Signals and systems/GF Fourier - Revision history

172.16.18.138 at 15:29, 4 November 2006

2006-11-04T15:29:25Z

← Older revision		Revision as of 08:29, 4 November 2006
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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 <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_m \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=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}</math>

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

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

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

	<math> = \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 ) </math>

	~~Changing variables~~
	*<math> c_0 = \alpha_0 \,</math>
	*<math> c_m = 2 \| \alpha_m\| \,</math>
	*<math> \Theta_m = \phi_m \,</math>

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

Fonggr: /* Changing Basis Functions */

2006-10-30T09:02:51Z

Changing Basis Functions

← Older revision		Revision as of 02:02, 30 October 2006
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	==Changing Basis Functions==		==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>		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_m \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=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}</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=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}</math>
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	<math> \alpha_{-m} = \alpha_m^ </math>		<math> \alpha_{-m} = \alpha_m^ </math>
	<math> u + u^ = 2 \Re [u] </math>		<math> u + u^ = 2 \Re [u] </math>
	*<math> \alpha_{m} = \| \alpha_m \|e^{j\~~theta~~ m} \,</math>		*<math> \alpha_{m} = \| \alpha_m \|e^{j\phi m} \,</math>

	<math> = \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\~~theta~~ m}e^{j2\pi mt/T}\right] = \alpha_0 + \sum_{m=1}^{\infty} \|\alpha_m\|2 \Re \left[e^{j(2\pi mt/T+\~~theta~~ m)}\right]= \alpha_0 + \sum_{m=1}^{\infty} \|\alpha_m\|2\cos \left (\frac{2\pi mt}{T} + \Theta_m \right ) </math>		<math> = \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 ) </math>

			Changing variables
			*<math> c_0 = \alpha_0 \,</math>
			*<math> c_m = 2 \| \alpha_m\| \,</math>
			*<math> \Theta_m = \phi_m \,</math>

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

	==Identities==		==Identities==

Fonggr: /* Changing Basis Functions */

2006-10-30T08:55:46Z

Changing Basis Functions

← Older revision		Revision as of 01:55, 30 October 2006
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	*<math> \alpha_{m} = \| \alpha_m \|e^{j\theta m} \,</math>		*<math> \alpha_{m} = \| \alpha_m \|e^{j\theta m} \,</math>

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

	==Identities==		==Identities==

Fonggr: /* Changing Basis Functions */

2006-10-30T08:50:50Z

Changing Basis Functions

← Older revision		Revision as of 01:50, 30 October 2006
Line 66:		Line 66:
	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>		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=1}^{\infty} \alpha_n e^{j2\pi nt/T}~~</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=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}</math>

	~~<math>~~ = \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}</math>

	<math> = \alpha_0 + \sum_{m=1}^{\infty} \left (\alpha_m e^{j2\pi mt/T} + \alpha_{-m} e^{-j2\pi mt/T}\right) </math>		<math> = \alpha_0 + \sum_{m=1}^{\infty} \left (\alpha_m e^{j2\pi mt/T} + \alpha_{-m} e^{-j2\pi mt/T}\right) </math>
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	<math> u + u^ = 2 \Re [u] </math>		<math> u + u^ = 2 \Re [u] </math>
	*<math> \alpha_{m} = \| \alpha_m \|e^{j\theta m} \,</math>		*<math> \alpha_{m} = \| \alpha_m \|e^{j\theta m} \,</math>

			<math> = \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\theta m}e^{j2\pi mt/T}\right] = \alpha_0 + \sum_{m=1}^{\infty} \|\alpha_m\|2 \Re \left[e^{j\theta m}e^{j2\pi mt/T}\right]</math>

	==Identities==		==Identities==

Fonggr: /* Changing Basis Functions */

2006-10-30T08:40:38Z

Changing Basis Functions

← Older revision		Revision as of 01:40, 30 October 2006
Line 66:		Line 66:
	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>		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>		<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=1}^{\infty} \alpha_n e^{j2\pi nt/T}</math>

			<math> = \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}</math>

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

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

			<math> \alpha_{-m} = \alpha_m^ </math>
			<math> u + u^ = 2 \Re [u] </math>
			*<math> \alpha_{m} = \| \alpha_m \|e^{j\theta m} \,</math>

	==Identities==		==Identities==

Fonggr: /* Changing Basis Functions */

2006-10-30T08:18:22Z

Changing Basis Functions

← Older revision		Revision as of 01:18, 30 October 2006
Line 66:		Line 66:
	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>		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} </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==		==Identities==

Fonggr: /* Changing Basis Functions */

2006-10-30T08:14:17Z

Changing Basis Functions

← Older revision		Revision as of 01:14, 30 October 2006
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	==Changing Basis Functions==		==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} </math>

	==Identities==		==Identities==

Fonggr: /* Identities */

2006-10-30T07:57:29Z

Identities

← Older revision		Revision as of 00:57, 30 October 2006
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	==Identities==		==Identities==
	<math>e^{j \theta} = \cos \theta + j \sin \theta \, </math>		<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>\sin x = \frac{e^{jx}-e^{-jx}}{2j} \,</math>

Fonggr: /* The Dot Product, Complex Conjugates, and Orthogonality */

2006-10-30T06:36:16Z

The Dot Product, Complex Conjugates, and Orthogonality

← Older revision		Revision as of 23:36, 29 October 2006
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	*<math> (a+jb)\hat i \cdot (a-jb)\hat i = a^2+b^2 </math>		*<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==		==Changing Basis Functions==

Fonggr: /* Identities */

2006-10-30T06:35:04Z

Identities

← Older revision		Revision as of 23:35, 29 October 2006
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	<math> \alpha_{-m} = \alpha^* \,</math>		<math> \alpha_{-m} = \alpha^* \,</math>

	~~<math> \int_{-\infty}^{\infty} \phi_n (t) \phi_m^* (t) dt = 0</math> Implies orthogonality~~

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