1. Phase Response of FIR Filters Victor M. Tseng Department of Bioengineering, University of Washington

2. Briefing of Presentation Theory behind FIR phase response Example Problem Application

3. Theory FIR Filters can have linear phase This happens when the impulse response is symmetric. The proof deserves mention. This is why we usually use symmetric time windows like the Chebyshev, Triangle, and Rectangle.

4. Proof of the Theorem The second derivative of the phase against frequency must be 0. We’ve got to solve the differential equation with : We can expand this into the real and imaginary components separately: How can we solve this?

5. Finish the Proof Remember If we have an impulse response, such that Then the equation can be solved. In other words, if the coefficients of the FIR filter are symmetric in the time domain, then the phase response will be linear. How can we describe this line? We use Group Delay

6. Theory Group Delay Group delay is the amount of phase that is added to the preceding frequency’s phase. Since it is linear over the passband, we only need to know. for a Q order filter. The phase at a certain frequency is

7. By the way How did they derive G? Remember the shift theorem: The axis of symmetry is So the phase shift is

8. Theory More on Group Delay The delay is only dependent on the sampling frequency and the order of the filter. Applies to a whole variety of shapes! Let’s look at an example, so we can see how it works out.

9. Example Phase response of a Chebyshev Window Problem: What does the phase response of a 64 order Chebyshev window with 50 dB sidelobe attenuation look like?

10. Example Phase response of a Chebyshev Window The Bode Plot

11. Example Phase response of a Chebyshev Window The Phase Plot, Wrapped

12. Example Phase response of a Chebyshev Window Which discontinuities can we keep, and which ones are wrapped? This depends on the relative sign of the real and imaginary components. If the CW angle is greater than 180º (-Re and +Im), then it will we wrapped around 360º. So we look at the sign of the components and subtract 360º wherever we see this case.

13. Example Phase response of a Chebyshev Window Sign of Re (smooth) and Im (dotted)

14. Applications Microscopy In phase contrast microscopy, we want 25% phase shift of diffracted light. The diffracted light passes through a ring in the phase plate which slows rays down by 50% wavelength. We want linear behavior as a function of angle. This optimizes contrast.

15. Application Pedals Guitar effects: The phaser applies a low-frequency oscillation to the phase. Notable uses of the phaser in rock music: Nirvana, “Smells Like Teen Spirit”, Solo The Eagles, “Life in the Fast Lane” Ashlee Simpson, “Boyfriend”

16. [Except from Animal Planet]: Well, that’s not exactly what I had in mind.