Phase Response of FIR Filters Victor M. Tseng Department of Bioengineering, University of...
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Transcript of Phase Response of FIR Filters Victor M. Tseng Department of Bioengineering, University of...
Phase Response of FIR Filters
Victor M. Tseng
Department of Bioengineering, University of Washington
Briefing of Presentation
Theory behind FIR phase response Example Problem Application
TheoryFIR 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.
Proof of the TheoremThe 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?
02
2
f
0
1
1
1
1arctan
2
2
2222
2
2
2
ffffff
0Im
ReRe
ImIm
Im
12...
ImRe
ImRe
ReIm
ReIm
Im
1ImRe
ReIm
Im
1
ImRe1
ImRe1
3
2
2
2
2
2222
f
XX
f
XX
df
X
X
f
XX
f
X
f
X
f
XX
f
X
f
X
fXf
XX
f
XX
XXX
XX
XX
Im
Re
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
i
Nhi
Nh
2
1
2
1
N
n
N
i N
nfifxihX
0 0
2cosRe
N
n
N
i N
nfifxihX
0 0
2sinIm
TheoryGroup 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
1f
sf
Q
fG
2
1
N
Gmfm
2
By the way How did they derive G?
Remember the shift theorem:
The axis of symmetry is
So the phase shift is
2
1N
nsymmetry
)(' 2
12 zHezHezHzzH
fNjfnjn symmetrysymmetry
TheoryMore 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.
ExamplePhase response of a Chebyshev Window
Problem: What does the phase response of a 64 order Chebyshev window with 50 dB sidelobe attenuation look like?
ExamplePhase response of a Chebyshev Window
The Bode Plot
ExamplePhase response of a Chebyshev Window
The Phase Plot, Wrapped
ExamplePhase 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.
ExamplePhase response of a Chebyshev Window
Sign of Re (smooth) and Im (dotted)
ApplicationsMicroscopy
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.
ApplicationPedals
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”
[Except from Animal Planet]: Well, that’s not exactly what I had in mind.