Eulers Formula and Negative Frequencies

1
Eulers Formula and Negative Frequencies
A Quick Review of Complex Numbers
z is complex, a and b are real, j is THE
imaginary number
a is the real part of z b is the imaginary
part z is the sum of its real and imaginary parts
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Vector in the Complex Plane
Consider a unit circle (a circle of radius 1)
centered at the origin of the complex number
plane. The horizontal axis is the real axis and
the vertical axis is the imaginary, or jw axis.
Also consider a vector of length one, with its
tail at the origin and its head somewhere on the
unit circle.
The vector makes an angle q with the x axis. If
we call the vector z, we can express it in either
polar or rectangular coordinates
jb
1
q
1
-1
a
-1
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Vector in the Complex Plane
a, the real part of z, can be found graphically
by projecting a vertical line from the head of z
down to the real axis. Notice that z, the real
axis, and the vertical line form a right
triangle, so the real part of z is given by the
definition of cosine
jb
1
z is a unit vector, meaning its length is 1
b
1
q
a
a
In a similar way, we can find b, the imaginary
part of z by projecting a horizontal line from
the head of z left to the jw axis
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Vector in the Complex Plane
Lets set q 0 degrees, and plot the imaginary
part of z. Obviously, the projection onto the jw
is zero.
jb
1
1
a
1
q 0
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Vector in the Complex Plane
Now lets start increasing q. Graphically, this
means holding the tail of the vector at the
origin while rotating the head counterclockwise
around the unit circle. Lets increase q to 30
degrees
jb
1

• 30
• deg.

b(30)
1
a
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Vector in the Complex Plane
Now lets increase q to 45 degrees
jb
1

• 45
• deg.

b(45)
1
a
7
Vector in the Complex Plane
Now increase q to 60 degrees
jb

• 60
• deg.

1
b(60)
1
a
8
Vector in the Complex Plane
Now increase q to 90 degrees
jb

• 90
• deg.

1
1
a
9
Vector in the Complex Plane
Next increase q to 180 degrees
jb
1

• 180
• deg.

1
a
10
Vector in the Complex Plane
Next increase q to 270 degrees
jb
1
1
a

• 270
• deg.

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Vector in the Complex Plane
And increase q to 360 degrees Notice that the
vector is back in the same position as when q was
0 degrees. If we continue to rotate the vector,
the resulting waveform repeats every 360 degrees.
jb
1
q 360 deg.
1
a
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Rotating Vector in the Complex Plane
Suppose the vector rotates at a constant rate.
For example, suppose the vector makes one
revolution around the unit circle every second.
The sine wave repeats once per second. It is
said to have a frequency of 1 cycle per second,
or 1 Hertz (Hz).
jb
1
1 rev./sec.
b(q)

• 180
• deg.

1
a
1
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Rotating Vector in the Complex Plane
A sine wave is an example of a periodic waveform,
a waveform that repeats itself after a fixed
(constant) period. The period of a sine wave is
the time it takes for the unit vector to complete
one revolution.
jb
1
1 rev./sec.

• 180
• deg.

1
a
1
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Rotating Vector in the Complex Plane
If f 1 Hz., it takes one second for the unit
vector to complete one revolution. The period is
one second. If f 10 Hz, the unit vector
completes 10 revolutions in one second one
revolution every 0.1 seconds. The period is 0.1
sec.
jb
1
1 rev./sec.

• 180
• deg.

1
a
1
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Rotating Vector in the Complex Plane
In general, if T represents the period of a sine
wave and f represents its frequency,
jb
1
1 rev./sec.

• 180
• deg.

1
a
1
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Rotating Vector in the Complex Plane
The rate of rotation of the unit vector can also
be expressed in degrees per second, or radians
per second. Each revolution the vector passes
through 360 degrees or 2p radians.
jb
1
1 rev./sec.

• 180
• deg.

1
a
1
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Angular Frequency
When expressed this way, the rate of rotation is
called angular rate, or angular frequency, or
radian frequency, and is usually represented by w
(lowercase omega). Its related to the speed in
revolutions per second (and frequency in Hz.) by
w 2pf
jb
1
1 rev./sec.

• 180
• deg.

1
a
1
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Angular Frequency
For any value of t, we can express the angle q as
So as the vector rotates at the rate wt, q is
constantly increasing by w radians per second.
Every time the vector completes a revolution
(increases by 2p radians), one cycle is completed.
y
1
1 rev./sec.

• 180
• deg.

1
x
1
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Eulers Formula
Recall that, in polar and rectangular
coordinates, z is given by
jb
1
For a unit-length vector such as the one we have
been discussing,
b
1
q
a
a
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Eulers Formula
Eulers formula (aka Eulers identity) was called
"the most remarkable formula in mathematics" by
Richard Feynman
If youve never seen Eulers formula before, it
probably looks strange enough to require proof.
jb
1
b
1
q
a
a
Nobel Prize in Physics, 1965. For more
information, read Genius by James Gleick, or
Surely Youre Joking, Mr. Feynman The
Adventures of a Curious Character by Richard P.
Feynman.
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Eulers Formula
ex, sin(x) and cos(x) may each be expanded into a
power series
jb
1
b
1
q
a
a
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Complex Numbers
Now, substitute jx for x in the first series
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Eulers Formula
A short time ago we found that our unit vector
could be written in polar or rectangular form as
we may now add
jb
1
If the vector had a length other than 1,
b
1
q
a
a
so a vector z in the complex plane, which is
equivalent to a complex number z, may be
expressed in polar form as
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Eulers Formula
Returning to the unit vector, Eulers formula
says
If we solve this for cosq, we find
jb
1
b
And solving for sinq,
1
q
a
a
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More Rotating Vectors
Now, lets take our unit-length vector and put it
in motion by making it rotate counterclockwise at
an angular rate of w radians per second. Now q
is an increasing function of time
2p/T Rad/sec.
jb
1
b
1
q
a
a
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More Rotating Vectors
Now that we have Eulers formula, the real
component of the rotating vector can be written
Which means that a cosine wave of unit amplitude
can be expressed as the real part of a rotating
vector. The rotating vector also has an imagnary
part. We could simply ignore it or throw it
away, but thats sloppy and inelegant, and not
quite kosher.
jb
2p/T Rad/sec.
1
b
1
q
a
a
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More Rotating Vectors
Heres a more palatable way to get rid of the
imaginary part Well add a second vector
rotating in the opposite (clockwise direction.
At any time t, the second vector will be at an
angle which is the additive inverse of the angle
of the first vector. The original vector is
jb
w Rad/sec.
1
The new, clockwise-rotating vector is
But
1
a
a
q-
So the new, clockwise-rotating vector may also be
written as
-b
-w Rad/sec.
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More Rotating Vectors
Adding the clockwise- and counterclockwise-rotatin
g vectors yields
jb
w Rad/sec.
1
The imaginary parts of the two vectors cancel
each other, so theres nothing to ignore. The
cosine waveform can be modeled as the sum of two
rotating vectors, or complex exponentials One
with frequency w, and the other with frequency
w. This is the negative-frequency part of the
cosine wave.
1
a
a
q-
-b
-w Rad/sec.
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Negative Frequency
To generate a sine wave, start with the original
vector, but instead of adding
add
jb
w Rad/sec.
-w Rad/sec.
1
1
wt
-wt
a
So a sine wave also can be modeled as the sum of
a positive-frequency part and a
negative-frequency part.