Question

# When plotted in the complex plane for , the function f () = cos() + j0.1...

When plotted in the complex plane for , the function f () = cos() + j0.1 sin(2) results in a so-called Lissajous figure

a. In MATLAB, create two row vectors fr and fi corresponding to the real and imaginary portions of f (), respectively, over a suitable number N samples of . Plot the real portion against the imaginary portion and verify the figure resembles a propeller.

b. Let complex constant w = x + jy be represented in vector form

Consider the 2 × 2 rotational matrix R:

Show that Rw rotates vector w by radians.
c. Create a rotational matrix R corresponding to 10° and multiply it by the 2 × N matrix f = [f r; f i];. Plot the result

to verify that the "propeller" has indeed rotated counterclockwise.

Given the matrix R determined in part c, what is the effect of performing RRf? How about RRRf? Generalize the result.

Investigate the behavior of multiplying f () by the function ej?.

When plotted in the complex plane for , the function f () = cos() + j0.1 sin(2) results in a so-called Lissajous figure

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