fourier, convolution and sampling

when can you use a fourier series

used when a function is periodic aka: repeats and has a constant interval of \Delta x

what are harmonic waves?

waves that have a whole number frequency multiple of each other

this also results in the wavelength being divided by that positive whole number

fourier series equation

F(x)=\frac{a_o}{2} \Sigma a_n cos(n\Omega x)+b_n sin(n\Omega x) where \Omega=\frac{2\pi}{\Delta x}

a_0=\frac{1}{\pi}\int_{-\pi}^{\pi}\!f(x)\,dx

a_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}\!f(x)\cos(n\Omega x)\,dx

b_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}\!f(x)\sin(n\Omega x)\,dx

what are the properties of a sine graph

at 0 sine = 0

when n is an interger of PI the value of sine = 0

the graph is NOT SYMMETRIC → IT IS ODD → imaginary

what are the properties of a cos graph

at x = 0 cos = 1

the graph is symmetric and therefor and even function → real function

when x is an even number the value of cos is either 1 or -1

how does increasing the number of terms effect the image in a fourier series

increasing the number of turns makes the image “shaper” making it resemble the actual function more closely

what is sin(nx) in exponential

\frac{(e^{inx}-e^{-inx})}{2i}

what is cos(nx) in exponential

\frac{e^{inx} + e^{-inx}}{2}

general fourier series equation:

→ when do you use it,

→ how is it produced

→ used when the period of the function is anything except of 2\pi

→ produced by substituting the exponential versions of sine and cos then grouping the like exponentials to each other

c_{n}=\frac{1}{P}\int_{-\frac{P}{2}}^{\frac{P}{2}}\!f\left(x\right)e^{\left(-inw_0x\right)}\,dx where w_0 = \frac{2\pi}{period}

f(T)=\Sigma c_{n}e_{}^{inw_0t}

fourier tranform

→ when is it used

F\left(u\right)=\int_0^{\infty}\!f\left(x\right)e^{-2\pi iux}\,dx

→ used for non-periodic functions

inverse fourier tranform

f(x)=\int_0^{\infty}\!F\left(u\right)e^{2\pi iux}\,dx

how is the width of the real funciton related to that of the fourier

→ what happens when you make the real pulse wider or larger

widthofreal=\frac{constant}{widthoffourier}

→ wider pulse = narrower fourier and less frequencies

→ narrower pulse = wider fourier and more frequencies

integral of a dirac delta function

\int_0^{\infty}\!\delta\left(x\right)\,dx=1

this is because the area under the dirac delta is always 1

integral of a function with a dirac delta function

\int_0^{\infty}\!f\left(X\right)\cdot\delta\left(x-a\right)\,dx=f\left(a\right)

picks out the value of the function when x = a

\int_0^{\infty}\!f\left(X\right)\cdot\delta\left(b\left(x-a\right)\right)\,dx=?

\int_0^{\infty}\!f\left(X\right)\cdot\delta\left(b\left(x-a\right)\right)\,dx\to\int_0^{\infty}\!f\left(X\right)\cdot\frac{1}{b}\delta\left(x-a\right)\,dx=\frac{1}{b}\int_0^{\infty}\!f\left(X\right)\cdot\delta\left(x-a\right)\,dx=\frac{1}{b}f\left(a\right)

what is the fourier of a dirac function

\int_0^{\infty}\!\delta\left(x-a\right)e^{-2\pi iux}\,dx=e^{-2\pi iua}