Frequency Domain Channelizer:A computationally efficient method for non-uniform channelization

Gereon A. Such GEREON.SUCH@UNIBW.DEMatthias G. Schraml MATTHIAS.SCHRAML@UNIBW.DEAndreas Knopp ANDREAS.KNOPP@UNIBW.DE

Bundeswehr University Munich, Chair of Signal Processing, Werner-Heisenberg-Weg 39, 85579 Neubiberg, Germany

Abstract

We present both, generalized theory and our im-plementation of a frequency domain channel-izer suited for multicarrier Software Defined Ra-dio receivers in terms of performance and ap-plicability. Frequency domain channelizers fillthe gap between multiple FFT/FIR-based chan-nelizers and the polyphase channelizer consid-ering dissimilar bandwidth carriers in a signal.Our open source implementation gr-FDC outper-forms a combination of Frequency Xlating FFTfilters at about ten channels and is introduced asan implementation example.

1. IntroductionThe demand for services in digital radio communicationrises in every part of the society. Prominent examplesare multimedia streaming direct-to-home, the internet ofthings and connected cars for autonomous driving. Cer-tain frequency bands are typically licensed to a certain typeof communication, e.g. for television broadcasting (ITU,2016).

However, the services often differs dramatically in the de-mand of bandwidth. Using a large bandwidth channel for anarrow-band service wastes the expensive frequency spec-trum resource. The scarcity in the radio frequency (RF)spectrum leads to developments in dynamic spectrum allo-cation. Transmitting in currently unused frequencies canmaximize the overall throughput of radio communicationservices. This is often done with Frequency Division Mul-tiple Access (FDMA) schemes. Hence, the spectrum canbe populated with many distinct carriers. These carriersare often non-uniform in their center frequencies and sym-bol rates. Applying a software defined radio (SDR) receiverwith a high-bandwidth radio front-end at the base stations

Proceedings of the 8 th GNU Radio Conference, Copyright 2018by the author(s).

to such a scenario may be more advantageous than one ra-dio front-end per carrier concerning flexibility or cost effi-ciency.

The extraction of many channels from a large bandwidth isa computationally expensive task. The polyphase channel-izer is an efficient method to extract equally spaced carri-ers (Harris et al., 2003). For non-uniform carrier frequen-cies with arbitrary bandwidths, the current approach is toapply a rotator and a filter per carrier. For a few carriersthis method is absolutely reasonable. For many carriers,however, the frequency domain channelizer (FDC) can bea very efficient option.

In this paper the theory of FDC is presented in Section 2alongside of our GNU Radio implementation gr-FDC inSection 3. Moreover it is compared to a Frequency XlatingFFT filter per carrier in Section 4.

2. Frequency Domain Channelization2.1. General Algorithm

An FDC generally transforms the complete received signalto the frequency domain (hence the name), extracts the cor-responding frequency bins of the desired spectral part andtransforms only this part back to the time domain. Since thetime signal at the output should be continuous, several pa-rameters must be considered as stated in Borgerding, 2006.The most important parameters are

Block length N ∈ N, (1)Overlap length L ∈ N with L < N, (2)

Decimation factor d ∈ Q, (3)

where N are positive integer values and Q is any fractionof two positive integers. These parameters are well knownfrom the frequency domain filter method called overlap-save (sometimes also called overlap-discard). Please referto a signal processing textbook like Harris, 1987 for in-formation about this method. Since the FDC is supposedto decimate the signal, d ≥ 1 is an additional require-ment. The decimated channel block lengthNd and the dec-imated overlap length Ld must have an integer length, i.e.

Frequency Domain Channelizer

Nd = N/d ∈ N and Ld = L/d ∈ N. A more practicalconsideration of the parameters is given in section 2.3.

Fig. 1 is the graphical presentation of the general FDC al-gorithm. The first step of an FDC is to divide the vector ofa digital complex baseband signal r with a sample rate fsin consecutive blocks of length N . Each block starts withan overlap of L samples from the last block, while the firstblock is padded with L zeros. These blocks rk are trans-formed to the frequency domain by an N -point Discrete-Fourier-Transformation (DFT) to

Rk = DFT (rk) (4)

in the second step, where k is the index of subsequentblocks.

. . .Input signal r

0L r [1:N − L] ← Block r1

r [N − 2L:2N − 2L]Block r2→

...

Block rk

1.B

lock

-wis

eO

verl

ap-s

ave

2. Block-wise N -point DFT

= Rk

w

3.E

xtra

ctio

nan

dW

indo

win

g

4. Block-wise Nd-point IDFT

c1 [Ld + 1:Nd] . . .

5. Discard Decimated Overlap andChannelized Signal Reconstruction

Figure 1. FDC structure.

The mixing and filtering operations and the decimation ofthe channelizer happens in the frequency domain in the

third step. Mixing the signal with a negative frequency inthe time domain is equivalent to the circular rotation of theDFT bins to the left. The rotation is given by a zero-basedoffset o, i.e. with o = 0 no rotation is performed. Hence,the baseband frequency is lowered by fo = ofs/N . How-ever, the possible center frequencies of the FDC are directlyrelated to the DFT block length N , so a larger block lengthgives a finer granularity for the selection of the mixing fre-quencies.

The filtering of the FDC, like in the overlap-save method,can use a maximum of L + 1 lowpass finite impulseresponse filter coefficients. These coefficients are zero-padded and transformed into the frequency domain with anN -point DFT. Since we assume that the coefficients forma suitable antialiasing filter for the selected decimation fac-tor, there are only Nd relevant bins in the DFT represen-tation of the lowpass filter, which we call windowing vec-tor w.

Hence, instead of rotating all the bins and then use only thefirst Nd bins for further processing, index based selectionof the bins of interest can be performed. So it is sufficientwith good accuracy to extract a contiguous section of Rk

with length Nd and discard the other bins. The extractedset of DFT bins is weighted with the Nd relevant bins ofthe windowing vector w. However, if the used lowpass fil-ter is not a good antialiasing filter, serious distortions canhappen and the output of the FDC is different to the outputof a channelizer based on a rotator and a filter working inthe time domain. In the FDC, the window w is typicallycomplex-valued which is explained in Section 2.2. The re-sulting output signal in the frequency domain is

Ck = w · Rk [o+ 1 : o+Nd] (5)

with (·) being the Hadamard product (element-wise multi-plication).

In the last steps, the time domain channelized and deci-mated signal c is obtained by the Nd-point inverse DFT(IDFT) of all channelized blocks Ck, discarding the over-lap, i.e. removing the first Ld samples of each ck, andreconstructing the full vector by the concatenation of allblocks:

ck = IDFT (Ck) [Ld + 1 : Nd] (6)

c =[c1, c2, · · · , cN−L

d

](7)

2.2. Subsequent Phase Shifts and its Compensation

Extracting a certain range of DFT bins in the third step inFig. 1 is equivalent to baseband mixing. So the channelized

Frequency Domain Channelizer

signal in the time domain corresponds to

c(n) = r(n) exp

(−j2πfon

fs

), (8)

where fo = ofs/N is the offset frequency selected by theoffset o.

Due to the overlap-save procedure some samples are re-peated in the blocks. However, the signal must be phasecontinuous even with these L samples which are repeatedand discarded afterwards. Hence the phase at the begin-ning of the second block (n = N −L+1) must be equal tothe phase at the beginning of the first block (n = 1). Thenecessary requirement for the phase difference is given by

exp

(−j2πo(N − L)

N

)= 1. (9)

Therefore, the argument of the phase difference must be aninteger multiple of 2π. With respect to the brevity of furtherequations we define the inverse relative overlap factor Ψ tobe

Ψ =N

L. (10)

The phase difference can be rewritten to

exp

(−j2π

(o− o

Ψ

))= exp

(j2π

o

Ψ

), (11)

since the offset o is an integer value. The requirement of (9)is fulfilled if the offset o is 0 or an integer multiple of theinverse relative overlap Ψ (Borgerding, 2006). Otherwisephase shifts occur on block transitions.

The first option to prevent the phase shifts is to ad-just the offset o accordingly to be an integer multiple ofN/GCD(N,L), where GCD(·) is the greatest common di-visor of both arguments. However, this can dramaticallyreduce the possible center frequencies the FDC can pro-cess, especially in cases of fractional values for Ψ or smallblock lengths N .

Another option is to compensate the block-wise phase dif-ference by replacing the windowing vector w in the thirdstep in Fig. 1 with

wk = w exp

(−j2π o

Ψ(k − 1)

). (12)

The windowing vector wk can be implemented by a look-up table, since the phase values repeat after a certain num-ber of frames. In general N/GCD(N, oL) vectors must besaved, where for the worst-case the offset is set to o = 1. Inmodern digital signal processing units or software definedradio environments, the additional memory requirement ismost likely irrelevant but has to be considered for every ap-plication.

2.3. FDC Parameter Considerations

We further analyze the parameters from Section 2.1, asthey crucially define the possible channel carrier frequen-cies and channel bandwidths, since these cannot be chosencontinuously for an FDC but are limited by the overlap-save method and possible software or algorithmic limita-tions for DFT block lengths.

In every case, N , Nd, L and Ld need to be integers, sincehalf samples do not exist and an interpolation is not consid-ered for performance reasons. Hence the maximum deci-mation factor dmax is the GCD(N,L). Therefore, all pos-sible decimation factors are given by

d =dmax

m, (13)

where m ∈ N and m ≤ dmax. All possible bandwidths ofthe channelized signals are fs,d = fs/d.

To assure a comfortable set of possible decimation factorsmatching to several bandwidths alongside an easy compu-tation of the (I)DFTs, we can limit some parameters to cer-tain values. If we choose N and L to be power-of-2 val-ues and the possible decimation rates to all power-of-2 val-ues up to L, the maximum of Ψ different windowing vec-tor sets have to be saved to prevent phase shifts on blocktransitions. This parameter limitation is considerable forthe Radix-2 Fast-Fourier-Transform (FFT) algorithm andits inverse to increase the performance. Even though thecommon FFTW3 library (Frigo & Johnson, 2003) allowsnon-power-of-2 transformation lengths on general purposeprocessors, performance gains can be acchieved by power-of-2 block lengths. Our FDC implementation takes advan-tage of this performance gain and restricts block lengthsand bandwidths to power-of-2 values.

3. Presenting gr-FDCOur multi-purpose implementation of an FDC is theGNU Radio gr-FDC module developed and shared onGitHub (Such, 2018). It contains three core signal pro-cessing blocks utilized by a wrapping Python hierarchicalblock, a custom waterfall block to visualize extracted chan-nels and additional auxiliary blocks. The three core blocksrepresent different operational modes to operate on a fre-quency domain vector, extract and process a specific band-width and reconstruct the channelized time signal.

• The first operational mode is the Throughput mode.The output is a constant stream of complex samplesrepresenting the time signal of a channel with con-stant carrier frequency and constant bandwidth. TheThroughput mode is meant to operate on constantstream channels.

Frequency Domain Channelizer

• The second block is the FixedCarrier mode, whichalso operates on a fixed carrier frequency and band-width, but it detects power differences in the channel,buffers and finally emits signal via message passingas PDU or stores each buffered signal in a separatefile. The FixedCarrier mode is designed to operate onTime Division Multiple Access (TDMA) channels. Itis possible to periodically emit the buffered channelsignal after a certain amount of blocks processed, tobe able to handle channels with long activation time,i.e. speech in close to realtime.

• The third block called Detection is not defined by acarrier frequency and a fixed bandwidth but a startand stop frequency, which define a segment. In thissegment, carriers are determined by rising and fallingedges of the power spectral density and a temporarychannel is tailored to it adjusting carrier frequency andbandwidth according to detected parameters. The ex-tracted channels are as well as in FixedCarrier modeemitted as a PDU via message passing or stored inseparate files. Like in FixedCarrier mode, the pre-emptive emission of the channel buffer is possible.

To reduce the necessary numerical complexity andavoid noise influence on determination of the carri-ers, especially if a carrier is still active, the number ofsamples for the power spectral density computation isreduced by averaging. The number of samples deter-mined by the parameter Minimum Channel Distancespecified by the developer. On the other hand, thisaveraging factor reduces the resolution to determinethe channel’s start and stop frequencies. This is whyin Fig. 2 the Detection channels suffer from an off-set. Nevertheless, a bandwidth puffer can be chosento securely acquire all channels.

• The hierarchical block Frequency Domain Channel-izer is a wrapping convenience block which takes careof parameter checks and connections. We stronglyrecommend to use this block instead of connecting theblocks yourself to avoid false parametrization. It ispossible to input a stream of complex or real samplesrepresenting the time signal, or, if you plan to imple-ment a high performance system, input the overlappedand Fourier transformed vector to be able to imple-ment the Fourier transformation on another machineor hardware.

• The visualizing block WaterfallMsgTagging is a PyQtblock displaying a waterfall diagramm and visualizingthe emitted channels based on the PDU meta data. Itis tailored to gr-FDC in terms of displaying and notdiscarding any blocks, which is why we did not de-rive it from QT GUI Waterfall Sink. The Throughputchannels are not visualized, since this mode does not

emit any PDUs. Fig. 2 illustrates the graphical out-put. Currently, only the frames on detected channelsare drawn, no further meta information is provided.

Figure 2. gr-FDC WaterfallMsgTagging Output

The wrapping channelizer and the waterfall block are writ-ten in Python and the signal processing blocks are writtenin C++ utilizing C++ STL in terms of data structures, pri-marily std::vector and std::deque, and in blockthreading in terms of std::thread. For numerical oper-ations, both the Vector Optimized Library of Kernels (FSF,2015) and FFTW3 (Frigo & Johnson, 2003) via GNU Ra-dio wrapper are used making gr-FDC efficient for live sys-tems.

All channelizing blocks operate on vector lengths of power-of-2. The main reasons for this are performance evaluationof the FFT and IFFT, avoidance of errors by design and al-gorithmic convenience to find a matching decimation fac-tor representing an integer extraction range according to afloating point bandwidth. The result is a slight carrier fre-quency deviation from the desired frequency and the band-width according to the used integer decimation factor. Ac-tual carrier frequency and bandwidth information are pro-vided in the meta data of each PDU or are printed on ini-tialization by the Throughput channels. For Throughputchannels, this is not a satisfying solution but the only rea-sonable one we see right now, since the actual extractionbandwidth and carrier frequency is determined on initial-ization. You can use these parameters to adjust subsequentblocks if they depend on the actual sample rate.

Since the carrier frequency and carrier phase usually haveto be synchronized in subsequent steps after the channel-ization, these deviations might be compensated in your ap-plication anyways, but to be fair compared to FrequencyXlating FFT blocks, a rotator might be applied on the chan-nelized data. If the parameters are set properly, the up-per bound for carrier frequency deviation f∆ is the equiva-lent of one sample of the frequency domain vector of blocklength N , and thus f∆ ≤ fs/N with sample rate fs.

Currently, all applied windows in the frequency domainare lowpass filters, and a matched filter has to be applied

Frequency Domain Channelizer

on the channelized signal. As stated in Section 5, for theThroughput and FixedCarrier modes, this is planned butnot implemented yet. Furthermore, the window on the timesignal before transforming the block-wise signal is rectan-gular to assure a reversible Fourier Transformation, whichleads to a leakage effect visible in the Waterfall diagram.Currently, we accept this since we do not want to sacri-fice performance, but consider several solutions to resolvethis, for instance by performing a convolution of a minimalBlackman-Harris window in the frequency domain.

4. Performance ComparisonCompared to a parallel Frequency Xlating FFT Filter com-bination, gr-FDC yields a significant performance gain,where performance means a reduction of computationalcomplexity without a loss of signal quality.

4.1. Computational Efficiency

For evaluation of the computational performance ofgr-FDC compared to a Frequency Xlating FFT Filter chan-nelizing combination, we used a sample file with samplerate fs = 8 MHz and a duration of T = 40 s, resultingin a length of 3.2 · 108 samples. This file was channelizedwith a parallel filter combination of K Frequency XlatingFFT Filters, each with a unique center frequency but al-ways with a lowpass of bandwidth B = 0.05 includinga transition width of 0.01 relative to the sample rate anddecimation factor of d = 40. The FDC operates on thesame carrier frequencies with the same bandwidth but witha block length of N = 4096 samples and a relative in-verse overlap of Ψ = 4. Internally, the decimation factoris lowered to d = 32 due to gr-FDC design limitationsgiving gr-FDC a slight penalty in terms of samples-to-be-processed. The simulations where performed throttle-lessand one after another measuring the necessary executiontime. All processing times below 40 s (the duration of thesample file) indicate a realtime channelization capability.Nonetheless, considering signal processing of each chan-nel, a sufficient performance buffer should always be con-sidered.

The simulation was performed on a desktop system withan Intel Core i5-2500K Quad Core processor, with 16 GBRAM and Ubuntu 18.04 operating system with GNU Ra-dio 3.7.12.0 built from GitHub source. Fig. 3 shows the re-sults of the simulation. In good approximation the process-ing time is linear dependent of the number of channelizerswith a break-even point at around 10 channelizers, wherethe gr-FDC outperforms the Frequency Xlating FFT chan-nelizers. With the used hardware, the maximum realtimecapability for an 8 Mega-Samples per second signal wouldbe in case of a Frequency Xlating FFT filter combination60 channelizers and for the gr-FDC 120 channelizers.

20 40 60 80 100 1200

20

40

60

80

Number of carriers

Proc

essi

ngtim

e[s

]

Xlating FFTFDC

Figure 3. gr-FDC and multiple Frequency Xlating FFT perfor-mance comparison per carrier

Keep in mind that this is one specific case and for differ-ent parameters, i.e. bandwidth, block length, relative in-verse overlap and necessary post-channelization frequencycorrection, this gain might be reduced. Furthermore, themaximum realtime capability in terms of used channelizersis rather a criteria for the computational capability than theFDC performance.

4.2. Signal Quality Validation

−0.2 −0.1 0 0.1 0.2−2.5

−2

−1.5

−1

−0.5

Normalized Frequency

Pow

erSp

ectr

alD

ensi

ty[d

B]

Xlating FFT FilterFDC

Figure 4. Power Spectral Density of gr-FDC and Frequency Xlat-ing FFT Filter

The blockwise fast convolution with integrated decimationin the frequency domain as proposed in Section 2 and im-plemented by gr-FDC is not equal to a straight forwardconvolution of the whole signal with an equivalent FIR fil-ter of the used window. This is due to multiplication ofthe window in the frequency domain representing a circu-lar convolution on each block in the time domain, which isequal to interference of samples from the beginning to the

Frequency Domain Channelizer

4 6 8 10 12 14 16

5

10

15

Source SNR [dB]

Cha

nnel

ized

SNR

[dB

]FDCXlating FFT Filter

Figure 5. Signal-to-noise ratio comparison of gr-FDC and Fre-quency Xlating FFT Filter

end of the block in the time domain and vice versa. Thisdiffers from the straight forward convolution in the time do-main as implemented by an FIR filter. Though this effectshould be canceled by the appropriate choice of an overlap,the quality of the resulting signal has to be verified againstconventional channelizing methods.

To confirm the similarity of gr-FDC and Frequency XlatingFFT Filter channelization concerning the signal-to-noiseratio (SNR), a single QPSK carrier is channelized, synchro-nized and the SNR is measured and compared.

For Figure 4, the QPSK carrier is channelized with an out-put rate of 4 samples per symbol. Also the complete band-width of the carrier is within the passband of the lowpassfilter. Since the magnitudes of the power spectral densi-ties of the common Frequency Xlating FFT Filter and ourgr-FDC are identical, the expected behavior of distortion-free channelization of the gr-FDC is confirmed.

Figure 5 illustrates the resulting SNRs for channelizationwith gr-FDC in Throughput-mode with block length ofN = 2048 and a Frequency Xlating FFT Filter, where eachSNR is plotted against the theoretical SNR. Obviously, themeasured SNRs do not differ besides numerical variations,which further indicates the low-loss properties of gr-FDC.

5. Conclusion and Future DevelopmentFrequency domain channelizing and our GNU Radio im-plementation gr-FDC offer a performant channelizationmethod for different use cases in live software defined ra-dio systems for broadband radio monitoring, Demand As-signed or Frequency Division Multiple Access system car-rier extraction. Main advantages include the variable chan-nel bandwidths and frequencies when multiple carriers arechannelized and its performance boost, since the forward

DFT only needs to be performed once independently of thenumbers of carriers. The main disadvantages are the limi-tation of the extraction bandwidth and carrier frequency tointeger steps and cannot be chosen continuously.

Future planned and considered features for gr-FDC in-clude:

• Root Raised Cosine Matched Filtering option insteadof lowpass windowing for FixedCarrier or Through-put channels

• Further GUI interaction and meta data visualizationfrom the waterfall display to monitor and control thechannelizer

• Code clean-up

• Further documentation besides GRC-doc string

We encourage you to implement the FDC concept for achannelizer customized to your needs, for example a chan-nel activation controlled by a systems broadcast and signal-ing channel, which we do not plan to implement in gr-FDC.

If gr-FDC suffices to your application, feel free to use,contribute, fork or customize it accordingly. gr-FDC is li-censed under GPL v3 and hosted on GitHub (Such, 2018).

ReferencesBorgerding, Mark. Turning Overlap-Save into a Multiband

Mixing, Downsampling Filter Bank. IEEE Signal Pro-cessing Magazine, 2006.

Frigo, Matteo and Johnson, Steven. FFTW3, 2003. URLhttp://www.fftw.org.

FSF. Vector Optimized Library of Kernels (Volk). FreeSoftware Foundation, 2015. URL http://libvolk.org.

Harris, F. J., Dick, C., and Rice, M. Digital receiversand transmitters using polyphase filter banks for wire-less communications. IEEE Transactions on MicrowaveTheory and Techniques, 51(4):1395–1412, April 2003.

Harris, Frederic J. Chapter 8 - Time Domain Signal Pro-cessing with the DFT. In Handbook of Digital SignalProcessing, pp. 633 – 699. Academic Press, San Diego,1987. ISBN 978-0-08-050780-4.

ITU. Radio Regulations. International Telecommunica-tions Union, 2016. URL http://www.itu.int/pub/R-REG-RR-2016.

Such, Gereon. gr-FDC, 2018. URL https://github.com/gereonsuch/gr-FDC.