• Lecture 22: Fourier transforms
ECEN 1400 Introduction to Analog and Digital Electronics
Lecture 22
Fourier transforms
Original image
The Fourier Transform represents a signal
not as v(t) but as a set of frequencies.
Low-pass
High pass
When you add all the frequencies up, you
get the original waveform v(t).
The same concepts are used in image
processing and photography
Very high pass
Robert R. McLeod, University of Colorado
191
Example 1: the δ function
• Lecture 22: Fourier transforms
ECEN 1400 Introduction to Analog and Digital Electronics
1 frequency at f=0
results in constant
2 frequencies
results in raised cosine
3 frequencies
4 frequencies
10 frequencies
add up at t=0
All frequencies add
to zero everywhere
except t=0
Robert R. McLeod, University of Colorado
192
Example 2: rectangle function
• Lecture 22: Fourier transforms
ECEN 1400 Introduction to Analog and Digital Electronics
1 frequency at f=0
results in constant
2 frequencies
results in raised cosine
3 frequencies
4 frequencies
10 frequencies
add up for |t| < 0.2
All frequencies add
to zero everywhere
except |t| < 0.2
Robert R. McLeod, University of Colorado
193
• Lecture 22: Fourier transforms
ECEN 1400 Introduction to Analog and Digital Electronics
What a filtering
capacitor does (1/3)
We put capacitors near our chip power supplies. Why?
Here’s the circuit, including the (very tiny) resistance of the wire
VIN
VOUT
ZR = R
ZC =
1
jω C
Some
integrated
circuit
This is just a voltage divider. Use the concept of “impedance” from last time (Z).
VOUT
1
1
1
jω C
jω C ⎛ jω C ⎞
= VIN
= VIN
=
V
IN
1
1 ⎜⎝ jω C ⎟⎠
jRCω + 1
R+
R+
jω C
jω C
So at zero frequency, the chip sees Vcc, but at high frequency, the voltage is reduced.
Robert R. McLeod, University of Colorado
194
• Lecture 22: Fourier transforms
ECEN 1400 Introduction to Analog and Digital Electronics
Fourier analysis of the
smoothing capacitor
VIN = DC power + a spike
Fourier transform (DC removed)
Voltage divider (filter function)
VOUT = VIN
1
jRCω + 1
Add these frequencies back
up to get VOUT
Robert R. McLeod, University of Colorado
Product of VIN(f) and filter function
195
• Lecture 22: Fourier transforms
ECEN 1400 Introduction to Analog and Digital Electronics
This is just a “low pass”
filter
VIN = square wave
Fourier transform
Voltage divider (filter function)
VOUT = VIN
1
jRCω + 1
Add these frequencies back
up to get VOUT
Robert R. McLeod, University of Colorado
Product of VIN(f) and filter function
196