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.