ELEMENTARY,
MY DEAR WATSON
PART II
AKA: SDRI seventh meetup
SEPTEMBER 2016
![](https://www.everplans.com/sites/default/files/blog-images/meetup-logo-750.jpg)
Software Defined Radio Israel
![](https://s3.amazonaws.com/media-p.slid.es/uploads/150163/images/2375048/elementary-my-dear-watson.png)
WHAT IS
THE CONNECTION BETWEEN
ALICE
AND
OPTIMUS PRIME?
![](https://s3.amazonaws.com/media-p.slid.es/uploads/150163/images/2375038/fb1f166f1fb35111db0ff730cdd6c263.jpg)
THE FANTASTIC STORY
OF
COMPLEX NUMBERS
NEXT TIME...
FOURIER
AND
OTHER TRANSFORMATIONS
![](http://vignette1.wikia.nocookie.net/deathbattlefanon/images/9/9f/Optimus_Prime_DOTM.png/revision/latest?cb=20160504232619)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/150163/images/3082595/Screenshot_from_2016-10-05_18-25-13.png)
![](http://s2.quickmeme.com/img/55/55904b89806c065ebfd009f10bf52fdffc562d68b8f368be1a485835f15cf7c7.jpg)
IT ALL BEGINS WITH SERIES'
\sum _{ n=0 }^{ \infty }{ { a }_{ n }={ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+{ a }_{ 4 }... }
∑n=0∞an=a1+a2+a3+a4...
\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) +... }
∑n=0∞fn(x)=f1(x)+f2(x)+f3(x)+f4(x)+...
OUR MOTIVATION:
DESCRIBE A FUNCTION WITH...OTHER FUNCTIONS
![](https://s3.amazonaws.com/media-p.slid.es/uploads/150163/images/2957914/Photo_30.8.2016__21_58_15.jpg)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/150163/images/2957917/Photo_30.8.2016__22_00_11.jpg)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/150163/images/2957918/Photo_30.8.2016__21_58_21.jpg)
AND
BECOME...
(AND THE OTHER WAY AROUND)
FOR
EXAMPLE:
all kinds of series'
- Geometric series
- Taylor series
- Laurent series
And more...
![](https://s3.amazonaws.com/media-p.slid.es/uploads/150163/images/2961046/daum_equation_1472669229395.png)
\sum_{n=0}^\infty \frac{ \, x^n}{n!}
∑n=0∞n!xn
\sum _{ n=-\infty }^{ \infty }{ { { a }_{ n }x }^{ n } }
∑n=−∞∞anxn
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] }
sN(x)=2a0+∑n=1N[ancos(P2πnx)+bnsin(P2πnx)]
A BASIC EXAMPLE:
{ s }_{ }\left( x \right) =1+3\cos { \left( x \right) }
s(x)=1+3cos(x)
{ a }_{ 0 }=2,\quad { a }_{ 1 }=3,\quad { a }_{ n>1 },b_{ n>0 }=0
a0=2,a1=3,an>1,bn>0=0
P(period)=2\pi
P(period)=2π
HOW DOES A SQUARE WAVE LOOKS LIKE?
![](https://upload.wikimedia.org/wikipedia/commons/0/0f/SquareWaveFourierArrows%2Crotated.gif)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Fourier_Series.svg/316px-Fourier_Series.svg.png)
![](http://fourier.eng.hmc.edu/e101/lectures/Fourier_Expansion_Example.gif)
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 }
ℑ{h(x)}(f)=∫−∞∞h(x)e−2πjxfdx
j=\sqrt { -1 }
j=√−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] }
Xk=∑n=0N−1xn[cos(−N2πnk)+jsin(−N2πnk)]
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
![](http://www.dspguide.com/graphics/F_12_2.gif)
dspguide.com
common transforms
![](https://s3.amazonaws.com/media-p.slid.es/uploads/150163/images/2961575/Photo_1.9.2016__0_28_15.jpg)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/150163/images/2961577/Photo_1.9.2016__0_26_23.jpg)
things we didn't talk about
but are very important
REAL LIFE ARE COMPLEX
THERE (SOME) MORE MATH INTO IT
Elementary, My dear Watson. Part 2
By raziele
Elementary, My dear Watson. Part 2
SDRI sixth meetup
- 1,124