Kurt's Assignment - Revision history

Kurt at 07:11, 2 November 2010

2010-11-02T07:11:01Z

← Older revision		Revision as of 00:11, 2 November 2010
Line 110:		Line 110:
	*finish the triangle wave derivation		*finish the triangle wave derivation
	*start sawtooth wave derivation		*start sawtooth wave derivation
			*put pictures in
	*implement triangle wave Fourier Series in OCTAVE		*implement triangle wave Fourier Series in OCTAVE
	*implement sawtooth wave Fourier Series in OCTAVE		*implement sawtooth wave Fourier Series in OCTAVE

Kurt at 07:08, 2 November 2010

2010-11-02T07:08:41Z

← Older revision		Revision as of 00:08, 2 November 2010
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	&= \frac{8H}{T^2}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}t\,\cos\left(n\omega_0t\right) \,dt + \frac{4H}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{2}{T}t+1\right)\cos\left(n\omega_0t\right)\,dt\\		&= \frac{8H}{T^2}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}t\,\cos\left(n\omega_0t\right) \,dt + \frac{4H}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{2}{T}t+1\right)\cos\left(n\omega_0t\right)\,dt\\
	&= \frac{8H}{T^2}\left[\frac{t}{n\omega_0}\sin(n\omega_0 t)+\frac{1}{n^2\omega_0 ^2}\cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{1}{4}T} + \frac{4H}{T}\left[\frac{-\frac{2}{T}t+1}{n\omega_0} \sin(n\omega_0 t) - \frac{2}{Tn^2\omega_0^2} \cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{3}{4}T}\\		&= \frac{8H}{T^2}\left[\frac{t}{n\omega_0}\sin(n\omega_0 t)+\frac{1}{n^2\omega_0 ^2}\cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{1}{4}T} + \frac{4H}{T}\left[\frac{-\frac{2}{T}t+1}{n\omega_0} \sin(n\omega_0 t) - \frac{2}{Tn^2\omega_0^2} \cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{3}{4}T}\\
	&= \frac{8H}{T^2}\left[\left[\frac{\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{1}{4}T\right)\right] - \left[\frac{-\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)\right]\right] + \frac{4H}{T}\left[\left[\frac{-\frac{2}{T}\left(\frac{3}{4}T\right)+1}{n\frac{2\pi}{T}} \sin\left(n\frac{2\pi}{T}\frac{3}{4}T\right) - \frac{2}{Tn^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{3}{4}T\right)\right] - \left[\frac{-\frac{2}{T}\left(-\frac{1}{4}T\right)+1}{n\frac{2\pi}{T}} \sin\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right) - \frac{2}{Tn^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right)\right]\right]\\		&= \frac{8H}{T^2}\left[\left[\frac{\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{1}{4}T\right)\right] - \left[\frac{-\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right)\right]\right] + \frac{4H}{T}\left[\left[\frac{-\frac{2}{T}\left(\frac{3}{4}T\right)+1}{n\frac{2\pi}{T}} \sin\left(n\frac{2\pi}{T}\frac{3}{4}T\right) - \frac{2}{Tn^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{3}{4}T\right)\right] - \left[\frac{-\frac{2}{T}\left(-\frac{1}{4}T\right)+1}{n\frac{2\pi}{T}} \sin\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right) - \frac{2}{Tn^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right)\right]\right]\\
	&= \frac{8H}{T^2}\left[\left[\frac{T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)+\frac{T^2}{n^2 4\pi^2}\cos\left(\frac{1}{2}n\pi\right)\right] - \left[-\frac{T^2}{8\pi n}\sin\left(-\frac{1}{2}n\pi\right)+\frac{T^2}{n^2 4 \pi^2}\cos\left(-\frac{1}{2}n\pi\right)\right]\right] + \frac{4H}{T} \left[\left[-\frac{T}{4\pi n}\sin\left(\frac{3}{2}n\pi\right)-\frac{T}{2\pi^2 n^2}\cos\left(\frac{3}{2}n\pi\right)\right] - \left[\frac{3T}{4\pi n}\sin\left(-\frac{1}{2}n \pi\right)-\frac{T}{2\pi^2 n^2}\cos\left(-\frac{1}{2}n\pi\right)\right]\right]\\		&= \frac{8H}{T^2}\left[\left[\frac{T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)+\frac{T^2}{n^2 4\pi^2}\cos\left(\frac{1}{2}n\pi\right)\right] - \left[-\frac{T^2}{8\pi n}\sin\left(-\frac{1}{2}n\pi\right)+\frac{T^2}{n^2 4 \pi^2}\cos\left(-\frac{1}{2}n\pi\right)\right]\right] + \frac{4H}{T} \left[\left[-\frac{T}{4\pi n}\sin\left(\frac{3}{2}n\pi\right)-\frac{T}{2\pi^2 n^2}\cos\left(\frac{3}{2}n\pi\right)\right] - \left[\frac{3T}{4\pi n}\sin\left(-\frac{1}{2}n \pi\right)-\frac{T}{2\pi^2 n^2}\cos\left(-\frac{1}{2}n\pi\right)\right]\right]\\
	&= \frac{8H}{T^2}\left[\frac{T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)-\frac{T^2}{8\pi n}\sin\left(-\frac{1}{2}n\pi\right)+\underbrace{\frac{T^2}{n^2 4\pi^2}\cos\left(\frac{1}{2}n\pi\right)-\frac{T^2}{n^2 4\pi^2}\cos\left(-\frac{1}{2}n\pi\right)}_\text{0}\right] + \frac{4H}{T}\left[-\frac{T}{4\pi n}\sin\left(\frac{3}{2}n\pi\right) - \frac{3T}{4\pi n}\sin\left(-\frac{1}{2}n\pi\right) - \underbrace{\frac{T}{2\pi^2 n^2} \cos\left(\frac{3}{2}n\pi\right) + \frac{T}{2\pi^2 n^2} \cos\left(-\frac{1}{2}n\pi\right)}_\text{0}\right]\\		&= \frac{8H}{T^2}\left[\frac{T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)-\frac{T^2}{8\pi n}\sin\left(-\frac{1}{2}n\pi\right)+\underbrace{\frac{T^2}{n^2 4\pi^2}\cos\left(\frac{1}{2}n\pi\right)-\frac{T^2}{n^2 4\pi^2}\cos\left(-\frac{1}{2}n\pi\right)}_\text{0}\right] + \frac{4H}{T}\left[-\frac{T}{4\pi n}\sin\left(\frac{3}{2}n\pi\right) - \frac{3T}{4\pi n}\sin\left(-\frac{1}{2}n\pi\right) - \underbrace{\frac{T}{2\pi^2 n^2} \cos\left(\frac{3}{2}n\pi\right) + \frac{T}{2\pi^2 n^2} \cos\left(-\frac{1}{2}n\pi\right)}_\text{0}\right]\\

Kurt at 07:06, 2 November 2010

2010-11-02T07:06:21Z

← Older revision		Revision as of 00:06, 2 November 2010
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	&= \frac{8H}{T^2}\left[\frac{t}{n\omega_0}\sin(n\omega_0 t)+\frac{1}{n^2\omega_0 ^2}\cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{1}{4}T} + \frac{4H}{T}\left[\frac{-\frac{2}{T}t+1}{n\omega_0} \sin(n\omega_0 t) - \frac{2}{Tn^2\omega_0^2} \cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{3}{4}T}\\		&= \frac{8H}{T^2}\left[\frac{t}{n\omega_0}\sin(n\omega_0 t)+\frac{1}{n^2\omega_0 ^2}\cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{1}{4}T} + \frac{4H}{T}\left[\frac{-\frac{2}{T}t+1}{n\omega_0} \sin(n\omega_0 t) - \frac{2}{Tn^2\omega_0^2} \cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{3}{4}T}\\
	&= \frac{8H}{T^2}\left[\left[\frac{\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{1}{4}T\right)\right] - \left[\frac{-\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)\right]\right] + \frac{4H}{T}\left[\left[\frac{-\frac{2}{T}\left(\frac{3}{4}T\right)+1}{n\frac{2\pi}{T}} \sin\left(n\frac{2\pi}{T}\frac{3}{4}T\right) - \frac{2}{Tn^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{3}{4}T\right)\right] - \left[\frac{-\frac{2}{T}\left(-\frac{1}{4}T\right)+1}{n\frac{2\pi}{T}} \sin\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right) - \frac{2}{Tn^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right)\right]\right]\\		&= \frac{8H}{T^2}\left[\left[\frac{\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{1}{4}T\right)\right] - \left[\frac{-\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)\right]\right] + \frac{4H}{T}\left[\left[\frac{-\frac{2}{T}\left(\frac{3}{4}T\right)+1}{n\frac{2\pi}{T}} \sin\left(n\frac{2\pi}{T}\frac{3}{4}T\right) - \frac{2}{Tn^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{3}{4}T\right)\right] - \left[\frac{-\frac{2}{T}\left(-\frac{1}{4}T\right)+1}{n\frac{2\pi}{T}} \sin\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right) - \frac{2}{Tn^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right)\right]\right]\\
	&= \frac{8H}{T^2}\left[\left[\frac{T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)+\frac{T^2}{n^2 4\pi^2}\cos\left(\frac{1}{2}n\pi\right)\right] - \left[\frac{T^2}{8\pi n}\sin\left(-\frac{1}{2}n\pi\right)+\frac{T^2}{n^2 4 \pi^2}\cos\left(-\frac{1}{2}n\pi\right)\right]\right] + \frac{4H}{T} \left[\left[-\frac{T}{4\pi n}\sin\left(\frac{3}{2}n\pi\right)-\frac{T}{2\pi^2 n^2}\cos\left(\frac{3}{2}n\pi\right)\right] - \left[\frac{3T}{4\pi n}\sin\left(-\frac{1}{2}n \pi\right)-\frac{T}{2\pi^2 n^2}\cos\left(-\frac{1}{2}n\pi\right)\right]\right]\\		&= \frac{8H}{T^2}\left[\left[\frac{T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)+\frac{T^2}{n^2 4\pi^2}\cos\left(\frac{1}{2}n\pi\right)\right] - \left[-\frac{T^2}{8\pi n}\sin\left(-\frac{1}{2}n\pi\right)+\frac{T^2}{n^2 4 \pi^2}\cos\left(-\frac{1}{2}n\pi\right)\right]\right] + \frac{4H}{T} \left[\left[-\frac{T}{4\pi n}\sin\left(\frac{3}{2}n\pi\right)-\frac{T}{2\pi^2 n^2}\cos\left(\frac{3}{2}n\pi\right)\right] - \left[\frac{3T}{4\pi n}\sin\left(-\frac{1}{2}n \pi\right)-\frac{T}{2\pi^2 n^2}\cos\left(-\frac{1}{2}n\pi\right)\right]\right]\\
	&= \frac{8H}{T^2}\left[\frac{T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)-\frac{T^2}{8\pi n}\sin\left(-\frac{1}{2}n\pi\right)+\underbrace{\frac{T^2}{n^2 4\pi^2}\cos\left(\frac{1}{2}n\pi\right)-\frac{T^2}{n^2 4\pi^2}\cos\left(-\frac{1}{2}n\pi\right)}_\text{0}\right] + \frac{4H}{T}\left[-\frac{T}{4\pi n}\sin\left(\frac{3}{2}n\pi\right) - \frac{3T}{4\pi n}\sin\left(-\frac{1}{2}n\pi\right) - \underbrace{\frac{T}{2\pi^2 n^2} \cos\left(\frac{3}{2}n\pi\right) + \frac{T}{2\pi^2 n^2} \cos\left(-\frac{1}{2}n\pi\right)}_\text{0}\right]\\		&= \frac{8H}{T^2}\left[\frac{T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)-\frac{T^2}{8\pi n}\sin\left(-\frac{1}{2}n\pi\right)+\underbrace{\frac{T^2}{n^2 4\pi^2}\cos\left(\frac{1}{2}n\pi\right)-\frac{T^2}{n^2 4\pi^2}\cos\left(-\frac{1}{2}n\pi\right)}_\text{0}\right] + \frac{4H}{T}\left[-\frac{T}{4\pi n}\sin\left(\frac{3}{2}n\pi\right) - \frac{3T}{4\pi n}\sin\left(-\frac{1}{2}n\pi\right) - \underbrace{\frac{T}{2\pi^2 n^2} \cos\left(\frac{3}{2}n\pi\right) + \frac{T}{2\pi^2 n^2} \cos\left(-\frac{1}{2}n\pi\right)}_\text{0}\right]\\
	&= \frac{8H}{T^2}\left[\frac{2T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)\right] + \text{integrate other side}\\		&= \frac{8H}{T^2}\left[\frac{2T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)\right] + \text{integrate other side}\\

Kurt at 06:53, 2 November 2010

2010-11-02T06:53:24Z

← Older revision		Revision as of 23:53, 1 November 2010
Line 105:		Line 105:
	legend(num2str(M) ' terms');		legend(num2str(M) ' terms');
	print("squarewave.png","-dpng") % Prints the plot to a png file called squarewave.png		print("squarewave.png","-dpng") % Prints the plot to a png file called squarewave.png


			TODO:
			*finish the triangle wave derivation
			*start sawtooth wave derivation
			*implement triangle wave Fourier Series in OCTAVE
			*implement sawtooth wave Fourier Series in OCTAVE

Kurt at 06:50, 2 November 2010

2010-11-02T06:50:43Z

← Older revision		Revision as of 23:50, 1 November 2010
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	&= 0		&= 0
	\end{align}</math>		\end{align}</math>



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	a_n &= \frac{2}{T}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}\left(\frac{4H}{T}t\right)\cos\left(n\omega_0t\right) \,dt + \frac{2}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{4H}{T}t+2H\right)\cos\left(n\omega_0t\right)\,dt\\		a_n &= \frac{2}{T}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}\left(\frac{4H}{T}t\right)\cos\left(n\omega_0t\right) \,dt + \frac{2}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{4H}{T}t+2H\right)\cos\left(n\omega_0t\right)\,dt\\
	&= \frac{8H}{T^2}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}t\,\cos\left(n\omega_0t\right) \,dt + \frac{4H}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{2}{T}t+1\right)\cos\left(n\omega_0t\right)\,dt\\		&= \frac{8H}{T^2}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}t\,\cos\left(n\omega_0t\right) \,dt + \frac{4H}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{2}{T}t+1\right)\cos\left(n\omega_0t\right)\,dt\\
	&= \frac{8H}{T^2}\left[\frac{t}{n\omega_0}\sin(n\omega_0 t)+\frac{1}{n^2\omega_0 ^2}\cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{1}{4}T} + \~~text~~{~~integrate other side~~}\\		&= \frac{8H}{T^2}\left[\frac{t}{n\omega_0}\sin(n\omega_0 t)+\frac{1}{n^2\omega_0 ^2}\cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{1}{4}T} + \frac{4H}{T}\left[\frac{-\frac{2}{T}t+1}{n\omega_0} \sin(n\omega_0 t) - \frac{2}{Tn^2\omega_0^2} \cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{3}{4}T}\\
	&= \frac{8H}{T^2}\left[\left[\frac{\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{1}{4}T\right)\right] - \left[\frac{-\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)\right]\right]		&= \frac{8H}{T^2}\left[\left[\frac{\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{1}{4}T\right)\right] - \left[\frac{-\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)\right]\right] + \frac{4H}{T}\left[\left[\frac{-\frac{2}{T}\left(\frac{3}{4}T\right)+1}{n\frac{2\pi}{T}} \sin\left(n\frac{2\pi}{T}\frac{3}{4}T\right) - \frac{2}{Tn^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{3}{4}T\right)\right] - \left[\frac{-\frac{2}{T}\left(-\frac{1}{4}T\right)+1}{n\frac{2\pi}{T}} \sin\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right) - \frac{2}{Tn^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\left(-\frac{1}{4}T\right)\right)\right]\right]\\
	&=		&= \frac{8H}{T^2}\left[\left[\frac{T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)+\frac{T^2}{n^2 4\pi^2}\cos\left(\frac{1}{2}n\pi\right)\right] - \left[\frac{T^2}{8\pi n}\sin\left(-\frac{1}{2}n\pi\right)+\frac{T^2}{n^2 4 \pi^2}\cos\left(-\frac{1}{2}n\pi\right)\right]\right] + \frac{4H}{T} \left[\left[-\frac{T}{4\pi n}\sin\left(\frac{3}{2}n\pi\right)-\frac{T}{2\pi^2 n^2}\cos\left(\frac{3}{2}n\pi\right)\right] - \left[\frac{3T}{4\pi n}\sin\left(-\frac{1}{2}n \pi\right)-\frac{T}{2\pi^2 n^2}\cos\left(-\frac{1}{2}n\pi\right)\right]\right]\\
			&= \frac{8H}{T^2}\left[\frac{T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)-\frac{T^2}{8\pi n}\sin\left(-\frac{1}{2}n\pi\right)+\underbrace{\frac{T^2}{n^2 4\pi^2}\cos\left(\frac{1}{2}n\pi\right)-\frac{T^2}{n^2 4\pi^2}\cos\left(-\frac{1}{2}n\pi\right)}_\text{0}\right] + \frac{4H}{T}\left[-\frac{T}{4\pi n}\sin\left(\frac{3}{2}n\pi\right) - \frac{3T}{4\pi n}\sin\left(-\frac{1}{2}n\pi\right) - \underbrace{\frac{T}{2\pi^2 n^2} \cos\left(\frac{3}{2}n\pi\right) + \frac{T}{2\pi^2 n^2} \cos\left(-\frac{1}{2}n\pi\right)}_\text{0}\right]\\
			&= \frac{8H}{T^2}\left[\frac{2T^2}{8\pi n}\sin\left(\frac{1}{2}n\pi\right)\right] + \text{integrate other side}\\
			&= \frac{2H}{\pi n}\sin\left(\frac{1}{2}n\pi\right) + \text{integrate other side}
			\end{align}</math>

			<math>\begin{align}
			test function &=
	\end{align}</math>		\end{align}</math>

Kurt at 01:59, 2 November 2010

2010-11-02T01:59:17Z

← Older revision		Revision as of 18:59, 1 November 2010
Line 66:		Line 66:

	<math>\begin{align}		<math>\begin{align}
	a_n &= \frac{1}{T}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}\left(\frac{4H}{T}t \right)\,dt + \frac{1}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{4H}{T}t+2H\right) \,dt\\		a_n &= \frac{2}{T}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}\left(\frac{4H}{T}t\right)\cos\left(n\omega_0t\right) \,dt + \frac{2}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{4H}{T}t+2H\right)\cos\left(n\omega_0t\right)\,dt\\
			&= \frac{8H}{T^2}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}t\,\cos\left(n\omega_0t\right) \,dt + \frac{4H}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{2}{T}t+1\right)\cos\left(n\omega_0t\right)\,dt\\
			&= \frac{8H}{T^2}\left[\frac{t}{n\omega_0}\sin(n\omega_0 t)+\frac{1}{n^2\omega_0 ^2}\cos(n\omega_0 t)\right]_{-\frac{1}{4}T}^{\frac{1}{4}T} + \text{integrate other side}\\
			&= \frac{8H}{T^2}\left[\left[\frac{\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}\frac{1}{4}T\right)\right] - \left[\frac{-\frac{1}{4}T}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)+\frac{1}{n^2\frac{4\pi^2}{T^2}}\cos\left(n\frac{2\pi}{T}-\frac{1}{4}T\right)\right]\right]
			&=
	\end{align}</math>		\end{align}</math>

Line 81:		Line 84:
	sum1=0; %Initialize the sum to <br>		sum1=0; %Initialize the sum to <br>
	%----------FOURIER SERIES----------%		%----------FOURIER SERIES----------%
	for m=1:1:M, %For m=1, increment by 1 until you get to M		for m=1:1:M, %For m=1, increment by 1 until you get to M
	if(m!=0)		if(m!=0)
	sum1 = sum1 + ((2/(pim))-(2/(pim))cos(mpi))sin(m2pi/Tt);		sum1 = sum1 + ((2/(pim))-(2/(pim))cos(mpi))sin(m2pi/Tt);
	end		end
	end<br>		end<br>
	%---------------PLOT---------------%		%---------------PLOT---------------%

Kurt at 01:29, 2 November 2010

2010-11-02T01:29:18Z

← Older revision		Revision as of 18:29, 1 November 2010
Line 55:		Line 55:

	<math>\begin{align}		<math>\begin{align}
	a_0 &= \frac{1}{T}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}\frac{4H}{T}t \,dt +		a_0 &= \frac{1}{T}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}\left(\frac{4H}{T}t \right)\,dt + \frac{1}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{4H}{T}t+2H\right) \,dt\\
			&= \frac{1}{T}\left[\frac{1}{2}\frac{4H}{T}t^2\right]_{-\frac{1}{4}T}^{\frac{1}{4}T} + \frac{1}{T}\left[-\frac{1}{2}\frac{4H}{T}t^2+2Ht\right]_{\frac{1}{4}T}^{\frac{3}{4}T}\\
			&= \frac{1}{T}\left[\frac{2H}{T}\left(\frac{1}{4}T\right)^2-\frac{2H}{T}\left(-\frac{1}{4}T\right)^2\right] + \frac{1}{T}\left[-\frac{2H}{T}\left(\frac{3}{4}T\right)^2+2H\left(\frac{3}{4}T\right)-\left(-\frac{2H}{T}\left(\frac{1}{4}T\right)^2+2H\left(\frac{1}{4}T\right)\right)\right]\\
			&= \frac{1}{T}\left[\underbrace{\frac{2H}{T}\frac{1}{16}T^2-\frac{2H}{T}\frac{1}{16}T^2}_\text{0}\right] + \frac{1}{T}\left[-\frac{2H}{T}\frac{9}{16}T^2+\frac{3}{2}HT+\frac{2H}{T}\frac{1}{16}T^2-\frac{1}{2}HT\right]\\
			&= 0 + \frac{1}{T}\left[-\frac{18H}{16T}T^2+\frac{3}{2}HT+\frac{2H}{16T}T^2-\frac{1}{2}HT\right]\\
			&= \frac{1}{T}\left[\underbrace{-\frac{18}{16}HT+\frac{24}{16}HT+\frac{2}{16}HT-\frac{8}{16}HT}_\text{0}\right]\\
			&= 0
			\end{align}</math>


			<math>\begin{align}
			a_n &= \frac{1}{T}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}\left(\frac{4H}{T}t \right)\,dt + \frac{1}{T}\int_{\frac{1}{4}T}^{\frac{3}{4}T}\left(-\frac{4H}{T}t+2H\right) \,dt\\

	\end{align}</math>		\end{align}</math>

Kurt at 00:54, 2 November 2010

2010-11-02T00:54:12Z

← Older revision		Revision as of 17:54, 1 November 2010
Line 4:		Line 4:
	<math>\begin{align}		<math>\begin{align}
	x(t) &= x(t+T) = a_0 + \sum_{n=1}^\infty a_n \cos(n\omega_0t) + b_n \sin(n\omega_0t)\\		x(t) &= x(t+T) = a_0 + \sum_{n=1}^\infty a_n \cos(n\omega_0t) + b_n \sin(n\omega_0t)\\
	a_0 &= \frac{1}{T}\int_0^T f(t) dt\\		a_0 &= \frac{1}{T}\int_0^T f(t)\, dt\\
	a_n &= \frac{2}{T}\int_0^T f(t)\cos(n\omega_0t) dt\\		a_n &= \frac{2}{T}\int_0^T f(t)\cos(n\omega_0t)\, dt\\
	b_n &= \frac{2}{T}\int_0^T f(t)\sin(n\omega_0t) dt\\		b_n &= \frac{2}{T}\int_0^T f(t)\sin(n\omega_0t)\, dt\\
	\end{align}		\end{align}
	</math>		</math>
Line 17:		Line 17:

	<math>\begin{align}		<math>\begin{align}
	a_0 &= \frac{1}{T}\int_0^{\frac{1}{2}T} H dt + \frac{1}{T}\int_{\frac{1}{2}T}^T -H dt\\		a_0 &= \frac{1}{T}\int_0^{\frac{1}{2}T} H\, dt + \frac{1}{T}\int_{\frac{1}{2}T}^T -H\, dt\\
	&=\frac{1}{T}\left[Ht\right]\bigg\|_{t=0}^{\frac{1}{2}T} - \frac{1}{T}\left[Ht\right]\bigg\|_{t={\frac{1}{2}T}}^T\\		&=\frac{1}{T}\left[Ht\right]\bigg\|_{t=0}^{\frac{1}{2}T} - \frac{1}{T}\left[Ht\right]\bigg\|_{t={\frac{1}{2}T}}^T\\
	&=\frac{1}{T}H\frac{1}{2}T-0 - \left[\frac{1}{T}HT - \frac{1}{T}H\frac{1}{2}T\right]\\		&=\frac{1}{T}H\frac{1}{2}T-0 - \left[\frac{1}{T}HT - \frac{1}{T}H\frac{1}{2}T\right]\\
Line 27:		Line 27:

	<math>\begin{align}		<math>\begin{align}
	a_n &= \frac{2}{T}\int_0^{\frac{1}{2}T} H\cos(n\omega_0t) dt + \frac{2}{T}\int_{\frac{1}{2}T}^T -H\cos(n\omega_0t) dt\\		a_n &= \frac{2}{T}\int_0^{\frac{1}{2}T} H\cos(n\omega_0t)\, dt + \frac{2}{T}\int_{\frac{1}{2}T}^T -H\cos(n\omega_0t)\, dt\\
	&=\frac{2}{T}\left[\frac{H}{n\omega_0}\sin(n\omega_0t)\right]_0^{\frac{1}{2}T} + \frac{2}{T}\left[\frac{-H}{n\omega_0}\sin\left(n\omega_0t\right)\right]_{\frac{1}{2}T}^T\\		&=\frac{2}{T}\left[\frac{H}{n\omega_0}\sin(n\omega_0t)\right]_0^{\frac{1}{2}T} + \frac{2}{T}\left[\frac{-H}{n\omega_0}\sin\left(n\omega_0t\right)\right]_{\frac{1}{2}T}^T\\
	&=\frac{2}{T}\left[\frac{H}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\frac{1}{2}T\right)-0\right] + \frac{2}{T}\left[-0+\frac{H}{n\frac{2\pi}{T}}\sin\left(n\frac{2pi}{T}\frac{1}{2}T\right)\right]\\		&=\frac{2}{T}\left[\frac{H}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\frac{1}{2}T\right)-0\right] + \frac{2}{T}\left[-0+\frac{H}{n\frac{2\pi}{T}}\sin\left(n\frac{2pi}{T}\frac{1}{2}T\right)\right]\\
Line 38:		Line 38:

	<math>\begin{align}		<math>\begin{align}
	b_n &= \frac{2}{T}\int_0^{\frac{1}{2}T} H\sin(n\omega_0t) dt + \frac{2}{T}\int_{\frac{1}{2}T}^T -H\sin(n\omega_0t) dt\\		b_n &= \frac{2}{T}\int_0^{\frac{1}{2}T} H\sin(n\omega_0t)\, dt + \frac{2}{T}\int_{\frac{1}{2}T}^T -H\sin(n\omega_0t)\, dt\\
	&=\frac{2}{T}\left[\frac{-H}{n\omega_0}\cos(n\omega_0t)\right]_0^{\frac{1}{2}T} + \frac{2}{T}\left[\frac{H}{n\omega_0}\cos(n\omega_0t)\right]_{\frac{1}{2}T}^T\\		&=\frac{2}{T}\left[\frac{-H}{n\omega_0}\cos(n\omega_0t)\right]_0^{\frac{1}{2}T} + \frac{2}{T}\left[\frac{H}{n\omega_0}\cos(n\omega_0t)\right]_{\frac{1}{2}T}^T\\
	&=\frac{2}{T}\left[-\frac{H}{n\frac{2\pi}{T}}\cos(n\frac{2\pi}{T}\frac{1}{2}T)+\frac{H}{n\frac{2\pi}{T}}\right] + \frac{2}{T}\left[\frac{H}{n\frac{2\pi}{T}}\cos(n\frac{2\pi}{T}T)-\frac{H}{n\frac{2\pi}{T}}\cos(n\frac{2\pi}{T}\frac{1}{2}T)\right]\\		&=\frac{2}{T}\left[-\frac{H}{n\frac{2\pi}{T}}\cos(n\frac{2\pi}{T}\frac{1}{2}T)+\frac{H}{n\frac{2\pi}{T}}\right] + \frac{2}{T}\left[\frac{H}{n\frac{2\pi}{T}}\cos(n\frac{2\pi}{T}T)-\frac{H}{n\frac{2\pi}{T}}\cos(n\frac{2\pi}{T}\frac{1}{2}T)\right]\\
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	===Triangle Wave===		===Triangle Wave===
	Like the Square wave, the DC component of the Triangle Wave is 0 by inspection. Also, since the triangle wave is odd, it is made up only by sine components.		Like the Square wave, the DC component of the Triangle Wave is 0 by inspection. Also, since the triangle wave is odd, it is made up only by sine components.

			<math>\begin{align}
			a_0 &= \frac{1}{T}\int_{-\frac{1}{4}T}^{\frac{1}{4}T}\frac{4H}{T}t \,dt +

			\end{align}</math>


Line 60:		Line 65:
	====Square Wave====		====Square Wave====
	clf; %Clear Figure		clf; %Clear Figure
	t=0:.01:10; %Limits of the graph		t=0:.01:10; %Limits of the graph
	T=2*pi %Definition of the period		T=2*pi %Definition of the period
	M=100 %Number of iterations to undergo		M=100 %Number of iterations to undergo
	sum1=0; %Initialize the sum to <br>		sum1=0; %Initialize the sum to <br>
	%----------FOURIER SERIES----------%		%----------FOURIER SERIES----------%
	for m=1:1:M, %For m=1, increment by 1 until you get to M		for m=1:1:M, %For m=1, increment by 1 until you get to M

Kurt at 23:54, 1 November 2010

2010-11-01T23:54:27Z

← Older revision		Revision as of 16:54, 1 November 2010
Line 57:		Line 57:
	===Sawtooth Wave===		===Sawtooth Wave===

	==OCTAVE Scripts==		==OCTAVE Scripts to Plot Fourier Series==
	====Square Wave====		====Square Wave====
	clf; %Clear Figure		clf; %Clear Figure

Kurt at 23:53, 1 November 2010

2010-11-01T23:53:41Z

← Older revision		Revision as of 16:53, 1 November 2010
Line 13:		Line 13:
	The four basic waveforms are Sine Waves, Square Waves, Triangle Waves, and Sawtooth Waves.		The four basic waveforms are Sine Waves, Square Waves, Triangle Waves, and Sawtooth Waves.

	==Square Wave==		===Square Wave===
	By inspection of the waveform, the DC component of the wave will be 0. Also, since the waveform is odd, a<sub>n</sub> will be 0. Here is the proof:		By inspection of the waveform, the DC component of the wave will be 0. Also, since the waveform is odd, a<sub>n</sub> will be 0. Here is the proof:

Line 28:		Line 28:
	<math>\begin{align}		<math>\begin{align}
	a_n &= \frac{2}{T}\int_0^{\frac{1}{2}T} H\cos(n\omega_0t) dt + \frac{2}{T}\int_{\frac{1}{2}T}^T -H\cos(n\omega_0t) dt\\		a_n &= \frac{2}{T}\int_0^{\frac{1}{2}T} H\cos(n\omega_0t) dt + \frac{2}{T}\int_{\frac{1}{2}T}^T -H\cos(n\omega_0t) dt\\
	&=\frac{2}{T}\left[\frac{H}{n\omega_0}\sin(n\omega_0t)\right]_0^{\frac{1}{2}T} + \frac{2}{T}\left[\frac{-H}{n\omega_0}\sin(n\omega_0t)\right]_{\frac{1}{2}T}^T\\		&=\frac{2}{T}\left[\frac{H}{n\omega_0}\sin(n\omega_0t)\right]_0^{\frac{1}{2}T} + \frac{2}{T}\left[\frac{-H}{n\omega_0}\sin\left(n\omega_0t\right)\right]_{\frac{1}{2}T}^T\\
	&=\frac{2}{T}\left[\frac{H}{n\frac{2\pi}{T}}\sin(n\frac{2\pi}{T}\frac{1}{2}T)-0\right] + \frac{2}{T}\left[-0+\frac{H}{n\frac{2\pi}{T}}\sin(n\frac{2pi}{T}\frac{1}{2}T)\right]\\		&=\frac{2}{T}\left[\frac{H}{n\frac{2\pi}{T}}\sin\left(n\frac{2\pi}{T}\frac{1}{2}T\right)-0\right] + \frac{2}{T}\left[-0+\frac{H}{n\frac{2\pi}{T}}\sin\left(n\frac{2pi}{T}\frac{1}{2}T\right)\right]\\
	&=\frac{2}{T}\left[\frac{TH}{2\pi n}\underbrace{\sin(n\pi)}_\text{0}\right] + \frac{2}{T}\left[\frac{TH}{2\pi n}\underbrace{\sin(n\pi)}_\text{0}\right]\\		&=\frac{2}{T}\left[\frac{TH}{2\pi n}\underbrace{\sin(n\pi)}_\text{0}\right] + \frac{2}{T}\left[\frac{TH}{2\pi n}\underbrace{\sin(n\pi)}_\text{0}\right]\\
	&=0		&=0
Line 51:		Line 51:
	<math>\text{Square Wave Fourier Series: }x(t) = x(t+T) = \sum_{n=1}^\infty \left(\frac{2H}{\pi n}-\frac{2H}{\pi n}\cos(n\pi)\right) \sin(n\omega_0t) </math>		<math>\text{Square Wave Fourier Series: }x(t) = x(t+T) = \sum_{n=1}^\infty \left(\frac{2H}{\pi n}-\frac{2H}{\pi n}\cos(n\pi)\right) \sin(n\omega_0t) </math>

	==Triangle Wave==		===Triangle Wave===
	Like the Square wave, the DC component of the Triangle Wave is 0 by inspection. Also, since the triangle wave is odd, it is made up only by sine components.		Like the Square wave, the DC component of the Triangle Wave is 0 by inspection. Also, since the triangle wave is odd, it is made up only by sine components.


	==Sawtooth Wave==		===Sawtooth Wave===

			==OCTAVE Scripts==
			====Square Wave====
			clf; %Clear Figure
			t=0:.01:10; %Limits of the graph
			T=2*pi %Definition of the period
			M=100 %Number of iterations to undergo
			sum1=0; %Initialize the sum to <br>
			%----------FOURIER SERIES----------%
			for m=1:1:M, %For m=1, increment by 1 until you get to M
			if(m!=0)
			sum1 = sum1 + ((2/(pim))-(2/(pim))cos(mpi))sin(m2pi/Tt);
			end
			end<br>
			%---------------PLOT---------------%
			plot(t,real(sum1),'b-')
			title('Fourier Series Representation of a Wave')
			xlabel('time (seconds)')
			ylabel('Function')
			grid on;<br>
			legend(num2str(M) ' terms');
			print("squarewave.png","-dpng") % Prints the plot to a png file called squarewave.png