|
|
| (4 intermediate revisions by one other user not shown) |
| Line 1: |
Line 1: |
| ==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 <math> x(t) = x(t+T)\, </math> for <math> T \neq 0 </math>.
| |
|
| |
| The exponential form of the Fourier series is defined as <math> x(t) = \sum_{n=-\infty}^\infty \alpha_n e^{{j2\pi nt}/T} \, </math>
| |
|
| |
| ==Determining the coefficient <math> \alpha_n \,</math> ==
| |
|
| |
| <math> x(t) = \sum_{n=-\infty}^\infty \alpha_n e^{{j2\pi nt}/T} \, </math>
| |
|
| |
| *The definition of the Fourier series
| |
|
| |
| <math> \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</math>
| |
|
| |
| *Integrating both sides for one period. The range of integration is arbitrary, but using <math> \int_{-T/2}^{T/2} </math> scales nicely when extending the Fourier series to a non-periodic function
| |
|
| |
| <math> \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</math>
| |
|
| |
| *Multiply by the complex conjugate
| |
|
| |
| <math> \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</math>
| |
|
| |
| *<math> \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}</math>
| |
|
| |
| ** Using L'Hopitals to evaluate the <math>\frac{T\cdot 0}{0}</math> case. Note that n & m are integers
| |
|
| |
| <math> \alpha_m = \frac{1}{T}\int_{-T/2}^{T/2} x(t) e^{{-j2\pi mt}/T} dt </math>
| |
|
| |
|
| |
|
| |
| ==Linear Time Invariant Systems==
| |
|
| |
| Must meet the following criteria
| |
|
| |
| *Time independance
| |
| *Linearity
| |
| **Superposition (additivity)
| |
| **Scaling (homogeneity)
| |
|
| |
| ==The Dot Product, Complex Conjugates, and Orthogonality==
| |
| [[Image:300px-Scalarproduct.gif |thumb|300px|right]]
| |
|
| |
| Geometrically, the dot product is a scalar projection of a onto b
| |
|
| |
| *<math> \vec a \cdot \vec b = \left | a \right \vert \left | b \right \vert \cos \theta</math>
| |
|
| |
| Arthimetically, multiply like terms and add
| |
|
| |
| *<math> (3,2,1)\cdot(5,6,7)=3\cdot5^*+2\cdot6^*+1\cdot7^*</math>
| |
|
| |
| Lets imagine that we are only have one dimension
| |
|
| |
| *<math> (a+jb)\hat i \cdot (a+jb)\hat i \ne a^2+b^2 </math>
| |
|
| |
| 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
| |
|
| |
| *<math> (a+jb)\hat i \cdot (a-jb)\hat i = a^2+b^2 </math>
| |
|
| |
| To test for orthogonality, take the complex conjugate of one of the vectors and multiply.
| |
|
| |
| *<math> \int_{-\infty}^{\infty} \phi_n (t) \phi_m^* (t) dt = 0</math>
| |
|
| |
| ==Changing Basis Functions==
| |
|
| |
| We'd like to change from <math> \sum_{n=-\infty}^{\infty} \alpha_n e^{j2\pi nt/T} </math> to <math> \sum_{m=0}^{\infty} c_n \cos \left (\frac{2\pi mt}{T}+\Theta_m \right) </math>
| |
|
| |
| <math> 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=0}^{\infty} \alpha_n e^{j2\pi nt/T}</math>
| |
|
| |
| ==Identities==
| |
| <math>e^{j \theta} = \cos \theta + j \sin \theta \, </math> Euler's identity linking rectangular and polar coordinates
| |
|
| |
| <math>\sin x = \frac{e^{jx}-e^{-jx}}{2j} \,</math>
| |
|
| |
| <math>\cos x = \frac{e^{jx}+e^{-jx}}{2} \,</math>
| |
|
| |
| <math> \left \langle \ Bra \mid Ket \ \right \rangle = Ket \cdot Bra </math>
| |
|
| |
| <math> \alpha_{-m} = \alpha^* \,</math>
| |
|
| |
| The dirac delta has an infinite height and an area of 1
| |