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# Phasors: Numerical Experiment (Interference Patterns)

Module by: Louis Scharf. E-mail the author

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Let's add two sinusoids whose amplitudes and frequencies are identical and whose phases are different:

x ( t ) = A cos ( ω t + φ ) + A cos ( ω t + φ + ψ ) . x ( t ) = A cos ( ω t + φ ) + A cos ( ω t + φ + ψ ) .
(1)

Show analytically that this sum has the phasor representation

X = 2 A c o s ( ψ 2 ) e j [ φ + ( ψ / 2 ) ] . X = 2 A c o s ( ψ 2 ) e j [ φ + ( ψ / 2 ) ] .
(2)

Interpret this finding. Then write a MATLAB program that computes and plots complex X on the complex plane as ψ varies from 0 to 2π2π and that plots magnitude, |X||X|, and phase, arg X argX, versus the phase angle ψ ψ. (You will have to choose ψ=n2πN,n=0,1,...,N-1ψ=n2πN,n=0,1,...,N-1, for a suitable N.N.) When do you get constructive interfelence and when do you get destructive interference? Now compute and plot x(t)x(t) versus t t (you will need to discretize t t) for several interesting values of ψ ψ. Explain your interference results in terms of the amplitude and phase of x(t)x(t) and the magnitude and phase of X X. Use the subplots discussed in Appendix 1 to plot all of your results together.

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