Modern radio astronomy backends make heavy use ofdigital signal pro- cessing (DSP)techniques. One of the most commonly used components in digital radio astronomy receivers is thefilterbank (FB), which is es- sentially a contiguous arrangement of filters.FBsare used to transform time-domain signals into their frequency-domain representations (Vaidy- anathan,1993), for the spectral analysis of signals (Boashash,2003;Stoica and Moses,2005) or as transmultiplexers (Fliege,1993). FBsare con- structed in many different schemes and are usually tailored to the spe- cific application. The broadest classification ofFBsdepending on their output lists three types: analysis, synthesis and full. Essentially, the analysisFBis where the signal is broken down into smaller parts and then filtered while the synthesisFBcombines smaller pieces to repro- duce the input signal. A fullFBconsists of both, the analysis and syn- thesisFBs. However, there exist many other classification schemes for
FBswhich are too numerous to list here.FBsare especially relevant for pulsar astronomy given the high data rates and the need to perform ef- ficient computations on these data, especially to remove the effect of
56 t i m i n g & p r o p e r t ie s o f r e c ycl e d p u l s a r s
theISM(i.e. to perform dedispersion).
The signals we receive from pulsars are continuous and time vari- able. These must be translated into discrete signals which can then be processed within filterbanks. As a result of this conversion from a con- tinuous function to its discrete version, as well as the fundamental nature of discrete algebra (and thereforeDSP), the recorded signal suffers from many artefacts. We present below an overview of the most important of these artefacts, along with their origins and mathematical quanti- fication. While a majority of the discussion deals with full reconstruc- tionFBsimplemented withinFPGAs, most present-day pulsar data re- cording systems implement only analysisFBsafter which the data are streamed over high-speed network links to recording computers where the data are dedispersed and stored. However, implementing optim- ised schemes for full-reconstructionFBsrequires effectively the same mathematical treatment. Hence we discuss the implementation of a full-reconstructionFBand only address the question of applicability to present-day pulsar data recording systems inSection 3.6.
The most common challenge of digital filter design is that of fitting complex designs into resource-limited devices likeFPGAsor micro- controllers. Apart from the limits on the available physical memory, these designs need to have minimum computational complexity to im- prove processing times and reduce power consumption. A commonly used scheme to this end is thePFB. This relies on the polyphase decom- position technique, introduced byBellanger et al.(1976) andVary(1979). It remains extremely popular due to the fact that it allows designers to perform the necessary computations at the “lowest rate permissible within the given context” (Vaidyanathan,1998).
Thanks to the polyphase structure and the existence of certain ‘Noble identities’ (Vaidyanathan,1993, see alsoFigure 3.1), it is possible to de- compose the discrete (or sampled) version of the continuous time-series of interest into small parts which can then be processed using limited resources. This allows designers to construct fast, compactDSPalgorithms which form the backbone of most astronomical receivers. The flow dia-
↓M H(z)
≡
H(zm) ↓MFigure 3.1: Noble identities are commut- ative relations which allow us to change the order of the up/down-sampling op- erations and the processing, without af- fecting the signal. H(z) is the filter transfer function and ↑↓M represents an interpolation or decimation opera- tion with factor M. Here the superscript
m denotes that the filters (shown only as
an example) on the right hand side of the equivalence symbol are special versions of those on the left hand side. The top- plot shows the equivalence relation for the action of down sampling, while the bottom plot shows that for up sampling. Derivations showing the validity of these relations can be found inVaidyanathan
(1993).
H(z) ↑M
≡
↑M H(zm)gram of a simple three-channel filterbank is shown inFigure 3.2. A digital signal x[n] is analysed, sub-processed and synthesised to pro- duce a reconstructed signal, x′[n]. This can be explained in the follow- ing manner; the sequence x[n] is first split into three parts by applying a decimate-by-three operation represented by↓ 3. The output of the decimator in each branch can be written as
x[n] ↓ 3 = xd[3n] (3.1) where d= 0, 1, 2. This implies that of the full sequence x[n] the decim-
a r te f a c t s i n p ol y ph a s e fi l te r b a n k s 57
ated sequence xd[n] retains only every 3rdsample starting from the sample number corresponding to the channel number. Translated to the fre- quency domain, this produces an infinite number of replicas of the in- put signal to appear at integer multiples of the input frequency. To be
precise, in terms of the z-transform1, this can be written as (seeEqua- 1Which is defined for a discrete sequence
x[n] of finite length k as X(z) = k n=0 x[n]z−n
where z = e𝔦𝜔is any complex number. See e.g.,Jury(1964)
tions (3.17)and(3.19)for hints on the derivation):
XM(z) = 1 3 2 m=0 X(z1/3Wm) m = 0, 1, 2. (3.2) In this case, these three terms make up a function which is periodic in 𝜔, the associated frequency for discrete-time signal x[n]. This is a basic property for any sequence which has been Fourier transformed (Op- penheim and Willsky,2013). For the first or 0thchannel the terms with
m= 1, 2 are called aliases and are removed by applying a low-pass filter (for real valued signals, while for complex valued signals this becomes a band-pass filter), represented here by H(z). This is often called the anti-aliasing filter. The order of this operation is also shifted as shown inFigure 3.2, using the first Noble identity ofFigure 3.1. This can be fol- lowed by a number of signal processing steps with the condition that any operation must either preserve the phase of the input signal or al- ter it only by a constant value. This is a fundamental requirement of the entirePFBas well.
x[n] H0(z) ↓3 x[0],x[3],x[6],... sub-processing ↑3 F0(z) H1(z) ↓3 x[1],x[4],x[7],... sub-processing ↑3 F1(z) H2(z) ↓3 x[2],x[5],x[8],... sub-processing ↑3 F2(z) Σ x′[n]
Figure 3.2: Schematic of a polyphase filterbank. A time domain signal x[n], where the index represents individual sample points, is filtered by the analysis filter and split into three parts by the decimators such that every third sample is retained by the decimator. The re- duced representations of the input sig- nal are then passed through the subpro- cessing blocks. Finally, the interpolat- ors pad the branch signals with zeros and the synthesis filters remove imaging artefacts, after which the branch signals are recombined the branch signals in or- der, to produce the reconstructed output
x′[n]
After the sub-processing is completed, the signal is now interpolated by the interpolator↑ 3, which inserts zero-valued samples into the sig- nal from each branch such that:
xd[n] ↑ 3 = x[n] for n = d, d + 3, d + 6, ...,
= 0 otherwise. (3.3)
The three streams are then added together to produce the reconstructed signal, x′[n] In the most general case, the z-transform of the output of a polyphase analysis-synthesisFBcan be represented by (Vaidyanathan,
1998):
X′(z) = T(z)X(z) + terms due to aliasing (3.4)
where X′(n) is the reconstructed signal represented in the time-domain,
X(z) is the z-transform of the digitised input signal and T(z) is the mat- rix operator representing the effect of the PFB. For example, inFig- ure 3.2T(z) contains the action of the filters Hm(z), Fm(z) and the sub- processing blocks. It can be shown that aliasing can be completely re-
58 t i m i n g & p r o p e r t ie s o f r e c ycl e d p u l s a r s
moved in specific cases (see e.g.,Crochiere and Rabiner,1976), by mak- ing a proper choice of synthesis filters F0(z), F1(z), etc (see also,Sec-
tion 3.4). Given a set of specifications, a further simplification can be made by using modulated filters, wherein a single prototype filter is mod- ulated by a real or complex function to obtain the analysis and syn- thesis filters.
Further, when T(z) is equal to a pure delay2, T(z) = cz−k, the output of 2The frequency response of T(z) is T(e𝔦𝜔) = T(e𝔦𝜔) e−𝔦𝜔whereT(e𝔦𝜔) = c is the amplitude response and e−𝔦𝜔 is the phase response. If the phase response is modified by a constant then the output resembles the input except for a constant delay in time.
thePFBis said to be ‘perfectly reconstructed’. The simplest design case is that of aperfect reconstruction (PR), two-channelquadrature modu- lated filterbank (QMF)3as demonstrated inSmith and Barnwell(1986)
3The original definition of theQMFis
that of a two-channel FB. It is worth noting that the math remains so sim- ilar even in the case of multi-channel FBsthat they are often also calledQMFs. However, in the discussion here, we spe- cify the number of channels to avoid am- biguity.
andMintzer(1985). However, in the most general case, it is non-trivial to satisfy thePRconditions, although in many cases it is possible to ob- tain a very close approximation ornear-perfect reconstruction (NPR).
Infinite impulse response filter (IIR)4filters can also be used since
4Digital filters can be classified into two
types,IIRandFIR.FIRfilters are those filters whose response to an input im- pulse is a finite duration signal (in other words, the set of filter coefficients is fi- nite) while forIIRfilters the response does not go to zero even after the in- put impulse has disappeared. In theIIR filter, a fraction of the input power is looped back into the filter by design, pre- venting its response from going to zero after a signal has appeared at its input. In this case, the length of the set of filter coefficients is theoretically infinite but in practice this can be limited to some large, finite value. Proper definitions
may be found inRabiner and Schafer
(1978).
they can be implemented using recursive methods but it can be difficult to design universally stableIIRfilters. Hence, mostPFBsare imple- mented usingFIRfilters, since these filters are easy to stabilise across wide bandwidths and can also satisfy the phase linearity requirements which are necessary forPRas discussed inSection 3.4.
In the following sections we first introduce the various sources of er- rors or artefacts that affectPFBsstarting with the artefacts that appear due to the digitisation itself inSection 3.2. The polyphase decomposi- tion is carried out for the two channelQMFand the primary sources of error are demonstrated inSection 3.4.