Introduction to Filters

1
Introduction to Filters

• Section 14.1-14.2

2
Application of Filter
Application Cellphone Center frequency 900
MHz Bandwidth 200 KHz
Adjacent interference
Use a filter to remove interference
3
Filters

• Classification
• Low-Pass
• High-Pass
• Band-Pass
• Band-Reject
• Implementation
• Passive Implementation (R,L, C)
• Active Implementation (Op-Amp, R, L, C)
• Continuous time and discrete time

4
Filter Characteristics
Not desirable. Alter Frequency content.
Must not alter the desired signal!
Affect selectivity
Sharp Transition in order to attenuate the
interference
5
Low-Pass Example
How much attenuation is provided by the filter?
6
Answer
How much attenuation is provided by the filter?
40 dB
7
High-Pass Filter
What filter stopband attenuation is necessary in
order to ensure the signal level is 20 dB above
the interference?
8
High-Pass Filter (Solution)
What filter stopband attenuation is necessary in
order to ensure the signal level is 20 dB above
the interference? 60 dB _at_60 Hz
9
Bandpass
10
Replace a resistor with a capacitor!
How do you replace a resistor with a switch and a
capacitor?
11
Resistance of a Switched Capacitor Circuit
(315A, Murmann, Stanford)
12
What is the equivalent continuous time filter?
13
Filter Transfer Function
(Increase filter order in order to increase
filter selectivity!)
14
Low Pass Filter Example
15
Adding a Zero



16
Complex Poles and Zero at the Origin
17
RC Low Pass (Review)
 
A pole a root of the denomintor 1sRC0?S-RC
18
Laplace Transform/Fourier Transform
 
(Laplace Transform)
Complex s plane
 
 
 
 
(Fourier Transform)
 
 
-p
p1/(RC)
Location of the zero in the left complex plane
 
 
19

• Rules of thumb (applicable to a pole)
• Magnitude
• 20 dB drop after the cut-off frequency
• 3dB drop at the cut-off frequency
• Phase
• -45 deg at the cut-off frequency
• 0 degree at one decade prior to the cut-frequency
• 90 degrees one decade after the cut-off frequency

20
RC High Pass Filter (Review)
 
A zero at DC. A pole from the denominator. 1sRC0
?S-RC
21
Laplace Transform/Fourier Transform
 
(Laplace Transform)
Complex s plane
 
 
 
 
(Fourier Transform)
 
 
-p
p1/(RC) Zero at DC.
Location of the zero in the left complex plane
 
 
22
Zero at the origin. Thus phase(f0)90
degrees. The high pass filter has a cut-off
frequency of 100.
23
RC High Pass Filter (Review)
 
R12(R1R2)/(R1R2) A pole and a zero in the left
complex plane.
24
Laplace Transform/Fourier Transform (Low
Frequency)
 
(Laplace Transform)
Complex s plane
 
 
 
 
 
(Fourier Transform)
 
 
-p
z1/(RC) p1/(R12C)
-z
Location of the zero in the left complex plane
 
25
Laplace Transform/Fourier Transform (High
Frequency)
 
(Laplace Transform)
Complex s plane
 
 
 
 
 
(Fourier Transform)
 
 
-p
z1/(RC) p1/(R12C)
-z
Location of the zero in the left complex plane
 
26
Stability Question
Why the poles must lie in the left half plane?
27
Answer
Recall that the impulse response of a system
contains terms such as . If , these terms grow
indefinitely with time while oscillating at a
frequency of