Signals and Systems EE235 Lecture 21 Leo Lam © 2010-2012.

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Signals and Systems EE235 Lecture 21 Leo Lam © 2010-2012

It’s here! Solve Given Solve Leo Lam © 2010-2012

Today’s menu Fourier Series Leo Lam © 2010-2012

Trigonometric Fourier Series Note a change in index Set of sinusoids: fundamental frequency w0 4 Leo Lam © 2010-2012

Trigonometric Fourier Series Orthogonality check: for m,n>0 5 Leo Lam © 2010-2012

Trigonometric Fourier Series Similarly: Also true: prove it to yourself at home: 6 Leo Lam © 2010-2012

Trigonometric Fourier Series Find coefficients: The average value of f(t) over one period (DC offset!) 7 Leo Lam © 2010-2012

Trigonometric Fourier Series Similarly for: 8 Leo Lam © 2010-2012

Compact Trigonometric Fourier Series Instead of having both cos and sin: Recall: Expand and equate to the LHS Harmonic Addition Theorem 9 Leo Lam © 2010-2012

Compact Trigonometric to est In compact trig. form: Remember goal: Approx. f(t)Sum of est Re-writing: And finally: Writing signals in terms of exponentials 10 Leo Lam © 2010-2012

Compact Trigonometric to est Most common form Fourier Series Orthonormal: , Coefficient relationship: dn is complex: Angle of dn: Angle of d-n: Writing signals in terms of exponentials 11 Leo Lam © 2010-2012

So for dn We want to write periodic signals as a series: And dn: Need T and w0 , the rest is mechanical 12 Leo Lam © 2010-2012

Harmonic Series Building periodic signals with complex exp. Obvious case: sums of sines and cosines Find fundamental frequency Expand sinusoids into complex exponentials (“CE’s”) Write CEs in terms of n times the fundamental frequency Read off cn or dn Writing signals in terms of exponentials 13 Leo Lam © 2010-2012

Harmonic Series Example: Expand: 14 Fundamental freq. Writing signals in terms of exponentials 14 Leo Lam © 2010-2012

Harmonic Series Example: Fundamental frequency: Re-writing: 15 w0=GCF(1,2,5)=1 or Re-writing: Writing signals in terms of exponentials dn = 0 for all other n 15 Leo Lam © 2010-2012

Harmonic Series Example (your turn): Write it in an exponential series: d0=-5, d2=d-2=1, d3=1/2j, d-3=-1/2j, d4=1 Writing signals in terms of exponentials 16 Leo Lam © 2010-2012

Harmonic Series Graphically: 17 One period: t1 to t2 All time (zoomed out in time) 17 Leo Lam © 2010-2012