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Introduction to Filters

Introduction to Filters. Section 14.1-14.2. Application of Filter. Application: Cellphone Center frequency: 900 MHz Bandwidth: 200 KHz. Use a filter to remove interference. Adjacent interference. Filters. Classification Low-Pass High-Pass Band-Pass Band-Reject Implementation

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Introduction to Filters

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  1. Introduction to Filters Section 14.1-14.2

  2. Application of Filter Application: Cellphone Center frequency: 900 MHz Bandwidth: 200 KHz Use a filter to remove interference Adjacent 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 @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 1+sRC=0→S=-RC

  18. Laplace Transform/Fourier Transform (Laplace Transform) Complex s plane (Fourier Transform) -p p=1/(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. 1+sRC=0→S=-RC

  21. 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

  22. Zero at the origin. Thus phase(f=0)=90 degrees. The high pass filter has a cut-off frequency of 100.

  23. RC High Pass Filter (Review) R12=(R1R2)/(R1+R2) 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 z=1/(RC) p=1/(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 z=1/(RC) p=1/(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

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