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Demystifying the PFB Lifting the hood to a key technique in the Radio Astronomers Toolbox Andrew van der Byl Signal Processing Engineer (CBF) [email protected] CASPER 2017
Demystifying the PFBLifting the hood to a key technique in the Radio Astronomers Toolbox
Andrew van der BylSignal Processing Engineer (CBF)
CASPER 2017
Keeping it simple: What you’re not going to get….
Outline: Polyphase Filter Bank
• PFB in the real world
• The why...
• Diving under the hood
• PFB: A CASPER Tool
Polyphase Filter: The CASPER tool
PFB in the real world
Polyphase Filter: The why…
Bonjour!
Part1: Lets do some translating…
PFB: Mixing things up...
H(z)
Digital Low-Pass
x[n]M-to-1
y[n,k]y[nM,k]
e-jθkn
Spectrum: Complex Filtered Outputf
Spectrum: Real Baseband FilterH0
f
fk
Spectrum: Down-sampled Output Signalfk
fs/M f-fs/M
fk fs/Mf
Spectrum: Translated Input Signalfk fs/M
Channel of interest
f
Spectrum: Input Signal
PFB: Making a switch....
H(z)
Digital Low-Passx[n]
e-jθkn
M-to-1
y[n,k]
y[nM,k]
Down convert 1st, LP filter 2nd
fk fs/M
Channel of interest
f
fkfs/M
Channel of interest
f
ffk fk
fs/Mf-fs/M
=H(ze-jθk)
Digital Band-Pass
x[n]
y[n,k]
e-jθkn
M-to-1y[nM,k]
BP filter 1st, down convert 2nd
fk fs/M
Channel of interest
f
fkfs/M
Channel of interest
f
fk
fs/Mf-fs/M
ffk
But wait....why down convert samples that are to be discarded?
H(ze-j2πk/M)
Digital Band-Passx[n]
y[n,k]
M-to-1
y[nM,k]
H(ze-jθk)
Digital Band-Passx[n]
y[n,k]
e-jθkn
M-to-1
y[nM,k]
H(ze-jθk)
Digital Band-Passx[n]
y[n,k]
e-jMθkn
M-to-1
y[nM,k]
But what about the filter?It is still at full rate!
Not for long!
When moving the resampler, the complex sinusoid is also
down-sampled
Limit center frequencies to integer multiples of the output sample rate
Part2: Lets do some transforming…(starting with the low-pass filter)
An interesting twist...
H[ZM]
M-to-1
y[n] y[nM]x[n]
Filter, then down sample
H[Z]
M-to-1
y[nM]x[n]
Down sample, then filter
Under what conditions will a filter operate on every M input samples?
Divy them up into M paths!
H0[ZM]
M-to-1
y[n] y[nM]
x[n]H1[ZM]
H2[ZM]
HM-1[ZM]
Z-1
Z-2
Z-(M-1)
How Noble…
An interesting twist...
H0[Z]M-to-1
y[nM]x[n]
H1[Z]
H2[Z]
HM-1[Z]
Z-1
Z-2
Z-(M-1)
M-to-1
M-to-1
M-to-1Move the down sampling stage
Synchronous switches
An interesting twist...
H0[Z]
y[nM]x[n]
H1[Z]
H2[Z]
HM-1[Z]
Input commutator
Each input sees every 1/M samples
One more step to complete the transformation to an M-path down converter…
This M-to-1 down samplingaliases to baseband the spectral terms residing at multiples of the
output sample rate
Introducing the Polyphase Filter...
y[nM,k]x[n]
HM-1[Z]
H0[Z]
H1[Z]
H2[Z]
ej2π0k/M
ej2π1k/M
ej2π2k/M
ej2π(M-1)k/M
This is the Polyphase Filter
Wait! You’ve broken Nyquist!
Cancelling the aliases...
y[nM,k]x[n]
HM-1[Z]
H0[Z]
H1[Z]
H2[Z]
ej2π0k/M
ej2π1k/M
ej2π2k/M
ej2π(M-1)k/M
This is the Polyphase Filter
Phase correction
Each path has a unique phase profile
We cancel the aliases
Déjà vu...
Wait a minute…This looks like a DFT!
y[nM,k]x[n]
HM-1[Z]
H0[Z]
H1[Z]
H2[Z]
ej2π0k/M
ej2π1k/M
ej2π2k/M
ej2π(M-1)k/M
The DFT performs the task of separating the channels after the polyphase filter
DFT defines the channel spacing(one-Mth of the
input sample rate)
And the taps...?
y[nM,k]x[n]
HM-1[Z]
H0[Z]
H1[Z]
H2[Z]
ej2π0k/M
ej2π1k/M
ej2π2k/M
ej2π(M-1)k/M
Extend the filter width (multiples of M)(multiples of the summation length)
These terms are periodic
I think I have seen this before…
Source: CASPER wiki
And the taps...?
- 3 - 2 - 1 0 1 2 3 4 F r e q u e n c y ( n o r m a l i z e d t o c h a n n e l c e n t e r )
- 1 2 0
- 1 0 0
- 8 0
- 6 0
- 4 0
- 2 0
0
Mag
nitu
de R
espo
nse
(dB)
F i l t e r B a n k F r e q u e n c y R e s p o n s e
F F T4 - t a p P F B8 - t a p P F B
The CASPER PFB: Lets take another look
y[nM,k]x[n]
HM-1[Z]
H0[Z]
H1[Z]
H2[Z]
ej2π0k/M
ej2π1k/M
ej2π2k/M
ej2π(M-1)k/M
Right, so what controls can we tweak?
Looking back…
• PFB in the real world
• The why...
• Diving under the hood
• PFB: A CASPER Tool
y[nM,k]x[n]
