leo lam © 2010-2011 signals and systems ee235. leo lam © 2010-2011 fourier transform q: what did...
TRANSCRIPT
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 the Fourier transform of the sinc function?
A: "You're such a square!"
Leo Lam © 2010-2011
Extra Fourier Transform
• Fourier Transform Examples
Leo Lam © 2010-2011
Fourier Transform:
4
• Fourier Transform
• Inverse Fourier Transform:
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
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)
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
Leo Lam © 2010-2011
Output determination Example
8
• Solve for y(t)
• Recall that:
• Partial fraction:• Invert:
1( )ate u t
a j
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)
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
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)
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
???
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!
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
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
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
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
Leo Lam © 2010-2011
Summary
• Fourier Transforms and examples• Next, and last: Sampling!