = sin(2*pi/4*t);


 Then do the following:

   (1) Plot the signal x(t)

   (2) Create the following transformations of the signal x(t), and
       plot each of them in the same figure but different plots using
       the subplot command:

       (a) y1(t) = x(2*t)
       (b) y2(t) = x(0.5*t)
       (c) y3(t) = x(t-1)
       (d) y4(t) = x(-t)
       (e) y5(t) = x(4-2*t)

       Give each plot a title to distinguish the original signal from its
       transformations.

   (3) Answer the following questions:

       (a) Explain the behavior you observe in 2(a) and 2(b).
       (b) Explain in words how the signal x(t) is transformed in 2(e).