1. Signals and Systems
EE235
Lecture 21
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2. It’s here!
Solve
Given
Solve
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3. Today’s menu
Fourier Series
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4. Trigonometric Fourier Series
Note a change in index
Set of sinusoids: fundamental frequency w0 4
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5. Trigonometric Fourier Series
Orthogonality check:
for m,n>0 5
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6. Trigonometric Fourier Series
Similarly:
Also true: prove it to yourself at home: 6
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7. Trigonometric Fourier Series
Find coefficients:
The average value of f(t) over one period (DC offset!) 7
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8. Trigonometric Fourier Series
Similarly for: 8
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9. Compact Trigonometric Fourier Series
Instead of having both cos and sin:
Recall:
Expand and equate to the LHS
Harmonic Addition Theorem 9
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10. 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
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11. 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
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12. So for dn We want to write periodic signals as a series: And dn:
Need T and w0 , the rest is mechanical 12
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13. 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
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14. Harmonic Series Example: Expand: 14 Fundamental freq.
Writing signals in terms of exponentials 14
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15. 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
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16. 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
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17. Harmonic Series Graphically: 17 One period: t1 to t2 All time
(zoomed out in time) 17
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