HO#2 System Arch 200７ (Fire Tom Wada)

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HO#2 System Arch 200７ (Fire Tom Wada) OUTLINE Periodic Signal Fourier series introduction Sinusoids Orthogonality Integration vs inner product 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

Consider any wave is sum of simple sin and cosine Periodic Tc 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

HO#2 System Arch 200７ (Fire Tom Wada) Periodic signal is composed of DC + same frequency sinusoid + multiple frequency sinusoids Frequency = 0 Hz Basic frequency fc=1/Tc 2 x fc 3 x fc 4 x fc 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

Spectrum of periodic signal frequency f (Hz) -5・fc -4・fc -3・fc -2・fc -fc fc 2・fc 3・fc 4・fc 5・fc There are only n * fc (n=integer) frequencies! 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

Another example (even rectangular pulse) 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

Increase the number of sum (1) 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

Increase the number of sum (2) 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

HO#2 System Arch 200７ (Fire Tom Wada) Fourier Jean Baptiste Joseph, Baron de Frourier France, 1778/Mar/21 – 1830/May/16 Fourier Series paper is written in 1807 Even discontinue function (such as rectangular pulse) can be composed of many sinusoids. Nobody believed the paper at that time. 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

HO#2 System Arch 200７ (Fire Tom Wada) Fourier Series If f(t) ‘s period is Tc… If we use complex exponential…, 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

HO#2 System Arch 200７ (Fire Tom Wada) Anyway, when you see the periodic signal, Please think it is just sum of sinusoids!!! 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

How we can divide f(t) into sinusoids? Filter Pass nω (Hz) Filter is used an and bn 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

If we integrate in [ 0 to Tc] 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

If we integrate in [ 0 to Tc] (2) a1 can be computed 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

If we integrate in [ 0 to Tc] (3) b1 can be computed 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

By changing multiplier, each coefficient computed One coefficient 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

Sinusoidal Orthogonality m,n: integer, Tc=1/f0 Orthogonal Orthogonal Orthogonal 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

Another Orthogonality (1) Vector inner product Orthogonal Θ＝90 degree 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

Another Orthogonality (2) n dimensional vector IF THEN A and B are Orthogonal. 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

is same as the N dim inner product Freq=nω(Hz) sinusoids are Orthogonal each other (n=integer) 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

Fourier Series Summary 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

Complex form Fourier Series Orthogonal 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)

HO#2 System Arch 200７ (Fire Tom Wada) HW2 [2-1]Compute the complex form Fourier Series coefficient cn for f(x). [2-2]Draw the Spectrum of f(t) when T0=0.04sec. 2.30 2019/5/6 HO#2 System Arch 200７ (Fire Tom Wada)