Fourier Series

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Fourier series

The Fourier series is used to analyze arbitrary periodic functions by showing them as a composite of sines and cosines.

A function is considered periodic if  x(t) = x(t+T)\, for  T \neq 0 .

The exponential form of the Fourier series is defined as  x(t) = \sum_{n=-\infty}^\infty \alpha_n e^{{j2\pi nt}/T} \,

Determining the coefficient  \alpha_n \,

 x(t) = \sum_{n=-\infty}^\infty \alpha_n e^{{j2\pi nt}/T} \,

  • The definition of the Fourier series

 \int_{-T/2}^{T/2} x(t)\, dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2} e^{{j2\pi nt}/T} dt

  • Integrating both sides for one period. The range of integration is arbitrary, but using  \int_{-T/2}^{T/2} instead of  \int_{0}^{T} scales nicely when extending the Fourier series to a non-periodic function

 \int_{-T/2}^{T/2} x(t) e^{{-j2\pi mt}/T} dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2} e^{{j2\pi nt}/T}e^{{-j2\pi mt}/T} dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2} e^{{j2\pi (n-m)t}/T} dt

  • Multiply by the complex conjugate

 \int_{-T/2}^{T/2} x(t) e^{{-j2\pi mt}/T} dt = \sum_{n=-\infty}^\infty \alpha_n \frac{Te^{{j2\pi (n-m)t}/T}}{{j2\pi (n-m)}} \bigg|_{-T/2}^{T/2} = \sum_{n=-\infty}^\infty \alpha_n T\delta_{n,m} = T\alpha_m

  •  \frac{Te^{{j2\pi (n-m)t}/T}}{{j2\pi (n-m)}} \bigg|_{-T/2}^{T/2} = T\frac{e^{j\pi(n-m)}-e^{-j\pi(n-m)}}{j2\pi(n-m)} = T \frac{\sin\pi(n-m)}{\pi(n-m)} =  \begin{Bmatrix} T, n=m \\ 0, n\ne m \end{Bmatrix} = T\delta_{n,m}
    • Using L'Hopitals to evaluate the \frac{T\cdot 0}{0} case. Note that n & m are integers

 \alpha_m = \frac{1}{T}\int_{-T/2}^{T/2} x(t) e^{{-j2\pi mt}/T} dt

Linear Time Invariant Systems

Must meet the following criteria

  • Time independance
  • Linearity
    • Superposition (additivity)
    • Scaling (homogeneity)

The Dot Product, Complex Conjugates, and Orthogonality

File:300px-Scalarproduct.gif

Geometrically, the dot product is a scalar projection of a onto b

  •  \vec a \cdot \vec b = \left | a \right \vert \left | b \right \vert \cos \theta

Arthimetically, multiply like terms and add

  •  (3,2,1)\cdot(5,6,7)=3\cdot5^*+2\cdot6^*+1\cdot7^*

Lets imagine that we are only have one dimension

  •  (a+jb)\hat i \cdot (a+jb)\hat i \ne a^2+b^2

In order to get the real parts and imaginary parts to multiply as like terms, we need to take the complex conjugate of one of the terms

  •  (a+jb)\hat i \cdot (a-jb)\hat i = a^2+b^2

To test for orthogonality, take the complex conjugate of one of the vectors and multiply.

  •  \int_{-\infty}^{\infty} \phi_n (t) \phi_m^* (t) dt = 0

Changing Basis Functions

We'd like to change from  \sum_{n=-\infty}^{\infty} \alpha_n e^{j2\pi nt/T} to  \sum_{m=0}^{\infty} c_m \cos \left (\frac{2\pi mt}{T}+\Theta_m \right)

 x(t) = \sum_{n=-\infty}^{\infty} \alpha_n e^{j2\pi nt/T} = \underbrace{ \sum_{n=-\infty}^{-1} \alpha_n e^{j2\pi nt/T} }_{n'=-n} + \; \alpha_0 \; + \sum_{n=1}^{\infty} \alpha_n e^{j2\pi nt/T} = \underbrace{\sum_{n'=1}^{\infty} \alpha_n e^{j2\pi nt/T}}_{m=n'} + \; \alpha_0 \; + \underbrace{\sum_{n=1}^{\infty} \alpha_n e^{j2\pi nt/T}}_{m=n}

 = \alpha_0 + \sum_{m=1}^{\infty} \left (\alpha_m e^{j2\pi mt/T} + \alpha_{-m} e^{-j2\pi mt/T}\right)

If we assume  x(t) \in \Re \ \forall \ m, then to make the imaginary parts cancel out

  •  \alpha_{-m} = \alpha_m^*
  •  u + u^* = 2 \Re [u]
  •  \alpha_{m} = | \alpha_m |e^{j\phi m} \,

 = \alpha_0 + \sum_{m=1}^{\infty} 2 \Re \left[\alpha_m e^{j2\pi mt/T}\right] = \alpha_0 + \sum_{m=1}^{\infty} 2 \Re \left[|\alpha_m| e^{j\phi m}e^{j2\pi mt/T}\right] = \alpha_0 + \sum_{m=1}^{\infty} |\alpha_m|2 \Re \left[e^{j(2\pi mt/T+\phi m)}\right]= \alpha_0 + \sum_{m=1}^{\infty} |\alpha_m|2\cos \left (\frac{2\pi mt}{T} + \phi_m \right )

Changing variables

  •  c_0 = \alpha_0 \,
  •  c_m = 2 | \alpha_m| \,
  •  \Theta_m = \phi_m \,

 = \sum_{m=0}^{\infty} c_m \cos \left (\frac{2\pi mt}{T}+\Theta_m \right)

Identities

e^{j \theta} = \cos \theta + j \sin  \theta \, Euler's identity linking rectangular and polar coordinates

\sin x = \frac{e^{jx}-e^{-jx}}{2j} \,

\cos x = \frac{e^{jx}+e^{-jx}}{2} \,

 <u|v> = \int_{-\infty}^\infty u^*(x) v(x) dx

 \alpha_{-m} = \alpha^* \,

The dirac delta has an infinite height and an area of 1