Digital Signal Processing Reference
In-Depth Information
The second row represents a
2
cos(2ft+
2
), again as a pair of phasors rotating
in opposite directions, as given by Euler's inverse formula. On the third row, we
have a visualization of the addition result (keeping the left and right sides separate).
Notice how similar the left and right sides are; they are complex conjugates of one
another.
7.9
Multiplying Phasors
We multiply two phasors by putting them in exponential form, multiplying their
amplitudes, and adding their angle arguments. To see how adding the angles works,
let's look at how a rotating phasor multiplied by another phasor and an amplitude
behave when we multiply them using Euler's formula. Here are a couple of equations
about adding the arguments of sinusoids [26], which we will need shortly.
cos(
1
+
2
) = cos(
1
) cos(
2
)sin(
1
) sin(
2
)
sin(
1
+
2
) = cos(
1
) sin(
2
) + sin(
1
) cos(
2
):
Rather than write 2f over and over again, we will replace it with !. With these
items in mind, we can simplify the following:
ae
j
e
j!t
= a(cos() + j sin())(cos(!t) + j sin(!t))
= acos() cos(!t) + jasin() cos(!t) + jacos() sin(!t) + j
2
asin() sin(!t)
= acos() cos(!t)asin() sin(!t) + jasin() cos(!t) + jacos() sin(!t)
= a(cos() cos(!t)sin() sin(!t)) + ja(cos() sin(!t) + sin() cos(!t))
= acos( + !t) + jasin( + !t):
We should see now how multiplying two exponentials simplies to adding their
arguments. Of course, a shortcut would be to replace e
c
e
d
with e
c+d
, or, more
specically, replace e
j
e
j!t
with e
j(+!t)
. Either way, we arrive at the same result.
If we multiply two complex exponents, z
3
= z
1
z
2
,
z
1
= r
1
e
j
1
= r
1
cos(
1
) + jr
1
sin(
1
)
z
2
= r
2
e
j
2
= r
2
cos(
2
) + jr
2
sin(
2
)
z
3
= (r
1
cos(
1
) + jr
1
sin(
1
))(r
2
cos(
2
) + jr
2
sin(
2
))
z
3
= r
1
r
2
(cos(
1
+
2
) + j sin(
1
+
2
)):
