4. Periodic signals: Complex valued sinusoidsNurgun Erdol

13Jan09

Review: Complex numbersA complex number x is a special combination of a pair of real numbers r and m as

In rectangular coordinates. An alternate way of expressing x is in polar coordinates as

where the magnitude and the phase . The example in Figure 1 shows the complex number 4+j3 in the complex plane with its real and imaginary parts of 4 and 3, respectively. The complex number 4+j3 is at a distance from the origin and

at an angle .

Figure 1. The complex plane showing the location and components of complex number 4+j3.

The complex number in can be written in another polar coordinate form as.

So for the example , we can write . To see that it conveys the same information as and , we write and note that ,

and .Equations and are equivalent for complex valued functions as well. Some examples are given below:

1. . You can map this on the complex plane, such as the one in Figure 1. You should consider its values for different values of t. Select points on values are shown in Figure 2. Note that since the period is 1, map to the same point at 0 degrees,

as do the values for at 120 degrees.

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tt

4+j3

4

j3

real

imaginary

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Let and . Each one is plotted versus t in Figure 3 and the points corresponding to those in Figure 2 are marked with *.

2. .

Mapping this on the complex plane for results in Figure 4. Note that since there is a phase shift of radians or 120 degrees. Note also that the magnitude is still 1 so all the points lie on a circle of radius 1.

and are plotted versus t in Figure 5 and the points corresponding to those in Figure 4 are marked with *.

3. . Note that the magnitude of this function is not 1. Why? Its polar plot in the complex plane for is given in Figure 6 and the plots, versus, time of the real part and imaginary part , the magnitude and phase of

are shown in Figure 7. Compare the results as they all relate to the same signal.

Recall that a real valued periodic function satisfies the property for all values of . A complex valued periodic function satisfies the same condition which means its real part

must be periodic and its imaginary part must be periodic also. Exercise:

1. Find the periods of the complex valued functions given above as examples. Validate your results against the given plots associated with the functions.

2. Write MATLAB programs to duplicate the graphs of Figure 2-Figure 7.

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Figure 2. for values of . Note the same points are covered after one cycle is complete.

Figure 3. Plot versus time of the real and imaginary parts of overlaid with points corresponding to those shown on the complex plane of Figure 2.

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Figure 4. for values of . Note the advanced starting point at radians or 120 degrees.

Figure 5. Plot of the real and imaginary parts of versus time overlaid with points corresponding to those shown on the complex plane of Figure 4.

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Figure 6 Track taken by the complex signal for .

Figure 7. Plots of the real and imaginary parts (left column) and magnitude and phase (right column) of the two frequency sinusoid .

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