Even and Odd Signals

Signals are sometimes classified by their symmetry along the time axis relative to the origin, t = 0. Even signals fold about t = 0, and odd signals fold about t = 0 but with a sign change. Simply put,

image0.jpg

To check the even and odd signal classification, use the Python rect() and tri() pulse functions to generate six aperiodic signals. Here’s the code for generating two of the signals:

In [759]: t = arange(-5,5,.005) # time axis for plots
In [760]: x1 = ssd.rect(t+2.5,3)+ssd.rect(t-2.5,3)
In [763]: x4 = ssd.rect(t+3,2)-ssd.tri(t,1)
+ssd.rect(t-3,2)

Check out the six signals, including the classification:

image1.jpg

To discern even or odd, observe the waveform symmetry with respect to t = 0. Signals x1(t), x4(t), and x6(t) are even; they fold nicely about t = 0. Signals x2(t) and x5(t) fold about t = 0 but with odd symmetry because the waveform on the negative time axis has the opposite sign of the positive time axis signal.

Signal x3(t) is neither even nor odd because a portion of the waveform, the triangle, is even about 0, while the rectangles are odd about 0. Taken in combination, the signals are neither even nor odd.

A single sinusoid in cosine form, without any phase shift, is even, because it’s symmetric with respect to t = 0, or rather it’s a mirror image of itself about t = 0. Mathematically, this is shown by the property of being an even signal:

image2.jpg

Similarly, a single sinusoid in sine form, without any phase shift, is odd, because it has negative symmetry about t = 0. Instead of an exact mirror image of itself, values to the left of t = 0 are opposite in sign of the values to the right of t = 0. This is mathematically an odd signal:

image3.jpg

If a nonzero phase shift is included, the even or odd properties are destroyed.

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