Design and Analysis Lab 2: Basic Signal Operations
Josh Wood
February 5, 2007
Introduction: In this lab, we used java applets posted on the instructor’s website that
demonstrate the basic operations of different signals. The first applet we used, titled
“Basic Signal Operations”, was used for the first 8 questions.
Question 1 read:
“Click on the “Basic Signal Operations” applet. Practice using the various controls on
this applet. What happens when you adjust the amplitude slider?”
By adjust the amplitude slider, the amplitude of the signal in the graph x(t) increases
accordingly.
Question 2 read:
“Describe what happens when you adjust the time shift slider on any of the signals x(t)?”
By adjusting the time shift slider, the signal’s location on the horizontal axis moves
according to the value on the slider—in this case, a 2 on the time shift slider shifted the
signal 2 places to the right, resulting in the equation x(t-2.0).
Question 3 read:
“Repeat #2 when the time shift slider is applied to the signal x(-t). Comment!”
By adjusting the time shift slider to the same value, but to the x(-t) signal, the signal’s
location on the horizontal axis moves in the opposite direction (to the left) by 2 places (2
being the value on the slider). This results in the equation x(-t-2.0).
Question 4 read:
“Plot the signal x(t) = u(t) and y(t) = u(t). The signal u(t) is the unit step function. Next,
plot 3x(t – 2) + 3y(-t – 2).”
This graph was achieved by setting both x(t) and y(t) to a unit step function. Then, by
using the Amplitude and Time Shift sliders as well as the add operation, the desired
equation was created.
Question 5 read:
“Use the controls on the applet to demonstrate that if y(t) = rect(t – 0.5, 1), then y(-t + 1)
= y(t).”
Here, x(t) was set to rect(t-0.5, 1), and y(t) was set to y(-t + 1). By comparing these
graphs, we notice that both graphs are identical, and the equality is true.
Question 6 read:
“If x(t) = u(t) and y(t) = rect(t – 0.5, 1), plot the graph of x(t + 1) + y(-t + 2).”
This graph was self explanatory; by using the appropriate functions for x(t) and y(t),
adjusting the slider values, and enabling the addition operation, the desired graph was
achieved.
Question 7 read:
“If x(t) = rect(t – 0.5, 1), then plot the graph of rect(t, 1).”
This graph was achieved by setting x(t) to the rect function, then adjusting the time shift
slider to the +0.5 position. This allowed the graph of rect(t, 1).
Question 8 Read:
“Plot the graph of the product exp(-t)u(t)sin(8t). To better understand what is happening
mathematically, try filling in the following table (the values can be obtained by
positioning the mouse over the graph of each signal, at which time the signal coordinates
will turn red):”
This graph was created by multiplying x(t) and y(t), where x(t) was the exponential
function, and y(t) was the given sine function. The resultant graph was then analyzed
point by point (by dragging the mouse on the curve and plotting points), using the
following table:
Question 9 read:
“Now bring up the applet called “Sinusoids”. Plot the signals cos(4t) and sin(4t).”
This question required a different java applet, called “Sinusoids”. The equations cos(4t)
and sin(4t) were created by adjusting the frequency slider to the 4 position.
Question 10 read:
“Find the smallest positive phase shift, Φ, such that sin(4t – Φ) = cos(4t). Insert a plot in
your write-up which confirms your answer.”
For this graph, the phase shift that makes this equality correct is 1.5, resulting in the
equation sin(4t-1.5) = cos(4t). The following values were compared from x(t) and y(t)
for comparison and confirmation.
Question 11 read :
“Explain the difference between a phase shift and a time shift in a sinusoidal signal!”
In a sinusoidal signal, there is no distinguishable difference between a phase shift and a
time shift. They both make the same adjustments to the signal.
Summary: In this lab, several signal operations that we had learned about in lecture
were demonstrated in real time using different java applets provided to us. This lab
allowed us to get more hands-on experience with different signals, as well as how
significant elements such as phase and time shift are to a signal.