IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. …

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 3, MARCH 2009 1113

Time-Varying FIR Transmultiplexers WithMinimum Redundancy

Cássio B. Ribeiro, Member, IEEE, Marcello L. R. de Campos, Senior Member, IEEE, andPaulo S. R. Diniz, Fellow, IEEE

Abstract—In this paper, we derive conditions for existence ofzero-forcing equalizers for FIR transmultiplexers. We extendtheoretical results from the literature for zero-forcing equalizationin transmultiplexer systems, and derive new conditions for a moregeneral configuration that includes filter-bank based systems withlong responses and time-varying filter banks. The time-varyingfilter banks can be used to model code division multiple accesssystems with long codes, as well as other practical problems. Theresults can be applied to both downlink and uplink scenarios,due to the general framework considered for the analysis. Thederived relations allow the use of relatively simple equalizers, leadto transmitters using small amount of redundancy, and also allowthe transmission through channels with long impulse responses.Experimental results obtained via computer simulations validatethe derived expressions. The simulations were carried out forzero-forcing design and also for least-squares and minimum meansquared error designs that use the relations derived for existenceof zero-forcing equalizers. The results show that the performanceof all equalizers improve if the zero-forcing conditions derived inthis paper are followed.

Index Terms—Code-division multiaccess, equalizers, transmul-tiplexers, zero-force equalizers.

I. INTRODUCTION

T HE recent demand for high-data rate wireless communi-cations has sparkled new research efforts aiming the max-

imum possible exploitation of channel capacity. An example isthe current development of fourth-generation mobile communi-cations systems, which requires a redefinition of the radio inter-face, impacting directly the multiple-access techniques and datamodulation schemes. Multiple input multiple output (MIMO)systems, together with orthogonal frequency division multiplex(OFDM) and filter-bank based systems, are expected to play animportant part in this development.

For filter-bank based transmultiplexers, which includeOFDM as a special case [1], finite impulse response (FIR)

Manuscript received January 21, 2008; revised October 18, 2008. First pub-lished November 25, 2008; current version published February 13, 2009. Theassociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Zhengyuan (Daniel) Xu. This work was supported in partby CNPq and by FAPERJ.

C. B. Ribeiro is with Nokia Research Center, Wireless Systems and ServicesLaboratory, FI-00045 Nokia Group, Finland (e-mail: [email protected]).

M. L. R. de Campos and P. S. R. Diniz are with the Electrical EngineeringProgram, Federal University of Rio de Janeiro, Rio de Janeiro, RJ, 21941-972,Brazil (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2008.2010007

filters are usually preferred to infinite impulse response filters,due to the difficulties in design and analysis of the latter. Designand performance of transmultiplexers based on FIR filter bankshave been investigated, e.g., in [2]–[7]. Previous works inthe literature have analyzed the behavior of communicationssystems based on filter banks [1], [8]–[12], but due to the dif-ferences in the analysis tools and in the assumptions consideredin each work, the results are sometimes complementary and/orconflicting. Nevertheless, the importance of redundant symbolsin the transmitted block for zero-forcing (ZF) equalization withFIR filter banks is consistently addressed in [1], [8]. The exactamount of redundancy depends on the channel realization andtransmultiplexer design.

In this paper, we derive conditions for ZF equalization withFIR transmultiplexers. Compared to related works in the liter-ature, the proposed analysis is more general, and it is not re-stricted to systems where intersymbol interference is explicityremoved [1] or time-invariant filters [8]–[10], [12]. As a result,we show that some conditions considered in [1] as necessaryfor ZF equalization are in fact sufficient. Consequently, the rela-tions derived in this paper justify the use of less complex equal-izers and transceivers with smaller amount of redundancy anddemonstrate how the transmission through channels with im-pulse response longer than the block length is possible.

As a byproduct, the obtained theoretical results are usefultools to address the problem of MultiUser Detection (MUD)in communications systems, in particular for those employingCode Division Multiple Access (CDMA). There are severalworks in the literature addressing the problem of MUD forshort-code CDMA systems [13], [14], but MUD for long-codeCDMA systems is a more complicated problem. Recent worksrelated to CDMA systems employing long codes include min-imum mean square error (MMSE) equalization and interferencecancellation [15]–[20], and ZF equalization relying on multiplereceiver antennas and oversampling at the receiver [15], [17],[21]. Long codes are widely used in modern third-generationcommunications systems (UMTS), hence, the need to developdetectors that are able to cope with the conditions presented bythese commercial systems. Using the framework presented inthis article, it is possible to analyze the communications systemas a MIMO system, and the long codes can be considered astime-varying short codes.

This paper is organized as follows: Section II describes thesystem model used throughout the paper. Section III sum-marizes the main related works available in the literature, inSection IV the proposed analysis framework is introducedand the conditions for zero-forcing equalization are presented,

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1114 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 3, MARCH 2009

Fig. 1. Communications system in time domain. Index � indicates that the corresponding signal is at symbol rate � , whereas index � indicates that the corre-sponding signal is at channel rate � with � � �� .

Section V shows how the analysis can be applied to othersystem configurations, and Section VI shows simulation resultsthat validate the analysis.

II. SYSTEM DESCRIPTION

In this section, we describe the basic communications systemmodel that will be used in this work. As will be shown in the fol-lowing sections, by properly defining the filters and the channelmodel the framework may characterize different applicationsand scenarios, like those with time-invariant and time-varyingfilter-bank transmitters and receivers, code division multiple ac-cess (CDMA) systems with short or long codes. It also charac-terizes both direct link and reverse link of wireless communica-tions systems.

The communications system is shown in Fig. 1, where, are the transmit filters, is the

channel model, , , are the receivefilters, and is the interpolation/decimation ratio. Indexindicates that the corresponding signal is at a rate timeshigher that of signals at index . Although we will consideronly SISO channels in the analysis, MIMO channels can alsofit the model.

Index in the definition of indicates that the filterresponse is time-varying at rate , i.e., the filter response is con-stant during the interval . The output of the ex-pander in Fig. 1 for the symbol transmitted by the th user attime instant is given by [22]

integerotherwise

(1)

and can be written as a function of as

(2)

where is the response of the transmit filter corre-sponding to th input symbol. The input and output in (2) are

at the same rate as , but the time variation of occursat the same rate as . Hence, we can rewrite (2) as

(3)

From (1), is nonzero iff is integer. Hence, we canwrite (3) as a function of the input as

(4)

where it was used the mapping .From (4) and Fig. 1, the output of the transmit filters (syn-

thesis filter bank) is given by

(5)

The transmitted signal is applied to a linear time-invariant(LTI) channel and is received together with additiveGaussian noise with zero mean , i.e.,

(6)

If the channel is also time-varying, this time variation can beincorporated into the response of the transmit filters, withabove representing the static component of the channel (cf.Section V).

The receiver estimates the transmitted symbols for the thuser by filtering the signal by and then decimatingthe output by , as shown in Fig. 1. This process is describedby

(7)


RIBEIRO et al.: TIME-VARYING FIR TRANSMULTIPLEXERS 1115

A compact representation in matrix form for the process isobtained by defining the vectors of transmitted and re-ceived symbols, and , respectively, as

(8)

(9)

and the vectors

(10)

(11)

(12)

(13)

Equations (5) and (7) can be rewritten as

(14)

and

(15)

where the elements of the matrix and of thematrix are

(16)

Define the matrix as

......

. . ....

(17)The received signal is then given by

(18)

where the convolution between and introduces inter-symbol interference.

III. RELATED WORKS

In [8], Lin and Phoong establish conditions for ZF equaliza-tion using FIR filter banks. Their system model is restricted totime-invariant transmit filters, and, hence, the results cannot beapplied to, e.g., CDMA systems with long codes. These resultshave been extended in [12] to a MIMO system. The assumptionsconsidered in [8] are

(a) the channel is the same for all users;

(b) the channel is an th-order FIR filter, with;

(c) the block length is greater than or equal to the number ofsymbols, i.e., ;

(d) the transmitter is time-invariant.Let us define congruous zeros as a set of distinct zeros

, such that .Based on these assumptions, and following a -domain anal-

ysis, the necessary and sufficient conditions for ZF equalizationderived in [8] are

(A) the redundancy must be greater than or equal to, where is the cardinal of the largest number

of congruous zeros with respect to [8].(B) the matrix with transmit filters must be full rank.If the transmit filters are time-varying, then the sufficient con-

ditions for ZF equalization no longer hold, as shown in [1].However, in [1] the resulting conditions for ZF equalization areunnecessarily restrictive. The assumptions considered in [1] are

(a) the channel is the same for all users;(b) the channel is an th order FIR filter, with

;(c) , , and are chosen such that and ;(d) , , are causal length- FIR fil-

ters, and , , are causal length-FIR filters, with integer.

Based on these assumptions and following a time-domainanalysis, Scaglione et al. concluded that if the transmit filtersare time-varying, the necessary and sufficient conditions for ZFequalization are

(A) the redundancy must be greater than or equal to 1.(B) the length in blocks of the receive filters must be greater

than or equal to ;(C) the matrix with the transmit filters must be full rank, with

additional design restrictions (see [1] for a detailed de-scription).

IV. ZERO-FORCING EQUALIZATION

We extend the results summarized in Section III by removingthe constraint on the length of the transmit filters and of thechannel impulse response. Also, the transmit filters can be time-varying. In this section, the conditions for ZF equalization arederived, first considering filters with short responses, i.e., filterswhose order is less than the expander rate. We then generalizethe results for a transmit system with long response.

Some assumptions that will be considered in the followingfor equalization are now introduced:

(a) the channel is modeled as an th-order FIR filter, with;

(b) the block length is greater than or equal to the number ofinput symbols, i.e., ;

(c) , , are causal length- FIRfilters, and , , are causallength- FIR filters, with integer .

Assumption (a) implies that matrices are zero, except for, with , where denotes the smallest



integer greater than or equal to . Assumption (c) implies thatmatrices are zero, except for , and matrices arezero, except for .

Based on these assumptions we have , andhence, (18) can be rewritten as

(19)

Let us define the vectors , and as

(20)

(21)

We can then write

...

(22)

where is , is ,and is , and given by

......

. . .. . .

...(23)

.... . .

...

(24)

(25)

Finally, we can write the relation between the transmitted andreceived symbols as

(26)

where the matrix is defined as

(27)

A. System With Short Transmit Filters and Single Channel

We will assume that the channel is the same for all users.From (26), ZF equalization is obtained if and

(28)

where is the delay in the reception. Without loss of generality,we will assume for the derivation of the conditions for

ZF equalization. The condition in (28) is satisfied if and onlyif the space spanned by the rows of contains the rows of

, i.e.

(29)

where denotes the space spanned by the columns of .Let us express the relation in (28) and (29) in terms of the null

space of , denoted as . Let us define the vectoras the th column of an identity matrix. Hence, we

can say that if there is a vector such that

(30)

then the condition expressed in (29) is satisfied if and only if.

Since it is not easy to characterize , it is better to workwith the nullspace of , that has a straightforward characteri-zation as a function of the zeros of . It is possible to verifythat

(31)

where are the roots of the polynomial , is the nullspace of , and is the direct sum operation. The first set onthe right-hand side (RHS) of (31) is due to the fact that ,defined in (17), has zero columns.

If , then is orthogonal to all vectors in .Hence, (29) is satisfied if and only if the vector that satisfies(30) also satisfies

(32)

Let us make the assumption that has full column rank.Without loss of generality we can write

(33)

where is , and the matrixpossesses full column rank. Since the ZF condition can be gen-eralized to , we can use a generalvector instead of vector in (30), and, hence, . From(24) and (33), we can rewrite (30) as

(34)where , and

defines a block-diagonal matrix.It can be shown (see Appendix A) that the problem of finding

a solution of (34) that is orthogonal to the vectors in the setis equivalent to finding the solution to

(35)



where the matrix is defined as

...... (36)

the matrix is definedas

.... . .

... (37)

is a vector of unknowns, and isan vector. It is convenient to also define the matrix

(38)

Equation (35) admits solution if and only if. Then, a necessary condition for existence of solution for (29)

is

(39)

In [8], [23], the authors show that if the transmit filters aretime-invariant, one condition that must be satisfied is

, where is the cardinal of the largest set of congruous zeros,as defined in Section III (see Appendix B). However, an impor-tant observation is that this condition is no longer necessary fortime-varying transmit filters. Defining as the number of dis-tinct transmit filters, ZF equalization is possible if and only if

and (see Appendix B). If is notexactly known, it suffices to consider the worst case for deter-mining . Hence, it is sufficient to have

(40)

In Appendix B it is shown that the transmit filters must bedesigned such that

(41)

has full rank. For example, a simple design that guarantees thathas full rank for the case is given by

(42)

It is interesting to observe that this simple design comprisestransmitters based on cyclic prefix and extension by zeros,which are very commonly used. Also, it is clear that thetransmit-filter design does not need channel state information.

The conditions derived so far constitute the necessary and suf-ficient conditions for ZF equalization, given the assumptions de-scribed at the beginning of this section. It should be noted thatthe result is more general and guarantees equalization with a re-ceiver of smaller length than in [1], meaning that the conditionsin [1] are only sufficient.

The results show that for time-varying transmit filters theamount of redundancy, , does not depend on the numberof congruous zeros, as already observed in [1]. The redundancyjust defines the lower bound for given by (39). This allowsthe system design to be independent of a particular channel.However, for systems where the channel order is greater thanthe block length, it is still necessary to guarantee that ,where is the cardinal of the largest set of congruous zeros, asdefined in Section III. This relation is not observed in [1] sincethe authors assume , and by definition .

For example, a CDMA system with long codes can be shownto be equivalent to the system in Fig. 1 with time-varyingtransmit filters (c.f. Section V-B). Hence, we conclude thatfor this system configuration ZF equalization will be alwayspossible, provided that the system is not at full capacity, i.e.,provided that . However, additional redundancy can beused to trade off system throughput and receiver complexity.

One can also observe that these conditions are also satisfiedby OFDM systems [24]. In OFDM systems the transmit filtersconsist of an inverse DFT operator followed by the insertion of acyclic prefix of length . The length of this cyclic prefix ischosen as , which corresponds to a worst case sce-nario, when all channel zeros would belong to a common groupof congruous zeros. Also, for a channel order it is assumedthat , implying that . Hence, from (39) ZF equal-ization is possible for , a well-known result for OFDMsystems.

B. System With Long Transmit Filters and Single Channel

In order to accommodate more general filter banks in thetransmitter, assumption (c) in Section II will be modified to

(c.1) , , are causal length-FIR filters, and , , are causallength- FIR filters, with and integers.

This modified assumption implies now that matrices arenull, except for .

Based on this assumption and assumptions a) and b) inSection II, we can write the input-output relation of the systemin a manner similar to (26)

(43)

where the matrix isgiven by (44) shown at the bottom of the next page.

Again, from (43), ZF equalization is obtained if and

(45)

Let us assume that ,, and let us define

(46)

It can be shown (see Appendix C) that a necessary conditionfor the existence of ZF solution is

(47)



TABLE INECESSARY AND SUFFICIENT CONDITIONS FOR ZF EQUALIZATION

Without loss of generality, we can write as

(48)

where the matrix is full rank, is, and is , .

It can be shown (see Appendix C) that if, then it is necessary to have different transmit filters,

with .Once the conditions on the design of the transmitter are met,

the ZF solution can be achieved by modifying the condition in(39) to (see Appendix C)

(49)

Hence, by allowing enough degrees of freedom to the designof the receiver it is possible to guarantee the perfect reconstruc-tion of the transmitted symbols. It is important to note from (49)that an increase in the length (in blocks) of the transmit filter by

does not correspond to an increase of the same amount inthe receive filters. The actual increase in the length of the receivefilters is . If the system is designed in orderto have high throughput, meaning that the ratiois large, then the complexity of the equalizer will be greatly in-creased. On the other hand, if the complexity of the receiveris an important issue, then by adding redundancy it is possibleto reduce the impact of the longer transmit filters, specially if

.This result can be used, e.g., in the design of discrete wavelet

multitone (DWMT) systems, where longer filters are used inthe transmitter in order to achieve higher spectral efficiency

[25]–[28]. This is a useful property in scenarios where narrow-band interference or jammers are likely to exist and bands suf-fering from interference can be “disconnected,” leading to sim-pler interference rejection. Similar strategies are considered forDMT systems, but with reduced efficiency, since only short fil-ters, defined by the IDFT transform in the transmitter, are used.By designing the system according to the conditions specifiedin this section, it is possible to use highly efficient filters in thetransmitter without making the receiver prohibitively complex.

The necessary and sufficient conditions for ZF equalizationare summarized in Table I. It should be noted that, on the trans-mitter side, only partial channel state information is necessary,specifically an upper bound on the channel order, and an upperbound on the number of congruous zeros. This information isused to specify the system dimensions. The transmit filter de-sign is independent of the particular channel realization.

C. Relation Between Load and Complexity

From the equations shown in Table I, it is not easy to infer howthe choices of filter length, block length, and amount of redun-dancy affect the complexity of the receiver. In this section somerelations are derived in order to provide some insight on howthese different parameters are related. The discussion includessome examples for particular choices of system parameters.

Start by defining the load of the communications system as. Then it is possible to rewrite (49) as a function of

as

(50)

and also as a function of , , , and as

(51)

Fig. 2 shows the behavior of the load as a function of theequalizer length , for block length , transmit-filter

.... . .

. . .... (44)



Fig. 2. Maximum load as a function of � for � � �� and � � �.

length and channel order. One can observe that even though both the transmitter and

the channel have responses longer than the block length, it ispossible to achieve ZF equalization using only one block at thereceiver, provided that the load is low. For example, forit is possible to have ZF equalization with , and for

ZF equalization is possible for . Also, we canobserve that for large values of , a more complex receiver doesnot lead to an equivalent increase in the maximum load of thesystem, whereas for small the load is largely affected by aslight increase in the complexity.

D. Equalizer Design

In the presence of noise, the ZF equalizer may performpoorly, since the receiver design does not take into account thecharacteristics of the noise. This problem is specially criticalif the signal-to-noise ratio (SNR) is low. In [23], the authorsnote that this problem with ZF equalizers can be avoided if thematrix is approximated by a least squares solution, or usinga pseudoinverse, or using a different approach like adaptivefiltering. In all these cases, the conditions found in the previoussections should be used as guidelines for proper design anddimensioning of the system. In this section we will describethe least squares equalizer and derive the minimum meansquared error (MMSE) equalizer for the system described inthe previous sections.

ZF-LS: Let us define the matrix , and thematrix as

(52)

(53)

The matrix that minimizes the quadratic error be-tween the received symbols and the transmitted symbols is

, where denotes the pseudoinverse [29]. An ap-proximation for is obtained using a finite number of symbols

(54)

where

(55)

with .ZF-MMSE: From (26), or equivalently (43), we can define

the mean squared error (MSE) in the estimation of the trans-mitted symbols as

(56)

where is the delay in the reception, assumed known. Beforecontinuing the derivation, let us define the following correlationmatrices:

(57)

(58)

(59)

Using the definitions in (57)–(59) in (56), the error can beexpressed as

(60)

where . Equation (60) can berewritten as

(61)

Differentiating (61) with respect to the elements of , weobtain

(62)

The MSE is minimized by the matrix such that (62) is equalto zero. Hence

(63)

Similar derivations for the MSE equalizer can be found in [1],[11], and [30].

V. APPLICATIONS

In this section, we discuss the conditions just derived forZF equalization applied to two scenarios: (i) when each trans-mitted signal experiences a different channel, and (ii) when long(scrambling) codes are used in code division multiple access(CDMA) systems.



Fig. 3. Communications system in time domain with distinct channels, where � �� and � �� .

A. System With Different Channels

The next step in the generalization of the communicationssystem is to consider that each transmitted signal will go througha different channel to the receiver, modeled by . As brieflydiscussed in [31], this system model can represent both down-link and uplink scenarios in a wireless communications system.

In order to accommodate this new definition of the channels,assumption (a) in Section IV must be changed to

(a.1) each channel is modeled as an th-order FIR filter,with and ;

Without loss of generality, in order to simplify the analysisit is considered in the sequel that the order of all channels is

.Multipath will distort the signal transmitted by each mo-

bile station in a different way because users are locatedin different positions inside a cell in a mobile commu-nications system. Hence, we will assume that all chan-nels can be factored as , where

. The system is representedin Fig. 3.

For time-invariant transmit filters we can define

(64)

For time-varying transmit filters, the equivalent filter to the as-sociation of and can also be represented as alength- FIR filter , ,[32]. The following assumption will be considered in additionto assumption (a.1) and assumptions (b) and (c) in Section IV:

(d) are causal length- FIR filters, for integer.

We observe that the conditions for ZF equalization of thesystem in Fig. 3 have already been investigated in Section IV-B.One more degree of generalization is allowed if we observe thatfor the definition of it is irrelevant if either the transmitfilters or the channel is time-varying. This means that the anal-ysis can also be applied to time-varying channels, assuming theycan be decomposed as in Fig. 3.

B. Multiuser Detection

In a communications system employing CDMA technology,the sequence of symbols from each user is multiplied by a code

at higher rate. This process can be represented by the transmul-tiplexer model used in this article.

Commercial CDMA systems (e.g., UMTS) usually employlong codes in order to help mitigate multipath distortion andinterference. A long code is a pseudorandom sequence leadingto the same spreading gain as a short code, but the former is

times longer, where is an integer. In these systems,the symbols of each user are spread by a short code and thenmultiplied at chip rate by a long code, that may also be calledscrambling code.

Let , , be the short code pertainingto the th user, and , , be the

-length scrambling (long) code of user . The transmittedsignal is given by

(65)where denotes the largest integer lower than .

This is equivalent to consider that the short code for each useris time varying, and that the variation is periodic with period

. This time-varying short-code is given by

(66)

Equation (65) can be rewritten as

(67)

Therefore, the MIMO system considered in this article can rep-resent this long-code CDMA system if we make

.Once the relation between the system considered in this ar-

ticle and CDMA systems with long codes is established, it isstraightforward to conclude that ZF-MUD is possible if the con-ditions in Table I are observed.

If the constraints in the complexity of the receiver are not con-sidered, i.e., if any finite value is acceptable, then the possi-bility of performing ZF-MUD is conditioned to the number ofunused codes . For short codes, it is also necessaryto take into account the number of congruous zeros, whereasfor long codes ZF-MUD is possible whenever the system is not



Fig. 4. BER as a function of SNR for � � �� and� � �, 9, 12, 14.

at full capacity, i.e., if . The complexity of the re-ceiver will depend directly on the difference , as notedin Section IV-C.

VI. SIMULATION RESULTS

This section presents some computer simulations in order toverify the theoretical results and highlight their usefulness. Theexperiments consist of averaging the outcome of 50 transmis-sions through an FIR channel, each one comprising 10 000BPSK modulated symbols for each , for

. The performance is measured in terms of bit error rate (BER).The transmitter consists of random codes for each user, eachcode following a Gaussian distribution with zero mean and unityvariance. The equalizer is designed as in Section IV-D using aLS criterion. The channel model is given by an FIR approxima-tion with 14 coefficients of the following transfer function

(68)

The results shown in Fig. 4 are the BER as a function of thesignal to noise ratio (SNR) for and ,9,12,14.Points for which are not shown in this plot and in theremaining figures. As one could expect, the larger the redun-dancy, the smaller the BER, indicating that the number of usersin the system must be controlled in order to achieve a target BERlevel.

Fig. 5 shows the BER as a function of SNR for ,, and , 2, 3, 4. From (39), is the minimum value

for , and, hence, a degradation in performance is expected asdecreases beyond this limit. This degradation occurs for ,3, however for there is a considerable loss.

In a second simulation, the transmit filters are designed usingthe first length-511 Gold sequences [33] to form the

matrix . Then, we make ,, where is the period of the time-varying .

The equalizer is designed as in Section IV-D using LS criterion.Fig. 6 shows the variation of the MSE with the redundancy

. The simulation was carried out for ,

Fig. 5. BER as a function of SNR for� � ��,� � ��, and� � �, 2, 3, 4.

Fig. 6. MSE as a function of � �� for a channel with a set of congruouszeros with 6 elements.

, for the downlinktransmissions and for the uplink transmissions.For the uplink transmissions the channels weregenerated randomly as a normalized Gaussian process.The channel model is a 14-coefficient FIR filter withzeros

.One can verify that for , the first six channel zeros

form a set of congruous zeros with six elements. Hence, fortime-invariant transmit filters (CDMA with short codes) ZFequalization is possible only if is greater than or equalto the number of elements in the set of congruous zeros with thelargest number of elements. For the channel being considered,ZF equalization will be possible for , as can beverified from Fig. 6 for both downlink and uplink directions.For time-varying transmit filters (CDMA with long codes) ZFequalization is possible for any value of , providedthat there are enough distinct transmit filters. This can be alsoverified from Fig. 6, both for downlink and uplink directions.



Fig. 7. BER as a function of SNR for a channel with a set of congruous zeroswith 6 elements. � � �� and � � ��. The remaining parameters are setas: “ZF” and “MMSE” use � � �� and � � �; “� �� , ZF” and“� �� , MMSE” use� � �� and� � �; and “� � �, ZF” and “� ��, MMSE” use � � �� and � � �. The symbol constellation is 16-QAM.The points with � � � in the simulations for “ZF” �� and“MMSE” �� are not shown in the figure.

Finally, Fig. 7 shows some results obtained using the MMSEequalizer derived in Section IV-D, which demonstrate that evenfor designs other than ZF, the conditions derived here improvesthe performance of the equalizer. The transmitted symbols aregenerated randomly from a 16-QAM constellation. The channelmodel is the same 14-coefficient FIR filter used in the previoussimulation, and the transmitter does zero-padding, i.e., inserts

zeros after each block. For this channel, ,hence, with , the conditions for ZF equalization are

and . In Fig. 7, we show that by not followingthese conditions, either by choosing or , leadsto a floor in the bit error rate (BER), whereas for the designsfollowing the ZF conditions the BER tends to zero as SNR in-creases.

VII. CONCLUSION

In this paper, conditions are derived for existence of zero-forcing equalizers in communication systems employing blocktransmissions. As compared to previous related works, the ob-tained relations allow a reduction in the length of the equalizerfilters, also showing that it is possible to perform zero-forcingequalization when the channels have impulse responses longerthan the block length. The results were extended for with re-sponse longer than the block length, implying that the resultsare also valid for filter-bank based systems.

The proposed framework is general enough to accommodatea variety of configurations, like CDMA systems with longcodes, systems with independent channels for each data stream,and time-varying channels. For the particular case of long-codeCDMA systems, it is shown that ZF-MUD is theoreticallypossible if the system is not at full capacity, i.e., if .Even though the ZF solution exists for all and such that

, the complexity of the receiver and its performance inthe presence of noise depend directly on the difference .

Hence, the conditions derived in this article shall serve as usefulguidelines for the design of communication systems, allowinga trade off between performance and receiver complexity.

If alternative design criteria are used, e.g., minimization ofsquared error or direct-bit-error rate minimization, the derivedconditions are still useful in order to avoid performance loss dueto errors in the reconstruction of the signal. This means thateven for very high SNR, it is possible to have a “BER floor”due to the nonexistence of the ZF solution. This “BER floor” isnonexisting by observing the conditions presented in this paper.

APPENDIX A

It is necessary to characterize that satisfies (34). Let us de-note each section of as ,

. From (34)

(69)where , , denotes eachsection of . Let us divide each vector as

(70)

where and , are and , respectively,. From (69) and (70), we have

(71)

The next step is to establish restrictions on the values of ,, and that guarantee orthogonality between and the

vectors in the set . Let us define

......

... (72)

where are the roots of the polynomial , as in (31).From (70), (71), and (72), (32) can be rewritten as

(73)

where

(74)

Therefore, ZF equalization is possible if and only if there ex-ists a vector such that

(75)

We want to guarantee the existence of solution for (75) regard-less of (and ), implying that the necessary and suf-ficient condition is that .



In order to analyze the structure of , let us define thematrix , given by

(76)

and the vectors and , given by

(77)

(78)

for , where represents the first elementsof th row of , and represents the subsequent elementsof the th row of . With these new matrices we can write (79)shown at the bottom of the page, where

. We observe that the RHS of(79) has a structure that is repeated for all its columns, exceptfor its first columns, which have a different structure ifthe . In order to avoid the extra complications and re-strictions imposed to the analysis by this nonuniform structure,we will constrain the solution of (75) to

(80)

where , , is the new vector ofunknowns.

Let us define the matrix

...... (81)

and the matrix

.... . .

... (82)

Assuming satisfies (80), we can rewrite (75) as

(83)

where is a vector formed with the first elements of .Equation (83) admits solution if and only if

. Then, a necessary condition for existence of solution is

(84)

APPENDIX B

Extra constraints must be met regarding the zeros of thechannel for the special case of time-invariant transmit filters,because special configurations of channel zeros may lead torank deficient matrix . Assume that there are zeros of

such that , , called congruous

zeros (see Section III). Hence, is given by

......

......

(85)

We can write the first rows of as

......

.... . .

...(86)

In (86), matrix

......

has at most independent columns. Conse-quently, if , matrix will not have full rowrank, and ZF equalization will not be possible.

......

... (79)



This dependency on the channel zeros is not an issue for time-varying transmit filters. Again, let us write the first rows of

as

...

.... . .

...(87)

Analyzing (87), we can verify that if we have at least dif-ferent matrices such that

(88)

then the left-hand side (LHS) of (87) possesseslinearly independent columns, with .

This implies that with (88) satisfied, has full row rank ifand only if , and, hence

(89)

Since it is necessary that , and by definition ,the condition for ZF equalization without knowing the exactnumber of congruous zeros is given by

(90)

APPENDIX C

The analysis is similar to the one developed in Appendix A fortransmit filters with short response, but since is not block di-agonal it is not sufficient that we assume has full columnrank. The relation can be written as

.... . .

...

. . ....

...

(91)

where the vector is . Let us assume for sim-plicity that , .Let us assume also that has full column rank. Hence, byanalyzing the lines of , we observe that

(92)

Let us define the matrix

... (93)

By comparing (93) with (91), we observe that

(94)

Assuming for simplicity that ,, the expression for the rank by columns is

given by . Hence, the transmitfilters must be designed such that

(95)

Even if the relation in (95) is satisfied, the existence of solu-tion in (91) is not guaranteed in general if

. In order to derive conditionsfor the design of the transmit filters that avoid reducing the rankof , let us write, without loss of generality

(96)

where the matrix is full rank, is, and is , . In

order to simplify the notation, we will assume, . If , we can write (91) as shown as

(97) at the bottom of the next page.By performing linear row operations on the LHS of (97), its

last lines can be obtained as

(98)

where , , is a scalar coefficient. The matrixin (98) must have full row rank in order to guarantee that thesolution to (97) exists.

The rank of (98) is less than or equal to ,since there are at most transmit filters. Hence, the rank of (98)is equal to if and only if . It is straightfoward



to verify that the restriction on can be generalized to.

Once we can guarantee the existence of solution to (91), weobserve that it can be split in two terms

......

. . ....

. . ....

.... . .

...

......

(99)

where it was considered that . As it will be shownlater, this relation is always satisfied. Since (99) must have asolution for any vector , after some algebraical manipulationswe conclude that (99) is equivalent to

(100)

(101)

where

.... . .

......

(102)

The relation is now split in two, and we must find con-ditions for the existence of a vector that simultaneously solves

(100) and (101). By performing a similar analysis to the one inAppendix A for transmitters with short response, the ZF condi-tion is satisfied if and only if there is a vector such that

(103)

where the rows of , of dimension ,form a base to , and , of dimension

, is given by

(104)

As in Appendix A, we will avoid dealing with the rows ofthat do not correspond to zeros of , which exist only if isnot a multiple of . Hence, we will restrict the solution to

(105)

Consequently, (103) is modified to

(106)

where and are as defined in (36) and (37), respectively.The solution of (100) has already been found in Appendix A,

hence our task is to find which restrictions should be added tothe solution of (100) in order to solve (100) and (101) simulta-neously. The difference between the number of columns and thenumber of rows of is the number of degrees of freedomin the solution of (106). The vector that satisfies (106) can berewritten as

(107)

where is one solution for (35). Thematrix spans the nullspace of

, and is a vector. From(105), we can write as

(108)

.... . .

......

. . ....

......

......

(97)



It is now possible to verify the conditions that the solutionof (100) must satisfy in order to be also a solution of (101).Substituting (108) in (101) and using , we have

(109)

For the equation above to admit solution for any , and, , it is necessary that the number of columns

of the matrix multiplying in the equation above be greater thanor equal to the number of rows, i.e.

(110)

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Cássio B. Ribeiro (M’00) was born in Brazil in 1977.He received the electronics engineering degree fromthe Federal University of Rio de Janeiro (UFRJ), Riode Janeiro, Brazil, in 2000, the M.Sc. degree in elec-trical engineering from COPPE/UFRJ in 2002, andthe D.Sc. degree from COPPE/UFRJ in 2007. He re-ceived the D.Sc. (Tech.) degree from Helsinki Uni-versity of Technology, Helsinki, Finland, in 2008.

In 2006, he worked as a researcher with Nokia In-stitute of Technology, Brazil. Since 2007, he has beenworking as a researcher with Nokia Research Center,

Finland. His research interests are in the fields of digital communication, cel-lular systems, multiple-antenna systems, and statistical signal processing.

Dr. Ribeiro received the Best Paper award from IEEE PIMRC 2005.



Marcello L. R. de Campos (SM’02) was born inNiterói, Brazil, in 1968. He received the engineeringdegree (cum laude) from the Federal University ofRio de Janeiro (UFRJ), Rio de Janeiro, Brazil, in1990, the M.Sc. degree from COPPE/UFRJ in 1991,and the Ph.D. degree from the University of Victoria,Victoria, BC, Canada, in 1995, all in electricalengineering.

In 1996, he was Postdoctoral Fellow with theDepartment of Electronics, School of Engineering,UFRJ, and with the Program of Electrical Engi-

neering, COPPE/UFRJ. From January 1997 to May 1998, he was an AssociateProfessor with the Department of Electrical Engineering (DE/3), Military Insti-tute of Engineering (IME), Rio de Janeiro. He is currently Associate Professorof the Program of Electrical Engineering, COPPE/UFRJ, where he served asDepartment Vice-Chair and Chair during 2004 and 2005, respectively. FromSeptember to December 1998, he was visiting the Laboratory for Telecom-munications Technology, Helsinki University of Technology, Espoo, Finland.In 2008, he visited Unik, the University Graduate Center of the University ofOslo, Oslo, Norway. His research interests include adaptive signal processing,statistical signal processing, signal processing for communications, mobile andwireless communications, and MIMO systems.

Dr. de Campos served as IEEE Communications Society Regional Directorfor Latin America in 2000 and 2001. In 2001, he received a Nokia Visiting Fel-lowship to visit the Centre for Wireless Communications, University of Oulu,Oulu, Finland.

Paulo S. R. Diniz (F’00) was born in Niterói, Brazil.He received the electronics engineering degree (cumlaude) from the Federal University of Rio de Janeiro(UFRJ), Rio de Janeiro, Brazil, in 1978, the M.Sc. de-gree from COPPE/UFRJ in 1981, and the Ph.D. de-gree from Concordia University, Montreal, Canada,in 1984, all in electrical engineering.

Since 1979, he has been with the Department ofElectronic Engineering, UFRJ. He has also been withthe Program of Electrical Engineering (GraduateStudies Dept.), COPPE/UFRJ, since 1984, where

he is presently a Professor. He served as Undergraduate Course Coordinatorand as Chairman of the Graduate Department. He is one of the three seniorresearchers and coordinators of the National Excellence Center in Signal Pro-cessing. From January 1991 to July 1992, he was a visiting Research Associatewith the Department of Electrical and Computer Engineering, University ofVictoria, Victoria, B.C., Canada. He also holds a Docent position with HelsinkiUniversity of Technology, Espoo, Finland. From January 2002 to June 2002, hewas a Melchor Chair Professor with the Department of Electrical Engineering,University of Notre Dame, Notre Dame, IN. His teaching and research interestsare in analog and digital signal processing, adaptive signal processing, digitalcommunications, wireless communications, multirate systems, stochasticprocesses, and electronic circuits. He has published several refereed papersin some of these areas and wrote the books Adaptive Filtering: Algorithmsand Practical Implementation (New York: Springer, Third Edition 2008), andDigital Signal Processing: System Analysis and Design (Cambridge, U.K.:Cambridge University Press, 2002), with E. A. B. da Silva and S. L. Netto.

Dr. Diniz received the Rio de Janeiro State Scientist award from the Gov-ernor of Rio de Janeiro state. He has served as the Technical Program Chairof the 1995 MWSCAS held in Rio de Janeiro. He is the General Co-Chair ofISCAS2011, and Technical Program Co-Chair of SPAWC2008. He has been onthe technical committee of several international conferences including ISCAS,ICECS, EUSIPCO, and MWSCAS. He has served as Vice President for region9 of the IEEE Circuits and Systems Society and as Chairman of the DSP tech-nical committee of the same Society. He is also a Fellow of IEEE for funda-mental contributions to the design and implementation of fixed and adaptivefilters and Electrical Engineering Education. He has served as an Associate Ed-itor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND

DIGITAL SIGNAL PROCESSING from 1996 to 1999, the IEEE TRANSACTIONS ON

SIGNAL PROCESSING from 1999 to 2002, and the Circuits, Systems and SignalProcessing Journal from 1998 to 2002. He was a distinguished lecturer of theIEEE Circuits and Systems Society from 2000 to 2001. In 2004, he served asdistinguished lecturer of the IEEE Signal Processing Society and received the2004 Education Award of the IEEE Circuits and Systems Society.


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