EECE 301 NS_09 Summary & Solutions_rev

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Binghamton University *

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301

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Electrical Engineering

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Oct 30, 2023

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NS-09 FS Spectrum Summary List What is the point of Fourier Series? Breaks a periodic signal into a sum of sinusoids Advantage… can analyze LTI system for each of those sinusoids individually using generalized “Phasor Method” Plotting the spectrum Plot magnitude | c k | vs k or k ω o (the latter is better!) Plot phase < c k vs k or k ω o (the latter is better!) Looking at the magnitude plot shows us The relative importance of the various frequency components for building the signal We can use Parseval’s Theorem to find power in frequency regions 1/6 This allows us to visualize the FS! Can visualize effect of a system on the Input!
NS-09 Fourier Series Problem 1. Plot the Spectrum for the “ sawtooth ” wave below: ( 1) 0 k k j c k k = 0 1 c = So… plot | c k | vs k o and c k vs k o We found these FS coefficients last time! with o = π rad/sec 2 t 1 1 3 5 3 5 2/6
NS-09 Fourier Series Solution 0 1 0 1 k c k k c = = Magnitude result for k ≠ 0: k c = ( 1) k j k = ( 1) k j k ( 1) 0 k k j c k k = 0 1 c = 3/6 1 k = ( 1) k j k =
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When k < 0: c k is j (neg)(-1) k When k > 0: c k is j (pos)(-1) k Since j (neg) = - /2 Since j (pos) = /2 …we get this Phase Results: = = even negative k odd positive k odd negative k even positive k k c k & & , 2 & & , 2 0 , 0 ( 1) 0 k k j c k k = 0 1 c = 4/6
0 1 0 1 k c k k c = = 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 2 𝜋 2 −𝜋 = = even negative k odd positive k odd negative k even positive k k c k & & , 2 & & , 2 0 , 0 5/6
This can also be done directly in MATLAB/Octave… (but… keep in mind that MANY times we DO need to be able to find magnitude/phase by hand!!!) k=-30:30; ck=j*( (-1).^k)./(pi*k); ck(31)=1; % insert correct value for c_0 subplot(2,1,1) stem(k,abs(ck)) xlabel('k\omega_o (rad/sec)') % Need to insert pi symbol "by hand" ylabel('|c_k|') grid subplot(2,1,2) stem(k,angle(ck)) xlabel('k\omega_o (rad/sec)') % Need to insert pi symbol "by hand" ylabel('<c_k (radians') grid 6/6
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