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…

Part1: Lets do some translating… Bonjour!

PFB: Mixing things up... Spectrum: Input Signal Channel of e -jθ k n Digital Low-Pass interest M-to-1 x[n] H(z) y[nM,k] y[n,k] f f k f s /M Spectrum: T ranslated Input Signal f f k f s /M Spectrum: Real Baseband Filter H 0 f Spectrum: Complex Filtered Output f k f Spectrum: Down-sampled Output Signal f k f -f s /M f s /M

PFB: Making a switch.... Down convert 1 st , LP fjlter 2 nd BP fjlter 1 st , down convert 2 nd e -jθ k n e -jθ k n Digital Band-Pass M-to-1 M-to-1 Digital Low-Pass y[nM,k] y[nM,k] x[n] = x[n] H(z) H(ze -jθ k ) y[n,k] y[n,k] Channel of Channel of interest interest f f f k f k f s /M f s /M Channel of Channel of interest interest f f f s /M f s /M f k f k f f f f k f k k f k f f -f s /M f s /M -f s /M f s /M

But wait....why down convert samples that are to be discarded? e -jθ k n M-to-1 Digital Band-Pass y[nM,k] x[n] H(ze -jθ k ) y[n,k] When moving the resampler, the complex sinusoid is also down-sampled e -jMθ k n M-to-1 Digital Band-Pass x[n] y[nM,k] H(ze -jθ k ) y[n,k] But what about the fjlter? Limit center frequencies It is still at full rate! to integer multiples of the output sample rate Not for long! M-to-1 Digital Band-Pass x[n] y[nM,k] y[n,k] H(ze -j2πk/M )

Part2: Lets do some transforming… (starting with the low-pass fjlter)

An interesting twist... Filter, then down sample Down sample, then fjlter M-to-1 M-to-1 H[Z M ] H[Z] x[n] x[n] y[n] y[nM] y[nM] Under what conditions will a fjlter operate on every M input samples? H 0 [Z M ] Divy them up into M paths! Z -1 H 1 [Z M ] M-to-1 x[n] Z -2 H 2 [Z M ] y[n] y[nM] How Noble… Z -(M-1) H M-1 [Z M ]

An interesting twist... Move the down M-to-1 sampling stage H 0 [Z] M-to-1 Z -1 H 1 [Z] x[n] M-to-1 y[nM] Z -2 H 2 [Z] M-to-1 Z -(M-1) H M-1 [Z] Synchronous switches

An interesting twist... Each input sees every 1/M samples Input commutator H 0 [Z] H 1 [Z] x[n] y[nM] H 2 [Z] H M-1 [Z] One more step to complete the transformation This M-to-1 down sampling to an M -path down converter… aliases to baseband the spectral terms residing at multiples of the output sample rate

Introducing the Polyphase Filter... e j2π0k/M This is the Polyphase Filter H 0 [Z] e j2π1k/M H 1 [Z] e j2π2k/M y[nM,k] x[n] H 2 [Z] e j2π(M-1)k/M H M-1 [Z] Wait! You’ve broken Nyquist!

Cancelling the aliases... e j2π0k/M This is the Polyphase Filter H 0 [Z] Each path has a e j2π1k/M unique phase profjle H 1 [Z] e j2π2k/M y[nM,k] x[n] H 2 [Z] e j2π(M-1)k/M H M-1 [Z] We cancel the aliases  Phase correction

Déjà vu... e j2π0k/M DFT defjnes the channel spacing H 0 [Z] ( one-M th of the input sample rate) e j2π1k/M H 1 [Z] e j2π2k/M y[nM,k] x[n] H 2 [Z] e j2π(M-1)k/M H M-1 [Z] Wait a minute…This looks like a DF The DFT performs the task of separating the channels after the polyphase fjlter

And the taps...? e j2π0k/M Extend the fjlter width (multiples of M ) These terms (multiples of the summation length) H 0 [Z] are periodic e j2π1k/M H 1 [Z] e j2π2k/M y[nM,k] x[n] H 2 [Z] e j2π(M-1)k/M H M-1 [Z] I think I have seen this before… Source: CASPER wiki

And the taps...? F i l t e r B a n k F r e q u e n cy R e sp o n se 0 F F T 4 - t a p P F B 8 - t a p P F B - 2 0 - 4 0 d B ) ( R e sp o n se - 6 0 u d e t M a g n i - 8 0 - 1 0 0 - 1 2 0 - 3 - 2 - 1 0 1 2 3 4 F r e q u e n cy ( n o r m a l i ze d t o ch a n n e l ce n t e r )

The CASPER PFB: Lets take another look e j2π0k/M H 0 [Z] e j2π1k/M H 1 [Z] e j2π2k/M y[nM,k] x[n] H 2 [Z] e j2π(M-1)k/M H M-1 [Z]

Right, so what controls can we tweak?

Looking back… • PFB in the real world y[nM,k] x[n] • The why... • Diving under the hood • PFB: A CASPER Tool

Questions & Comments Andrew van der Byl avanderbyl@ska.ac.za