Blind OFDM Channel Estimation Using FIR …singer/pub_files/Blind_OFDM...1136 IEEE TRANSACTIONS ON...

1136 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 3, MARCH 2007

[17] L. Cottatellucci and M. Debbah, “On the capacity of MIMO Rice chan-nels,” in Proc. 42nd Allerton Conf. Communication, Control and Com-puting, Monticello, IL, Sep./Oct. 2004.

[18] R. R. Müller, “On the asymptotic eigenvalue distribution of concate-nated vector–valued fading channels,” IEEE Trans. Inf. Theory, vol.48, no. 7, pp. 2086–2091, Jul. 2002.

[19] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ.Press, 1998.

[20] S. Kay, Fundamentals of Statistical Signal Processing, EstimationTheory. Englewood Cliffs, NJ: Prentice-Hall, 1993, vol. 1, PrenticeHall Signal Processing Series.

[21] V. L. Girko, Theory of Stochastic Canonical Equations. Dordrecht,The Netherlands: Kluwer Academic, 2001, vol. 1.

[22] ——, Theory of Stochastic Canonical Equations. Dordrecht, TheNetherlands: Kluwer Academic, 2001, vol. 2.

Blind OFDM Channel Estimation Using FIR Constraints:Reduced Complexity and Identifiability

Seongwook Song, Member, IEEE, andAndrew C. Singer, Member, IEEE

Abstract—In this correspondence, blind channel estimators exploitingfinite alphabet constraints are discussed for orthogonal frequency-divisionmultiplexing (OFDM) systems. Considering the channel and data jointly,a joint maximum-likelihood (JML) algorithm is described, along withidentifiability conditions in the noise-free case. This approach enablesdevelopment of general identifiability conditions for the minimum-dis-tance (MD) finite alphabet blind algorithm of Zhou and Giannakis. Boththe JML and MD algorithms suffer from high numerical complexity,as they rely on exhaustive search methods to resolve a large number ofambiguities. We present a substantially more efficient blind algorithm, thereduced complexity minimum distance (RMD) algorithm, by exploitingproperties of the assumed finite-length impulse response (FIR) channel.The RMD algorithm exploits constraints on the unwrapped phase of FIRsystems and results in significant reductions in numerical complexity overexisting methods. In many cases, the RMD approach is able to completelyeliminate the exhaustive search of the JML and MD approaches, whileproviding channel estimates of the same quality.

Index Terms—Blind channel estimation, finite-length impulse response(FIR), orthogonal frequency-division multiplexing (OFDM).

I. INTRODUCTION

Orthogonal frequency-division multiplexing (OFDM) is a popularmodulation technique used in a variety of high-rate digital commu-nication standards, including digital audio and video broadcasting(DAB, DVB) [1], [2] and high-speed broadband wireless local areanetworks (IEEE 802.11a and HIPERLAN/2) [3], [4]. Similar to othertransmission schemes, such as single-carrier time-division multipleaccess (TDMA) and code-division multiple access (CDMA), OFDMsystems require knowledge of the channel impulse response, or

Manuscript received May 20, 2005; revised November 28, 2006.S. Song is with Samsung Electronics Co., Ltd., Suwon, Gyeonggi, 442-600,

Korea (e-mail: [email protected]).A. C. Singer is with the University of Illinois at Urbana Champaign, Urbana,

IL 61801 USA (e-mail: [email protected]).Communicated by X. Wang, Associate Editor for Detection and Estimation.Color versions of Figures 3–6 in this correspondence are available online at

http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2006.890774

channel state information (CSI), in order to perform equalization. Forthe purpose of channel estimation, training data may proceed or beinserted into a data block at a modest expense of system throughputand various pilot-aided channel estimation algorithms have beendeveloped [5]–[8].

Either to reduce the overhead of training data, or for applicationssuch multicasting or eavesdropping, for which training is either incon-venient or unavailable, blind algorithms aim to perform equalizationin the absence of training sequences or pilots, and a number of blindchannel estimation algorithms have been proposed and investigated inthe literature [9], [10]. For OFDM systems, the cyclic prefix can alsobe effectively exploited for blind identification. Correlation-matchingmethods have been developed in [11] and [12], by taking advantage ofthe cyclostationarity of the transmitted OFDM signal. In [13], the struc-ture induced in the autocorrelation matrix of the received signal due tothe cyclic prefix is used to develop a subspace-based blind identifica-tion algorithm. However, this method fails to guarantee identifiabilityfor channels with nulls located on subcarriers (discrete Fourier trans-form (DFT) bins of the associated channel impulse response that areidentically zero). A subspace algorithm that guarantees channel identi-fiability was proposed for OFDM systems with zero padding in [14] atthe expense of structural changes to the transmitter and receiver. As iscommon to many subspace approaches, numerical complexity can be-come an issue due to the necessary matrix multiplications and singularvalue decompositions (SVD) of matrices of degree proportional to thenumber of subcarriers used [11], [12], restricting their use.

Joint channel and data estimation methods have also been employedin a number of scenarios, including that shown in [15], in which statis-tical sampling methods are used for single-carrier systems with turbo-equalization-based receivers, and [16] in which sequential Monte Carlo(SMC) methods are studied for OFDM systems. Even though suchSMC methods can be shown to be asymptotically optimal for certaincriteria, they suffer a number of potentially major drawbacks. First,they typically require high numerical complexity, due to their itera-tive nature. Second, their performance can be sensitive to the amountand quality of prior knowledge available about the channel, which isoften rather limited. Decision-directed approaches use quantized out-puts from the equalizer or the decoder, as does the blind algorithm in[17], [18], which makes use of the output of an error correction de-coder to refine a channel estimate starting from a coarse initial channelestimate. This comes at the expense of numerical complexity, storage,and latency involved in transferring the data between the additional en-coding and decoding.

finite alphabet constraints are often employed in decision-directedstrategies, where hard decisions (quantization of equalizer or decoderoutputs to valid symbols) facilitate channel estimation when such sym-bols are largely correct. The existence of a finite alphabet constraint onthe transmitted data has been applied to enable factorization of the datamatrix in multiple-input multiple-output blind channel estimation forsingle-carrier communication systems [19], [20]. Recently, in [21] and[22], a minimum distance (MD) blind algorithm exploiting finite al-phabet constraints was proposed for OFDM systems and shown to havea number of benefits, including guaranteeing channel identifiability, re-gardless of the existence of null subcarriers, and requiring fewer sam-ples for convergence under phase-shift keying (PSK) signaling. How-ever, the MD algorithm relies on exhaustive search techniques for re-solving ambiguities for each subcarrier. As such, the numerical com-plexity becomes exponential in the number of subcarriers, making itrapidly become intractable for systems with large numbers of subcar-riers. To combat the complexity of these approaches that rely on ex-haustive search, in [21], a number of reduced complexity modifications

0018-9448/$25.00 © 2007 IEEE

Authorized licensed use limited to: University of Illinois. Downloaded on July 27,2010 at 06:27:23 UTC from IEEE Xplore. Restrictions apply.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 3, MARCH 2007 1137

Fig. 1. Baseband model of OFDM systems.

have been introduced, such as the modified minimum distance (MMD)algorithm that performs an exhaustive search over a subset of ambigu-ities on prespecified subcarriers, and the phase-directed MD method.

The organization of this correspondence is as follows. We first deriveand discuss a joint maximum-likelihood (JML) estimate of the channeland data, which exploits constraints imposed by the finite transmissionalphabet in an OFDM system. Such a joint estimation algorithm wassuggested in [21] for OFDM, leaving issues such as channel identifi-ability open. After stating identifiability conditions for the JML algo-rithm, we will use similar methods to show identifiability conditionsfor the MD algorithm, under less restrictive conditions than those pre-viously known [21], [22]. Also, we present an overview of several blindOFDM channel estimation schemes and propose a new reduced com-plexity minimum distance (RMD) technique, building on the work of[21], by exploiting finite-length impulse response (FIR) channel con-straints. The RMD algorithm inherits the search dimensionality reduc-tion benefits (from the JML approach) of the finite alphabet blind al-gorithm of [21], with a substantial additional reductions in overall al-gorithmic complexity and without the need for an initial channel es-timate. This makes the algorithm particularly attractive for practicalsystems, even when a large number of subcarriers is in use. For mildlyfrequency-selective channels (i.e., those with at most one spectral nullin the equivalent complex baseband channel response), the RMD algo-rithm completely removes the exhaustive search procedure of the MDapproach and its variants, through phase unwrapping. Section IV dis-cusses the FIR constraints used in the derivation of the JML and RMDblind algorithms detailed in Section V. Identifiability conditions are ad-dressed in Section VI. Simulation results and comparisons are made inSection VII, followed by concluding remarks in Section VIII.

II. NOTATION

• xxx:m: Element-wise mth power of xxx.• [AAA]i;j : ith row and jth column element of AAA.• FFFN;M :N�M DFT matrix, where [FFFN;M ]p;q = exp �j 2�pq

N,

0 � p < M , 0 � q < N .• IIIN : N � N identity matrix• 111N : N � 1 vector with 1 as its entries• (�)�: Conjugate operator.• (�)H : Hermitian operator.• (�)T : Transposition operator.• Arg[�]: Principal value of the phase.• arg[�]: Continuous phase or unwrapped phase.• R(AAA): Range space of AAA.

• N (AAA): Null space of AAA.• kxxxk2

pxxxHxxx.

• kAAAk maxkxxxk =1 xxxHAAAxxx.

• diag(a1; . . . ; an): Diagonal matrix with a1; . . . ; an as its diag-onal elements.

• cov[xxx] E[xxxxxxH ] � E[xxx]E[xxxH ], where xxx is a vector.• (a � b)[n]: Convolution of sequences a[n] and b[n].• (b �L b)[n]: Convolution of b[n] with itself L times.• Bc: Set complement.• A n B: Set difference, i.e., A n B = A \ Bc.• vec(AAA): Vectorization of AAA.• AAA � BBB: Khatri–Rao product.

III. SYSTEM MODEL

A canonical model for an OFDM system is depicted in Fig. 1. Theinformation sequence b[n] is M -ary mapped to an element of the al-phabet A, and a stream of data s[k] 2 A is concatenated into a lengthN signal vector, sss (s[0]; s[1]; . . . ; s[N � 1])T 2 AN , through se-rial-to-parallel conversion. Signal vectors sss are modulated with the in-verse discrete Fourier transform (IDFT) and produce uuu 1

NFFFHN;Nsss 2

CN . After insertion of a cyclic prefix uuucyc 2 CN , serially converteddata from uuup = ((uuuTcyc uuuT )T 2 CN+N ) is transmitted througha length-L time-invariant FIR channel with impulse response h[n]. Atthe receiver, the received signal x[n] is converted, through serial-to-par-allel conversion, to a vector xxx 2 CN+N

xxx = Huuup + vvv + nnn (1)

where vvv denotes interference from neighboring OFDM symbols, andnnn 2 CN+N is complex circularly symmetric white Gaussian noisewith cov[nnn] = N

NIIIP , and H 2 C(N+N )�(N+N ) is the convolution

matrix associated with the channel response h[0]; h[1]; . . . ; h[L � 1].The interference vvv in (1) can be eliminated by removing the cyclicprefix and any additional guard time fromxxx, and wheneverNc � L�1,it is possible to write the output from the cyclic prefix removal xxxc interms of a circulant matrix Hc 2 CN�N with (h[0]; h[1]; . . . ; h[L �1]; 0; . . . ; 0) as its first column

xxxc =Hcuuu+ nnnc (2)

=1

NFFFHN;Ndiag(�h[0]; �h[1]; . . . ; �h[N � 1])FFFN;Nuuu+ nnnc (3)

where nnnc 2 CN is a truncated version of nnn, f�h[k]gN�1k=0 is the N -pointDFT of (h[0]; h[1]; . . . ; h[L � 1]; 0; . . . ; 0)T , and in (3), it was usedthat a circulant matrix can be diagonalized by a DFT matrix [29]. Thisis essentially equivalent to noting that the cyclic prefix maps the effect



of the channel to a circular convolution (rather than a linear convolu-tion), and is then equivalent to multiplication of the DFT of the channelimpulse response with that of the data block. This is processed with theN -point DFT to result in yyy (y[0]; y[1]; . . . ; y[N � 1])T , which canbe written

yyy = FFFN;Nxxxc = diag(�h[0]; �h[1]; . . . ; �h[N � 1])sss+www (4)

where www (w[0]; w[1]; . . . ; w[N � 1])T = FFFN;Nnnnc withcov[www] = N0IIIN . In effect, the frequency-selective channel is mappedinto N single-tap (non-frequency-selective) channels, and given �h[k],the transmitted symbol s[k] is recovered by simply dividing y[k] by�h[k], dispensing with time-domain equalization.

IV. FINITE ALPHABET AND FIR CHANNEL CONSTRAINTS

A. Finite Alphabet Constraints

In the context of blind OFDM channel estimation, finite alphabetconstraints were effectively used in [21], [22] yielding certain benefitsas compared to other methods, such as fast convergence for M -PSKsignaling (i.e.,M -ary PSK). Note that, trivially, any symbol s[k] drawnfrom a finite alphabet A of size Q will satisfy the following equation:

Q�1

j=0

(s[k]� aj) = sQ[k] + �1s

Q�1[k] + � � �+ �Q = 0 (5)

where aj 2 A, and �1; . . . ; �Q are determined by the constellationA [21], [22]. The smallest index J such that �J 6= 0 is less than orequal to Q and furthermore J = Q for Q-PSK signaling (i.e., Q-aryPSK) and J = 4 for quadrature amplitude modulation (QAM) with Qsignaling points. As a result of this property, it can be shown [21] that�hJ [k] can be computed or estimated from the product �h[k]s[k] withoutexplicit knowledge of s[k]. For Q-PSK signaling, with js[k]j = 1,s[k] 2 fej2��=QgQ�1�=0 , �hQ[k] can be obtained simply by �hQ[k] =(�h[k]s[k])Q. In this correspondence, for simplicity, we focus on PSKsignaling.

B. FIR Channel Constraints

In practical OFDM systems, the number of subcarriers N inone OFDM symbol is designed to be significantly larger than thechannel length [1], [2], [4]. Thus, the range space of the possiblechannel impulse responses is restricted to lie in a smaller L-di-mensional space than the N -dimensional space of its N -point DFT�hhh (�h[0]; . . . ; �h[N � 1])T .

Property 1: The relationship between the time-domain CSI and fre-quency-domain CSI is given simply through the DFT matrix

�hhh = FFFN;Lhhh (6)

where hhh = (h[0]; h[1]; . . . ; h[L � 1])T .

Thus, the estimate of �hhh needs to lie in R(FFFN;L), a constraint thathas been effectively used in many training-based channel estimatorsfor the purpose of denoising [17], [18], [23].

Property 2: For�hhh:Q (�hQ[0]; . . . ; �hQ[N � 1])T

and

FFF q FFFN;QL�Q+1

the vector �hhh:Q can be expressed as �hhh:Q = FFF q��, where �� =(�[0]; �[1]; . . . ; �[QL�Q])T and �[n] = (h �Q h)[n].

Property 3: Let �hhha (j�h[0]j2; . . . ; j�h[N�1]j2)T , then �hhha = FFF a��,where

�� = (�[�L+ 1]; �[�L+ 2]; . . . ; �[L� 2]; �[L� 1])T

with �[n] = (h � ~h)[n], where ~h[n] = h�[�n], and

[FFF a]p;q = e�j

1 � p � N; 1 � q � 2L� 1: (7)

The following property of the discrete-time Fourier transform (DTFT)of the FIR channel plays an important role in reducing the dimension ofthe search to resolve the multidimensional phase ambiguities that arisein the MD blind algorithm of [21].

Property 4: Let R(z) and R(!) jR(!)jEj� (!) be the z-trans-form of the finite-length sequence r[n], and the DTFT of r[n], respec-tively, where �e(!) arg[r(!)]. Then, �e(!) can be written such thatit is continuous for 2 f! j jR(!)j 6=0g.

Remark: Discontinuities in the unwrapped phase arise wheneverphase shifts of � occur due to zero crossings of the magnitude of theDTFT ofh[n], �h(!), and thus possible positions of such discontinuitiescan be detected by examining the frequencies ! for which j�h(!)j = 0.

V. BLIND CHANNEL ESTIMATION WITH FINITE ALPHABET

CONSTRAINTS

A. JML Blind Algorithm

For OFDM systems, joint detection and estimation yields the JMLalgorithm discussed below. In general, JML algorithms can be consid-ered as the expectation a maximization algorithm that takes iterations toreach the maximum-likelihood (ML) estimate. However, since one ofthe parameters of interest sss is in the finite sets, we can obtain a uniqueML estimate of hhhML(sss) for each sss. Assuming that sss0 is transmitted,the received vector can be written as

yyy = diag(sss0)FFFN;Lhhh+www: (8)

Since www is drawn from a complex circular white Gaussian noiseprocess, the ML criterion leads to the following minimization:

(sss?; hhh?) = arg minsss2A ;hhh2C

J0ML(sss; hhh) kyyy � diag(sss)FFFN;Lhhhk

22:

(9)When sss is known, the ML estimate for hhh is given by

hhhML(sss) = FFFHN;LSSS

HSSSFFFN;L

�1

FFFHN;LSSS

Hyyy (10)

where SSS = diag(sss). Replacing hhh by its estimate hhhML(sss), we obtain

J0ML(sss) = III � SSSFFFN;L FFF

HN;LSSS

HSSSFFFN;L

�1

FFFHN;LSSS

Hyyy

2

2

= kyyyk22 � kPPPF (sss)yyyk22; (11)

where

PPPF (sss) = SSSFFFN;L FFFHN;LSSS

HSSSFFFN;L

�1

FFFHN;LSSS

H: (12)

Therefore, the search for transmitted data sss0 can be restated as

sss? = arg min

sss2AJ0ML(sss) = arg max

sss2AkPPPF (sss)yyyk

22: (13)

Using sss?, we finally obtain the channel estimate

hhh?

hhhML(sss?)

= FFFHN;Ldiag(sss

?H diag(sss?)FFFN;L)�1FFFHN;Ldiag(sss

?H)yyy:

(14)

The JML blind algorithm can be summarized by the following equa-tions:

sss? = arg max

sss2AJML(sss) kPPPF (sss)yyyk

22 (15)



�hhh?=FFFN;LhhhML(sss

?): (16)

In Section VI, it will be shown that channel identifiability using theJML algorithm is guaranteed for N � 2L, in the presence of noise-free observations. Since the minimization in (15) is carried out by ex-haustive search, the numerical complexity of the algorithm using asingle OFDM symbol increases rapidly with jAjN , and rapidly be-comes impractical for multiple OFDM symbols. However, the MD fi-nite alphabet blind algorithm in [21] effectively used multiple OFDMsymbols to average out additive noise without increasing the numer-ical complexity of the resulting ambiguity resolution. Unlike subspacemethods, which also perform denoising, channel identifiability of theMD algorithm is guaranteed even when the channel has nulls on sub-carriers. The remaining scalar ambiguity is further restricted to haveunit amplitude and phase values belonging to a finite set, which can beeasily resolved.

B. MD Blind Algorithm

This subsection will review the MD algorithm and many of itsproperties [21], on which the RMD algorithm is based. First, considerthe noiseless case and assume that we are given an accurate estimateof j�h[k]j and �hQ[k], k 2 f0; 1; . . . ; N � 1g, which can be obtainedby using the finite alphabet properties in Section IV or could also beobtained through knowledge of higher order statistics of s[k], i.e.,E[sQ[k]]. The channel estimation problem can then be formulated asfollows.

Problem 1: Given j�h[k]j and �hQ[k], k 2 f0; 1; . . . ; N � 1g, find alength L channel vector hhh consistent with these 2N constraints.

Sincearg[�h[k]] =

arg[�hQ[k]]

Q+

2�

Qm[k]; m[k] 2 f0; . . . ; Q� 1g

the channel can be represented in the frequency domain as �h[k] =

c?k(�hQ[k]) where

c?k 2 � 1; e ; . . . ; e

(Q�1):

This can be expressed in a matrix form as �hhh = diag(ccc?)~hhh where~hhh ((�hQ[0]) ; . . . ; (�hQ[N � 1]) )T

and

ccc? = (c?0; c

?1; . . . ; c

?N�1) 2 �N

is a vector of complex constants to compensate for the phase ambigui-ties in each subcarrier, such that the Qth roots can be taken arbitrarily.

Additionally, since hhh is assumed FIR, any estimate of �hhh, say �hhh, should

satisfy �hhh 2 R(FFFN;L). Therefore, a channel estimate is determinedonce a consistent phase ambiguity vector ccc� is found. While we aresearching for an ambiguity vector that is consistent with the imposedconstraints, we can equivalently formulate this search through the fol-lowing optimization, to facilitate later operation under noisy observa-tions:

ccc? = arg min

ccc2�JQ(ccc) k�hhh:Q � �hhh(ccc):Qk22 (17)

�hhh(ccc) (PPPdiag(ccc)~hhh) (18)

wherePPP 1NFFFN;LFFF

HN;L denotes a projection matrix ontoR(FFFN;L).

If N � QL� Q + 1, the cost function in (17) can be replaced usingProperty 2 and Parseval’s relation with

Jq(ccc) k�� (ccc)k22 (19)

where��(ccc) = ((h �Q h)[0]; (h �Q h)[1]; . . . ; (h �Q h)[QL�Q])T

h[n] is the channel estimate corresponding to ccc, and �� is a solution toFFF q�� = hhh:Q. Note that when N < QL�Q+ 1, in order for the costfunctions to be equivalent, �� and ��(ccc) would need to be time-aliased toN samples in duration [24]. Alternatively, for the MD blind algorithm,we could use a cost function similar to that of the JML algorithm in(15), such as

ccc? = arg max

ccc2�JP (ccc) kPPPF (ccc)~hhhk

22: (20)

Since �N is a finite set, ccc? can be found by exhaustive search. Whilethese three criteria are equivalent in the noise-free case (in that they areeach minimized to a value of zero for a consistent ambiguity vector),under noisy observations, they may yield different solutions, in general.For example, as we will see, (17) and(19) can be substantially improvedthrough denoising, while(20) cannot.

In the noise-free case, the uniqueness of the channel estimate ob-tained by this method is shown in [21] and [22]. Since the MD algo-rithm searches over�N , the search complexity grows as j�jN , makingit rapidly become impractical as the number of subcarriers increases.Alternatively, the MMD algorithm [21] uses only a small subset of sub-carriers, say NMMD � N for the exhaustive search, yielding a numer-ical complexity of O(QN ). However, this savings comes at theexpense of additional estimation error.

C. Reduced Complexity MD (RMD) Algorithm

In the MMD algorithm, the reduction in numerical complexity isachieved simply by neglecting data from a substantial fraction of avail-able subcarriers. In contrast, the RMD algorithm uses data from nearlyall subcarriers and simultaneously reduces the dimensionality of the setover which the optimization is carried out by exploiting the continuityof the phase of the frequency response of an FIR channel. As a result,the RMD blind algorithm maintains (or nearly maintains) the perfor-mance of the MD algorithm with significantly reduced complexity.

From the continuity of the unwrapped phase of �h(!), we obtain thefollowing.

Property 5: Let �e(!) arg[�h(!)], �e(!) arg[�hQ(!)], ands (!l; !u) � be an interval within which �h(!) 6= 0. Thenfor all ! 2 s, it follows that �e(!) = � (!)

Q+ �0, where �0 2

f0; 2�Q; . . . ; 2�

Q(Q � 1)g.

Proof: Consider �hQ(!) = j�h(!)jQejQ� (!). It then follows that�e(!) = � (!)

Q+ m(!)2�

Q, m(!) 2 f0; 1; . . . ; Q � 1g. Assume

that there exists some !� such that m(!�) 6= m(!), for some ! 2s n!

�. This would cause a discontinuity of �e(!) for ! 2 s, whichcontradicts that �e(!) and �e(!) are both continuous functions for all! 2 s from Property 4.

Property 5 indicates that for the MD algorithm in an interval s

(!l; !u) � , only one single-phase ambiguity exists, which enablessignificant reduction in search dimensionality. Let N f! 2 :j�h(!)j = 0g, and n N = I

k=1(!k;l; !k;u), then an exhaustivesearch over �N can be simplified to one over the smaller set �I bycollecting subcarriers which lie in the same constant phase ambiguityinterval, since the phase ambiguity is constant within each interval.Therefore, it is possible to write

ccc = (ct0; . . . ; ct0; ct1 . . . ; ct(I�1); . . . ; ct(I�1))T 2 CN (21)

where cti 2 � denotes the phase ambiguity shared by subcarriers inith interval. Also, we can find a matrix GGG 2 CL�I such that ccc = GGGcccr ,where cccr (ct0; ct1; . . . ; ct(I�1))

T 2 CI . Then (17) can be written

ccc? = GGGccc

?r ; ccc

?r = arg min

ccc 2�JQ(cccr) (22)



Fig. 2. Magnitude of the frequency response of h[n] = �[n] + �[n � 2].

from which I ambiguities can be resolved up to only one scalar am-biguity. Consequently, by monitoring the zero crossings of j�h(!)j topartition the intervals, the dimensionality of the resulting search can bedramatically reduced. Considering that a scalar ambiguity is inherentto all blind estimation schemes, the dimensionality of the search spacecan be further reduced to I � 1 by fixing one of the phase ambiguitiesat an arbitrary value, e.g., ct0 = 1, yielding

ccc?r = arg minccc 2�

JQ(cccr) (23)

where cccr = (1; ct1; . . . ; ct(I�1))T . Notice that the search can be com-

pletely dispensed with for I = 1, i.e., when the channel has no spectralnulls. For channels with a small number of taps, i.e., those with shorttemporal region of support, or relatively mild channels with at mostone null in their frequency response, the RMD algorithm becomes veryattractive, since it need only search for one or a small number of ambi-guities. Additionally, two such intervals can usually be combined intoone due to the periodicity of the DTFT, by appropriately choosing theinitial frequency for phase unwrapping, thereby completely removingthe exhaustive search for channels with at most one spectral null. Notethat for channels with real-valued discrete-time baseband impulse re-sponses, the search dimension can also be halved, due to phase antisym-metry constraints [24]. Table I summarizes the numerical complexityof each blind algorithm discussed.

Example: Figs. 2 and 3 illustrate the magnitude and phase ofthe channel h[n] = �[n] + �[n � 2], the DTFT of which, �h(!) =2e�j cos !

2, has nulls at ! = �=2, and 3�=2. We can see that

arg[�h4(!)]=4 differs from arg[�h(!)] by the same phase shifts withinthe interval ! 2 ��

2; �2

and ! 2 �

2; 3�2

. By identifying the phaseambiguities for these two disjoint intervals, we can recover arg[�h(!)]from arg[�h4(!)]=4. Here, we also note that by starting the phase

TABLE INUMERICAL COMPLEXITY OF FINITE ALPHABET BLIND ALGORITHMS

unwrapping from the subcarrier right next to the null subcarrier, e.g.,unwrapping the phase from !0 = 3�

2to (!0 + 2�)mod2� = �

2, can

help to further reduce the search dimension.

D. Phase Unwrapping

The RMD algorithm benefits from the relationship between the con-tinuity of the unwrapped phase of �hQ(!) and �h(!) in reducing its nu-merical complexity, however, in practice, it is not usually possible toobtain a continuous expression for �hQ(!). An arbitrary number of fre-quency samples, however, could be obtained using the DFT of �� fromProperty 2, to aid in this process.

Phase unwrapping is, in general, a well-studied problem and issuesthat arise from associated phase unwrapping applications in the pres-ence of limited data are well addressed in the literature [26], [27].Consider the N -point DFT of an FIR channel, where the frequencyspacing between consecutive frequency samples is 2�=N . In general,there is an ambiguity of an integer multiple of � in the phase valuesof the DFT at each frequency sample !k = 2�k=N . This is primarilydue to the possible presence of zeros in the vicinity of the unit circle,which are we refer to as “sharp” zeros, and which produce large phasechanges across adjacent frequency samples. This can occur whenever asteep phase change of more than �=Q occurs between two consecutivesubcarriers in �h[k], which results in a phase shift of more than � for



Fig. 3. Phase of the frequency response of h[n] = �[n] + �[n � 2].

arg[�hQ[k]]. Therefore, if the number of frequency samples is not suffi-cient, unwanted phase jumps may be induced into the unwrapped phaseof �hQ[k], though none of the consecutive subcarriers need be identi-cally zero, and the effect of any unresolved phase distortion may persistin any channel estimator, even at high signal-to-noise ratio (SNR), forthe corresponding interval. Thus, the frequency response would needto be sampled densely enough to guarantee that no phase change largerthan �=Q occurred between consecutive samples within the same in-terval, to avoid unwanted phase shifts of ��.

Given the maximum channel length, the phase shift between sub-carriers can be made sufficiently small by increasing the number ofsubcarriers used. Since the number of subcarriers N is usually signifi-cantly larger than the channel length in many OFDM standards and thephase change between neighboring subcarriers is limited, often, settingR = 1 is sufficient. With this setting, all of the simulation results inSection VII show that the RMD algorithm produces accurate channelestimates. However, when the maximum channel length can be larger(with a longer temporal region of support), and to improve the sharpzero detection process, we may be inclined to oversample the frequencyresponse, by using an RN -point fast Fourier transform (FFT) of ��,where R denotes the oversampling rate. Here, selection of the over-sampling rate R depends on the channel length and subcarrier spacing.Typically, a longer channel has more frequency response zeros and assuch, this would imply that in sharp zero detection the frequency re-sponse should be more densely sampled. Note that such oversamplingwould have no impact on the search dimensionality used to resolve thephase ambiguities in each subcarrier.

E. Detection of Sharp Zeros

The locations of and the number of sharp zeros, which specify the in-tervals over which phase ambiguities are constant, can be deduced fromthe shape of j�h(!)j. Since only a finite number of samples of j�h(!)j are

available for a given size DFT, it may be difficult to determine whetheror not two consecutive subcarriers belong to the same interval, in gen-eral. However, due to FIR channel constraints (for L � N ), j�h(!)jcannot change arbitrarily rapidly. Thus, two consecutive subcarrierswith sufficiently large power must be in the same interval. For example,for a single zero placed between DFT samples, the magnitude of theDFT will change by no more than j1 � e�j�=N j = 2 sin(�=2N) orapproximately �

Nfrom one sample to another. As such, given assump-

tions about the distributions of zeros of the channel, a threshold canbe selected for locating potential sharp zeros based on the magnitudeof adjacent DFT samples. However, when the estimate of j�h[k]j is ob-tained in the presence of noise, the estimate of j�h[k]j will be biasedthrough the magnitude operation impacting accurate localization of thezeros.

In the RMD algorithm, each block of data between consecutive sharpzero candidates is assumed to share the same constant complex expo-nential ambiguity as in Fig. 4. Sharp zero detection and localization canbe carried out by the following two-step procedure:

(1) As will be discussed later, a denoised estimate of the channelmagnitude is obtained by projecting the frequency responsemagnitude estimate onto a (2L � 1)-dimensional subspacespanned by the DFTs of all length-(2L � 1) FIR channels,using Property 3 and (25). This results in an estimate of thechannel magnitude with L � 1 local minima in the frequencydomain, each of which are candidate sharp zeros. As such, thisprojection greatly simplifies the problem of locating these (atmost) L � 1 local minima, such that any suitable fast searchprocedure can be used.

(2) Given these L�1 local minima, we can identify L�1 disjointintervals over which the phase ambiguities must be constant. Inaddition, a further reduction in the number of sharp zeros can bemade by introducing a predetermined sharp-zero threshold �n,



Fig. 4. Sharp zero detection.

such that local minima with sufficiently large magnitude andwhich cannot be sharp zeros, can be ignored in the sharp zerosearch procedure.

For example, we can find three distinct local minima in Fig. 4, andthus three disjoint intervals I0, I1, I2. While a lower value of �n couldreduce the number of search intervals and the associated dimensionalityof the search space, this also increases probability of missing one of thesharp zeros, particularly at low SNR, at which a relatively large positivebias occurs. Therefore, it is desirable to choose �n sufficiently large toavoid missing sharp zeros. Note that no matter how large �n is taken,the ambiguity search dimensionality will never exceedL�1, since thisis the maximum number of local minima present in the magnitude ofan L � 1th-order channel.

Moreover, since phase unwrapping is a nonlinear operation, its ac-curacy does not necessarily improve with SNR, e.g., phase unwrap-ping is susceptible to significant errors for subcarriers with small j�h[k]jcompared with the background noise level. Such errors become rela-tively large and pose performance limitations at high SNR, and thusdata obtained from subcarriers on which the estimate of j�h[k]j2 is lowerthan a noise-floor threshold, �th, are not used for channel estimation.Since these errors are dominant at higher SNRs, the value of �th canbe chosen to be relatively small.

equalizer outputs.

F. Estimation of �hq[k] and j�h[k]j

The RMD algorithm assumes knowledge of j�h[k]j and �hQ[k], which,in general, need to be estimated at the receiver. As the channel lengthbecomes large, the number of possible nulls in the frequency responsealso increases, allowing for the possibility of a considerable number ofsubcarriers to undergo severe attenuation. In this event, an estimate of�hQ[k] can be affected by the channel and receiver noise and can lead

to errors in the exhaustive search of (17). By averaging the estimatesof �� and �� from each of, say, Nb consecutive data frames, the qualityof the estimate, in terms of the SNR, can be improved by 10 logNb

(decibels) [21]. With the aid of Property 2, and as in [21], least squaresestimates of �� and its DFT �hhh:Q can be readily obtained

��LS =1

NFFFHq yyy:

Q; �hhh:Q = FFF q��LS : (24)

In the RMD algorithm, j�h[k]j is used to determine the number of in-tervals, and can be obtained by computing j�hQ[k]j . Since �� alwayslies in a lower dimensional space than ��, further denoising can be per-formed by projecting the estimate j�h[k]j2 onto R(FFF a) using Prop-erty 3. Least squares (LS) estimates of �� and �hhha can be obtained from

��LS = FFFHa yyya;

�hhha =1

NFFF a��LS (25)

where yyya consists of an estimate of j�h[k]j2 obtained through the oper-

ation j � j2=Q for the entries of �hhh:Q.

G. Least squares (LS) Approach

Finally, an estimate of �h[k] which differs term by term from �h[k]only by the selection of one ofQ phase ambiguities, can be constructed

from refined estimates �ha[k] of j�h[k]j2 and �hQ

[k] of �hQ[k] as

~�h[k] = �ha[k]ej

: (26)

Then, an efficient search algorithm for the RMD algorithm can be de-veloped based on an LS criterion. Let

V kjj~�h[k]j2 < �th; k = 0; 1; . . . ; N � 1

~�hhhp (~�h(k1); . . . ;~�h(kN ))T 2 CN



TABLE IISUMMARY OF THE RMD ALGORITHM

and the associated ambiguity vector be cccp 2 CN , whereNp = N�jVj

and k1; k2; . . . ; kN 2 V . Then, in the noiseless case, ~�hhhp and hhh shouldsatisfy that

diag(cccp)~�hhhp = FFFZhhh (27)

where FFFZ 2 CN �L is a sparse DFT matrix with rows correspondingto the subcarriers that are in V deleted from FFFN;L. For each cccpconsidered in the search, the LS solution hhh to (27) is found byhhh = AAAydiag(cccp)

~�hhhp, where AAAy = FFFHZFFFZ

�1FFFHZ .1 The matrix GGGp

can be defined such that cccp = GGGpcccr , and an estimate for hhh associatedwith cccp can be written

hhhLS(cccr) =vec AAAydiag(cccp~�hhhp) (28)

=~�hhhT

p �AAAycccp =~�hhhT

p �AAAy GGGpcccr (29)

=MMMcccr (30)

�hhhLS(cccr) =FFFN;LhhhLS(cccr) (31)

where MMM = (~�hhhT

p � AAAy)GGGp 2 CL�I , and it was used in the secondline that vec(AAAdiag(ddd)BBB) = (BBBT �AAA)ddd[29]. Here, MMM can be com-puted and stored in advance, and hhhLS(cccr) can be computed by L� Ielementwise complex multiplications for each cccr . Since �h(z) is an(L� 1)th-order polynomial and has at most L� 1 roots, the numberof intervals I satisfies I � L� 1, which in turn leads to a reduction innumerical complexity. The resulting RMD algorithm is summarized inthe Table II.

VI. CHANNEL IDENTIFIABILITY

In this section, we discuss channel identifiability of finite alphabetbased blind algorithms using the following definition.

Definition 1: Let �h[k] be the N -point DFT of the time-domainchannel response. Then, �h[k] is identifiable if all simultaneous channelestimates (i.e., the vector of estimates for all subcarriers) can bewritten as q�h[k] for all k =2 Z , where q is a complex constant, andZ = fkj�h[k] = 0; k = 0; 1; . . . ; N � 1g.

We refer to identifiability conditions for an algorithm as a set of suf-ficient conditions such that, when satisfied, all channels (meeting such

1Specially for jVj = 0,AAA = FFF , and an N -point FFT can be used.

conditions) become identifiable by that algorithm. For notational con-venience, we will use the following definition for an ambiguity weightof a finite alphabet vector.

Definition 2: Let sss = (s0; s1; . . . ; sN�1)T 2 CN , where sk 2

fa0; a1; . . . ; aQ�1g and Q � N . If mi is the number of entries withsk = ai, then the ambiguity weight of sss is defined

wt(sss) = maxi=0;1;...;Q�1

(mi): (32)

For example, if sss = (�1 �1 1 1 1 )T , wt(sss) = 3. It is alsopossible to restate channel identifiability as follows: �hhh is identifiableif all simultaneous estimates obtained can be written diag(sss)�hhh withwt(sss) = N .

The following proposition states a sufficient condition for channelidentifiability of a finite alphabet blind algorithm.

Proposition 1: Let �hhh 2 R(FFFN;L) and sss = (s0; s1; . . . ; sN�1)T

with sk 6= 0 for all k. If diag(sss)�hhh 2 R(FFFN;L) only for sss with sk =c0 6= 0, k =2 Z , then channel identifiability is guaranteed.

Proof: Consider that �sss = (�s0; �s1; . . . ; �sN�1)T was sent, i.e., yyy =

diag(�sss)�hhh from (8), then we can see from (15) and (16) that JML(sss) =

kdiag(�sss)�hhhk22 and �hhh =c

jc j�hhh for sss with sk = c0�sk 6= 0 for k =2

Z . However, the rest of sss satisfies �hhh =2 R(diag(sss)FFFN;L) from thenecessary condition, and thus JML(sss) < kdiag(�sss)�hhhk22. Consequently,we can identify the channel �hhh up to a complex scalar c0 through theJML algorithm.

The following proposition is useful for the proof of channel identi-fiability.

Proposition 2: Let FFFM;N (fff0; fff

1; . . . ; fffN�1) 2 CM�N , and

define

UUUFFFM;N

FFFM;Ndiag(sss)2 C2M�N (33)

where sss = (s0; s1; . . . ; sN�1)T and sk is nonzero for all k. If K =

wt(sss), then it follows that

rank(UUU) = min(2M;N;M +N �K): (34)

Proof: See Appendix A.

With the aid of Propositions 1 and 2, it is possible to find conditionsfor which channel identifiability is guaranteed.



Theorem 1: ForN � 2L, the JML blind algorithm in (15) and (16),identifies the channel uniquely up to a scalar complex exponential.

Proof: See Appendix B.

As for the MD blind algorithm, channel identifiability is addressedin [22].

Theorem 2: If N � QL � Q + 1, identifiability of an(L � 1)th-order channel �h(z) is guaranteed from the receiveddata �hQ[k], k = 0; 1; . . . ; N � 1, up to a scalar complex exponential.

Proof: See [22].

Theorem 2 implies that the MMD blind algorithm requires at leastNMMD � QL � Q + 1 subcarriers to resolve the associated ambi-guities. however, with the aid of Theorem 1, we can show the sameidentifiability result under a less restrictive constraint for Q > 2.

Corollary 1: If N � 2L, identifiability of an (L � 1)th-orderchannel �h(z) is guaranteed from the received data �hQ[k],k = 0; 1; . . . ; N � 1, up to a scalar complex exponential.

Proof: For each subcarrier, there exists a phase ambiguity in afinite set A = f1; . . . ; e

j (Q�1)g which can be viewed as a quater-

nary phase-shift keying (QPSK) signal with a phase shift by �=Q. Ifsuch ambiguities are resolved, the channel is identifiable up to a com-plex exponential. Since this is a special case of Theorem 1, the resultis immediate.

For example, the JML algorithm for 4-PSK signaling per-forms a search over ambiguity vectors whose entries are inA = fe ; ej ; ej ; e g. Note that the MD algorithm, given

(�hhh4[k])

1=4, performs precisely the same search for the ambiguity only

over the set A0 = f1; ej ; ej�; ej g. Therefore, we may regard theMD algorithm as the JML algorithm for 4-PSK signaling over A0.

Note that we have used a weak condition that ambiguities are lim-ited to nonzero complex values. Using stronger conditions that ambi-guities are in a finite set and letting hhh be jointly distributed accordingto some continuous probability density function on CL, we can obtainthe identifiability result in a probability 1 sense under a more relaxedrequirement on N .

Corollary 2: If N > L and hhh is jointly distributed according tosome continuous probability density function on CL, and �hhh = FFFN;Lhhh,then identifiability of an (L � 1)th-order channel �h(z) is guaranteedwith probability 1 from the received data �hQ[k], k = 0; 1; . . . ; N � 1,up to a scalar complex exponent.

Proof: See Appendix C

The RMD algorithm is derived from the MD algorithm and thus in-herits all identifiability results of Corollaries 1 and 2. Several exam-ples of channel identifiability with different identifiability conditionsare further discussed in Appendix D.

VII. SIMULATIONS AND RESULTS

In this section, we simulate the MMD, and RMD algorithms andcompare their performance in terms of mean-square error (MSE) andsymbol error rate (SER). Performance of the pilot-aided ML channelestimator equipped with comb pilots with different sized payload isalso presented. In accordance with IEEE 802.11a and HIPERLAN/2,the number of subcarriers is set to N = 64, through which a 4-DPSKsignal is transmitted. Each set of simulations was run over more than1000 realizations of a length four unit FIR channel (L = 4) withGaussian taps that are identically distributed and mutually indepen-dent. In this scenario, the MD blind algorithm using all subcarriers isprohibitive due to its high numerical complexity O(jAjN ) = O(464)and thus, the MMD algorithm that searches over combinations of phase

ambiguities of NMMD = 8 equispaced subcarriers are used, instead,while an estimate for �hQ[k] is obtained using all available data fromNsubcarriers. Notice that even for this case, the numerical complexity ofthe MMD algorithm O(48) is still high. Note that Theorem 1 guaran-tees channel identifiability from 2L = 8 = NMMD. In general, �n and�th for the RMD algorithm need to be normalized by the channel en-ergy. Simulating channels with unit energy, i.e., E[khhhk22] = 1, amongall detected local minima, those with j�hj2 smaller than �n = 0:5 areconsidered as candidate sharp zeros, and data with energy lower than�th = 0:01 are all discarded. The four-tap channel has at most threezeros, and therefore the RMD algorithm has at most two intervals overwhich the exhaustive search is carried out. This in turn implies that theambiguity can be resolved by testing at most 16 combinations. In addi-tion, taking the relatively slow variations of this channel into account,we set the oversampling rate to be R = 1.

In many standards such as IEEE 802.11a and IEEE 802.16a, anumber of OFDM symbols constitute one data frame, and hence itis possible to use several OFDM symbols for channel estimation. Inthis experiment, we show the performance of the RMD algorithmand compare it with the MMD blind algorithm when Nb OFDMsymbols are available. For comparison, a pilot-aided channel estimatorwith Np = 8 comb pilots inserted in each OFDM symbol and allNbNp available data samples used in the ML algorithm is shown.We tested the RMD and MMD algorithm for one OFDM symbolNb = 1 and different channel order estimates Le ranging from 3 to5. As shown in Figs. 5 and 6, using nearly all available subcarriers,the RMD algorithm provides MSE and SER performance comparableto that of the pilot-aided estimator with Np = 8. Figs. 5 and 6 alsoshow that the MMD and RMD algorithms are relatively robust to thechannel order overestimation. However, in case that the channel orderis underestimated, the channel estimators are significantly degraded;the MMD, and RMD algorithms use the projection onto the reducedsignal space and obtain the improved estimates on the desired channel,therefore, the underestimated channel order causes the loss of thechannel information.

VIII. CONCLUSION

In this correspondence, we have presented a reduced-complexity(RMD) blind channel estimation algorithm for OFDM systems, basedon FIR channel constraints. Specifically, exploiting the continuity ofthe unwrapped phase of the FIR channel frequency response dramati-cally reduces the number of resulting phase ambiguities for such blindalgorithms. Additionally, we have discussed channel identifiability forvarious finite alphabet blind algorithms and derived sufficient condi-tions for identifiability in several cases. As such, when the numberof subcarriers is larger than or equal to twice the channel length, i.e.,N � 2L, the JML, MD, and RMD algorithms guarantee identifiability,to within a complex scalar gain. Identifiability with probability 1 isguaranteed for the less restrictive condition that N > L. Simulationresults are shown that indicate that with modest numerical complexity,the RMD algorithm provides SER performance comparable to that ofa pilot-aided channel estimator.

APPENDIX

A. Proof of Proposition 2

Case 1: M � N . The matrix UUU contains the full column rankmatrix FFFM;N , and rank(UUU) � min(2M;N) = N , which leads torank(UUU) = N .

We first illustrate a few useful facts about the matrix UUU that will behelpful in the following proof. Consider s0 = s1 = � � � = sK�1



Fig. 5. MSE performance comparison of the RMD algorithm, L = 4, Q = 4, N = 64,N = 1.

Fig. 6. SER performance comparison of the RMD algorithm, L = 4, Q = 4, N = 64,N = 1.

and sk 6= s0 for k = K; . . . ; N � 1, without loss of generality, andpartition the matrix UUU as

UUU=UUU0

2C

UUU1

2C

UUU2

2C

:

(35)

Note that the column vectors ofUUU0 = (uuu0; uuu1; . . . ; uuuM�1) 2 C2M�M

constitute linearly independent vectors; if M < N , an arbitrary col-lection of m � M column vectors of the DFT matrix FFFM;N are lin-early independent, and hence uuuk (fffTk skfff

Tk )

T , where fffk is the kthcolumn vector of the matrix FFFM;N , are also linearly independent vec-tors.



For M < N , we consider four different cases as follows.Case 2: M < K � N � K +M . For this part of the proof, we

show that there are exactlyM +N �K linearly independent columnsin the matrix UUU , and thus rank(UUU) = M +N �K . Since N �K �M , UUU2 consists of linearly independent column vectors, and columnvectors of UUU1 (uuuM ; . . . ; uuuK�1) 2 C2M�(K�M) are not linearlyindependent of those of UUU0 from the condition that s0 = s1 = � � � =sK�1. In addition, we can show that column vectors ofUUU2 are linearlyindependent from those in UUU0. For uuuk , k0 2 fK;K + 1; . . . ; N �1g, not linearly independent of uuu0; uuu1; . . . ; uuuM�1, then there exists aunique nontrivial zzz 2 CM such that

uuuk = UUU0zzz: (36)

From the definition of uuuk and (36), it follows that fffk = FFF 0zzz

and sk fffk = diag(s0; s1; . . . ; sM�1)FFF 0zzz, where FFF 0 =(fff0; fff1; . . . ; fffM�1). Combining these two equations and noting thatfffk is nonzero, we obtain that sk fffk = diag(s0; . . . ; sM�1)fffk ,and thus sk = s0. This contradicts the assumption that sk 6= s0 fork0 2 fK;K + 1; . . . ; N � 1g. Consequently, column vectors in UUU0

should be linearly independent with each column vector in UUU2, andhence rank(UUU) = rank(UUU0) + rank(UUU2) = M +N �K .

Case 3: M < K � N andN > K+M . SinceN > K+M >2M , it is immediate that rank(UUU) � min(2M;N) = 2M . Also, wecan find a 2M � 2M submatrix in UUU satisfying conditions of Case 2,and hence rank(UUU) � 2M , which proves that rank(UUU) = 2M .

Case 4: K � M < N � 2M . In this case, K �M < 0 andthe matrix UUU1 does not exist, and UUU2 contains N �M � M linearlyindependent columns. The proof is complete if we show that theN�Mlinearly independent columns inUUU2 are also linearly independent fromthose inUUU0. Assume otherwise thatuuuk , k0 2 fM;M+1; . . . ; N�1gis not linearly independent fromUUU0, then for some nontrivial zzz, uuuk =UUU0zzz, which leads to fffk = FFF 0zzz andsk fffk =diag(s0; s1 . . . ; sM�1)FFF 0zzz=diag(s0; s1; . . . ; sM�1)fffk :

Since each entry of fffk is nonzero, this requires that s0 = s1 = � � � =sM�1 = sk , i.e., wt(sss) > M , which contradicts the assumption thatwt(sss) � M . Consequently, the columns of UUU2 are linearly indepen-dent from those of UUU0 and hence rank(UUU) = rank(UUU0) + rank(UUU2)= N .

Case 5: K � M and N > 2M . From the result of Case 4, wecan find a 2M�2M full-rank matrix inUUU , and thus, rank(UUU) � 2M .Considering that rank(UUU) � min(2M;N) = 2M , we obtain thatrank(UUU) = 2M .

Combining the result from each case yields that rank(UUU) =min(2M;N;M + N �K).

B. Proof of Theorem 1

Proposition 1 states that if �hhh 6= 0 is not identifiable, we can find atleast one ambiguity vector sss with sm 6= sn for some m;n =2 Z suchthat

�hhh 2 R(FFFN;L); diag(sss)�hhh 2 R(FFFN;L): (37)

Partitioning the DFT matrix FFFN;N = FFFN;L FFF?

N;L , we can rewriteconstraints in (37) as

FFF?N;LH�hhh = 0; FFF?N;L

Hdiag(sss)�hhh = 0: (38)

In a matrix form, UUU �hhh = 0, where

UUU =FFF?N;L

H

FFF?N;LHdiag(sss)

2 C2(N�L)�N : (39)

Denote K = wt(sss), for which we consider two different cases.Case 1: K � N � L. In this case, Proposition 2 implies that UUU

is a full column rank matrix and �hhh = 0. Therefore, all nontrivial �hhh 6= 0

are identifiable for ambiguity vectors sss with K � N � L.Case 2: K > N � L. By Proposition 2, UUU is a rank-deficient

matrix and thus generates a K �N + L-dimensional null space, andhence there exist (possibly) multiple ambiguity vectors achieving themaximum of JML(sss). However, we can show that such multiple ambi-guity vectors differ only in k 2 Z , and hence, based on Proposition 1,produce valid channel estimates. Without loss of generality, let s0 =s1 = � � � = sK�1 and sk 6= s0 for k = K + 1; . . . ; N � 1, then theproof is complete if we can show that all k = K+1; . . . ; N � 1 2 Z .If �hhh 2 N (UUU), then from (39) we have that

K�1

k=0

sk�h[k]fffk +

N�1

k=K

sk�h[k]fffk = 0 (40)

and simultaneously

N�1

k=0

�h[k]fffk = 0 (41)

where fffk 2 CN�L is the kth column vector ofFFF?N;LH

, i.e.,FFF?N;LH

=

(fff0; fff1; . . . ; fffN�1) 2 C(N�L)�N Combining (40) and (41), we ob-tain that

N�1

k=K

(s0 � sk)�h[k]fffk = 0: (42)

Since ffffK ; fffK+1; . . . ; fffN�1g constitute a set of linearly independentvectors, and sk 6= s0 for k = K;K + 1; . . . ; N � 1, we have that�h[k] = 0 for all k = K;K + 1; . . . ; N � 1, i.e., Z = fK;K +1; . . . ; N � 1g, which completes the proof.

C. Proof of Corollary 2

Let T = f�hhh j �hhh = FFFN;Lhhh;8hhh 2 CLg. An unidentifiable channel,say �hhhu 2 T , should satisfy thatUUU �hhhu = 0. Then, channel identifiabilityfor N � 2L is immediate from Corollary 1. For the case L < N <2L, we have that rank(UUU) = min(2(N � L); 2N � K � L) anddim(N (UUU))= N �min(2(N �L); 2N �K �L) � 2L�N < L.Accordingly, an unidentifiable channel hhhu lies in a space of dimensionstrictly less than L. Since our stochastic setting defines a continuousprobability density function on the set CL, the total measure induced ona proper lower dimensional subset of T is zero. Consequently, since thefinite union of sets of zero measure is also of measure zero, it followsthat the probability of an ambiguous solution is zero.

D. Examples of Channel Identifiability

Example 1: Consider N = 2, L = 2, and QPSK signaling. Thenthe DFT matrix in (8) is given as

FFF 2;2 =1 1

1 �1: (43)

With diag(sss)diag(sssH) = III2 for QPSK signaling and FFF 2;2FFFH2;2 =

FFFH2;2FFF 2;2 = 2III2, we obtain that PPPF (sss) = III2, and thus can write (15)as

JML(sss) = yyyHPPPF (sss)yyy = yyyHyyy = �hhhH�hhh: (44)

This implies that JML(sss) is independent of sss, and thus it is not possibleto resolve the ambiguity vector.



Example 2 (Theorem 1): Consider N = 4, L = 2. Then uniden-tifiable channel should satisfy UUU�hhh = 0 for some sss with wt(sss) < 4,where

FFF 4;2 =

1 1

1 �j

1 �1

1 j

; FFF?

4;2 =

1 1

�1 j

1 �1

�1 �j

UUU =FFF?

4;2

H

FFF?

4;2

Hdiag(sss)

2 C4�4: (45)

Note that rank(UUU) = min(4; 4; 6�wt(sss)) from Proposition 2. There-fore, UUU is a full-rank matrix for all sss with wt(sss) � 2. For wt(sss) = 3,without loss of generality, consider an ambiguity vector s0 = s1 =s2 = 1 and s3 6= 1. From Proposition 2, rank(UUU) = 3, and UUU gen-erates nontrivial null vectors. This implies that the projection used inJML algorithm fails to detect the wrong ambiguity s3 for some �hhh sat-isfying

UUU �hhh =

1 �1 1 �1

1 �j �1 j

1 �1 1 �s31 �j �1 js3

�h0�h1�h2�h3

= 0: (46)

However, we can show that the null vector �hhh of UUU associated with thewrong ambiguity vector sss requires �h3 = 0, and hence the wrong am-biguity s3 does not affect the final channel estimate. Subtracting thethird row from the first row in (46), we have �h3(�1+s3) = 0, and theassumption of s3 6= 1 leads to �h3 = 0.

Example 3 (Corollary 2): Consider N = 4, L = 3, and hhh is aGaussian random vector with cov[hhh] = III3), which satisfies the identi-fiability condition in Corollary 2. From (38), an unidentifiable channelshould satisfy that FFF?4;3

H�hhh = 0 and FFF?4;3Hdiag(sss)�hhh = 0 for some sss

with wt(sss) < 4, i.e., UUU�hhh = 0, where

FFF 4;3 =

1 1 1

1 �j �1

1 �1 1

1 j �1

; FFF?

4;3 =

1

j

�1

�j

UUU = FFF?

4;3

HFFF?

4;3

Hdiag(sss) 2 C2�3: (47)

Since UUU is a wide matrix, we can always find unidentifiable chan-nels described by a null vector of UUU for any sss. However, withdim(N (UUU)) = 1 < dim(�hhh) = 3 and continuity of the probabilitydensity function of hhh, it follows that Pf�hhh 2 N (UUU)g = 0 for any sss.Moreover, since sss 2 A3 which is a finite set, the event of unidentifiablechannels occurs with probability 0, and thus the channel is identifiablewith probability 1.

REFERENCES

[1] Radio Broadcasting System, Digital Audio Broadcasting (DAB) toMobile, Portable, and Fixed Receivers, European TelecommunicationStandard Institute, Sophia-Antipolis, Valbonne, France, 1995–1997,Std. ETS 300 401.

[2] Digital Broadcasting System Television, Sound, and Data Services;Framing Structure, Channel Coding, and Modulation Digital TerrestrialTelevision, European Telecommunication Standard Institute, Sophia-Antipolis, Valbonne, France, 1996, Std. ETS 300 744.

[3] Wireless LAN Medium Access Control (MAC) and Physical Layer(PHY) specifications, IEEE, Piscataway, NJ, 1999, Std. IEEE 802.11a.

[4] Broadband Radio Access Networks (BRAN): High Performance RadioLocal Area Networks (HIPERLAN) Type 2: System Overview, Eu-ropean Telecommunication Standard Institute, Sophia-Antipolis, Val-bonne, France, 1999, Std. ETS 101 683 114.

[5] R. Negi and J. Cioffi, “Pilot tone selection for channel estimation in amobile OFDM system,” IEEE Trans. Consumer Electron., vol. 44, no.3, pp. 182–188, Aug. 1998.

[6] M. Morelli and U. Mengali, “A comparison of pilot-aided channel es-timation methods for OFDM systems,” IEEE Trans. Signal Process.,vol. 49, no. 8, pp. 3065–3073, Dec. 2001.

[7] Y. Li, “Pilot-symbol-aided channel estimation for OFDM in wirelesssystems,” IEEE Trans. Veh. Technol., vol. 49, no. 4, pp. 807–815, Jul.2000.

[8] O. E. Edfors et al., “OFDM channel estimation by singular value de-composition,” IEEE Trans. Commun., vol. 46, no. 7, pp. 931–939, Jul.1998.

[9] L. Tong and S. Perreau, “Multichannel blind identification: From sub-space to maximum likelihood methods,” Proc. IEEE, vol. 86, no. 10,pp. 1951–1968, Oct. 1998.

[10] G. B. Giannakis, Y. Hua, P. Stoica, and L. Tong, Signal Processing Ad-vances in Wireless and Mobile Communications. Englewood Cliffs,NJ: Prentice-Hall PTR, 2001.

[11] R. W. Heath, Jr and G. B. Giannakis, “Exploiting input cyclostation-arity for blind channel identification in OFDM systems,” IEEE Trans.Signal Process., vol. 47, no. 3, pp. 848–856, Mar. 1999.

[12] H. Bölcskei, R. W. Heath, Jr, and A. J. Paulraj, “Blind channel identifi-cation and equalization in OFDM-based multiantenna systems,” IEEETrans. Signal Process., vol. 50, no. 1, pp. 96–109, Jan. 2002.

[13] B. Muquet, M. de Courville, and P. Duhamel, “Subspace-based blindand semi-blind channel estimation for OFDM systems,” IEEE Trans.Signal Process., vol. 50, no. 7, pp. 1699–178, Jul. 2002.

[14] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filter-bank precoders and equalizers-Part II: Blind channel estimation, syn-chronization and direct equalization,” IEEE Trans. Signal Process., vol.47, no. 7, pp. 2007–2022, Jul. 1999.

[15] X. Wang and R. Chen, “Blind turbo equalization in Gaussian andimpulsive noise,” IEEE Trans. Veh. Technol., vol. 50, no. 4, pp.1092–1105, Jul. 2001.

[16] Z. Yang and X. Wang, “A sequential Monte Carlo blind receiver forOFDM systems in frequency-selective fading channels,” IEEE Trans.Signal Process., vol. 50, no. 2, pp. 271–280, Feb. 2002.

[17] V. Mignone and A. Morello, “CD3-OFDM: A novel demodulationscheme for fixed and mobile receiver,” IEEE Trans. Commun., vol. 44,no. 9, pp. 1144–1151, Sep. 1996.

[18] P. Frenger and A. Svensson, “Decision directed coherent detection inmulticarrier systems on Rayleigh fading channels,” IEEE Trans. Veh.Technol., vol. 48, no. 2, pp. 490–498, Mar. 1999.

[19] A. J. Van Der Veen, S. Talwar, and A. Paulaj, “Blind identification ofFIR channels carrying multiple finite alphabet signals,” in Proc. IEEEInt. Conf. Acoustics, Speech, and Signal Processing , Detroit, MI, May1995, vol. 2, pp. 1213–21616.

[20] S. Talwar, M. Viberg, and A. Paulraj, “Blind estimation of multipleco-channel digital signals using an antenna array,” IEEE SignalProcess. Lett., vol. 1, pp. 29–31, Feb. 1994.

[21] S. Zhou and G. B. Giannakis, “Finite-alphabet based channel esti-mation for OFDM and related multicarrier systems,” IEEE Trans.Commun., vol. 49, no. 8, pp. 1402–1414, Aug. 2001.

[22] ——, “Long codes for generalized FH-OFDMA through unknownmultipath channels,” IEEE Trans. Commun., vol. 49, no. 4, pp.721–733, Apr. 2001.

[23] M. D. Trott, S. Benedetto, R. Garello, and M. Mondin, “Rotationalinvariance of trellis codes-Part I: Encoders and precoders,” IEEE Trans.Inf. Theory, vol. 42, no. 3, pp. 751–765, May 1996.

[24] A. V. Oppenheim and R. W. Schafer, Digital Signal Processing. En-glewood Cliffs, NJ: Prentice-Hall, 1998.

[25] Q. Dai and E. Shwedyk, “Detection of bandlimited signals over fre-quency selective Rayleigh fading channels,” IEEE Trans. Commun.,vol. 42, no. 2/3/4, pp. 941–950, Feb./Mar./Apr. 1994.

[26] H. A. -Nash, “Phase unwrapping of digital signals,” IEEE Trans.Acoust., Speech Signal Process., vol. 37, no. 11, pp. 1693–1702, Nov.1989.

[27] T. F. Quatieri and A. V. Oppenheim, “Iterative techniques for minimumphase signal reconstruction from phase or magnitude,” IEEE Trans.Acout., Speech, Signal Process., vol. ASSP-29, pp. 1187–1193, Dec.1981.

[28] M. Alles and S. Pasupathy, “Coding for the discretely phase ambiguousadditive white Gaussian channel,” IEEE Trans Inf. Theory, vol. 39, no.2, pp. 347–357, Mar. 1993.

[29] T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithmsfor Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 2000.


Blind OFDM Channel Estimation Using FIR …singer/pub_files/Blind_OFDM...1136 IEEE TRANSACTIONS ON...

Documents

Transcript of Blind OFDM Channel Estimation Using FIR …singer/pub_files/Blind_OFDM...1136 IEEE TRANSACTIONS ON...