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The Functions e^x and e^jθ: Numerical Experiment (Approximating e^jθ)

Module by: Louis Scharf. E-mail the author

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This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

We have demonstrated that the function ejθejθ has two representations:

  1. ejθ=limn→∞(1+jθn)nejθ=limn→∞(1+jθn)n ; and
  2. ejθ=limn→∞∑k=0n(jθ)kk!ejθ=limn→∞∑k=0n(jθ)kk!.

In this experiment, you will write a MATLAB program to evaluate the two functions f n f n and S n S n for twenty values of n n:

  1. fn=(1+jθn)n,n=1,2,...,20fn=(1+jθn)n,n=1,2,...,20; and
  2. S n = ∑ k = 0 n ( j θ ) k k ! , n = 1 , 2 , ... , 20 k S n = ∑ k = 0 n ( j θ ) k k ! , n = 1 , 2 , ... , 20 k .

Choose θ=π/4(=pi/4)θ=π/4(=pi/4). Use an implicit for loop to draw and plot a circle of radius 1. Then use an implicit for loop to compute and plot f n f n and an explicit for loop to compute and plot S n S n for n=1,2,...,100n=1,2,...,100. You should observe plots like those illustrated in Figure 1. Interpret them.

Figure 1: Plots for Convergence;(a) of fn, and (b) of Sn
(a) (b)
Figure one part a is a cartesian graph with horizontal axis labeled, real, and vertical axis labeled imaginary. Both axes range in value from 0 to 1 in increments of .2. A large curve beginning at the top-left corner of the graph, (0, 1), begins decreasing at a decreasing rate in a bowed curve until it is nearly vertical and terminates at the bottom-right corner of the graph, (1, 0). In the middle of this curve are a number of small dots, starting approximately at (0.6, 0.7). They are closely spaced together and follow a path that is slightly positive in slope. The dots then become more spread out and the last dot is placed at (1, 0.8).Figure one part b is a cartesian graph with horizontal axis labeled, real, and vertical axis labeled imaginary. Both axes range in value from 0 to 1 in increments of .2. A large curve beginning at the top-left corner of the graph, (0, 1), begins decreasing at a decreasing rate in a bowed curve until it is nearly vertical and terminates at the bottom-right corner of the graph, (1, 0). In the middle of this curve are a number of small dots, starting approximately at (0.6, 0.7). They are closely spaced together around that point except for one at (0.6, 0.8) and one at (1, 0.8).

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