1. Signals and Systems
EE235
Lecture 23
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2. Today’s menu
Fourier Transform
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3. 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! 3
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4. Fourier Transform: Fourier Formulas compared: 4 Fourier transform
Fourier transform coefficients:
Fourier transform
(arbitrary signals)
Fourier series (Periodic signals):
Fourier series coefficients:
and 4
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5. Fourier Transform (example):
Find the Fourier Transform of
What does it look like?
If a <0, blows up
phase
varies with 
magnitude
varies with  5
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6. Fourier Transform (example):
Fourier Transform of
Real-time signals magnitude: even phase: odd
magnitude
phase 6
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7. Fourier Transform (Symmetry):
Real-time signals magnitude: even – why?
magnitude
Even magnitude
Odd phase
Useful for checking answers 7
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8. 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 8
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9. Fourier Transform (example):
Fourier Transform of
F(w) is purely real
F(w) for a=1 9
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10. Summary
Fourier Transform intro
Inverse etc.
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11. Fourier Transform (delta function):
Fourier Transform of
Standard Fourier Transform pair notation 11
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12. Fourier Transform (rect function):
Fourier Transform of
Plot for T=1? t
-T/ T/2 1
Define 12
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13. 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: 13
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14. Fourier Transform - Inverse relationship
Inverse relationship between time/frequency 14
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15. Fourier Transform - Inverse
Inverse Fourier Transform (Synthesis)
Example: 15
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16. 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 16
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17. Fourier Transform Properties
A Fourier Transform “Pair”: f(t)  F()
Re-usable!
time domain Fourier transform
Scaling
Additivity
Convolution
Time shift 17
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18. 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) 18
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19. FT Properties Example:
Find FT for:
We know the pair:
So: 
G() 19
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20. More Transform Pairs: More pairs: time domain Fourier transform 20
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21. 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 21
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22. Periodic signals: Transform from Series
We see this pair:
More generally, if X(w) has equally spaced impulses:
Then:
Fourier Series!!! 22
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23. Periodic signals: Transform from Series
If we know Series, we know Transform
Then:
Example:
We know:
We can write: 23
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24. Summary
Fourier Transform Pairs
FT Properties
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