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Deterministic and Random Signal Classifications

A signal is classified as deterministic if it’s a completely specified function of time. A good example of a deterministic signal is a signal composed of a single sinusoid, such as

image0.jpg

with the signal parameters being:

image1.jpg

A is the amplitude, f0 is the frequency (oscillation rate) in cycles per second (or hertz), and

image2.jpg

is the phase in radians. Depending on your background, you may be more familiar with radian frequency,

image3.jpg

which has units of radians/sample. In any case, x(t) is deterministic because the signal parameters are constants.

Hertz (Hz) represents the cycles per second unit of measurement in honor of Heinrich Hertz, who first demonstrated the existence of radio waves.

A signal is classified as random if it takes on values by chance according to some probabilistic model. You can extend the deterministic sinusoid model

image4.jpg

to a random model by making one or more of the parameters random. By introducing random parameters, you can more realistically model real-world signals.

To see how a random signal can be constructed, write

image5.jpg

where

image6.jpg

corresponds to the drawing of a particular set of values from a set of possible outcomes. Relax; incorporating random parameters in your signal models is a topic left to more advanced courses.

To visualize the concepts in this section, including randomness, you can use the IPython environment with PyLab to create a plot of deterministic and random waveform examples:

In [234]: t = linspace(0,5,200)
In [235]: x1 = 1.5*cos(2*pi*1*t + pi/3)
In [237]: plot(t,x1)
In [242]: for k in range(0,5): # loop with k=0,1,…,4
 ...: x2 = (1.5+rand(1)-0.5))*cos(2*pi*1*t + pi/2*rand(1)) # rand()= is uniform on (0,1)
 ...: plot(t,x2,'b')
 ...:

The results are shown here, which uses a 2xPyLab subplot to stack plots.

image7.jpg

Generate the deterministic sinusoid by creating a vector of time samples, t, running from zero to five seconds. To create the signal, x1 in this case, these values were chosen for the waveform parameters:

image8.jpg

For the random signal case, A is nominally 1.5, but a random number uniform over (–0.5, 0.5) is added to A, making the composite sinusoid amplitude random. The frequency is fixed at 1.0, and the phase is uniform over

image9.jpg

Five realizations of

image10.jpg

are created using a for loop.

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