# Introduction to Filters

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```					Introduction to Filters

Section 14.1-14.2
Application of Filter

Application: Cellphone      Adjacent       Use a filter to remove
Center frequency: 900 MHz   interference   interference
Bandwidth: 200 KHz
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
Filter Characteristics
Not desirable.
Alter Frequency content.

Must not alter
the desired signal!                          Affect selectivity
Sharp Transition
in order to attenuate
the interference
Low-Pass Example

How much attenuation is provided by the filter?

How much attenuation is provided by the filter? 40 dB
High-Pass Filter

What filter stopband attenuation is necessary in order
to ensure the signal level is 20 dB above the interference?
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 @60 Hz
Bandpass
Replace a resistor with a
capacitor!

How do you replace a resistor with a switch and a capacitor?
Resistance of a Switched
Capacitor Circuit

(315A, Murmann, Stanford)
What is the equivalent
continuous time filter?
Filter Transfer Function

(Increase filter order in order to increase filter selectivity!)
Low Pass Filter Example
Complex Poles and Zero at the
Origin
RC Low Pass (Review)

A pole: a root of the denomintor
1+sRC=0→S=-RC
Laplace Transform/Fourier
Transform

(Laplace Transform)                                      Complex s plane

(Fourier Transform)

-p
p=1/(RC)

Location of the zero in the left complex
plane
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
RC High Pass Filter (Review)

A zero at DC.
A pole from the denominator.
1+sRC=0→S=-RC
Laplace Transform/Fourier
Transform

(Laplace Transform)                                      Complex s plane

(Fourier Transform)

-p
p=1/(RC)
Zero at DC.
Location of the zero in the left complex
plane
Zero at the origin.
Thus phase(f=0)=90 degrees.
The high pass filter has a cut-off frequency of 100.
RC High Pass Filter (Review)

R12=(R1R2)/(R1+R2)
A pole and a zero in the left complex plane.
Laplace Transform/Fourier
Transform (Low Frequency)

(Laplace Transform)                                      Complex s plane

(Fourier Transform)

-p
z=1/(RC)                           -z
p=1/(R12C)
Location of the zero in the left complex
plane
Laplace Transform/Fourier
Transform (High Frequency)

(Laplace Transform)                                      Complex s plane

(Fourier Transform)

-p
z=1/(RC)                           -z
p=1/(R12C)
Location of the zero in the left complex
plane
Stability Question

Why the poles must lie in the left half plane?

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

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