Introduction to Filters Section 14.1-14.2. Application of Filter Application: Cellphone Center frequency: 900 MHz Bandwidth: 200 KHz Adjacent interference.

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Presentation transcript:

Introduction to Filters Section

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

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 Must not alter the desired signal! Sharp Transition in order to attenuate the interference Not desirable. Alter Frequency content. Affect selectivity

Low-Pass Example How much attenuation is provided by the filter?

Answer 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 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

Adding a Zero

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 p=1/(RC) (Fourier Transform) (Laplace Transform) -p Location of the zero in the left complex plane Complex s plane

Rules of thumb: (applicable to a pole) Magnitude: 1.20 dB drop after the cut-off frequency 2.3dB drop at the cut-off frequency Phase: deg at the cut-off frequency 2.0 degree at one decade prior to the cut-frequency 3.90 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 p=1/(RC) Zero at DC. (Fourier Transform) (Laplace Transform) -p Location of the zero in the left complex plane Complex s 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) R 12 =(R 1 R 2 )/(R 1 +R 2 ) A pole and a zero in the left complex plane.

Laplace Transform/Fourier Transform (Low Frequency) z=1/(RC) p=1/(R 12 C) (Fourier Transform) (Laplace Transform) -p Location of the zero in the left complex plane Complex s plane -z

Laplace Transform/Fourier Transform (High Frequency) z=1/(RC) p=1/(R 12 C) (Fourier Transform) (Laplace Transform) -p Location of the zero in the left complex plane Complex s plane -z

Stability Question Why the poles must lie in the left half plane ?

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