designing anaylsis and synthesis filterbanks in gnu radio

Designing Anaylsis and Synthesis Filterbanksin GNU Radio

Thomas W. RondeauUniversity of Pennsylvania & Rondeau Research

Shelburne, [email protected]

Timothy J. O’SheaThe Hume Center

Virginia TechArlington, [email protected]

Index Terms—GNU Radio, software radio, SDR, dy-namic spectrum access, DSA, filters, polyphase filter-banks, analysis filterbanks, synthesis filterbanks, dsp

Abstract—This paper provides a look at the polyphase analy-

sis and synthesis filterbanks and filter design tools inGNU Radio. The filterbanks are two powerful blocksfor performing signal detection, collection, and analy-sis operations that are of increasing interest to the cog-nitive radio (CR) and dynamic spectrum access (DSA)communities. We look at how to design and use thefilterbanks to channelize, synthesize, and reconstructsignals. We use a few simple examples for explanatorypurposes but which are readily available to experimentwith and learn from. The processes, descriptions, andtools used in this paper are designed to allow others toadapt filtering and filterbank tools of GNU Radio forother, more complex purposes and applications.

I. INTRODUCTION

Cognitive radio (CR) and dynamic spectrum ac-cess (DSA) problems generally involve discovery andthen use or reuse of spectrum. After the identificationof available spectrum, the DSA system will seek totransmit and receive in it. These spectrum opportuni-ties, however, may be close to other users, and so werisk receiving or creating adjacent channel interfer-ence with other users. Even in the discovery process,we desire efficient structures for isolating and testingareas of spectrum. Polyphase analysis and synthesisfilterbanks are useful tools for these problems. Andwhile there is a nice body of literature about how thefilterbanks themselves work, the design of the filtertaps is still not as widely understood.

GNU Radio as a software radio platform provides anumber of tools to build, use, and analyze polyphasefilterbanks for use in DSA systems. We have the prin-ciple tools necessary to construct analysis and syn-thesis channelizers as well as a large suite of filter de-

sign tools. Use of these, however, is not necessarilystraight-forward.

In this paper, we discuss the principles of usingthe filterbank blocks, the FIR filter design tools, andhow to build the necessary prototype filters requiredto effectively use polyphase filterbanks in GNU Ra-dio. Design decisions for each step are explored toenable further use of these tools and techniques.

As an extension to the filter design process, we willalso look into the process of building and using per-fect reconstruction filters in GNU Radio. A recon-struction filter allows us to channelize spectrum withan analysis filterbank and then recombine any set ofchannels with a synthesis filterbank with no loss ofsignals split between channels. By combining theanalysis and synthesis filterbanks like this, we cancreate arbitrary bandwidth channelizers.

A companion webpage1 is available to downloadthe flowgraphs and filters discussed in this paper.

II. ASSUMPTIONS

As we cover the topics of polyphase analysis andsynthesis filterbanks in GNU Radio, we recognizethat they are fairly advanced topics in signal process-ing. Using these tools requires quite a bit of knowl-edge to appropriately set the parameters for an appli-cation. Here, we will give a very basic understandingof the theory and properties of the filterbank tools,but we will mostly leave the real theoretical details toother papers referenced through the text that discussthese concepts in-depth. Instead, we want to providecontext about the use of these filterbanks in GNU Ra-dio to help enable their use elsewhere. As such, weassume a basic literacy of the GNU Radio2, such asthe what it is, what it does, and the basic principles of

1http://www.trondeau.com/examples/2014/1/23/pfb-channelizers-and-synthesizers.html

2http://gnuradio.org

8th Karlsruhe Workshop on Software Radios

12

a flowgraph and a GNU Radio block. We also assumea basic familiarity here with digital filtering [1].

When discussing specific GNU Radio blocks, wemention the full C++ namespace of the block to makeit easy to find the entry in the GNU Radio Manual3.

We further assume that the radio front-end has suf-ficient selectivity and dynamic range in its out-of-band filtering and digitization to allow the desired sig-nal to be negligibly distorted by hardware. Given thiscondition, we can do the rest of our signal analysis,recovery, and isolation using software filters.

With the highly dynamic nature of software filters,we expect these rapidly reconfigurable and efficientchannelizer techniques to be widely applicable and anappealing option in the development of next genera-tion cognitive and dynamic spectrum radio systems.

III. POLYPHASE FILTERBANKS: ANALYSIS AND

SYNTHESIS FILTERS

We begin with the basic structures of the analysisand synthesis filterbanks. The one is basically the in-verse of the other. Both involve first the design andthen the partition of a prototype filter, where the parti-tioning occurs over a bank of smaller filters, hence thename filterbank. There is then the filtering of the sig-nal through the bank of filters and combining the filteroutput such that they separate or combine the signalsinto the different channels, practically done using adiscrete Fourier transform. Details of the filterbankstructures can be found in [2]. Here, we focus on thedevelopment and understanding of the prototype fil-ter.

One of the fundamental concepts required whenunderstanding what is going on in the channelizer fil-terbank is the ability to analyze the filters in terms ofphase. In Figure 1, we have created a prototype filterand partitioned it among four filters (M = 4). Thepartitioned filters are simply taking every M = 4 tapstarting at tap m where m is the channel number. Wecan think of this as each of the channels is the samefilter with a phase delay of m2π

M .A consequence of the partitioning process is that

we are down-sampling the filter. Each partitioned fil-ter is now at 1/M the rate of the original filter. Thisclues us in to both the nature of the operations beingperformed as well as the design methods for makingthe prototype filter. In the application of the filter-bank, we successively insert samples into the filters,

3http://gnuradio.org/doc/doxygen

which means that each filter is processing a down-sampled version of the signal. Because of the down-sampling of the signal, we design the prototype filterat the full signal sample rate. In a channelizing filter-bank, the sample rate for the design process is the in-put rate, and for a synthesis filterbank, we work withthe output sample rate. This value turns out to be thehighest sample rate the system will ever operate at.

Analysis, or channelizer, filterbanks bring in a widespectrum and split it into equally-space, equal band-width channels. Think of splitting the US FM broad-cast band from 88 to 108 MHz into the one hundred200 kHz FM stations using a single filterbank. Eachstation is 200 kHz wide and separated by 200 kHz foreach channel. Instead of filtering and moving fromits RF channel to baseband for each channel, we pro-cess all one hundred channels at the same time in thechannelizer.

The channelization process uses the concept ofaliasing. While we generally work hard to avoid alias-ing, in the channelizing filterbank, we are aliasing ina very controlled way such that we can pull out theindividual channels. We know from sampling theorythat when we down-sample a signal by a factor of Mwe also create M Nyquist zones that are aliased ontobaseband. We typically avoid the effects of aliasingby removing energy from the parts of the spectrumthat will alias by using a low pass or band pass filter.If we filter out enough of the signal, the aliased zoneswill not fold in and distort our desired signal.

In the channelization process, we want thosealiases. Recall that the filterbanks use the same fil-ter with a different phase. When filtering, each of thealiased zones is processed by each arm of the filter-bank, which has its own phase. At this point, the out-put of each filterbank’s arms is a set of aliases fromthe different channels. We can then despin by apply-ing a correction factor for filterm of exp(j2πkm/M)for a given aliased zone k. The correction factor is re-lated to the phase difference between each of M filterarms. If we despin with this function for a given kover all arms of the filterbank and sum them together,we constructively add the alias from that channelwhile destructively canceling all other aliases fromthe other channels. If we think about that correctionfactor and summing all arms together over every armand every channel, we make the leap to see that this isactually a discrete Fourier transform (DFT) operationthat we can efficiently implement using fast Fouriertransform (FFT) algorithms. This topic is given muchgreater coverage with figures explaining the aliasing


13

Fig. 1. Prototype filter partitioned into four filterbanks. Each filterbank filter is a phase-delayed version of the same filter.

in fred harris’ book on the subject [2].The synthesis filter behaves the same way in re-

verse. It uses an FFT to spin the data properly for thealiases to be separated and then filtered through thefilterbanks. The output samples are then successivelytaken from the filterbanks.

Later, we will modify these structures to allow usto handle signals where the sampling rate is twice thechannel bandwidth. The changes necessary to thesestructures is important and complicated. The workin [3], [4] explains the required changes.

IV. PROTOTYPE FILTER DESIGN IN GNU RADIO

In analysis and synthesis filterbanks, each channelis 1/M th the total bandwidth of the received or trans-mitted, respectively, bandwidth. We either break thesignal apart in the channelizer or put them together inthe synthesizer. We do the same with the prototypefilter. By partitioning the filter, we split the originalfilter into M pieces. So building the prototype meansbuilding a filter capable of filtering a single channelat the full bandwidth (i.e., the sample rate we receiveor transmit on) of all M channels. This is importantto understand that in a polyphase filterbank, the pro-totype filter is designed at the total bandwidth of allM channels.

GNU Radio has tools for building filters that wewill use to construct the prototype filter, including agraphical program called gr filter design. In this ap-plication, we set the type of filter we want, which willbe a windowed low pass FIR filter. We set the samplerate, the end of the pass band, and the start of the stopband along with the stop band attenuation (in dB).The pass band and stop band parameters are speci-fied in units relative to the sample rate. Using the realsample rate of the full channel in samples per second,we can also specify the pass band and stop band pa-rameters in Hertz. An alternative approach is to set

TABLE ICHANNELIZER PARAMETERS

Parameter ValueType FIRStyle Low PassWindow Hann WindowSample Rate 20 MHzFilter Gain 1End of Pass Band 125 kHzStart of Stop band 225 kHzStop Band Atten. 60 dB

these values using a normalized sample rate or 1 andmake the other parameters relative to this.

The other parameter we set for this type of filter isthe window type. In this section and in Figure 1, weuse a Hann window because its parameters are wellunderstood [1]. In latter parts of the paper, we foundit easier to build perfect reconstruction filters usingthe Blackman-harris window [5].

Continuing with our broadcast FM example, weknow that the full spectrum bandwidth is 20 MHz andeach channel is 200 kHz. We make a lowpass filterbased on this. However, we will not design this filterto have the full bandwidth of 200 kHz. The issue isthat in the process of channelizing, we will alias bandstogether at the channel bandwidth, so filtering beyondthe channel boundaries ends up aliasing in pieces ofthe neighboring channels. Instead, we will select pa-rameters that will filter out the edges of our band witha transition period that ends roughly with the channel.The final parameters we select for this experiment areshown in Table I.

We made the start of the stop band slightly beyondthe channel in order to give us a filter with a smallernumber of taps and knowing that any aliasing withthis will be negligible to the FM quality. Even still,


14

the resulting filter consists of 1091 taps, which seemsquite long. However, since we separate this filteramong a filterbank of 100 channels, each channel willactually consist of only 11 taps (i.e., d1091/100e).Computationally, these become very light-weight fil-ters, especially if we can parallelize the computations.A bigger concern may be memory usage, depend-ing on the platform on which the filterbank is imple-mented.

The synthesis filter design issues are the same asthe analysis filter. We can easily take in 100 channelsof frequency modulated audio and synthesize themtogether to broadcast 100 channels of FM.

V. USING A POLYPHASE CHANNELIZER

In the next section, we will look more heavily intousing the synthesis filterbanks. Here, though, we con-tinue with the channelizer example from the proto-type filter we built in Table I.

The GNU Radio flowgraph is shown in Fig-ure 2. The signal is taken from a USRP4,sampled at 20 Msps, and fed to the channel-izer (gr::filter::pfb channelizer ccf ) after some gainadjustment (gr::analog::agc2 cc). In this flow-graph, we take a single channel out for FM de-modulation and simply discard the rest. We usethe Busports5 concept in the GNU Radio Com-panion to handle the large number of connec-tions. The FM demodulation is done with aWBFM receive block (wfm rcv Python block ingr::analog), and this is followed up by a resampler(gr::filter::pfb arb resampler ccf ) to get the samplerates correct for the audio sink (gr::audio::sink). Amultiplier (gr::blocks::multiply const cc) is in therefor volume control.

Setting the receiver to a center frequency of97.9 MHz, we channelize the entire FM broadcastspectrum. The output of a slightly modified flow-graph from Figure 2 where we plot the output of twochannels is shown in Figure 3. The input spectrum isshown along with channels 5 and 10. We only showthe two channels here for space even though all 100channels are available.

VI. DESIGNING RECONSTRUCTION FILTERS

For the FM example, and other signals that haveequal channel bandwidths and spacing, the standard

4http://ettus.com5http://gnuradio.org/redmine/projects/

gnuradio/wiki/Busports

Fig. 2. GNU Radio flowgraph for channelizing the FM broadcastband and pulling out a single channel for FM demod.

analysis and synthesis filterbanks work well. But forcases becoming more relevant in developing ideas ofDSA and cognitive radio, we may not have channelsof equal spacing or bandwidth. For these cases, weneed to modify the standard filterbank structures toallow for arbitrary bandwidth signals. Again, muchof this work on the design of the filterbanks and theprototype filters is already published. We require fewmodifications on the filterbank designs, some modifi-cations on how we partition the filters, and very spe-cific design requirements of the prototype filter. Pa-pers [3], [4] provide a good overview of the chan-nelizer and synthesizer changes required to supportthe reconstruction and [6] provides details on the re-quired changes to handling the filter partitioning inthe updated synthesis filterbank.

First, the changes in the analysis and synthesis fil-terbank comes from the need to be able to completelystitch together two channels already split up by achannelizer filterbank without any loss. That meansthat, unlike before, we must design the filters suchthat the roll-off of the filter hits −6 dB just at thechannel edge. Previously when channelizing, we de-signed the filter to make sure that it was within thebandwidth of the channel to prevent aliasing. Here,we have a requirement that when combining adjacentchannels, the sum of the signals at the edges will re-combine with the correct energy, which would intro-duce adjacent channel aliasing that we need to avoid.

To overcome the aliasing problem, we redesign thefilterbanks to produce channels of the same band-width but now at twice the sample rate. With this, wemove the adjacent channels far outside of our transi-tion width of the filter, and so the signal remaining inthe adjacent channels is specifically designed to hold


15

Fig. 3. Channels 5 (98.9 MHz) and 10 (99.9 MHz) displayed out of all 100 channels of the input FM band spectrum centered at 97.9MHz.

the part of the current channel that spills over into theadjacent channel, but unaliased. From here, we cansee that synthesizing these two channels, using thesame prototype filter, will combine the signals suchthat the edges of both channels add together to createno loss between them.

Following from the channelizer now producingchannels at twice the sample rate, we must alsochange the synthesis filterbanks to perform in a simi-lar way. This structure takes in the channels at twicethe sample rate and produces a signal that combinesthe channels now at twice the output sample rate.That is, synthesize N channels of 2fs produces a sig-nal at 2fsN .

To show how the reconstruction works, we willuse an impulse as the input to the reconstruc-tion filter, which is made up of a channelizer(gr::filter::pfb channelizer ccf ) and synthesis filter-bank (gr::filter::pfb synthesizer ccf ) connected di-rectly to each other as shown in Figure 4. The pro-totype filters for both filterbanks are the same exceptfor a gain factor of M/2 (M is the number of chan-nels) of the synthesis taps so that the output signal isat the same level as the input signals.

GNU Radio’s firdes filter design program lets usfairly easily design the reconstruction filters we need.The firdes tool has a low pass 2 function to designa low pass filter with the gain, bandwidth, transitionwidth, window function, and out-of-band attenuationas design parameters. To hit the required 6 dB roll-offat the channel’s edge, we just need to set the band-

Fig. 4. Diagram of the reconstruction filter to split a signal intomany pieces and reassemble them again.

width of our prototype filter equal to the bandwidthof the channel as the tool defines this as the end ofthe pass band. We then have the window, attenuation,and transition band as degrees of freedom to tweakour filter.

In our analysis here, we will look at six channelswith a sample rate of 2 kHz per channel. We loseno generality in the approach, but these few num-ber of channels allows use to plot and visualize theprocess much easier than a large number of channels.Given this, our filter design parameters are shown inTable II.

Passing an impulse through this structure will re-sult in an impulse out at the receiver at twice the sam-ple rate. While this is an arbitrary example with aclean, known signal, the important consequence ofthis structure and the prototype filter design is thatthe resulting signal has been segmented and put backtogether with almost no change to the signal. Theoutput, shown in Figure 5, shows that the signal is at


16

TABLE IIIMPULSE CHANNELIZER PARAMETERS

Parameter ValueSample Rate 12 kHzFilter Gain 1∗ or 3∗∗

End of Pass Band 1000 HzStart of Stop band 400 HzStop Band Atten. 80 dBWindow Blackman-harris [5]∗ channelizer∗∗ synthesizer

Fig. 5. Output of the reconstruction filter. Each channel filter isdisplayed in the top figure while the reconstructed signal is shownin the bottom figure.

twice the original sampling rate with very minor passband distortion due to the filters.

Next, we show a more complex example that indi-cates better how this concept can be used. We stickwith the impulse as the input signal to give a cleanPSD of how the reconstruction produces perfect sig-nals at the output. Instead of synthesizing all six chan-nels back together, we use two synthesis filterbanks tocombine the first four and last two channels. The de-sign is shown in Figure 6 and the results in Figure 7.Not shown in these figures is the close-up of the re-constructed pass band; however, the two, four, andfull six reconstructed channel examples all have thesame maximum ripple of 5× 10−4 dB. While the rip-ple is dependent on the original filter, the results donot change with the number of channels, making thetechnique scalable to large numbers of channels.

In this example, the more general use of the recon-struction filter becomes more obvious. Any signal canbe broken up and reconstructed with a set of filters of

Fig. 6. Diagram of the reconstruction filter to split a signal intomany pieces and reassemble into a set of four and two channels.

Fig. 7. Output of the reconstruction filter where an impulse issegmented into two signals of four and two channels.

this style, which means that we can now more eas-ily channelize signals of different bandwidths and atdifferent frequencies. One restriction comes into thedesign process, though. To work at twice the samplerate, both the channelizer and synthesizer must havean even number of filters or channels. This may meanusing one more channel than required for some appli-cations.

A. Channel Mappings

The GNU Radio channelizer and synthesis filter-banks have the ability to control the location of theoutput channels using a channel map, or permuta-tion, through the set channel map function call. Thismechanism provides a mapping of where each inputchannel will end up at the output of the filterbank. Forthe channelizer, the default mapping is that channel0 is the 0 Hz channel, or the middle of the incomingfrequency band. Channel numbers increase positivelyin the spectrum and wrap around to the negative halfof the spectrum atM/2. For an even number of chan-


17

nels, theM/2 channel spans the positive and negativeedges of the spectrum.

The channel map function takes a vector of num-bers that is specified by an input channel value thatwill map to the output channel as the index of the ar-ray. This setup gives us flexibility in where and howto direct the channel positions, including being ableto direct the same channel to multiple outputs.

The synthesis filterbank does the same: the indexof the array is the output channel number and thevalue at that index is the input channel. Figure 8demonstrates this. Remember that the output of theoversampling synthesis filterbank is at two times thesampling rate, which gives us twice the number ofchannels. By default, the channel mapping is fromchannel 0 to M for M input channels. All of the in-coming channels from the channelizer are mapped di-rectly to the corresponding output index value. Thatis why in the above graphs we saw the impulse cov-ering the positive half of the spectrum; the other halfof the channels produced by the synthesis filterbankhad no input and therefore produce 0 on the output.In Figure 8, we are using the mapping sequence [10,11, 0, 1, 2, 3]. The channels are mapped in order oftheir output from the channelizer, but we have usedthis mapping technique to shift the entire spectrumdown by two channels.

B. Reconstructing FM

In this final example, we channelize part of the FMbroadcast band into ten 100 kHz channels, select sixof these to recombine, and demodulate the FM to au-dio. The GNU Radio flowgraph is shown in Figure 9that uses a USRP B2006 to pull in a 1 MHz signalcentered at 101.3 MHz. There is a strong FM sta-tion at 101.5 MHz. We channelize the signal intoten channels using the parameters in Table III. Thefirst six of these channels, covering the positive halfof the spectrum, go into the synthesis filterbank. Weterminate the other four channels for this experiment.The synthesizer parameters are shown in Table IV andare nearly the same as for the channelizer. Instead,though, this filter is designed at 600 ksps because weare only combining six 100 kHz channels together.The gain parameter is the half of the total numberof channels in the channelizer, which normalizes theoutput. For convenience, we decimate the output bytwo using a very inexpensive half-band decimatingfilter. The rest of the flowgraph performs the FM de-modulation and some signal processing to allow us

6http://ettus.com

TABLE IIIFM CHANNELIZER PARAMETERS

Parameter ValueSample Rate 1 MHzFilter Gain 1End of Pass Band 50 kHzStart of Stop band 20 kHzStop Band Atten. 80 dBWindow Blackman-harris [5]

TABLE IVFM SYNTHESIZER PARAMETERS

Parameter ValueSample Rate 600 kHzFilter Gain 5End of Pass Band 50 kHzStart of Stop band 20 kHzStop Band Atten. 80 dBWindow Blackman-harris [5]

to adjust the volume of the audio as well as resampleit to match the required rate of the audio subsystem,just as we did above with the FM channelizer. Andthe synthesizer filterbank’s channel map to set the re-sulting signal at DC is [10, 11, 0, 1, 2, 3].

At different places in the flowgraph, we use fre-quency sinks to visualize the PSD of the signal. Fig-ure 10 shows the output of these instrumentationtools, which shows the recovered signal with the stan-dard FM in the middle as well as the HD radio side-bands. On the input spectrum, we added the channelboundaries of the ten channels to show how it wassplit up in the channelizer. In this case, we only re-quire five of the channels to reconstruct the signal, butwe use six to meet the requirement that the filterbanksmust work with an even number of channels.

VII. CONCLUSION

From the details of this paper, we should under-stand the basics of how to design a polyphase filter-bank prototype filter for analysis and synthesis filter-banks. This information gives us the basic knowl-edge of how to build filters with GNU Radio and usethem in the GNU Radio polyphase filterbank blocks.We also discussed the added challenges of design-ing filterbank filters that allow perfect reconstruction,which we can used to break apart and then recombinechannels as we need to in such a way that any signalson channel boundaries are reconstructed perfectly.


18

Fig. 8. Channel mapping in the synthesis filterbank using [10, 11, 0, 1, 2, 3] as a method of moving the spectrum down two channels.

Fig. 9. GNU Radio flowgraph for channelizing and reconstructing a US broadcast FM radio station.

Fig. 10. Output of the reconstruction of the FM signal. This figure shows the input spectrum, the six highlighted channels sent to thesynthesis filterbank, and the output of the filterbank and decimate-by-2 filter.


19

The details of how the filterbanks work is thesubject of many other texts and papers referencedthroughout. GNU Radio provides one of the few areaswhere the code, documentation, and examples existto allow us to study and understand all of the aspects,features, and trade-offs of using these structures.

While this paper focused on the analysis and syn-thesis filterbanks, the same filter design principles areequally valid for any of the other polyphase filterbanktools we have in GNU Radio. Included among theseare the arbitrary resampler and the clock timing syn-chronization blocks. In both of these filterbanks, wecan set the quantization error by selecting the numberof filters we use, which is set to 32 by default. In theterms used above, the number of channels is the num-ber of filters in the filterbank. We then use the sameconcepts of designing the prototype filter by under-standing this simple shift in nomenclature.

REFERENCES

[1] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time Signal Processing (2Nd Ed.). Upper Saddle River, NJ,USA: Prentice-Hall, Inc., 1999.

[2] fred harris, Multirate Signal Processing For CommunicationSystems. Upper Saddle River, NJ: Prentice Hall, 2004.

[3] E. Venosa, X. Chen, and fred harris, “Polyphase analysis fil-ter bank down-converts unequal channel bandwidths with ar-bitrary center frequencies - design i,” in SDR’10-WinnComm,2010.

[4] X. Chen, E. Venosa, and fred harris, “Polyphase analysis fil-ter bank up-converts unequal channel bandwidths with arbi-trary center frequencies - design ii,” in SDR’10-WinnComm,2010.

[5] f. j. harris, “On the use of windows for harmonic analysiswith the discrete fourier transforms,” in Proc. IEEE, vol. 66,Jan. 1978.

[6] f. j. harris, C. Dick, X. Chen, and E. Venosa, “Wideband 160-channel polyphase filter bank cable tv channeliser,” in IETSignal Processing, 2010.


20

designing anaylsis and synthesis filterbanks in gnu radio

Documents