1. Signals and Systems
EE235
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2. Motivation
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3. Fourier Series: Dirichlet Conditon
Condition for periodic signal f(t) to exist has exponential series:
Weak Dirichlet:
Strong Dirichlet (converging series): f(t) must have finite maxima, minima, and discontinuities in a period
All physical periodic signals converge
Weak Dirichlet: Otherwise you can’t solve for the coefficients! 3
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4. End of Fourier Series We have accomplished: Next: Fourier Transform 4
Introduced signal orthogonality
Fourier Series derivation
Approx. periodic signals:
Fourier Series Properties
Next: Fourier Transform 4
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5. Fourier Transform: Introduction
Fourier Series: Periodic Signal
Fourier Transform: extends to all signals
Recall time-scaling: 5
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6. Fourier Transform: Recall time-scaling: 6 Fourier Spectra for T,
Fourier Spectra
for T,
for 2T, 6
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7. Fourier Transform: Non-periodic signal: infinite period T 7
Fourier Spectra
for T,
for 2T, 7
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8. Fourier Transform: Fourier Formulas:
For any arbitrary practical signal
And its “coefficients” (Fourier Transform):
F(w) is complex
Rigorous math derivation in Ch. 4 (not required reading, but recommended)
Time domain to
Frequency domain
Weak Dirichlet: Otherwise you can’t solve for the coefficients! 8
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9. Fourier Transform: Fourier Formulas compared: 9 Fourier transform
Fourier transform coefficients:
Fourier transform
(arbitrary signals)
Fourier series (Periodic signals):
Fourier series coefficients:
and 9
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10. Fourier Transform (example):
Find the Fourier Transform of
What does it look like?
If a <0, blows up
phase
varies with 
magnitude
varies with  10
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11. Fourier Transform (example):
Fourier Transform of
Real-time signals magnitude: even phase: odd
magnitude
phase 11
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12. Fourier Transform (Symmetry):
Real-time signals magnitude: even – why?
magnitude
Even magnitude
Odd phase
Useful for checking answers 12
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13. Fourier Transform/Series (Symmetry):
Works for Fourier Series, too!
Fourier series
(periodic functions)
Fourier transform
(arbitrary practical signal)
Fourier coefficients
Fourier transform coefficients
magnitude: even & phase: odd 13
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14. Fourier Transform (example):
Fourier Transform of
F(w) is purely real
F(w) for a=1 14
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15. Summary
Fourier Transform intro
Inverse etc.
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16. Fourier Transform (delta function):
Fourier Transform of
Standard Fourier Transform pair notation 16
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17. Fourier Transform (rect function):
Fourier Transform of
Plot for T=1? t
-T/ T/2 1
Define 17
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18. Fourier Transform (rect function):
Fourier Transform of
Observation:
Wider pulse (in t) <-> taller narrower spectrum
Extreme case:
<->
Peak=pulse width (example: width=1)
Zero-Crossings: 18
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19. Fourier Transform - Inverse relationship
Inverse relationship between time/frequency 19
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20. Fourier Transform - Inverse
Inverse Fourier Transform (Synthesis)
Example: 20
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21. Fourier Transform - Inverse
Inverse Fourier Transform (Synthesis)
Example:
Single frequency spike in w: exponential time signal with that frequency in t
A single spike in frequency
Complex exponential in time 21
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22. Fourier Transform Properties
A Fourier Transform “Pair”: f(t)  F()
Re-usable!
time domain Fourier transform
Scaling
Additivity
Convolution
Time shift 22
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23. How to do Fourier Transform
Three ways (or use a combination) to do it:
Solve integral
Use FT Properties (“Spiky signals”)
Use Fourier Transform table (for known signals) 23
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24. FT Properties Example:
Find FT for:
We know the pair:
So: 
G() 24
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25. More Transform Pairs: More pairs: time domain Fourier transform 25
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26. Periodic signals: Transform from Series
Integral does not converge for periodic fns:
We can get it from Fourier Series:
How? Find x(t) if
Using Inverse Fourier: So 26
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27. Periodic signals: Transform from Series
We see this pair:
More generally, if X(w) has equally spaced impulses:
Then:
Fourier Series!!! 27
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28. Periodic signals: Transform from Series
If we know Series, we know Transform
Then:
Example:
We know:
We can write: 28
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29. Summary
Fourier Transform Pairs
FT Properties
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