# ELEMENTARY, MY DEAR WATSON

PART II

AKA: SDRI seventh meetup

SEPTEMBER 2016

Software Defined Radio Israel

# THE CONNECTION BETWEEN

ALICE

# AND

OPTIMUS PRIME?

# OF

## COMPLEX NUMBERS

NEXT TIME...

### FOURIER

AND

OTHER TRANSFORMATIONS

## IT ALL BEGINS WITH SERIES'

\sum _{ n=0 }^{ \infty }{ { a }_{ n }={ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+{ a }_{ 4 }... }
$\sum _{ n=0 }^{ \infty }{ { a }_{ n }={ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+{ a }_{ 4 }... }$
\sum _{ n=0 }^{ \infty }{ { f }_{ n }\left( x \right) ={ f }_{ 1 }\left( x \right) +{ f }_{ 2 }\left( x \right) +{ f }_{ 3 }\left( x \right) +{ f }_{ 4 }\left( x \right) +... }
$\sum _{ n=0 }^{ \infty }{ { f }_{ n }\left( x \right) ={ f }_{ 1 }\left( x \right) +{ f }_{ 2 }\left( x \right) +{ f }_{ 3 }\left( x \right) +{ f }_{ 4 }\left( x \right) +... }$

# OUR MOTIVATION:

## DESCRIBE A FUNCTION WITH...OTHER FUNCTIONS

AND

BECOME...

(AND THE OTHER WAY AROUND)

FOR

EXAMPLE:

## all kinds of series'

• Geometric series

• Taylor series

• Laurent series

And more...

\sum_{n=0}^\infty \frac{ \, x^n}{n!}
$\sum_{n=0}^\infty \frac{ \, x^n}{n!}$
\sum _{ n=-\infty }^{ \infty }{ { { a }_{ n }x }^{ n } }
$\sum _{ n=-\infty }^{ \infty }{ { { a }_{ n }x }^{ n } }$

## FOURIER SERIES

SERIES THAT IS COMBINED FROM SINES AND COSINES

{ s }_{ N }\left( x \right) =\frac { { a }_{ 0 } }{ 2 } +\sum _{ n=1 }^{ N }{ \left[ { a }_{ n }\cos { \left( \frac { 2\pi nx }{ P } \right) + } { b }_{ n }\sin { \left( \frac { 2\pi nx }{ P } \right) } \right] }
${ s }_{ N }\left( x \right) =\frac { { a }_{ 0 } }{ 2 } +\sum _{ n=1 }^{ N }{ \left[ { a }_{ n }\cos { \left( \frac { 2\pi nx }{ P } \right) + } { b }_{ n }\sin { \left( \frac { 2\pi nx }{ P } \right) } \right] }$

A BASIC EXAMPLE:

{ s }_{ }\left( x \right) =1+3\cos { \left( x \right) }
${ s }_{ }\left( x \right) =1+3\cos { \left( x \right) }$
{ a }_{ 0 }=2,\quad { a }_{ 1 }=3,\quad { a }_{ n>1 },b_{ n>0 }=0
${ a }_{ 0 }=2,\quad { a }_{ 1 }=3,\quad { a }_{ n>1 },b_{ n>0 }=0$
P(period)=2\pi
$P(period)=2\pi$

## FOURIER TRANSFORM

• FOURIER SERIES COEFFICIENTS REPRESENT THE "ENERGY" OF THEIR CORRESPONDING "FREQUENCY"
• THE COMBINATION OF ALL THE COEFFICIENTS IS THE "SPECTRAL CONTENT" OF THE ORIGINAL FUNCTION (OR SIGNAL)
• THE PROCESS OF FINDING THE COEFFICIENTS IS CALLED "FOURIER TRANSFORM"

## HOW TO PERFORM A FOURIER TRANSFORM?

SIGNALS IN NATURE ARE CONTINUOUS IN TIME AND VALUE, AND THEREFORE WE DEFINE THE CONTINUOUS FOURIER TRANSFORM - CFT:

\Im \left\{ h\left( x \right) \right\} \left( f \right) =\int _{ -\infty }^{ \infty }{ h\left( x \right) { e }^{ -2\pi jxf }dx }
$\Im \left\{ h\left( x \right) \right\} \left( f \right) =\int _{ -\infty }^{ \infty }{ h\left( x \right) { e }^{ -2\pi jxf }dx }$
j=\sqrt { -1 }
$j=\sqrt { -1 }$

## BUT IN THE REAL WORLD...

- SIGNALS ARE NOT REALLY CONTINUOUS, BUT RATHER A VECTOR OF VALUES ( = DISCRETE TIME) WHICH ARE FROM A FINITE SET OF VALUES (SAY, A BYTE SIZE)

- THEREFORE, THE FOURIER TRANSFORM SHOULD BE ABLE TO HANDLE  THOSE SIGNALS TOO. HENCE: DISCRETE FOURIER TRANSFORM - DFT

X_{ k }=\sum _{ n=0 }^{ N-1 }{ x_{ n }\left[ \cos { \left( -\frac { 2\pi nk }{ N } \right) + } j\sin { \left( -\frac { 2\pi nk }{ N } \right) } \right] }
$X_{ k }=\sum _{ n=0 }^{ N-1 }{ x_{ n }\left[ \cos { \left( -\frac { 2\pi nk }{ N } \right) + } j\sin { \left( -\frac { 2\pi nk }{ N } \right) } \right] }$

## DFT vs. FFT

• DFT is very easy to understand, however it takes time to  evaluate
• FFT (Fast Fourier Transform) is an algorithm that can evaluate DFT very fast - given that the vector size is a power of 2

dspguide.com

# things we didn't talk about

## but are very important

REAL LIFE ARE COMPLEX

THERE (SOME) MORE MATH INTO IT

By raziele

# Elementary, My dear Watson. Part 2

SDRI sixth meetup

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