leo lam © 2010-2011 signals and systems ee235. leo lam © 2010-2011 fourier transform q: what did...

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Leo Lam © 2010-2011 Signals and Systems EE235

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Page 1: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Signals and Systems

EE235

Page 2: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Fourier Transform

Q: What did the Fourier transform of the arbitrary signal say to the Fourier transform of the sinc function?

A: "You're such a square!"

Page 3: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Extra Fourier Transform

• Fourier Transform Examples

Page 4: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Fourier Transform:

4

• Fourier Transform

• Inverse Fourier Transform:

Page 5: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Low Pass Filter

5

Consider an ideal low-pass filter with frequency response

w0

H(w)

• What is h(t)? (Impulse response)

Looks like an octopus centeredaround time t = 0 Not causal…can’t build a circuit.

-3 -2 -1 0 1 2 3-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Page 6: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Low Pass Filter

6

Consider an ideal low-pass filter with frequency response

w0

H(w)

• What is y(t) if input is:

• Ideal filter, so everything above is gone:

• y(t)

Page 7: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Output determination Example

7

• Solve for y(t)

• Convert input and impulse function to Fourier domain:

• Invert Fourier using known transform:

1( 1) ( 1)

1 j

/ 41 1 1( ) cos( / 4)

1 2 2je y t t

j

Page 8: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Output determination Example

8

• Solve for y(t)

• Recall that:

• Partial fraction:• Invert:

1( )ate u t

a j

Page 9: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Describing Signals (just a summary)

9

• Ck and X(w) tell us the CE’s (or cosines) that are needed to build a time signal x(t)– CE with frequency w (or kw0) has magnitude |Ck| or |

X(w)| and phase shift <Ck and <X(w)– FS and FT difference is in whether an uncountably

infinite number of CEs are needed to build the signal.

-B B w

t

x(t)

X(w)

Page 10: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Describing Signals (just a summary)

Leo Lam © 2010-2011

• H(w) = frequency response– Magnitude |H(w)| tells us how to scale cos amplitude– Phase <H(w) tells us the phase shift

-100 -80 -60 -40 -20 0 20 40 60 80 100-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-100 -80 -60 -40 -20 0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

magnitude phase

p/2

- /p 2

H(w)cos(20t) Acos(20t+f)

A

f20 20

Page 11: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Example (Fourier Transform problem)

• Solve for y(t)

• But does it make sense if it was done with convolution?

11

0 5-5w

F(w) transfer function H(w)

01-1w

0 5-5w

=Z(w) =0 everywhere

0 5-5w

Z(w) = F(w) H(w)

Page 12: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Example (Circuit design with FT!)

• Goal: Build a circuit to give v(t) with an input current i(t)

• Find H(w)• Convert to differential equation• (Caveat: only causal systems can be physically

built)

12

???

Page 13: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Example (Circuit design with FT!)

• Goal: Build a circuit to give v(t) with an input current i(t)

• Transfer function:

13

???

)(

)()(

I

VH

Inverse transform!

Page 14: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Example (Circuit design with FT!)

• Goal: Build a circuit to give v(t) with an input current i(t)

• From:

• The system:• Inverse transform:

• KCL: What does it look like?

14

???

)(

)()(

I

VH

Capacitor

Resistor

Page 15: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Fourier Transform: Big picture

• With Fourier Series and Transform:• Intuitive way to describe signals & systems• Provides a way to build signals

– Generate sinusoids, do weighted combination• Easy ways to modify signals

– LTI systems: x(t)*h(t) X(w)H(w)– Multiplication: x(t)m(t) X(w)*H(w)/2p

15

Page 16: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Fourier Transform: Wrap-up!

• We have done:– Solving the Fourier Integral and Inverse– Fourier Transform Properties– Built-up Time-Frequency pairs– Using all of the above

16

Page 17: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Bridge to the next class

• Next class: EE341: Discrete Time Linear Sys• Analog to Digital• Sampling

17

t

continuous in time

continuous in amplitude

n

discrete in timeSAMPLING

discrete in amplitudeQUANTIZATION

Page 18: Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to

Leo Lam © 2010-2011

Summary

• Fourier Transforms and examples• Next, and last: Sampling!