The_3D_Spectrum
Tayler discovered that an alternative ways to express
a function was to use a power series. This new way
of looking at sine and cosine functions lead to the
discovery that they were related to the exponental
function.
COS(X) = 1  X^2/2 + X^4/(4*3*2) +..etc
SIN(X) = + X + X^3/(3*2) +..etc
EXP(X) = 1 + X + X^2/2 + X^3/(3*2) + X^4/(4*3*2) +..etc
By using complex numbers, Euler was able to find
the following relationship.
EXP(j*X) = COS(X) j*SIN(X)
In the form below , this relationship effectly translates
a time function into a spectrum function.
COS(X) = EXP(j*X)/2 +EXP(j*X)/2
SIN(X) = j*EXP(j*X)/2 j*EXP(j*X)/2
A sine wave really needs only three data points to
completely define it. By looking at a spectrum, if
you know the amplitude and frequency, you still can't
relate it to time. You need to know when the sinewave
crosses zero in terms of time. In other words, you
need to know phase information.
Suppose you have a 5Hz signal that has a peak amplitude
of 3volts. That is saying that the X term will be 2*PI
for every 1/5th of a second. This is expressed below.
function(time) = 3*SIN(2*PI*5Hz*time) <ignoring phase
It is traditional to replace a the 2*PI*Freq term
with a single unit call Omega or "w".
w = 2*PI*Freq
If the amplitude is expressed as the term AMP, then..
function(time) = AMP*SIN(2*PI*5Hz*time) <ignoring phase
^
/\
 ____
 __/ \ _
 _/ \_
_ _/ _ _ _ _ _ _ _ _ _ \_ _ _ _ _ _ _ _ _ _ _ _ \
 \_ _/ /
 \__ __/
 \__ ___/
 
To know phase, one has to know in time when the
sinewave crosses zero. If at time "zero" the signal
is zero, you have a "SINE" wave. Now with the three
data points..(Frequency, Amplitude, Phase), you
know the signal in terms of voltage versus time.
function(time) = AMP*SIN(2*PI*5Hz*time)
It is common to express the magnitude and phase
of something in terms of a complex number. The
ratio to the real to imaginary values of give the
phase while the combine distance of the real and
imaginary parts give the magnitude.
REAL
^ X = valueR +jvalueI
/\

valueR ... __ Magnitude
 /
 / :
 / __ Phase
 /_: IMAGINARY
/___:____________\
: /
jvalueI
By adding one more dimension to represent frequency,
the Euler relationship shows an alternative way to
view Frequency, Amplitude, and Phase. Remmenber
SIN(X) = j*EXP(j*X)/2 j*EXP(j*X)/2
So the function of time can be viewed as a function
of frequency. The sine wave is mapped as two vectors
along the jw axis.
function(time) = AMP*SIN(2*PI*5Hz*time)
AMP*SIN(w*t) = j*AMP*EXP(j*w*t)/2  j*AMP*EXP(j*w*t)/2
REAL MAGNITUDE
^
/\

\ 
\ 
\ 
\ 
\ 
/______\ < j*AMP*EXP(j*w*t)/2
\ \ 
\ 
\  IMAGINARY MAGNITUDE
\__________________\
\ /
\
\
j*AMP*EXP(j*w*t)/2> \______\ <Magnitude = AMP
\ / <Direction = Phase
\ <Position = Freq
\
_V
J_OMEGA
These vectors always come in pairs ( jw and jw).
Where a vector is, how large it is, and which way
it is pointed tells frequency, amplitude, and phase.
If the phase of the signal happened to be a COSINE,
then the 3D spectrum would look like...
(Amplitude)
REAL MAGNITUDE
^
/\

\ 
\ 
\ ^ 
\ /\ 
\  
\ 
AMP*EXP(j*w*t)/2 \ 
\  (90degree Amplitude)
\  IMAGINARY MAGNITUDE
\__________________\
\ ^ /
\ /\
\ 
\
AMP*EXP(j*w*t)/2 \
\
\
\
_V
J_OMEGA (frequency)
3DS_Amplitude_Modulation_COS
________________\_ _ _\_ _ _\ 1001KHz at 0 degrees
/ / / 1000KHz at 0 degrees
999KHz at 0 degrees
_
/
/ 1001KHz leads 45 degree
________________\/ 1000KHz at 0 degrees
/\ 999KHz lag by 45 degrees
\
_\
^
/\
 1001KHz leads 90 degree
________________\ 1000KHz at 0 degrees
/ 999KHz lag by 90 degrees

\/
V
_
/\
\ 1001KHz leads 135 degree
________________\ 1000KHz at 0 degrees
/ 999KHz lag by 135 degree
/
/_
1001KHz leads 180 degree
______/ _ _/ _ _\ 1000KHz at 0 degrees
\ \ / 999KHz lag by 180 degree
^ (Amplitude)
/\ REAL MAGNITUDE
\  ^
\ ^  /\
\ /\ 
\   
\  
\  ^ 
\/\ 
\  
\ 
\ 
~etc~ ^
\  /\ IMAGINARY
\_________________\
\ ^  /
\ /\ MAGANITUDE
~etc~ 
\ 
\  ^
\/\
\ 
\
\
_V
J_OMEGA (frequency)
3DS_Amplitude_Modulation_SIN
^
/\

____ ____ Sine lags 90
cosine __/  _\/_ \__
_/  _/ \_ \_
_ _ / _ _ _ _ /_ _ _ _ \_ _ _ _ _ \_ _ _ _ _ _ _ \
 \_ \ _/ /
 \__ __/
 \__ ___/

_ _
_/     _
_        \
/          _
           \ _
            _ /  
              __ _/   
                     
                      
                _    
             / _  
           _/  
           \_
        _/
\_      
_ _ /
^
/\
 1001KHz lags 90 degree
________________\ 1000KHz at 0 degrees
/ 999KHz leads by 90 degrees

\/
V
_
/
/ 1001KHz lags 45 degree
________________\/ 1000KHz at 0 degrees
/\ 999KHz leads by 45 degrees
\
_\
________________\_ _ _\_ _ _\ 1001KHz at 0 degrees
/ / / 1000KHz at 0 degrees
999KHz at 0 degrees
^ (Amplitude)
/\ REAL MAGNITUDE
\  ^
\  /\
\  
\  
/____\  
\ \  
\ 
\ 
\____\
\ /
~etc~ ^
\  /\ IMAGINARY
\_________________\
\  /
\  MAGANITUDE
~etc~ 
/____\ 
\ \ 
\
\
\____\
\ /
_V
J_OMEGA (frequency)
3DS_Phase_Modulation
^
/\

____ ____ Sine lags 90
cosine __/  _\/_ \__
_/  _/ \_ \_
_ _ / _ _ _ _ /_ _ _ _ \_ _ _ _ _ \_ _ _ _ _ _ _ \
 \_ \ _/ /
 \__ __/
 \__ ___/

_________________\
/ 1000KHz at 0 degrees
 1001KHz lags 90 degrees
\/ 999KHz lags 90 degrees
V


\/
V
__________________\ 1000KHz at 0 degrees
/\ 1001KHz lags 45 degrees
/ \ 999KHz lags 135 degrees
/_ _\
___________/_ _ _\_ _ _\ 1000KHz at 0 degrees
\ / / 1001KHz lags 0 degrees
999KHz lags 180 degree
_ _
\ /
\ / 1001KHz leads 45 degrees
________________\/ 1000KHz at 0 degrees
/ 999KHz lags 225 degree
^
/\
 1001KHz leads 90 degrees
 1000KHz at 0 degrees
^ 999KHz lags 270 degree
/\

________________\
/
^ (Amplitude)
/\ REAL MAGNITUDE
\  ^
\  /\
\  
\  
/____\  
\ \  
\ 
\ 
/____\ 
\ \ 
~etc~ ^
\  /\ IMAGINARY
\_________________\
\  /
\  MAGANITUDE
~etc~ 
\___\
\  /
\
\
\____\
\ /
_V
J_OMEGA (frequency)
FM_2_PM
_ _ _ _ _ _ _ _ _ _ _ _
                       
                       
                       
                       
 _ _ _ _ _ _ _ _ _ _ _ 
Pure sign wave (no FM)
_ _ __ __ _ _ _ _ _ _
              [[     
                   
                   
                   
 _ _ __ _ _ _ _L_ _ _
A Moving sign wave
Frequency Modulate a 1kHz carrier with a 1Hz signwave
Freq(t) = [ 1000 + sin(2*PI*t) ]
Signal= sin( (Freq(t))*2*PI*t) <FM format
at time = 1/4 The Frequency is 1001 Hz
at time = 3/4 The Frequency is 999 Hz
Another way to think of it is Phase Modulation...
Signal= sin( (1000)*2*PI*t +Phase(t)) < PM format
the phase will be Integral of FM ...
_
/
Phase(t) =  2*PI*sin(2*PI*t)*dt = cos(2*PI*t)
_/
So 1Hz peak FM over 1Hz will generate 1radian peak of PM
And to convert FM to PM, just integrate
and
One Hz FM at One Hz equals One radian PM
The PM will lag the FM which causes it.
^
/\
Freq=1001Hz Phase= +1radian
____ ____
__/  _\/_ \__
_/  _/ \_ \_
_ _ / _ _ _ _ /_ _ _ _ \_ _ _ _ _ \_ _ _ _ _ _ _ \
_/ \_ \_ _/ /
__/  \__ \__ __/ TIME
__/  \__ ___/___
  
Phase= 1radian Freq=999Hz