Tutorial 6  Fourier Analysis Made Easy Part 2
Charan Langton, complextoreal.com
Page
1
Intuitive Guide to Principals of Communications
Tutorial 6 
Fourier Analysis Made Easy
Part 2
Complex representation of Fourier series
cos
sin
jwt
e
wt
i
wt
(1)
Bertrand Russell called this equation “the most beautiful, profound an
d subtle expression
in mathematics.”. Richard Feyman., the noble laureate said that it is “the most amazing
equation in all of mathematics”. In electrical engineering, this enigmatic equation is
equivalent in importance to F = ma.
This perplexing looking equation was first developed by Euler (pronounced Oiler) in the
early1800’s. A student of Johann Bernoulli, Euler was the foremost scientist of his day.
Born in Switzerland, he spent his later years at the University of St. Petersburg in Russia.
He perfected plane and solid geometry, created the first comprehensive approach to
complex numbers and is the father of modern calculus. He was the first to introduce the
concept of log x and e
x
as a function and it was his efforts that made the use of e, i and
pi
the common language of mathematics. He derived the equation e
x
+ 1 = 0 and its more
general form given above. Among his other contributions were the consistent use of the
sin, cos functions and the use of symbols for summation. A father of 13, he was a prolific
man in all aspects, in languages, medicine, botany, geography and all physical sciences.
The secret to this equation lies in understanding that sinusoids are a special case of a
general polynomial function of the form
e
jwt
in Euler’s equatio
n is a decidedly confusing concept. What exactly is the role
of
j
in e
jwt
? We know that it stands for
1
but what is it doing here? Can we visualize
this function?
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Tutorial 6  Fourier Analysis Made Easy Part 2
Charan Langton, complextoreal.com
Page
2
Before we continue the discussion of Fourier Series and its complex
re
presentation, let’s first try to make sense of e
jwt
as it relates to signal processing.
Take any real number, say 3, and plot it on a XY plot as in Fig 1a. Multiply this
number by j, so it becomes 3j.
Where do we plot it now? Herein lies our answer to what
multiplication with j does.
Figure 1a
 Relationship of real
Figure 1b
–
Multiplication with j
and imaginary numbers
represents a phase shift
The number stays exactly the same, 3j is the same as 3, except that multiplication
with j shifts the phase of this number by +90
o
.
So instead of an Xaxis number, it
becomes a Yaxis number. Each subsequent multiplication rotates it further by 90
o
in the
XY plane as shown in Figure 1b. 3 become 3j, then 3 and then 3j and back to 3 doing a
complete 360 degree turn. Division by j means the opposite. It shifts the phase by 90
o
.
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 Spring '11
 Tutorials
 Complex number, Charan Langton, Analysis Made Easy

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