FREQUENCY-HOPPED GENERALIZED MC-CDMA FOR MULTIPATH
`AND INTERFERENCE SUPPRESSIONy
`
`Shengli Zhou, Georgios B. Giannakis
`Dept. of ECE, Univ. of Minnesota, 200 Union Street SE, Minneapolis, MN 55455
`Emails: fgeorgios,szhoug@ece.umn.edu
`Ananthram Swami
`Army Research Laboratory, AMSRL-IS-TA, 2800 Powder Mill Rd, Adelphi, MD 20783
`Email: aswami@arl.mil
`
`ABSTRACT
`
`Generalized multi-carrier (GMC) CDMA transceivers
`equipped with frequency hopping are developed in this pa-
`per. Through judicious code design, multiuser interference
`(MUI) is eliminated deterministically and symbol recovery
`is guaranteed, relying on a minimum number of redundant
`subcarriers. This, together with subcarrier frequency hop-
`ping (FH), improves BER performance in the presence of
`frequency-selective multipath fading channels and enhances
`resistance to narrow-band interference. Blind channel es-
`timation methods are developed with guaranteed channel
`identifiability, regardless of the locations of the zeros of
`the FIR channel. Performance analysis and simulation re-
`sults illustrate the merits of the proposed FH-GMC-CDMA
`transceivers relative to competing OFDMA and multicar-
`rier CDMA alternatives.
`
`I. INTRODUCTION
`
`A variant of spread-spectrum (SS) signaling, called
`frequency-diversity spread spectrum (FD-SS) was recently
`proposed and shown to be more resistant
`than direct-
`sequence SS (DS-SS) to partial-band interference (PBI) [3].
`FD-SS, with disjoint frequency support for each subcarrier,
`is in fact the analog counterpart of the OFDM spread spec-
`trum (OFDM-SS) proposed in [6] and the underlying multi-
`carrier spread spectrum (MC-SS) for MC-CDMA [8] sys-
`tems with overlapping subcarriers. Assuming digital imple-
`mentations via DFTs, both OFDM-SS and MC-SS systems
`can be seen as special cases of a unifying framework [2].
`In MC-CDMA, users transmit simultaneously using the
`entire system bandwidth; in the down-link, user separation
`is achieved by use of orthogonal spreading codes. How-
`ever, in the up-link orthogonality is lost due to multipath,
`resulting in MUI and performance degradation as the sys-
`tem load increases. In [5], FH-OFDMA was proposed for
`uplink CATV transmission, and was shown to achieve MUI
`elimination. However, it suffers from frequency-selective
`fading and requires extra diversity to ameliorate the effects
`of channel nulls.
`
`In this paper, we generalize MC-CDMA and FH-
`OFDMA, and propose a generalized MC-CDMA (GMC-
`CDMA) system. By judicious transceiver design, MUI is
`eliminated deterministically. Use of redundant subcarriers
`guarantees recovery of the symbols, regardless of the loca-
`tions of the zeros of the multipath channel. Further, GMC-
`CDMA permits blind estimation of the channel, without
`imposing any restrictions on the zeros. Resistance to both
`narrow-band interference (NBI) and PBI is guaranteed.
`Section II provides a general model, with Section III
`specializing it to existing MC-CDMA and FH-OFDMA.
`Frequency-hopped GMC-CDMA transceivers are devel-
`oped in Section IV and simulation results are provided in
`Section V.
`
`II. DISCRETE TIME SYSTEM MODEL
`
`smn
`
`^smn
`
`smi
`
`S/P
`
`Cm
`
`K
`
`P
`
`^smi
`
`yi
`
`P/S
`
`GT
`
`m
`
`K
`
`P
`
`IFFT
`cyclic Prefix
`
`discard prefix
`FFT
`
`umi
`
`umn
`
`P/S
`
`hmn
`
`P
`
`xi
`
`P
`
`MUI
`
`xn
`
`S/P
`
`n
`
`n
`
`Fig. 1. Discrete time system model
`The baseband discrete-time equivalent transmitter and re-
`ceiver model for the mth user is depicted in Fig. 1, where
`m  ; M (cid:0)  and M is the number of users. The informa-
`tion stream smn, at symbol rate =Ts, is first parsed into
`K-long blocks, smi := siK; siK + ; : : : ; siK +
`K (cid:0) T , spread by the P  K code matrix Cm, and then
`IFFT processed to obtain the P  vector FHCmsmi.
`Here, P is the block spreading length, and F is the P  P
`FFT matrix with m; n entry  =p P  exp(cid:0)jmn= P .
`To avoid channel-induced inter symbol (block) interfer-
`ence, we use a cyclic prefix (CP) (as in conventional OFDM
`systems, e.g., [1]). The received signal, after conversion to
`baseband and receive filtering, is sampled at the chip rate,
`to yield
`
`M (cid:0)
`
`hmlumn (cid:0) l + n + n;
`
`(1)
`
`LX l
`X m
`
`=
`
`=
`
`y Work in this paper was supported by ARL grant no. DAAL01-98-Q-
`0648 and NSF Wireless Initiative grant no. 99-79443
`
`xn =
`
`1
`
`0-7803-6521-6/00/$10.00 (c) IEEE
`
`One-E-Way Ex. 2002
`Sony Corporation v. One-E-Way, Inc.
`IPR2016-1639
`
`

`
`III. SPECIAL CASES
`In this section, we discuss two special cases of our gen-
`eral model obtained by setting K = (no symbol blocking):
`MC-CDMA and FH-OFDMA.
`
`A. MC-CDMA
`The block spreading matrix Cm reduces to a spreading
`vector cm, and the receiver matrix Gm reduces to a vector
`m. Therefore, (3) reduces to
`gT
`
`yi = ~DmHcmsi +
`
`M (cid:0)
`X
`=;=m
`
`~DHcsi + wi:
`
`To gain insight into how MC-CDMA copes with frequency-
`selective channels, we focus on the single user case, i.e.,
`M = , and assume that wi is white (no MUI and no
`PBI). The Maximum Ratio Combiner is known to achieve
`the maximum output SNR:
`
`
`
`m = cHmDHmH = ~hHDHmc ;
`
`Gm = gT
`where Dmc = diagcm; cm ; : : : ; cm P (cid:0) ,
`~hm = Hm ; : : : ; Hm P (cid:0) T . Define the Vander-
`monde matrix V with l + ; k + th entry Vl+ ;k+ =
`exp(cid:0)jlk= P ; hence, ~hm = Vhm, and the output SNR
`is given by
`
`(5)
`
`
`
`
`
`
`
`
`
`
`
`SN R = hHmVHDHmcDmcVhm s= v = hHmhm s = v;
`
`
`
`
`
`which reveals that frequency diversity due to transmitting
`replicas through P subcarriers comes only from the L +
`channel taps. With P L + , the vector ~hm is corre-
`lated and it does not provide more diversity than the original
`L +  channel taps. More explicitly, for single user MC-
`SS, we do not need to exploit all the P subcarriers. It suf-
`fices to choose P  L + ; if P = P is an integer, we select
`P equispaced subcarriers out of the P subcarriers; vector
`cm will have P non-zero entries with amplitude =p P .
`
`mVHDmcDmcVhm = hHmhm, which
`We verify that hH
`indicates that choosing L P P subcarriers results in
`the same performance as choosing P subcarriers. Since an
`L + tap FIR channel can have a maximum of L nulls, we
`note that the symbol recovery is not guaranteed if we choose
`fewer than L + subcarriers. We summarize this discussion
`below:
`Result 1: If the FIR channel has order L, and the addi-
`tive noise is white, full diversity gain is achieved by MC-SS
`by using more than L equispaced subcarriers. Using fewer
`subcarriers leads to loss of diversity gain; using more sub-
`carriers does not improve performance.
`Thus, MC-SS exploits the full frequency diversity of the
`multipath channel, as long as P  L + . However, for
`multiple access, each user utilizes all available subcarriers
`and MUI occurs as soon as M  . The separation between
`
`2
`
`where hm‘ is the overall channel (transmit and receive
`filters, and propagation channel) encountered by the mth
`user’s signal, n is the filtered PBI, n is the filtered
`additive Gaussian noise (AGN), and L is the maximum or-
`der of FIR channels for all users, i.e., hml = ;l 
`; L;m. Additive noise t is assumed to have flat
`power spectral density N= over the system bandwidth B.
`PBI t is assumed to occupy "B bandwidth ( " )
`with large power density I=.
`In order to avoid inter-block interference (IBI), we as-
`sume that the prefix length is longer than the maximum
`channel order for all users:
`(a0) P (cid:0) P  L.
`To cast (1) into a convenient matrix-vector form, we
`define the P  vector xmi := xmiP ; xmiP +
` ; : : : ; xmiP + P (cid:0) T , and similarly i and i;
` 
`m with
`define the P  P Toeplitz channel matrices Hm; H
`k; lth entries hmk (cid:0) l and hmk (cid:0) l + P , respectively.
`Because of (a0), we can write (1) as
`
` 
`
`xi = PM (cid:0)
`m umi (cid:0)  + i + i;
`m= Hmumi + H
`where the second term represents IBI. For convenience, we
`define wi := i + i as the equivalent (colored) ad-
`ditive noise vector.
`At the receiver, the CP is removed by dropping the first
`P (cid:0) P elements of xi, thus eliminating IBI. After FFT
`processing, we have
`
`F ~HmFHCmsmi + ~wi;
`
`(2)
`
`M (cid:0)
`
`X m
`
`=
`
`yi =
`
`where ~wi is the filtered interference and noise vector, and
`~Hm is the resulting channel matrix. ~Hm is a P  P circulant
`matrix with its k; lth entry given by hmk (cid:0) l mod P 
`(see also [7]). Since ~Hm is circulant, ~DmH := F ~HmF
`is a diagonal matrix;
`the diagonal elements, Hm p,
`are values of the channel frequency response Hmz =
`l= hmlz(cid:0)l evaluated at z = p = expjp= P ,
`PL
`p = ; ; : : : ; P (cid:0) . Therefore, we can rewrite (2) as
`
`~DmHCmsmi + ~wi:
`
`(3)
`
`M (cid:0)
`
`X m
`
`=
`
`yi =
`
`Subsequently, for the mth user, multiplication by the equal-
`ization matrix Gm yields symbol block estimates ^smi as
`
`^smi = Gmyi:
`
`(4)
`
`Instead of adding a CP to avoid IBI, we can add trailing
`zeros as in [7].
`Equation (3) generalizes both conventional MC-CDMA
`and coherent FH-OFDMA.
`
`

`
`Result 1, we would like to choose equispaced subcarriers for
`each user such that mp+ ;q+ = if p = m + qM; q =
`; ; : : : ; P (cid:0) . Thanks to the non-overlapping frequency
`allocation, the corresponding subcarrier selector matrices
`m are mutually orthogonal by construction.
`Since m
`has a single non-zero (unity) entry per column, it can be
`readily verified that ~DmHm = mDmH, where
`DmH := diagHm m;; : : : ; Hm m;J(cid:0) . This fact,
`together with the orthogonality of , allows us to simplify
`(4) for user m to [7]:
`
`M (cid:0)
`
`^smi = (cid:0)mT
`m
`
`DHsi + (cid:0)mT
`m ~wi
`
`X 
`
`=
`= (cid:0)mDmHmsmi+T
`m ~wi := (cid:0)m ~ymi
`where ~ymi is the J  MUI-free vector corresponding to
`user m. Equation (8) reveals that MUI is eliminated deter-
`ministically regardless of the multipath channels.
`To guarantee recovery of the K symbols in smi re-
`gardless of the signal constellation, the matrix DmHm
`in (8) must have full rank, regardless of the mth chan-
`nel. Hence, we require rankDmHm = K, m 
`; M (cid:0) , so that zero forcing (ZF) equalization based
`If the
`on (cid:0)m = DmHmy will recover ^smi.
`noise covariance matrix R ~w ~w is known, we can ap-
`ply the MMSE receiver (cid:0)m = H
`mDH
`mHR ~w ~w +
`mH(cid:0) . For small K, the ML solution
`DmHmH
`mDH
`^smi = argminsmi kymi (cid:0) DmHmsmik is af-
`fordable.
`Since each user’s channel can have at most L zeros,
`DmH can have at most L zero diagonal entries. The
`above rank condition will be satisfied if any J (cid:0) L = K
`rows of m are linearly independent. To meet this rank re-
`quirement, each column of m should have at least L +
`non-zero entries, i.e., each symbol is transmitted over L +
`or more subcarriers, so that the frequency diversity of the
`multipath channel is fully exploited.
`Notice that this rank condition is not a condition on the
`channels; instead, it is a guideline for designing m. For
`example, we can choose m with entries:
`ml+ ;k+ =  =pJ (cid:0)k
`m;l:
`Since the signature frequencies are distinct, any K rows of
`the matrix will yield a full rank matrix.
`How do we choose the symbol block size K? If N :=
`Ts=Tc denotes the spreading gain, then we need
`
`(8)
`
`(9)
`
`J =
`
`KN (cid:0) L
`M
`
` K + L = K 
`
`LM + 
`N (cid:0) M
`
`(10)
`
`to achieve MUI elimination and guarantee symbol recovery
`in the presence of unknown multipath. Recall that M is the
`number of users, N the spreading gain, L the FIR channel
`length, and J the number of subcarriers (P = KN here).
`
`3
`
`users relies on having a distinct P  spreading code cm
`(e.g., Walsh-Hadamard code) for each user. Then choosing
`P L + serves to suppress the MUI by decreasing the
`MUI level or possible NBI (colored noise).
`Because of MUI, MC-CDMA suffers from the near-
`far problem and thus degraded performance in comparison
`with the single-user case. This motivates our MUI-resilient
`GMC-CDMA system in Section IV.
`
`B. FH-OFDMA
`For OFDMA, we set K = ; cm = em is the mth
`Euclidean basis vector and has only one non-zero element.
`Thus, gm = em and the equivalent model is
`
`ymi := gT
`myi = H msmi + wmi ;
`
`(6)
`
`which reveals that MUI is eliminated.
`However, FH-OFDMA suffers from frequency-selective
`fading. Specifically, when the channel has zeros close to
` m, the symbol will suffer from severe fading. To avoid
`consistent fading, frequency hopping is proposed (coding
`or fast hopping is beyond the scope of this paper). Specif-
`ically, cm is changed frequently (and thus depends upon a
`time index i) according to some prescribed hopping pattern.
`Multiple users are allowed to transmit information simulta-
`neously using different hopping sequences, and MUI can be
`avoided if no frequencies are employed by two users at the
`same time. For example, we can set cmi = em+i mod P .
`However, symbol recovery is not guaranteed and fre-
`quency diversity is not exploited in FH-OFDMA because
`it uses only a single subcarrier. We saw in Result 1 that
`L + subcarriers are needed to fully exploit the frequency
`diversity of the order L FIR channel.
`GMC-CDMA system considered next overcomes the lim-
`itations due to MUI, NBI, PBI, and multipath effects.
`
`IV. GMC-CDMA TRANSCEIVER DESIGN
`We will transmit K symbols per block, using P  P (cid:0) L
`subcarriers such that J := P =M is an integer. Thus, each of
`the M users can be assigned a distinct set of J subcarriers.
`We denote by m;q,  q  J(cid:0) , the J distinct subcarriers
`assigned to the mth user.
`To unravel the attractive features of MUI resilience pro-
`gressively, we factor our spreading and despreading matri-
`ces fCm; GmgM (cid:0)
`m= in the following forms:
`Cm = mm; Gm = (cid:0)mT
`m ;
`
`(7)
`
`with each matrix factor playing a different role: m is a
`J  K matrix that linearly maps the K information symbols
`of the ith block smi to J (J K) symbols msmi;
`these in turn are mapped to the user’s J signature subcarri-
`ers via the P J selector matrix m, to yield mmsmi.
`We have mp+ ;q+ = if m;q = expjp= P , and 0
`otherwise; note that we will have J non-zero entries. Per
`
`

`
`Increasing K increases the value of J (cid:0) K = KN (cid:0)
`M (cid:0) L=M, which indicates more freedom in choosing the
`redundant mapping matrix m.
`We illustrate the advantages of GMC-CDMA over MC-
`CDMA, via a simple example.
`Example 1: Suppose we design an under-loaded system
`with spreading gain N =  for M =  users, and
`channel order L = . The system load is approximately
` =  . Even in AWGN channel, MC-CDMA will
`exhibit BER larger than Qp =: due to MUI [4]. For
`GMC-CDMA, we have K  from (10). If we choose the
`minimum K = , so that J =   L + , then MUI is
`eliminated; further, each user fully exploits the frequency
`diversity of the channel (see Result 1), and obtains the best
`performance that can be achieved by MC-SS with a single
`user. Therefore, GMC-CDMA considerably outperforms
`MC-CDMA in this setting.
`As the system load increases, N (cid:0) M decreases and
`K is needed [c.f. (10)]. As K increases, the redun-
`dancy J (cid:0) K increases, and we have more flexibility in our
`design. For example, with J = K +L, instead of using the
`linear precoder m, we can use a nonlinear precoder (e.g.,
`a block channel encoder) to correct L faded symbols. We
`summarize our observations as follows:
`Result 2: Since the block size K can be varied, GMC-
`CDMA can improve its performance depending on system
`load. When the system load is M  P =L + , by set-
`ting K = and J = b P =Mc, each GMC-CDMA user
`achieves performance equivalent to a single-user MC-SS
`system. When M increases, K and J must increase accord-
`ing to (10) in order to maintain MUI elimination and symbol
`recovery guarantees. Increasing K increases J (cid:0) K, which
`in turn allows channel encoding (over the GF) to be used in-
`stead of (or joint with) the m-spreading (over the complex
`field).
`
`A. Frequency-hopping capability
`
`To randomize channel effects and avoid consistent fading
`on subcarriers, each user can be assigned different subcarri-
`ers at different time intervals. Thus, our GMC-CDMA sys-
`tem has frequency-hopping capabilities. In such a system,
`the frequency selector matrix, mi, varies with the block
`(time) index i. We consider a simple example for illustrative
`purposes.
`Example 2: The p + ; q +  entries of mi are non-
`zero, and equal to unity, if
`
`p = m + i + qM  mod P ; q  ; J (cid:0) :
`Hopping the frequency selector matrix m prevents a user
`from being consistently hit by channel nulls; by varying the
`precoding matrix m from block symbol to symbol, we in-
`crease system security by providing additional coding.
`
`(11)
`
`B. NBI suppression
`Assume that the PBI has bandwidth B = N=Ts. If we
`design our system as in (11), each subcarrier has bandwidth
` = P Tc, and the distance between two consecutive subcar-
`riers is P =J= P Tc = N=JTs. So the number of subcar-
`riers that will be hit by PBI is NI = N=Ts=N=JTs =
`J. We can treat the subcarriers hit by the strong PBI as
`having encountered a deep fade; thus symbol recovery is
`guaranteed if J  K + L + NI, regardless of the power of
`the PBI. GMC-CDMA gains resistance to PBI jamming, by
`narrowing the bandwidth of each subcarrier (from =Ts to
`N= P Ts) and increasing the spacing between the subcar-
`riers; for a jammer to be effective, it must know the user’s
`subcarrier set. In the case of NBI (e.g., single tone jammer),
`at most one of the J (J ) subcarriers will be hit, and the
`symbols can be recovered from the other J (cid:0) interference-
`free subcarriers.
`
`C. Blind channel estimation
`GMC-CDMA system design facilitates blind channel es-
`timation at the receiver, obviating the need for bandwidth-
`consuming training sequences. Especially if we select the
`mapping matrix m to satisfy J  K + L, and any
`K rows of m span the CK row vector space, e.g., as
`in (9), then we can use the subspace-based blind chan-
`nel estimation algorithm of [2].
`If we choose m :=
`KJ(cid:0)KT , where VJ(cid:0)KK is the Vandermonde
`IK; VT
`matrix constructed through J (cid:0) K distinct points other than
`fexpjk=KgK
`k= , then any K rows of m are linearly
`independent, and we can estimate the channel blindly along
`the lines of the finite alphabet (FA) based approach of [9].
`
`V. SIMULATIONS
`Because of space limitations, we do not present the re-
`sults of our theoretical performance analysis; instead, we
`show several numerical results.
`Test Case 1 (multipath effects): Here, N =  and L = 
`(3-ray channels), so that the maximum number of users in
`FH-OFDMA is N (cid:0) L = . We compare MC-CDMA,
`FH-OFDMA, and GMC-CDMA under four different loads:
`M = ; ; ;  (12.5% to 75%). Each MC-CDMA user
`picks up one Walsh-Hadamard sequence of length 16. For
`GMC-CDMA, we use the minimum K according to (10),
`so that the M; J; K triplets are ; ; , ; ; , ; ; ,
` ; ; . Fig. 2 shows that GMC-CDMA outperforms
`MC-CDMA and FH-OFDMA considerably. As long as the
`number of users M   and K = , each user of GMC-
`CDMA achieves the best performance that MC-CDMA can
`achieve in a single-user setting. Because of MUI, MC-
`CDMA suffers from near-far effects, and the BER curve
`levels off at high SNR (the floor effect). With MUI free
`reception, FH-OFDMA has constant performance regard-
`less of system load. However, it suffers from deep channel
`fading due to the lack of frequency diversity.
`
`4
`
`

`
`GMC−CDMA
`MC−CDMA
`
`PBI bandwidth percentage
`
`0/16
`2/16
`4/16
`6/16
`8/16
`
`5
`
`10
`
`15
`
`20
`
`25
`
`Eb/N0
`
`100
`
`10−1
`
`10−2
`
`10−3
`
`10−4
`
`10−5
`
`0
`
`Bit Error Rate
`
`Fig. 3. Comparisons under partial band interference
`
`Subspace based
`FA based
`known channel
`
`5
`
`10
`
`15
`
`20
`
`25
`
`Eb/N0
`
`Fig. 4. Blind channel estimation
`
`100
`
`10−1
`
`10−2
`
`Bit Error Rate
`
`10−3
`
`10−4
`
`0
`
`An idea whose time has come,” IEEE Communications Magazine,
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`
`Test Case 2 (PBI suppression): Assume that MC-CDMA
`is on the down-link, where all users experience the same
`channel so that MUI can be eliminated by the orthogonality
`of spreading codes after channel equalization. Here, we set
`N = ; L = ; M =  load. For GMC-CDMA,
`we set K = ; J = , so that J (cid:0) K (cid:0) L = , indicating
`that we can afford to have at most two subcarriers hit by the
`PBI. In Fig. 3, we show the performance of the two systems
`as the relative bandwidth of the strong PBI (I N) is var-
`ied: " = =  no interference; = ; = ; = ; = .
`For GMC-CDMA, NI = ; ; ; ;  subcarriers are
`hit by the strong PBI. For both MC-CDMA and GMC-
`CDMA, we assume that the receiver can detect the presence
`of strong interference and remove the contaminated subcar-
`riers. As shown in Fig. 3, GMC-CDMA outperforms MC-
`CDMA when  = . When the PBI occupies half the
`system bandwidth, GMC-CDMA becomes worse because
`now J (cid:0) K L + NI, so that symbol recovery is no
`longer guaranteed. However, even when = = (J= sub-
`carriers will be hit by the PBI), we can increase K; J so
`that symbol recovery can be guaranteed, e.g., with the pair
`K; J =  ; .
`Test Case 3 (Blind channel estimation): The test system
`has spreading gain N = , M =  users, and chan-
`nel order L = . We design GMC-CDMA with J = ,
`
`K = , m = IK; VTKJ(cid:0)KT , where the last two
`rows are Vandermonde vectors constructed from the roots
`expj=K and expjK+ =K. BPSK signals are
`used, and the channel is estimated from 20 symbol blocks,
`using the subspace-based and FA-based blind channel esti-
`mators. Fig. 4 shows the BER curves corresponding to these
`estimators; we note that the performance is close to the ideal
`BER, corresponding to known channel.
`
`GMC−CDMA
`
`MC−CDMA
`
`FH−OFDMA
`
`M=2
`
`M=4
`
`M=8
`
`M=12
`
`5
`
`10
`
`15
`
`20
`
`25
`
`Eb/N0
`
`Fig. 2. Comparisons under multipath channels
`
`100
`
`10−1
`
`10−2
`
`10−3
`
`10−4
`
`10−5
`
`0
`
`Bit Error Rate
`
`REFERENCES
`J. A. C. Bingham, “Multicarrier modulation for data transmission:
`
`[1]
`
`5