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

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

19. 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: 1.-45 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

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 p=1/(RC) Zero at DC. (Fourier Transform) (Laplace Transform) -p Location of the zero in the left complex plane Complex s 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) R 12 =(R 1 R 2 )/(R 1 +R 2 ) A pole and a zero in the left complex plane.

24. 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

25. 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

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