Leo Lam © 2010-2013 Signals and Systems EE235. Transformers Leo Lam © 2010-2013 2.

Leo Lam © 2010-2013

Signals and SystemsEE235

Leo Lam © 2010-2013

Transformers

2

Leo Lam © 2010-2013

Today’s menu

• Fourier Transform• Loads of examples

Leo Lam © 2010-2013

Fourier Transform:

4

• Fourier Transform

• Inverse Fourier Transform:

Leo Lam © 2010-2013

Another angle of LTI (Example)

• Given graphical H(w), find h(t)

• What does this system do? What is h(t)?

• Linear phase constant delay

5

)5()( tth

magnitude

w

w

phase

0

0

1

Slope=-5

5)( jeH

Leo Lam © 2010-2013



• What does this system do (qualitatively

• Low-pass filter. No delay.

6

magnitude

w

w

phase

0

0

1

Leo Lam © 2010-2013



• What does this system do qualitatively?

• Bandpass filter. Slight delay.

7

magnitude

w

w

phase

0

1

Leo Lam © 2010-2013

Summary

• Fourier Transforms and examples

Leo Lam © 2010-2013

Low Pass Filter

9

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

Leo Lam © 2010-2013

Low Pass Filter

10

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)

Leo Lam © 2010-2013

Output determination Example

11

• 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

Leo Lam © 2010-2013

Output determination Example

12

• Solve for y(t)

• Recall that:

• Partial fraction:• Invert:

1( )ate u t

a j

Leo Lam © 2010-2013

Describing Signals (just a summary)

13

• 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)

Describing Signals (just a summary)

Leo Lam © 2010-2013

• 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

Leo Lam © 2010-2013

Example (Fourier Transform problem)

• Solve for y(t)

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

15

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)

Leo Lam © 2010-2013

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)

16

???

Leo Lam © 2010-2013



• From:

• The system:• Inverse transform:

• KCL: What does it look like?

18

???

)(

)()(

I

VH

Capacitor

Resistor

Leo Lam © 2010-2013

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

19

Leo Lam © 2010-2013

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

20

Leo Lam © 2010-2013

Bridge to the next class

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

21

t

continuous in time

continuous in amplitude

n

discrete in timeSAMPLING

discrete in amplitudeQUANTIZATION

Leo Lam © 2010-2013 Signals and Systems EE235. Transformers Leo Lam © 2010-2013 2.

Documents

Transcript of Leo Lam © 2010-2013 Signals and Systems EE235. Transformers Leo Lam © 2010-2013 2.