J Sign Process SystDOI 10.1007/s11265-009-0348-1

Dynamic Frequency-Band Reallocation and Allocation:from Satellite-Based Communication Systemsto Cognitive Radios

Amir Eghbali · Håkan Johansson · Per Löwenborg ·Heinz G. Göckler

Received: 14 October 2008 / Revised: 29 January 2009 / Accepted: 29 January 2009© 2009 Springer Science + Business Media, LLC. Manufactured in The United States

Abstract This paper discusses two approaches for thebaseband processing part of cognitive radios. Theseapproaches can be used depending on the availability of(i) a composite signal comprising several user signals or,(ii) the individual user signals. The aim is to introducesolutions which can support different bandwidths andcenter frequencies for a large set of users and at the costof simple modifications on the same hardware platform.Such structures have previously been used for satellite-based communication systems and the paper aims tooutline their possible applications in the context ofcognitive radios. For this purpose, dynamic frequency-band allocation (DFBA) and reallocation (DFBR)structures based on multirate building blocks are in-troduced and their reconfigurability issues with respectto the required reconfigurability measures in cognitiveradios are discussed.

Keywords Satellite-based communication systems ·Cognitive radios · Multirate systems ·Transmultiplexers · Filter banks

A. Eghbali (B) · H. Johansson · P. LöwenborgDivision of Electronics Systems, Department of ElectricalEngineering, Linköping University, Linköping, Swedene-mail: amire@isy.liu.se

H. Johanssone-mail: hakanj@isy.liu.se

P. Löwenborge-mail: perl@isy.liu.se

H. G. GöcklerDigital Signal Processing Group (DISPO),Ruhr-Universität Bochum, 44780 Bochum, Germanye-mail: goeckler@nt.ruhr-uni-bochum.de

1 Introduction

One perspective in communication systems design isto increase the spectrum utilization by introducing theconcept of cognitive radios. A cognitive radio is anetwork of intelligent co-existing radios which sensetheir environment to find available frequency slots (alsoreferred to as white spaces or spectrum holes [1]) and,then, modify their transmission characteristics to usethat particular frequency slot. Figure 1 illustrates theoverlay1 spectrum sharing scenario [4] or the oppor-tunistic spectrum access [2], in which secondary userscan occupy those frequency slots that are not beingoccupied by primary (licensed) users. In this way, anefficient use of the limited available frequency spectrumis achieved. One of the main tasks in a cognitive radiois consequently the spectrum mobility [5], also referredto as dynamic frequency allocation [1] or dynamic spec-trum allocation [6], where users may constantly changetheir frequency of operation. In this paper, we will usethe term dynamic frequency-band allocation (DFBA).Being dynamic means that the transmission parame-ters, e.g., bandwidth, center frequency, transmissionpower, communication standard, etc., may vary withtime [5]. Specifically, this is frequently referred to asthe reconfigurability of the radio so that the hardwareof a cognitive radio can be reprogrammed not only totransmit and receive at various frequencies but also touse different technologies [7]. A simple example couldbe to assume a user transmitting text (or video) in

1The underlay spectrum sharing, also referred to as ultra wide-band [2], exploits spread spectrum techniques and users transmitat certain portions of spectrum which are regarded as noise bythe licensed users [3]. This scenario is not the focus of this paper.

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Figure 1 Overlay approach for spectrum sharing in a cognitiveradio.

frequency slot f1 (or f2) and at time t1 (or t2). Thegeneral block diagram of a cognitive radio is shown inFig. 2.

Another perspective in communication systems de-sign is that satellites with high-gain spot beam antennas,on-board signal processing, and switching will play animportant complementary role in future digital com-munication systems which aim at supporting variouswideband services accessible to everybody everywhere[8–11]. For this purpose, the European space agency(ESA) has proposed three major network structuresfor broadband satellite-based communication systems[11] in which satellites communicate with users throughmultiple spot beams. This requires an efficient use ofthe limited available frequency spectrum by satellite on-board signal processing [8–10, 12–14]. Although thesetwo perspectives arise from different needs, both try tomake an efficient use of the limited available frequencyspectrum.

Similar to the satellite-based communication systemswhich require both on-ground DFBA and on-boarddynamic frequency-band reallocation (DFBR), the adhoc-based or infrastructure-based cognitive radios canalso utilize these tasks separately. Specifically, in the adhoc-based networks, individual users can utilize DFBAwhile DFBR can be performed by the base stationsof an infrastructure-based network. However, DFBAcan also be deployed by the individual users operatingin an infrastructure-based network. The main aim is

Figure 2 General block diagram of a cognitive radio composedof digital to analog converters (DACs); analog to digital convert-ers (ADCs); baseband processing; and radio frequency part.

thus to dynamically change the operating frequency ofsecondary users so that they can utilize those parts offrequency spectrum which are not occupied by primaryusers. Furthermore, DFBR is applicable to compositesignals comprising several users with different (andfixed) bandwidths but it can not change the bandwidthof individual users. On the other hand, DFBA is ap-plicable to individual users and, thus, makes it possibleto change the bandwidth of individual users as well.

Both DFBA and DFBR can, in principle, be realizedstraightforwardly by making use of bandpass inter-polation/decimation with variable conversion factorsand center frequencies. In other words, this requiresinterpolators/decimators with variable parameters.These blocks can be constructed using variable up-samplers/downsamplers as well as bandpass filters withvariable center frequencies and bandwidths. However,for large sets of variable conversion factors, the imple-mentation complexity (and cost) of this straightforwardapproach becomes intolerably high. The reason is thatto cover a large set of interpolation/decimation factors,one requires a large set of filters with different ordersand characteristics. Thus, the implementation becomesexpensive and it is therefore paramount to developmore efficient realizations for DFBA and DFBR. Forexample, it was envisaged in [11] that the complex-ity reductions required for DFBR networks comprisetwo orders of magnitude. Complexity reduction canbe achieved by flexible and reconfigurable structureswhich have the same functionality as the straightfor-ward realizations but that do not require any systemredesign. In other words, the efficient solutions must becapable of performing various tasks by simple modifica-tions. Furthermore, these modifications should not re-quire any hardware or complexity changes. Specifically,one must be able to reprogram the same hardware inorder to cover a different functionality.

Recent advances in development of efficient DFBAand DFBR realizations for satellite-based communica-tion systems show that substantial complexity reduc-tions are feasible in comparison with straightforwardsolutions. For cognitive radio systems, such efficient re-alizations have hitherto not been reported. However, asthe satellite-based communication and cognitive radiosystems are closely related with respect to DFBA andDFBR, one can fathom that realization techniques forsatellite-based communication systems can also be usedfor cognitive radio systems provided that appropriatemodifications are made.

As both DFBR and DFBA process baseband digitalsignals, this paper focuses on the baseband processingpart of cognitive radios. We only consider the oper-ations related to changing the center frequency and

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bandwidth of user signals. Similar to bent-pipe satellitepayloads [12], these do not require any modulation/demodulation, coding/decoding, etc.

1.1 Contribution of the Paper

The paper shows that efficient DFBA and DFBR re-alizations for cognitive radios can indeed be achievedby appropriate use and modification of the realizationsdeveloped for satellite-based communication systems.The paper provides a brief survey of the recently pro-posed DFBR and DFBA solutions and it discussesthe necessary reconfiguration and parameter selectionissues when adopting these solutions for cognitive ra-dios. We will discuss two basically different approachesreferred to as Approach I and Approach II for thebaseband processing block of cognitive radios. Theyare appropriate for different cases depending on theavailability of (i) a composite signal comprising severaluser signals, or (ii) the individual user signals. In bothcases, users can have different bandwidths which canalso vary with time.

Approach I makes use of DFBR networks and isappropriate when dealing with a composite signal com-prising several users with different bandwidths. In thiscase, all users can simultaneously be reallocated to theirdestinations (or available frequency slots in cognitiveradios) through a DFBR network [13–24].

On the other hand, Approach II assumes that theindividual user signals are available making it appro-priate to use a DFBA network directly. For thispurpose, transmultiplexer-based (TMUX-based) ap-proaches [25–29] capable of generating different usersignals with different bandwidths will be discussed.These approaches support systems which dynamicallymove between different access technologies along theavailable spectrum pool [5].

An alternative method in dealing with compositesignals is to first divide, by the use of a filter bank(FB), the composite signal into its corresponding usersignals and, then, use Approach II on each user signal.However, it is more efficient to directly use Approach Iin one step and, consequently, we will not discuss thisalternative. As will be illustrated later, combinationsof Approaches I and II can also be used to achieve anincreased degree of freedom to allocate and reallocateuser signals.

1.2 Paper Outline

Section 2 introduces Approach I and discusses theDFBR networks based on oversampled complex mod-ulated FBs, tree-structured FBs, and modified discrete

Fourier transform (DFT) FBs. Furthermore, the arith-metic complexity of two recently proposed DFBR net-works is discussed. In Section 3, Approach II whichis based on TMUXs is introduced and some of theemerging solutions in this context are outlined. In ad-dition, each of Sections 2 and 3 provide discussionson reconfigurability (and modification) of the two ap-proaches with respect to their application in cognitiveradios. Finally, some concluding remarks are providedin Section 4.

2 Approach I: Use of DFBR Networks

To perform DFBR, it is assumed that signals fromseveral users, e.g., mobile handsets in a cellular phonenetwork or computers in a wireless local area network(WLAN), have been added into a composite signal ata main station, e.g., a base station in a cellular phonenetwork or an access point in a WLAN. Thus, it is upto the main station to find possible available frequencyslots and reallocate each user to one of those availableslots. In a dynamic communication system, users canoccupy any bandwidth at any time which means thatthe width of the required frequency slots may vary withtime. Hence, the main station has a similar functionalityto a bent-pipe satellite payload [12] with its idea ofoperation shown in Fig. 3. Here, composite signalshaving a number of users with different bandwidthsare processed by the DFBR network and the users arereallocated to new frequency slots. These slots couldbe different antenna beams of a satellite payload ordifferent cells in a cellular network. As an example,multiple antennas of a satellite payload perform sig-nal filtering in a spatial domain rather than in thefrequency domain. This is similar to the techniques uti-lizing multiple antennas which are introduced for cog-nitive radios in, e.g., [30]. In addition, the use of DFBR

Figure 3 Approach I: DFBR networks process composite signalsto reallocate users from one composite input signal to anothercomposite output signal.

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networks could be useful for the centralized2 coop-erative3 cognitive radio networks [31] or they can beconsidered as secondary base stations in licensed bandcognitive radio networks (http://ieee802.org/22/). In li-censed band networks, the DFBR can coexist with theprimary networks and opportunistically operate in anoverlay transmission manner.

In general, the DFBR network can be a multi-inputmulti-output (MIMO) network in the sense that it canhave a number of composite input and output signalswhere any user in any of the composite input signalscan be reallocated to any frequency slot in any of thecomposite output signals. Each composite signal canbe seen as a frequency division multiplexed (FDM)signal comprising several users having different band-widths and center frequencies. Consequently, from theuser point of view, it has used another frequency slotthrough the use of the DFBR networks. The dynamicnature of the DFBR networks makes it possible forusers to occupy any suitable frequency slot in a time-varying manner.

One way of performing DFBR is to make use ofmultirate digital signal processing structures, i.e., dec-imation and interpolation, to generate frequency shiftsof user signals. To do so, a FB divides the input signalinto a number of granularity bands (GBs) separated bya guardband. Any user can occupy any rational numberof GBs. After deriving the user subbands, a channelswitch performs the required frequency shifts on sub-bands. Finally, the frequency-shifted user subbands arecombined to construct the shifted user signals.

To exemplify, for frequency-band reallocation ofuser X1 in Fig. 4, there is a need for bandpass filtersas shown in Figs. 4c and 4g. Here, a composite sig-nal comprising users X0, X1, . . . , XM−1 is present atthe main station and user X1 must be reallocated toanother frequency slot. If the user signal requires awider band, channel combiners can combine severalfrequency bands. For example, assume combining theseparate frequency bands of X0 and X1 to get a wideruser signal. These frequency bands can either be con-tiguous or separate requiring contiguous or fragmentedDFBR [6]. The separate frequency bands can be con-sidered as a multi-spectrum transmission. Specifically,as white spaces are mostly fragmented [32], user signals

2Here, a centralized entity, i.e., the DFBR network in this case,controls the spectrum allocation and access. However, the distrib-uted (ad hoc) cognitive networks do not have an infrastructure.3Non-cooperative networks do not share the interference mea-surements of each node (user) with the others resulting inless spectrum utilization and less communication requirementsbetween nodes.

Figure 4 Principle of DFBR networks using multirate buildingblocks and variable bandpass filters (a–h).

can be transmitted in several non-contiguous frequencybands.

In a general dynamic communication system, userscan have the possibility of being reallocated to anyavailable and suitable4 frequency slot (see Figs. 4dand 4h). If the number of users is limited, the DFBRsolutions based on conventional FBs, e.g., [15, 17],may be applicable. In this case, to support DFBR ofseveral users having different bandwidths, there wouldbe a need for either several sets of filters or onlinefilter design. This could result in a high implementationcost and complexity. Thus, low complexity approaches

4The frequency slot depends on spatial and temporal parameterssuch as the number of slots available, user movement, and activityof primary users, etc. [5] but the operation of the DFBR networkis independent of these parameters.

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are required so that DFBR is performed with reason-able implementation complexity. In addition, differenttelecommunication standards may have different re-quirements on the approximation of perfect reconstruc-tion (PR). Hence, the DFBR solutions must be able toapproximate PR as close as possible.

2.1 User Bandwidth vs. Multiplexing Bandwidth

As discussed above, the DFBR networks divide usersignals into a number of GBs and they perform thefrequency shifts on these GBs. In other words, as theDFBR networks utilize FBs, the multiplexing band-width must be an integer multiple of the GB. Specifi-cally, in the DFBR networks of [13–24] one could havemultiplexing bandwidths of (i) the size q GB whereq ∈ N, or (ii) the size 2q GB where q ∈ N. However, thisdoes not mean that the individual user bandwidths arealso limited to these values. In other words, users canhave bandwidths which are rational multiples of a GB.Thus, the DFBR networks can perform frequency shiftson users whose bandwidths are, in general, rationalmultiples of a GB. This way, the users can have dif-ferent bandwidths and, hence, bandwidth-on-demandfor a dynamic communication system is supported. Theonly issue here is to ensure that users do not share aGB which, according to Fig. 5, can easily be achievedby allowing some extra guardband. However, the extraguardband would affect the efficiency in spectrum uti-lization resulting in a trade off as will be discussed inSection 3.3. Consequently, in subsequent discussions, adistinction is made between the terms user bandwidthand the multiplexing bandwidth. Specifically, a mul-tiplexing bandwidth contains a user bandwidth, withthe spectral width of rational multiples of a GB, andsome extra guardband. The next subsections will out-line some of the emerging low complexity solutionsfor DFBR. These solutions are based on oversampled

Figure 5 Formulation of user and multiplexing bandwidth (a–c).

Figure 6 Principle of N-channel DFBR network utilizing fixedanalysis filters and adjustable synthesis filters.

complex modulated FBs, tree-structured FBs, andmodified DFT (MDFT) FBs.

2.2 DFBR Network Based on Oversampled ComplexModulated FBs

In [13], a new class of DFBR networks based on finite-length impulse response (FIR) variable oversampledcomplex modulated FBs for bent-pipe payloads was in-troduced and an efficient implementation structure wasderived5. Furthermore, it was proved that the systemcan approximate PR as close as desired via a properdesign. As shown in Fig. 6, the DFBR network is com-posed of fixed analysis filters Hk(z), k = 0, 1, . . . , N−1which split the input signal x(n) into N subbands.Through downsampling/upsampling by M as well as ad-justable synthesis filters Gk(z), the required frequencyshifts of the subbands and recombination of subbandsare performed. To avoid high implementation cost ofadjustable filters, the DFBR network is instead realizedusing fixed synthesis filters and a channel switch, asshown in Fig. 7.

As shown in Fig. 8, the efficient N-channel DFBRnetwork assumes that the complex input signal isdivided into Q GBs where [13]

Q = NA

, A > 1, Q, A ∈ N, (1)

and the GBs are separated by a guardband of 2� = 2πεQ

with 0 < ε≤1. Any user bandwidth can have any ratio-nal number of GBs meaning that the users can havedifferent bandwidths and, through this, bandwidth-on-demand is supported. To suppress aliasing and, at the

5The system in [18] uses the same architecture as [13] except thatit has infinite-length impulse response (IIR) filters.

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Figure 7 Efficientimplementation of theN-channel DFBR network inFig. 6 using oversampledcomplex modulated FBs.

same time, to be able to shift the GBs by all possiblevalues of 2πq

Q , the value of M should be a multiple ofQ. Furthermore, to attenuate the aliasing terms by thestopband attenuation of the filters, the passbands andtransition bands of the shifted terms should not overlapwhich results in

M≤ N

1 + N�π

< N, N = ML, (2)

where L is the number of FB channels per GB. Fur-ther details on the prototype filter P(z) as well as thecomplex multipliers αk, βk, and γk can be found in[13, 33, 34]. It must be noted that the values of Q,M, N, and L are the fundamental system parametersand having defined these, the complex multipliers arederived easily.

To process real input signals, [33, 34] propose twosystems, i.e., System I and System II, which are shownin Fig. 9. System I consists of a complex DFBR

Figure 8 Formulation of guardband and GB for DFBR networks(a–d).

network as well as two Hilbert transformers6 whereasSystem II utilizes a real DFBR network7 and eliminatesthe need for the Hilbert transformers. It is shown in [33]that System II is suitable for cases with large numberof users which often arise in practice. Furthermore,the real DFBR has less total processing units, i.e., realadders and multipliers, than the complex DFBR net-work [33]. Specifically, for high prototype filter orders,the real DFBR network performs roughly 10 − 15%more operations in one time unit. For high prototypefilter orders and in both real and complex DFBR net-works, the total arithmetic complexities (in one timeunit) are asymptotically equal. To get the highest spec-trum utilization efficiency for DFBR in systems withlarge number of users, the per-time-unit arithmeticcomplexity of System II will always be less than 50%of that of System I [33].

2.3 DFBR Network Based on Tree-Structuredand MDFT FBs

The principle of these approaches is best explainedby the use of Fig. 10. Each user is assigned an in-dividual multiplexing bandwidth8 comprising q GBswhere either q = 2k, k ∈ N or q ∈ N. Depending on theparticular solution chosen, the incoming composite sig-nals, which contain several users having different userbandwidths, are decomposed into a number of GBs orin subband signals thereof [14, 19, 20]. These signals,

6Here, the analytic representation [35] of the real input signalmust be processed by the complex DFBR network and thefrequency multiplexed result should then be converted to a realsignal for retransmission.7The DFBR network in Fig. 7 has complex multipliers αk, βk,γk and, hence, it is a complex system by nature. However, real(complex) DFBR refers to two variants of Fig. 7 having real(complex) input/output signals. In other words, the complexmultipliers are present in both the real and complex DFBRs.8As discussed before, a multiplexing bandwidth can contain auser bandwidth and some extra guardband.

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Figure 9 Two systems to process real signals as proposed in [33](a, b).

available at the output ports of the bench of frequencydemultiplexer FBs (FDMUX), are cross-connected byan on-board channel switch to a corresponding benchof frequency multiplexer FBs (FMUX) to form theretransmitted signals. In other words, the decomposeduser signals are cross-connected by the on-board chan-nel switch to the FMUX for retransmission in the newfrequency bands.

To perform DFBR, the FDMUX always decomposesall incoming FDM signals into subband signals with thespectral width of a GB [19, 24] or even half of a GB[14, 20]. Therefore, after cross-connection, each FMUXhas to combine two functionalities due to the followingscenarios:

– Some of the user signals require mere recombi-nation with other user signals which necessitatesa simple cascade of an FDMUX and an FMUX

(FDFMUX) where only their passband/stopbandallocations must be matched [14, 21].

– Other wideband user signals also require near per-fect reconstruction (NPR) of the associated sub-band signals which calls for power complementarymatching of FDMUX and FMUX according to thesubband coder (SBC) approach [36].

Moreover, both functionalities are arbitrarily mixedand spread over all FDMUX and FMUX FBs. Con-sequently, each FB cascade must likewise be usable asan FDFMUX or an SBC FB. An important issue isthat the transition between both functionalities mustbe feasible by simple modifications which would notinterrupt the system operation and, thus, the DFBR canbe performed on the fly. In this regard, two solutionscan be outlined.

A first solution to the above procedure applies two-channel NPR FBs [21–23] that are extended to treestructures by successively interleaving two-channel FBs[19, 24]. Both critical sampling and oversampling by twoare efficiently feasible. Here, the multiplexing band-width is restricted to power-of-two multiples of a GB[19, 24]. Consequently, potential on-board switching isrestricted and only center frequencies that fit into thepattern of the tree-structured SBC-FDFMUX FB canbe realized for the reallocated wideband signals.

A second solution overcomes the restriction of theallowable multiplexing bandwidth. Here, the FDMUXand FMUX are based on the one-stage NPR MDFT

Figure 10 Principle of DFBRnetworks using FMUXand FDMUX as discussedin [19, 20, 24].

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FB [19, 20, 37–39], which applies maximal decimationand efficient polyphase implementation. This solutionis robust, since aliasing compensation is structurally im-posed. In contrast to [38], modifications are introducedin order to easily combine the structurally imposedaliasing compensation and the desired switching func-tionality within the NPR MDFT SBC-FDFMUX FB.This represents a straightforward complex modulatedDFT approach for arbitrary FB channel numbers withthe desired full switching flexibility.

2.4 Arithmetic Complexity

In this section, we will consider the arithmetic complex-ity of the DFBR networks discussed in the previoussubsections. To keep uniformity in the type of signalsbeing reallocated by these networks, we will focus onthe solutions which process complex signals. Further-more, we will consider solutions which do not imposeany restrictions on the multiplexing bandwidth. Conse-quently, we choose solutions defined in [13] and [20]and compute their arithmetic complexity in terms ofreal additions and multiplications. Comparing the realoperations gives a more accurate measure of arithmeticcomplexity [40]. In this paper, we assume that one com-plex multiplication requires 4 real multiplications and 2real additions whereas one complex addition requires2 real additions. Additionally, we use the arithmeticcomplexity of the DFT defined in [41].

Consequently, the DFBR network in [13] requires2Nlog2 N + 2(4 + 2S + 5N) real multiplications and6Nlog2 N + 2(4 + 2S − N) real additions. Here, S is thelength of the prototype filter with N being the numberof FB channels [33]. In a similar manner, the maximally

decimated N-channel DFBR system of [20], shown inFig. 11, requires

– 2N complex analysis and synthesis filters of lengthS operating on complex sequences resulting in 2NScomplex multiplications and 2N(S − 1) complexadditions.

– 2N adders operating on complex sequences result-ing in 2N complex additions.

– N complex multipliers WnN operating on complex

sequences resulting in N complex multiplications.

For the straightforward realization of Fig. 11, we re-quire 2N(4S + 1) real additions and 8NS real multi-plications. Obviously, using efficient realizations, thesecomplexity figures can drastically be reduced. For ex-ample, according to [42], the MDFT FB can be realizedby two DFT polyphase FBs where, for special proto-type filter orders, this can further be reduced to onepolyphase DFT FB [43].

Another issue is the fact that, both DFBR networksin Figs. 7 and 11 produce samples in N parallel branchesbut the output commutators retain every Mth sample.Thus, the arithmetic complexity figures must be dividedby M to get the per-sample number of real additionsand multiplications. However, this is common for boththe structures and, hence, one can ignore this division.Due to the similarity between these two DFBR struc-tures, the total arithmetic complexity of the efficientrealizations would be roughly equal. Discussion on theperformance of these approaches in approximating PRis outside the scope of this paper and the interestedreader is referred to [13] and [20].

Figure 11 N-channel DFBRnetwork proposed in [20]where the position of realand imaginary parts alternatefrom one branch to other.Here, it is assumed thatN is odd.

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2.5 Reconfigurability

A crucial point in the context of cognitive radios is theability of the system to be reconfigured as simply aspossible. Specifically, the cognitive radio must be ableto adjust its operating parameters without any modi-fications of the hardware components [7]. A cognitiveradio is built on the platform for a software definedradio in which the processing is mainly performed inthe digital domain [2]. There are several reconfigurableparameters which can be incorporated into the cog-nitive radio such as operating frequency, modulationmethod, transmission power, communication standard,etc. These make the cognitive radio capable of adapt-ing to new radio interfaces and standards; supportingnew applications and services; and incorporating newupdates, etc. [1]. In the context of adaptable operatingfrequency (or flexible frequency carrier tuning [6]), acognitive radio should be capable of changing its oper-ating frequency based on the information about the ra-dio environment. However, this should not impose anyrestrictions on the system throughput and hardware.

The DFBR networks can provide this capabilitywithout requirements to change the hardware. Specif-ically, the DFBR networks can easily perform any fre-quency shift of any user having any bandwidth, usinga channel switch. This switch only directs differentFB channels to their desired outputs and requires noarithmetic complexity. Furthermore, the channel switch

Figure 13 Channel switch configuration for two scenarios ofFBR in Fig. 12 (a, b).

can seamlessly perform this task and imposes no restric-tions on system throughput. In addition, the system pa-rameters are determined and fixed only once and, then,the reconfigurable operation is performed by reconfig-uring the channel switch. Here, the user bandwidths arepredetermined but can be arbitrary which means that ageneral dynamic asymmetric MIMO communication issupported. The user bandwidths can change before thesignals arrive at the DFBR networks and the DFBRnetwork makes a hand off by changing the operatingfrequency.

To illustrate these facts, Figs. 12 and 14 show twocases where, respectively, 4 and 3 users have occupiedthe whole frequency band between [0, 2π ] and, further-more, they have been reallocated to new positions inthe frequency spectrum. To generate these user signals,

Figure 12 Input patternand the reallocated outputsusing the channel switchconfigurations in Fig. 13(a–c).

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Figure 14 Input patternand the reallocated outputsusing the channel switchconfigurations in Fig. 15(a–c).

the multimode TMUX structure of [25, 26] has beenused. This TMUX will be discussed later in Section 3.In Fig. 12a, the user signals {X0, X1, X2, X3} occupy,respectively, user bandwidths of size {1, 2.9, 3.6, 1.9}GBs where each GB has a width of 2π

Q − 2επQ with Q =

10 ans ε = 0.125. Similarly, in Fig. 14a, the user signals{X0, X1, X2} occupy {1, 6.9, 1.9} GBs respectively. Ascan be seen, the user signals (or the spectrum holes) canoccupy any rational number of GBs and, thus, there isno restriction on the system operation. The only issueis that the users should not share a GB. As notedin Section 2.1 and will be formulated in Section 3.3,this can easily be taken care of by adding some extraguardband so that no GB is shared. This difference inthe amount of guardbands between different users caneasily be recognized by looking at Figs. 12 and 14.

To show the reconfigurability of the DFBR net-works, two scenarios of a channel switch for the inputsignal patterns of Figs. 12a and 14a are illustrated inFigs. 13 and 15. The reallocation of signals in Fig. 12ausing the channel switch with scenarios I and II ofFig. 13, is respectively shown in Figs. 12b and 12c. Sim-ilarly, the two reallocated outputs of Fig. 14a based onthe channel switches of Fig. 15, are shown in Figs. 14band 14c. As can be seen, the DFBR networks can bereconfigured by just modifying the position of the chan-nel switch. In other words, they can perform any re-allocation scenario without the need to change thehardware, filter coefficients, or the system parameters.

These illustrative examples use the DFBR networkdefined in [13] but other DFBR networks can also beused.

In these examples, the DFBR network is assumed tooperate on the same antenna beam but by having sev-eral DFBR networks, users can be reallocated betweendifferent antenna beams as well. This only requires aduplication of DFBR networks and a channel switch ca-pable of directing user signals between different DFBRnetworks. Such a channel switch would not require anyarithmetic complexity.

It is noted that every branch of the channel switchesin Figs. 13 and 15 represents the operation of two FBchannels as each GB contains two FB channels. To bemore specific, the values of N, M, and L in (2) are 20,10, and 2 respectively.

Figure 15 Channel switch configuration for two scenarios ofFBR in Fig. 14 (a, b).

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2.6 Modifications

To use the DFBR networks in cognitive radios, somemodifications need to be considered. These modifica-tions are mainly related to the choice of the systemparameters so that the reconfigurable operation doesnot require hardware changes. For different system pa-rameters, the implementation complexity of the systemmay be different. However, once the parameters arechosen, the implementation complexity remains con-stant and the system can be easily reconfigured on thesame hardware platform.

The DFBR networks divide the input signal intoGBs and any user bandwidth can occupy any rationalnumber of GBs. Hence, the width of a GB must bechosen such that its bandwidth is proportional to that ofthe spectrum holes. To be more specific, one requires tochoose a value for the BGB = 2π(1−ε)

Q , in Fig. 8, such thatthe bandwidth of any spectrum hole can be representedas a rational multiple of BGB. This way, any user signalwith any user bandwidth can occupy the spectrum holesand it is only required to control the channel switch sothat any frequency shift can be performed.

3 Approach II: Use of Transmultiplexers

Assuming that each user terminal has the ability toadjust its operating frequency and bandwidth, aTMUX-based solution can be used. In a dynamic com-munication system, at any time, any user can haveany bandwidth. For example, at three different timeinstances, a user can decide to send either text, audio,or video. Furthermore, it can occupy any availablefrequency slot. The basic idea of this type of communi-cation is depicted in Fig. 16 where different bandwidthsand center frequencies can be generated using multi-rate signal processing techniques. These TMUXs can

Figure 16 Approach II: TMUXs used to perform DFBA. Atany time t, each user can decide its bandwidth B and operatingfrequency f .

Figure 17 Principle of TMUXs using multirate building blocks(a–d).

also be regarded as the time-spectrum blocks [44] whichare capable of transmitting any amount of data at anytime interval and on any portion of the spectrum. Asshown in Fig. 17, the interpolation part represents thetransmitter where a variable filter places the desireduser signal at the required location in the frequencyspectrum. The receiver, i.e., the decimation part, is de-signed accordingly so that the input signal is recoveredwith reasonable and controllable levels of error.

Similar to the problem with straightforward DFBRsolutions, one can use conventional nonuniformTMUXs, e.g., [45–49], to place a large set of users(with different bandwidths) at different locations in thefrequency spectrum. Assuming a dynamic communica-tion system, these conventional TMUXs would requireeither predesign of different filters or online design offilters. This becomes inefficient when simultaneouslyconsidering the increased number of communicationscenarios and the desire to support dynamic communi-cations. Therefore, it is vital to develop low-complexityDFBA approaches which dynamically support differentcommunication scenarios and require reasonable im-plementation complexity as well as design effort.

In this context, TMUX structures of the generalform shown in Fig. 18 are introduced. In the synthesisFB (SFB), the system Cp performs interpolation byrational ratio Rp whereas the system Cp in the analysisFB (AFB) performs decimation by rational ratio Rp.These blocks make it possible to transmit and receivebaseband signals having arbitrary bandwidths througha common channel.

The main difference between the approaches in[25–29] lies in the implementation of the SRC blocks

J Sign Process Syst

Figure 18 General model of a multimode TMUX where systemsCp and Cp perform rational sampling rate conversion (SRC).

Cp and Cp. However, they all utilize the Farrow struc-ture9 and aim to design the system in such a way that,at any time, any user can transmit/receive data streamswith any bandwidth and centered at any point in thefrequency spectrum. Furthermore, they all use variablefrequency shifters which are achieved by modifyingsome multiplier values. Thus, the reconfiguration ofthese TMUXs can simply be performed by modificationof the variable multipliers resulting from (i) the Far-row structure, and (ii) the variable frequency shifters.Similar to the networks based on orthogonal frequencydivision multiplexing (OFDM), these TMUXs can re-move harmful interference in any frequency slot whichis occupied by licensed users. To do so, certain fre-quency shifters are fed with zeros and, consequently,that particular frequency band is not used.

The TMUX in [25, 26] generates a GB throughinteger interpolation. As users can have bandwidthswhich are rational multiples of a GB, a Farrow-basedfilter performs rational decimation. To place the usersin appropriate positions in the frequency spectrum,variable frequency shifters are utilized. In the AFB, thereceived signal is first frequency shifted such that thedesired signal can be processed in the baseband. Then,a Farrow-based interpolator followed by an integerdecimation is used to obtain the desired signal. Here,the Farrow structures in the AFB and SFB have dif-ferent complexities resulting in different sets of Farrowsubfilters. To get around this, [28] introduces anotherTMUX in which the Farrow subfilters have equal com-plexities. Furthermore, the lowpass filter resulting fromthe integer SRC is also removed.

The TMUX in [28] uses the Farrow structureto implement integer interpolators and decimators.

9The Farrow structure is composed of a set of fixed linear-phasefilters and variable multipliers and it performs arbitrary rationalSRC [50–52].

Furthermore, it combines additional downsamplers(upsamplers) with integer interpolators (decimators)to construct a general10 TMUX capable of performinggeneral rational SRC. In the SFB, the TMUX generatesthe required user bandwidths through integer interpo-lation followed by a downsampling. In the AFB, anupsampling followed by integer decimation is needed toobtain the desired signal. Here, the variable frequencyshifters have the same functionality as those in [25, 26].

3.1 Reconfigurability

As discussed before, the cognitive radio must be re-configured to adjust its operating parameters withoutany modifications of the hardware components. Twoof these operating parameters are the operating fre-quency and the bandwidth. If a cognitive radio supportsdynamic bandwidth-on-demand, users must be able tooccupy any portion of the frequency spectrum which iscentered around any point in the available frequencyspectrum.

As noted before, the DFBR networks partially pro-vide this capability by reallocating any user with anyuser bandwidth to any point in the frequency spectrum.However, they have no control over the user bandwidthitself. Here, the user bandwidths are predeterminedbut they can be arbitrary, i.e., rational multiples ofa GB, so that a general dynamic asymmetric MIMOcommunication is supported.

In contrast, the TMUX-based approaches providesolutions which add reconfigurability to the user band-width as well. Furthermore, these solutions bring flex-ible receiver signal filtering [6] which is performed byutilizing the Farrow structure. This can change theuser bandwidths (and the receiver filters) by modify-ing the values of some multipliers. In other words,these solutions provide another type of reconfigurabil-ity (compared to DFBR networks) which is achieved bymodification of some multiplier values.

As can be seen from Figs. 12 and 14, the TMUXallows different number of users, e.g., 4 and 3, withdifferent user bandwidths, i.e., any rational multiple ofa GB, to occupy the whole frequency band between[0, 2π ]. This requires the TMUXs to be fully recon-figurable, and this reconfiguration must be as simpleas possible. The TMUXs discussed in this section canprovide this full reconfigurability by one set of fixed

10An integer SRC variant of the TMUX in [28] is proposed in[27]. However, it can be considered as a subclass of the TMUXdiscussed in this paper.

J Sign Process Syst

filters and some variable multipliers. This shows thepotential of reconfiguring the TMUX without any hard-ware change and using one set of fixed filter coefficients.Having one set of fixed filter coefficients results in thefact that the filter design problem needs to be solvedonly once and offline.

3.2 Modifications

Similar to the modifications of DFBR networks incognitive radios, one mainly requires specific choice ofthe system parameters to eliminate the need for anyhardware change while having simple reconfigurability.

Regarding DFBA, there are two different ways ofperforming SRC which could be useful in different sce-narios. The TMUX in [25, 26] generates a GB throughinteger interpolation by, e.g., W, resulting in BGB =2π(1+ρ)

W where ρ is the roll-off. Then, any rational multi-ples of BGB can be created using the Farrow structurewhich performs rational SRC. Similar to the discus-sion above, one can determine the system parame-ters according to the bandwidth of the spectrum holesand, consequently, any user may occupy any rationalnumber of spectrum holes.

On the other hand, the TMUX in [28] assumes noGBs and it allows users to occupy any portion of thespectrum. To do so, it utilizes the Farrow structure toperform general rational SRC by, e.g., Rp = Ap

Bp. This

allows one to cover a large set of user bandwidths.Here, one can also assume a GB of size BGB = 2π(1+ρ)

Ap

and, then, users can have bandwidths of size Bp BGB

where [28]

2 ≤ Bp ≤ Ap

1 + ρ, {Ap, Bp} ∈ N. (3)

Examples for a set of 232 SRC ratios achieved by Ap ={2, 3, . . . , 30} and (3) are provided in [28]. This showsthat the filters need to be designed only once and offlineand, then, the TMUX is easily reconfigurable.

3.3 Choice of Frequency Shifters

As discussed in Section 2.5, to perform a hand off with-out loss of information, the DFBR network requiresthat the users do not share a GB. Consequently, toperform a lossless reallocation on users having differentbandwidths, one requires to (i) generate appropriateFDM input patterns, and (ii) determine proper pa-rameters for the DFBR networks. To generate input

patterns, the reconfigurability of TMUXs in Fig. 18 canbe used. Specifically, after generating user signals withdesired bandwidths, the frequency shifts by ωp, p =0, 1 . . . , P − 1 can be computed in a way to allow someextra guardband between the users. Assuming thatusers have user bandwidths which are rational, e.g., Rp,multiples of the GB, the value of subcarrier ωp for userp must be computed as

ωp =p−1∑

k=0

Fk + Fp

2(4)

where Fp = �Rp� 2πQ , p = 0, 1, . . . , k with �x� being the

ceiling of x. Here, Fk is the multiplexing bandwidthand using the ceiling operation, one allows some extraguardband so that the users do not share a GB11. Thisformulation applies to the case when DFBA and DFBRare simultaneously used. However, the idea of users notsharing a GB is independent of whether DFBA andDFBR are used simultaneously or not. Furthermore, inpractice, one may anyhow require some extra guard-band to allow for the design margins as well.

As an illustrative example, in Figs. 12 and 14 theusers occupy Rp = {2.9, 3.6, 1.9, 6.9} GBs which neces-sitates an extra guardband of Ep = {0.1, 0.4, 0.1, 0.1}multiples of 2π

Q . This results in a decreased spectrumefficiency as some parts of the spectrum are not used bythe DFBR networks. Mathematically, for a set of valuesRp, p = 0, 1, . . . , P − 1, about

ηdec. =2πQ

∑P−1p=0 (�Rp� − Rp)

2π=

∑P−1p=0 �Rp� − Rp

Q(5)

percent of the spectrum in [0, 2π ] is not used by theDFBR network. For the examples of Figs. 12 and14, about 6% and 2% of the total spectrum becomesunused as it is reserved as additional guardband.

To decrease these percentages, one can increase Qby, e.g., K times, which would, in turn, decrease ηdec..In this case, (5) becomes

ηdec. =∑P−1

p=0 �KRp� − KRp

KQ. (6)

However, increasing Q would increase the order of theprototype filter as well. For each K, the prototype filterof the DFBR network would have a transition band

11If the multiplexing bandwidth is limited to be a power-of-twoof a GB, the same idea can be utilized but the amount of extraguardband would be larger.

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Figure 19 Trade off ofspectrum efficiency andarithmetic complexity.a Decrease in spectrumefficiency versus per-samplearithmetic complexity.b Trend of spectrumefficiency for different Kin (6). Here, the DFBRnetwork in [13] is used whereQ = 10 and L = 2.

0 5 10 15

310

320

330

340

350

η dec.

(%)R

eal o

pera

tions

(a)

Rp = 1,2.9,3.6,1.9

Rp = 1,6.9,1.9

Rp = 1.75,1.25,2,3.5

1 2 3 4 5 6 7 8 9 100

5

10

15

K

η dec.

(%

)

(b)

of 2� = 2πεKQ [13, 33]. As the order of a linear-phase

FIR filter is inversely proportional to the width of thetransition band [35], there is a trade off between thespectrum efficiency and the arithmetic complexity.

With a K-fold increase in Q, the length of theprototype filter S and the number of FB channels Nincrease. Figure 19 shows the trend (and trade off)in spectrum efficiency with respect to the per-samplearithmetic complexity for the DFBR network discussedin Section 2.4. Here, the examples of Figs. 12 and 14 aswell as the example in [26] with Rp = {1.75, 1.25, 2, 3.5}are considered and K = 1, 2, . . . , 10. As can be seen,increasing S and N increases the per-sample arithmeticcomplexity but results in a smaller ηdec.. It is also seenthat the values of Rp mainly determine the maximumand minimum amount of ηdec. and, hence, for every setof Rp, one can determine the optimal K such that thevalues of ηdec. and the per-sample arithmetic complexityare in the range of interest.

4 Conclusion

In this paper, two approaches for the baseband process-ing part of cognitive radios based on DFBR and DFBA

are discussed. They can support different bandwidthsand center frequencies for a large set of users and areeasily reconfigurable.

In case of DFBR networks, composite FDM signalscomprising several users with different bandwidths areprocessed and the users are reallocated to new centerfrequencies. These networks are applicable to cognitiveradios with multiple antennas [30], centralized coopera-tive cognitive radios [31], and secondary base stations inlicensed band cognitive radios (http://ieee802.org/22/).

In case of DFBA networks, each user controls itsoperating frequency and bandwidth and, at any time,any user can have any bandwidth. These networks canbe regarded as the time-spectrum blocks [44] whichtransmit any amount of data at any time interval andon any portion of the available spectrum pool.

The reconfigurability of DFBA and DFBR can easilybe performed either by a channel switch (in case ofDFBR networks) or a set of variable multipliers (incase of DFBA networks).

The examples in Figs. 12 and 14 show the increasedflexibility to allocate and reallocate any user to any cen-ter frequency which is achieved by simultaneous utiliza-tion of DFBA and DFBR. In this case, the individualusers can decide to occupy any available frequency slot

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and, furthermore, any user can also be reallocated againby a base station in which a composite signal is present.

Basically, utilizing any of Approaches I and II incognitive radios only requires modifications imposed bythe special choice of system parameters. After choosingthe parameters, one must design the filters to satisfy anydesired level of error and, then, the same hardware canbe reconfigured in a simple manner.

References

1. Haykin, S. (2005). Cognitive radio: Brain-empowered wire-less communications. IEEE Journal on Selected Areas inCommunications, 23(2), 201–220.

2. Zhao, Q., & Sadler, B. M. (2007). A survey of dynamicspectrum access. IEEE Signal Processing Magazine, 24(3),79–89.

3. Huang, J., Berry, R. A., & Honig, M. L. (2005). Spectrumsharing with distributed interference compensation. In Proc.IEEE DySPAN, November 2005, pp. 88–93.

4. Cabric, D., ÓDonnell, I. D., Chen, M. S. W., & Brodersen,R. W. (2006). Spectrum sharing radios. IEEE Circuits andSystems Magazine, 6(2), 30–45.

5. Akyildiz, I. F., Lee, W. Y., Vuran, M. C., & Mohanty, S.(2006). Next generation/dynamic spectrum access/cognitiveradio wireless networks: A survey. Computer NetworksJournal (Elsevier), 50, 2127–2159.

6. Leaves, P., Moessner, K., Tafazolli, R., Grandblaise, D.,Bourse, D., Tonjes, R., et al. (2004). Dynamic spectrum allo-cation in composite reconfigurable wireless networks. IEEECommunications Magazine, 42(5), 72–81.

7. Jondral, F. K. (2005). Software-defined radio—basics andevolution to cognitive radio. EURASIP Journal on WirelessCommunication and Networking, 5, 275–283.

8. Farserotu, J., & Prasad, R. (2000). A survey of future broad-band multimedia satellite systems, issues and trends. IEEECommunications Magazine, 38(6), 128–133.

9. Wittig, M. (2000). Satellite on-board processing for multi-media applications. IEEE Communications Magazine, 38(6),134–140.

10. Re, E. D., & Pierucci, L. (2002). Next-generation mobilesatellite networks. IEEE Communications Magazine, 40(9),150–159.

11. Arbesser-Rastburg, B., Bellini, R., Coromina, F., Gaudenzi,R. D., del Rio, O., Hollreiser, M., et al. (2002). R&D direc-tions for next generation broadband multimedia systems: AnESA perspective. In Proc. 20th AIAA int. commun. satellitesyst. conf. exhibit, Montreal, Canada.

12. Nguyen, T., Hant, J., Taggart, D., Tsang, C.-S., Johnson,D. M., & Chuang, J.-C. (2002). Design concept and method-ology for the future advanced wideband satellite system. InProc. IEEE military commun. conf., MILCOM 2002 (Vol. 1,pp. 189–194). USA.

13. Johansson, H., & Löwenborg, P. (2007). Flexible frequency-band reallocation networks using variable oversampledcomplex-modulated filter banks. EURASIP Journal onAdvanced Signal Processing, 2007 (Article ID 63714), 15.

14. Abdulazim, M. N., & Göckler, H. G. (2006). Flexible band-width reallocation MDFT SBC-FDFMUX filter bank for fu-

ture bent-pipe FDM satellite systems. In 9th int. workshop onsignal process. for space commun.s (SPSC 2006), Noordwijk,Netherlands.

15. Boucheret, M. L., Mortensen, I., & Favaro, H. (1999). Fastconvolution filter banks for satellite payloads with on-boardprocessing. IEEE Journal on Selected Areas in Communica-tions, 17(2), 238–248.

16. Göckler, H. G., & Felbecker, B. (1999). Digital on-boardFDM-demultiplexing without restrictions on channel alloca-tion and bandwidth. In 7th int. workshop digital sign. process.techn. for space appl. Noordwijk.

17. Chiassarini, G., & Gallinaro, G. (1995). Frequency do-main switching: Algorithms, performances, implementationaspects. In 7th Tyrrhenian int. workshop digital comm.Viareggio, Italy.

18. Rosenbaum, L., Johansson, H., & Löwenborg, P. (2006).Oversampled complex-modulated causal IIR filter banks forflexible frequency-band reallocation networks. In Proc. XIVEur. signal process. conf. Florence, Italy.

19. Abdulazim, M. N., & Göckler, H. G. (2005). Efficient digitalon-board de- and remultiplexing of FDM signals allowingfor flexible bandwidth allocation. In Proc. 23rd int. comm.satellite systems conf. Rome, Italy.

20. Abdulazim, M. N., Kurbiel, T., & Göckler, H. G. (2007).Modified DFT SBC-FDFMUX filter bank systems for flex-ible frequency reallocation. In Proc. EURASIP 15th Eur.signal process. conference (EUSIPCO 2007) (pp. 60–64).Poznan, Poland.

21. Göckler, H. G., & Eyssele, H. (1992). Study of on-boarddigital FDM-demultiplexing for mobile SCPC satellite com-munications (part I and II). European Transactions onTelecommunications (ETT), 3, 7–30.

22. Göckler, H. G., & Abdulazim, M. N. (2005). Joint oversam-pling FDM demultiplexing and perfectly reconstructing SBCfilter bank for two channels. In Proc. Eur. signal process. conf.Antalya, Turkey.

23. Abdulazim, M. N., & Göckler, H. G. (2005). Design optionsof the versatile two-channel SBC-FDFMUX filter bank. InProc. Eur. conf. circuit theory design. Cork, Ireland.

24. Göckler, H. G., & Abdulazim, M. N. (2007). Tree-structuredMIMO FIR filter banks for flexible frequency reallocation.In Proc. int. symp. image signal process. analysis. Istanbul,Turkey.

25. Eghbali, A., Johansson, H., & Löwenborg, P. (2007). Anarbitrary bandwidth transmultiplexer and its application toflexible frequency-band reallocation networks. In Proc. Eur.conf. circuit theory design. Seville, Spain.

26. Eghbali, A., Johansson, H., & Löwenborg, P. (2008). Amultimode transmultiplexer structure. IEEE Transactions onCircuits and Systems II, 55(3), 279–283.

27. Eghbali, A., Johansson, H., & Löwenborg, P. (2008).A Farrow-structure-based multi-mode transmultiplexer. InProc. IEEE int. symp. circuits syst. Seattle, Washington,USA.

28. Eghbali, A., Johansson, H., & Löwenborg, P. (2008). Aclass of multimode transmultiplexers based on the Farrowstructure. IEEE Transactions on Circuits and Systems I,(in press).

29. Eghbali, A., Johansson, H., & Löwenborg, P. (2009). On thefilter design for a class of multimode transmultiplexers. InProc. IEEE int. symp. circuits syst. Taipei, Taiwan.

30. Cabric, D., & Brodersen, R. W. (2005). Physical layer de-sign issues unique to cognitive radio systems. In Proc.IEEE personal indoor and mobile radio communications(PIMRC).

J Sign Process Syst

31. Ganesan, G., & Li, Y. G. (2005). Cooperative spectrum sens-ing in cognitive radio networks. In Proc. IEEE DySPAN(pp. 137–143).

32. Yuan, Y., Bahl, P., Chandra, R., Chou, P. A., Ferrell, J. I.,Moscibroda, T., et al. (2007). KNOWS: Cognitive radio net-works over white spaces. In Proc. IEEE DySPAN (pp. 416–427).

33. Eghbali, A., Johansson, H., & Löwenborg, P. (2008). Flexiblefrequency-band reallocation: Complex versus real. Circuits,Systems, and Signal Processing, (in press).

34. Eghbali, A., Johansson, H., & Löwenborg, P. (2007). Flexiblefrequency-band reallocation MIMO networks for real signals.In Proc. int. symp. image signal process. analysis. Istanbul,Turkey.

35. Mitra, S. K. (2006). Digital signal processing: A computerbased approach. New York: McGraw-Hill.

36. Vaidyanathan, P. P. (1993). Multirate systems and filter banks.Englewood Cliffs: Prentice-Hall.

37. Fliege, N. J. (1994). Modified DFT polyphase SBC filterbanks with almost perfect reconstruction. In Proc. IEEE int.conf. acoust. speech, signal process. (pp. 149–152). Adelaide,Australia.

38. Karp, T. (1997). Modifizierte DFT-filterbänke. Ph.D. disserta-tion, Dbüsseldorf, VDI-Verlag.

39. Karp, T., & Fliege, N. J. (1999). Modified DFT filter bankswith perfect reconstruction. IEEE Transactions on Circuitsand Systems II, 46(11), 1404–1414.

40. Heideman, M. T., & Burrus, C. S. (1986). On the number ofmultiplications necessary to compute a length-2n DFT. IEEETransactions on Acoustics, Speech, and Signal Processing,ASSP-34, 91–95.

41. Mansour, M. F. (2006). On the odd-DFT and its applicationsto DCT/IDCT computation. IEEE Transactions on SignalProcessing, 54, 2819–2822.

42. Fliege, N. J. (1993). Computational efficiency of modifiedDFT-polyphase filter banks. In Proc. 27th Asilomar conf. sig-nals, syst., and computers (pp. 1296–1300). Asilomar.

43. Karp, T., & Fliege, N. J. (1996). Computationally efficientrealization of MDFT filter banks. In Proc EURASIP eur.signal process. conf. Trieste, Italy.

44. Yuan, Y., Bahl, P., Chandra, R., Moscibroda, T., Narlanka, S.,& Wu, Y. (2007). Allocating dynamic time-spectrum blocks incognitive radio networks. In Proc. ACM MobiHoc.

45. Ho, C. Y.-F., Ling, B. W.-K., Liu, Y.-Q., Teo, K.-L., & Tam,P. K.-S. (2005). Optimal design of nonuniform FIR transmul-tiplexer using semi-infinite programming. IEEE Transactionson Signal Processing, 53(7), 2598–2603.

46. Xie, X. M., Chan, S. C., & Yuk, T. I. (2006). Design oflinear-phase recombination nonuniform filter banks. IEEETransactions on Signal Processing, 54(7), 2809–2814.

47. Ding, Y. S. F., & Chen, T. (2006). 2-norm based recur-sive design of transmultiplexers with designable filter length.Circuits, Systems and Signal Processing, 25(4), 447–462.

48. Chen, T., Qiu, L., & Bai, E.-W. (1998). General multi-rate building structures with application to nonuniform filterbanks. IEEE Transactions on Circuits and Systems II, 45(8),948–958.

49. Liu, T., & Chen, T. (2001). Design of multichannel nonuni-form transmultiplexers using general building blocks. IEEETransactions on Signal Processing, 49(1), 91–99.

50. Farrow, C. W. (1988). A continuously variable digital de-lay element. In Proc. IEEE int. symp. circuits syst. (Vol. 3,pp. 2641–2645). Espoo, Finland.

51. Johansson, H., & Löwenborg, P. (2003). On the design ofadjustable fractional delay FIR filters. IEEE Transactions onCircuits and Systems II, 50(4), 164–169.

52. Johansson, H., & Gustafsson, O. (2005). Linear-phase FIRinterpolation, decimation, and M-th band filters utilizing theFarrow structure. IEEE Transactions on Circuits and SystemsI, 52(10), 2197–2207.

Amir Eghbali was born in Urmia, West Azerbaijan, Iran in1980. He received the B.Sc. degree in Electrical Engineeringfrom Iran University of Science and Technology, Tehran in 2003,and the M.Sc. degree in Electrical Engineering from LinköpingUniversity in 2006. He is currently working toward the Ph.D.degree in Electrical Engineering at the Division of ElectronicsSystems, Department of Electrical Engineering, Linköping Uni-versity, Sweden. He co-authored a paper which received the bestregular paper award in IEEE EWDTS 2008. His research inter-ests include digital and multirate signal processing, applicationspecific integrated circuits, and VLSI design.

Håkan Johansson research encompasses design and implemen-tation of efficient and flexible signal processing (SP) systems,mainly for communication applications. During the past decade,Prof. Johansson has developed many different SP algorithms forvarious purposes, including filtering, sampling rate conversion,signal reconstruction, and parameter estimation. He has devel-oped new estimation and compensation algorithms for errors inanalog circuits such as compensation of mismatch errors in time-interleaved analog-to-digital converters and mixers. He is one of

J Sign Process Syst

the founders of the company Signal Processing Devices that sellsthis type of advanced signal processing. Prof. Johansson is theauthor or co-author of 4 textbooks and over 100 internationaljournal and conference papers. He is the co-author of threepapers that have received best paper awards and he has publishedone invited paper in IEEE Transactions and one invited chapterin a book on multirate systems. Prof. Johansson has served/servesas Associate Editor for IEEE Trans. on Circuits and Systems Iand II, IEEE Trans. Signal Processing, and IEEE Signal Process-ing Letters, and he is a member of the IEEE Int. Symp. Circuits.Syst. DSP track committee.

Per Löwenborg (S’00–M’03) was born in Oskarshamn, Sweden,in 1974. He received the Master of Science degree in AppliedPhysics and Electrical Engineering and the Licentiate, and Doc-toral degrees in Electronics Systems from Linköping University,Sweden, in 1998, 2001, and 2002, respectively. Dr. Löwenborg’sresearch interests are within the field of theory, design, and im-plementation of analog and digital signal processing electronics,in particular architectures and circuit techniques for filters, fil-ter banks, and data converters. Dr. Löwenborg is the author ofsome 70 international journal and conference papers, one textbook, and holds 3 patents and is currently an Associate Editorfor IEEE Signal Processing Letters. He is a co-founder of theelectronics company Signal Processing Devices Sweden AB andwas awarded the 1999 IEEE Midwest Symposium on Circuits and

Systems best student paper award and the 2002 IEEE NordicSignal Processing Symposium best paper award. He is currentlyworking as R&D Manager at Signal Processing Devices and holdsa part time position as Assistant Professor at the Department ofElectrical Engineering at Linköping University, Sweden.

Heinz G. Göckler was born in Pforzheim, Germany. He receivedthe Dipl.-Ing. Degree in Electrical Engineering from UniversitätFridericiana, Karlsruhe, Germany, in 1968, and the Dr.-Ing. De-gree from Friedrich-Alexander-Universität, Erlangen, Germany,in 1989. From 1970 to 1996 he had been with AEG-Telefunken,ANT and finally Bosch Telecom, Backnang, Germany, in SpaceCommunications and, most time, in the Advanced DevelopmentDepartment. Since 1996 Dr. Göckler has been a professor forDigital Signal Processing and the head of the Digital SignalProcessing Group at the Department of Electrical Engineer-ing and Information Technology of Ruhr-Universität, Bochum,Germany. The main research interests of Dr. Göckler are in thefield of digital signal processing and filtering, multirate process-ing, filter banks and hypercomplex digital signal processing. Hehas published roughly 100 papers, about 50 patents, and onebook on Multirate Systems. Moreover, he has been editor andguest editor for various journals and conferences. Dr. Göckleris a member of the German Informationstechnische Gesellschaft(ITG) and the European Association for Signal Processing(EURASIP).