Demystifying the PFB Lifting the hood to a key technique in the - - PowerPoint PPT Presentation

Demystifying the PFB

Lifting the hood to a key technique in the Radio Astronomers Toolbox

Andrew van der Byl

Signal Processing Engineer (CBF) avanderbyl@ska.ac.za

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 Output

f

Spectrum: Real Baseband Filter H0

f

fk Spectrum: Down-sampled Output Signal fk

fs/M

f

• fs/M

fk fs/M f

Spectrum: T ranslated Input Signal

fk fs/M Channel of interest

f

Spectrum: Input Signal

PFB: Making a switch....

H(z)

Digital Low-Pass

x[n]

e-jθkn

M-to-1

y[n,k]

y[nM,k]

Down convert 1st, LP fjlter 2nd

fk fs/M

Channel of interest

f fk fs/M

Channel of interest

f f f

k fk

fs/M

f

• fs/M

=

H(ze-jθk) Digital Band-Pass

x[n] y[n,k]

e-jθkn

M-to-1

y[nM,k]

BP fjlter 1st, down convert 2nd

fk fs/M

Channel of interest

f fk fs/M

Channel of interest

f fk

fs/M

f

• fs/M

f f

k

But wait....why down convert samples that are to be discarded?

H(ze-j2πk/M)

Digital Band-Pass

x[n] y[n,k]

M-to-1

y[nM,k]

H(ze-jθk)

Digital Band-Pass

x[n] y[n,k]

e-jθkn

M-to-1

y[nM,k]

H(ze-jθk)

Digital Band-Pass

x[n] y[n,k]

e-jMθkn

M-to-1

y[nM,k]

But what about the fjlter? 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 fjlter)

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 fjlter Under what conditions will a fjlter 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-1

Move 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 sampling aliases to baseband the spectral terms residing at multiples of the

• utput 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 profjle

We cancel the aliases 

Déjà vu...

Wait a minute…This looks like a DF

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 fjlter

DFT defjnes 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 fjlter 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

1 2 3 4

F r e q u e n cy ( n o r m a l i ze d t

• ch a n n e l

ce n t e r )

• 1 2 0
• 1 0 0
• 8 0
• 6 0
• 4 0
• 2 0

M a g n i t u d e R e sp o n se ( d B ) F i l t e r B a n k F r e q u e n cy R e sp o n se

F F T 4 - t a p P F B 8 - 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]

Questions & Comments

Andrew van der Byl avanderbyl@ska.ac.za