Abstract
Recently, zeropadding orthogonal frequency division multiplexing (ZPOFDM) has been proposed as an alternative solution to the traditional cyclic prefix (CP)OFDM, to ensure symbol recovery regardless of channels nulls. Various ZPOFDM receivers have been proposed in the literature, trading off performance with complexity. In this paper, we propose a novel lowcomplexity (LC) receiver for ZPOFDM transmissions and derive an upper bound on the bit error rate (BER) performance of the LCZPOFDM receiver. We further demonstrate that the LCZPOFDM receiver brings a significant complexity reduction in the receiver design, while outperforming conventional minimum meansquare error (MMSE)ZPOFDM, supported by simulation results. A modified (M)ZPOFDM receiver, which requires the channel state information (CSI) knowledge at the transmitter side, is presented. We show that the MZPOFDM receiver outperforms the conventional MMSEZPOFDM when either perfect or partial CSI (i.e., limited CSI) is available at the transmitter side.
Index TermsZeropadding, orthogonal frequency division multiplexing (OFDM), equalization.
I. Introduction
The growing demand for high data rate services for wireless multimedia and internet services has led to intensive research efforts on highspeed data transmission. A key challenge for highspeed broadband applications is the dispersive nature of frequencyselective fading channels, which causes the socalled intersymbol interference (ISI), leading to an inevitable performance degradation. An efficient approach to mitigate ISI is the use of orthogonal frequency division multiplexing (OFDM) that converts the ISI channel with additive white Gaussian noise (AWGN) into parallel ISIfree subchannels by implementing inverse fast Fourier transform (IFFT) at the transmitter and fast Fourier transform (FFT) at the receiver side [1]. It has been shown that OFDM is an attractive equalization scheme for digital audio/video broadcasting (DAB/DVB) [2,3] and has successfully been applied to highspeed modems over digital subscriber lines (DSL) [4]. Recently, it has also been proposed for broadband television systems and mobile wireless local area networks such as IEEE802.11a and HIPERLAN/2 (HL2) standards [5].
The IFFT precoding at the transmitter side and insertion of CP enable OFDM with very simple equalization of frequencyselective finite impulse response (FIR) channels. To avoid interblock interference (IBI) between successive FFT processed blocks, the CP is discarded and the truncated blocks are FFT processed so that the frequencyselective channels are converted into parallel flatfaded independent subchannels. In this way, the linear channel convolution is converted into circular convolution, and the receiver complexity both in equalization and the symbol decoding stages is reduced [6]. However, since each symbol is transmitted over a single flat subchannel, the multipath diversity is lost along with the fact there is no guarantee for symbol detectability when channel nulls occur in the subchannels. As an additional countermeasure, coded OFDM (COFDM), which is fading resilient, has been adopted in many standards [7]. Trelliscoded modulation (TCM) [8] and convolutional codes [9,10] are typical choices for errorcontrol codes. Interleaving together with TCM enjoys lowcomplexity Viterbi decoding while enabling a better trade off between bandwidth and efficiency. However, designing systems that achieve diversity gain equal to code length is difficult considering the standard design paradigms for TCM. Linear constellation precoded OFDM (LCPOFDM) was further proposed in [11] to improve performance over fading channels, where a real orthogonal precoder is applied to maximize the minimum product distance [12,13] and channel cutoff rate [14], while maintaining the transmission rate and guaranteeing the symbol recovery. In [15,16], the LCP is combined with spacetime block coding (STBC) and channel coding in order to effectively improve the spacetime diversity order of the multiple antenna system. The aforementioned LCP systems are based on Hadamard matrices and can employ the simple MMSE linear decoder at the receiver side, which has low complexity compared to the maximum likelihood (ML) decoders currently used in such systems. Subcarrier grouping was proposed in [11] to enable the maximum possible diversity and coding gain. However, the LCP that is used within each subset of subcarriers is in general complex.
To ensure symbol recovery regardless of channel nulls, ZeroPadding (ZP) of multicarrier transmission has been proposed to replace the generally nonzero CP. Unlike CPOFDM, ZPOFDM guarantees symbol recovery and FIR equalization of FIR channels. Specifically, the zero symbols are appended after the IFFT processed information symbols. In such way, the single FFT required by CPOFDM is replaced by FIR filtering in ZPOFDM that adds to the receiver complexity. To tradeoff BER performance for extra savings in complexity, two lowcomplexity equalizers for ZPOFDM are derived in [17]. Both schemes are developed based on the circularity of channel matrix while reducing complexity by avoiding inversion of a channel dependent matrix. However, the BER performance of the two suboptimal receivers proposed in [17] is not as good as MMSEZPOFDM.
Adapting the transmitter designs to the propagation channel facilitates performance improvement. Depending on the CSI available to the transmitter, various parameters such as power levels, constellation size, and modulation are adjustable by the channeladaptive transmissions [18][20]. The challenge in a wireless setting is on whether and what type of CSI can be practically made available to the transmitter, where fading channels are randomly varying. Apparently, this is less of an issue for discrete multitone (DMT) systems in wireline links, which has been standardized for digital subscriber line modems. In both singleinput singleoutput (SISO) and multipleinput multipleoutput (MIMO) versions of DMT, perfect CSI is assumed to be available at the transmitter [21]. Although it is reasonable for wireline links, we can only justify the assumption of perfect CSIbased adaptive transmissions developed for SISO [22] and MIMO OFDM wireless systems [23] when the fading is sufficiently slow. Since noCSI leads to robust but rather pessimistic designs and perfect CSI is mostly impractical for wireless links, recent efforts geared toward exploitation of partial CSIbased transmissions. In this work, we assume limited CSI knowledge. In particular, we assume that only small number of channel taps are available at the transmitter.
In this paper, we propose a novel reducedcomplexity receiver design for ZPOFDM transmissions (LCZPOFDM) that not only outperforms MMSEZPOFDM (confirmed through simulation results) but also uses lowcomplexity computational methods that bring a significant power saving to the proposed receiver. We show that linear processing collects both multipath and spatial diversity gains, leading to significant improvement in performance, while maintaining low complexity. Note that the proposed LCZPOFDM receiver consists of two stages, i.e., a lowcomplexity suboptimal MMSE receiver at first stage followed by linear operations at second stage to investigate the underlying available diversity. A MZPOFDM receiver for transmitter designs based on perfect, as well as limited CSI knowledge, is further included, which significantly outperforms MMSEZPOFDM. BER upper bounds are presented for the twostage LC receiver. The performance improvement of the LCZPOFDM and MZPOFDM receivers as compared with the conventional MMSEZPOFDM receiver is further corroborated through simulation results.
The rest of this correspondence is organized as follows. We start in Section II by describing our model and assumptions. In Section III, the reducedcomplexity MMSEbased receiver is proposed. Computational complexity analysis and numerical results are presented in Sections IV and V, respectively. The paper is concluded in Section VI.
II. System model
We consider a singletransmit singlereceive antenna system over a frequencyselective fading wireless channel. The channel impulse response (CIR) of the jth information block is modeled as an FIR filter with coefficients h^{j }= [h^{j}[0],..., h^{j}[L]]^{T}, where L denotes the corresponding channel memory length. The random vectors h^{j }are assumed to be independent zeromean complex Gaussian with two choices of powerdelay profile: Uniform powerdelay profile with variance 1/(L + 1), and exponentially decaying powerdelay profile
The jth M ×1 information block
where
Therefore, the received block symbol is given by
where N = M + D; H is the N × N lower triangular Toeplitz filtering matrix with its first column being [h^{j}[0],..., h^{j}[L] 0,..., 0 ]^{T}; H_{IBI }is the N × N upper triangular Toeplitz filtering matrix that captures the interblock interference (IBI), with its first row being [0,..., 0 h^{j}[0],..., h^{j}[L] ]; and
The N × M submatrix H_{0}, which corresponds to the first M columns of H, is Toeplitz and is guaranteed to be invertible, which assures symbol recovery regardless of channel zero locations. In this case, due to its channelirrespective symbol detectability property, ZPOFDM is able to exploit fully the underlying multipath diversity [1]. Assuming the symbols
It should be emphasized that channel inversion of a N × N matrix cannot be precomputed due to its dependence on channel that brings about an extra implementation cost for the MMSEZPOFDM receiver. Specifically, equalization in (5) is computationally costly, as the pseudo inverse of the Toeplitz matrix H_{0 }is required, whose precomputation is impossible due to the channel coefficients' variations from block to block. Thus, from an implementation point of view, a lowcomplexity ZPOFDM is desirable. This observation motivates the subsequent suboptimal lowcomplexity ZPOFDM equalizers, such as ZPOFDMFASTZF/MMSE and the ZPOFDMOLA [17] originating from the overlapandadd (OLA) method. Thus, complexity is always an issue when dealing with ZPOFDM systems, as compared to the computationally efficient FFTbased CPOFDM systems. Its worth mentioning that ZPOFDMOLA is analogous to CPOFDM; thus, it incurs identical complexity and is not able to fully exploit the underlying multipath diversity. To overcome these limitations, our proposed scheme targets practical ZPOFDM receiver that exploits full multipath diversity while maintaining low complexity.
III. Reducedcomplexity mmsebased receiver
The proposed receiver includes a suboptimal channel independent MMSE equalizer in the feed forward stage, followed by linear processing operations in the feedback stage. Specifically, the feed forward stage exploits the circulant structure of the channel matrices,
where H′ is N × N circulant matrices with entries [H′]_{k,l }= h((k  l) mod N); Λ is a diagonal matrix whose (n, n) element is equal to the nth DFT coefficient of h. Due to the trailing zeros, the last D columns of H in (4) do not affect the received block. Thus, the Toeplitz matrix H can be replaced by the N × N circulant matrix H′ and (4) can be written as
The H′ matrix takes advantage of FFT to yield a set of flat fading channels that can be equalized easily. The ZPOFDM receiver output will be
where
After applying the Npoint FFT Q_{N }to
The (m, n) th element of VV^{H }is equivalent to
The resulting data streams detected from (11) are fed into a minimum Euclidean distance decoder, yielding
where 1 ≤ p, q ≤ N, then, (7) can be rewritten as
Under the high SNR assumption, we can safely assume that
where
Combining each diversity branch in (14), we have
The signal in (16) is then fed into a maximum likelihood decoder to recover the transmitted symbols. It is observed from (16) that the maximum achievable diversity order is given by L + 1. This illustrates that our proposed receiver is able to fully exploit the underlying spatial and multipath diversity gains relying on simple linear processing operations only.
IV. MZPOFDM RECEIVER
In this section, we propose a MZPOFDM receiver that has access to CSI at the transmitter side, such that, at the transmitter side, the zeropadded information symbols are multiplied by the set of matrices C_{2,l }and sent through the channel, processed by the circulant matrix H′ and finally multiplied by the the set of matrices C_{1,l }at the receiver side to exploit the underlying multipath diversity. For l = 1,..., L, the matrices C_{1,l }and C_{2,l }are generated as following [25].
where
Note that C_{1,l }and C_{2,l }are built such that Λ_{l }= C_{1,l }ΛC_{2,l}, where Λ_{l }= Q^{H}H_{l}Q. Using this property, we can drive their corresponding circulant matrices Ψ_{1,l }and Ψ_{2,l }using Ψ_{1,l }= QC_{1,l}Q^{H }and Ψ_{2,l }= QC_{2,l}Q^{H}. Thus exploiting the circulant structure of these matrices we have
Thus, multiplying the zeropadded symbols at the transmitter side with Ψ_{2,l }and the output at the receiver side by Ψ_{1,l}, for each diversity branch we can rewrite (7) as following
Note that for l = 1,..., L + 1, both C_{1,l }and Ψ_{1,l }have unit power and thus do not lead to noise enhancement. More importantly, it can be seen that (20) is similar to (14) in LCZPOFDM proposed in Section III, in the sense that we are exploiting the underlying multipath diversity from each diversity branch.
V. Analytical bounds on the ber performance for the lowcomplexity receiver
The twostage receiver consists of the suboptimal MMSEZPOFDM at the first stage and the linear operations at the second stage. The BER can then be written as
where
In order to find each of
The upper bound for the conditional
where
Table 1. Minimum squared distance for different modulation types with average power normalized to one
where
Assuming that no error is propagated from the first stage to the second stage, the second stage involves maximum likelihood estimation of the originally transmitted symbols from the L branch outputs
Assuming that there is error propagated from the first stage to the second stage, we can rewrite (14) as following
where n_{1 }has variance ε_{MMSE }and n_{2,l }has variance
where
VI. Computational complexity analysis
In this section, we focus on the computational complexity of the proposed receiver compared to the conventional MMSEZPOFDM receiver. The computational complexity is measured based on the number of the complex multiplications and complex divisions. Since complex divisions can be implemented in various different ways, we did not decompose them into complex multiplication. Note that it is not trivial to elaborate explicit complexity expressions for different OFDM systems in terms of the system's parameters. Specifically, system's complexity depends on the parameters such as N, M, D, and L, each of which are decisive in finding the number of the complex multiplications involved. Finding a closedform expression in terms of the aforementioned system parameters is rather difficult; therefore, in this work, we have derived expressions for the approximate multiplicative complexity counts and have used Matlab simulations to find the exact number of complex multiplications implemented. To do so, we have used the minimal multiplicative bounds [26] to find the arithmetic complexity of FFT of different block lengths. Besides, to implement the FFT of length W decomposable to the product of two prime numbers a and b, we have implemented a FFTs of size b or b FFTs of size a instead, without inquiring any additional operations such as multiplications by twiddle factors [27].
Following (5), we derive the approximate computational complexity count for the conventional MMSEZPOFDM as below
where [·] stands for the approximate operation count involved in the operation implementation. (28) requires total of N^{3 }+ N^{2 }(N + M + 2) + M^{2 }+ M (N + 1) operations.
Similarly, we derive the approximate computational complexity count for the lowcomplexity receiver using (11) and (16) as following
Therefore, the total number of operations required to implement the lowcomplexity receiver is N ((M + 6) L + M) + M (L + 1).
We also derive the approximate computational complexity count for the MZPOFDM receiver that includes the additional complexity brought to the transmitter side by multiplying the zeropadded information symbols by the set of matrices C_{2,l}, as well as the computational complexity at the receiver side. Thus, using [19] we have
which shows the additional complexity cost at the transmitter side per diversity branch and
which shows the additional complexity cost at the receiver side. Therefore, the total number of operations required to implement the lowcomplexity receiver is N (NL + L + M) + M.
The overall computational complexity for the LCZPOFDM receiver, MZPOFDM receiver, and the conventional MMSEZPOFDM receiver is summarized in Table 2 for the case of M = 64 and N = 80 and scenarios with different combinations of channel memory lengths:
Table 2. Comparison of overall computational complexity
Scenario (1) L + 1 = 5
Scenario (2) L + 1 = 10
Note that the computational complexity of MZPOFDM receiver at the transmitter side does not include the CSI estimation. The computational complexity of the LCZPOFDM receiver and the MZPOFDM receiver versus the MMSEZPOFDM receiver in terms of the number of the complex multiplications is illustrated in Figure 1, which is plotted against time, assuming that the channel estimate is updated at every t seconds. We would like to emphasize that the MMSE equalizer in MMSEZPOFDM requires the inversion of an N × N Toeplitz channel dependent matrix, which has to be computed at every t seconds with a complexity of the order O(N^{2}). Note that as the channel memory lengths of the underlying channel links increase, the complexity of MMSEZPOFDM, MZPOFDM, and the LCZPOFDM receiver increases.
Figure 1. Computational complexity of the LCZPOFDM receiver compared to MMSEZPOFDM.
It is observed from Figure 1 that for scenarios one and two with reasonable memory length for the underlying channels, the aggregate complexity of the LCZPOFDM receiver is insignificant compared to the conventional MMSEZPOFDM as well as MZPOFDM, as channel parameters change with time.
VII. Numerical results
In this section, we present MonteCarlo simulation results for the proposed receiver assuming a quasistatic Rayleigh fading channel.
Figure 2 illustrates the symbol error rate (SER) performance of the proposed receiver for L+1 = 2. We assume 64QAM, M = 64, and N = 80 as is practiced in the context of HL2 systems. Our simulation results indicate that LCZPOFDM receiver outperforms the conventional MMSEZPOFDM equalizer by ≈ 3.3 dB without error propagation (EP) from the first stage and by ≈ 0.5 dB with EP from first stage at SER = 10^{3}. We are also including the SER performance of the MZPOFDM receiver with perfect CSI, as well as partial CSI (limited CSI is available, e.g., only one of the two channel taps are available at the transmitter side). As is illustrated in Figure 2, the MZPOFDM receiver with partial CSI outperforms the conventional MMSEZPOFDM by ≈ 4 dB at SER = 10^{3}, when l = 1. The system's SER performance for 64QAM AWGN channel is provided as a reference curve.
Figure 2. SER performance of MMSEZPOFDM, LCZPOFDM receiver with and without EP, and MZPOFDM in the context of HL2 (L + 1 = 2).
In Figure 3, we provide further results on the performance of the LCZPOFDM receiver for L + 1 = 6. Our results indicate that the LCZPOFDM receiver outperforms the MMSEZPOFDM equalizer by ≈ 5 dB without EP and by ≈ 0.75 dB with EP at SER = 10^{7}. Moreover, we are including the SER performance of the MZPOFDM receiver with partial CSI (e.g., only two or three of the six channel taps are available at the transmitter side) when l = 2, 3. It can be seen that the proposed receiver outperforms the MMSEZPOFDM receiver by ≈ 4 dB at SER = 10^{6}, when l = 3.
Figure 3. SER performance of MMSEZPOFDM, LCZPOFDM receiver with and without EP, and MZPOFDM in the context of HL2 (L + 1 = 6).
In Figure 4, the SER performance of the LCZPOFDM receiver for L+1 = 2, 4, 6 is presented, assuming 64QAM modulation in the context of HL2, and is compared with the theoretical values presented in Section V. As is illustrated in Figure 4, both the theoretical curves and their corresponding simulated versions come with similar slopes. As an example, when L+1 = 4, the theoretical curve differs from its simulated counterpart by ≈ 2 dB.
Figure 4. SER performance of LCZPOFDM receiver compared to theoretical.
VIII. Conclusion
We propose a novel reducedcomplexity MMSEbased receiver for ZPOFDM systems. We show that, by incorporating linear processing techniques, our MMSEbased receiver is able to collect full antenna and multipath diversity gains. Simulation results demonstrate that our LCZPOFDM receiver outperforms the conventional MMSEZPOFDM receiver, while maintaining much less complexity by avoiding channel dependent matrix inversion. By using linear processing techniques that require minimum computational complexity, the communication power is minimized at no additional hardware cost. We also propose a modified receiver that outperforms the conventional MMSEZPOFDM.
The authors declare that they have no competing interests.
IX. Endnotes
Notation:
Note
^{0}The work of S. Muhaidat was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under grant RGPIN372049.
References

Z Wang, GB Giannakis, Linearly precoded or coded OFDM against wireless channel fades? Proceedings of the IEEE Workshop on Signal Processing Advances for Wireless Communications Taiwan, China (2001)

WW Lu, Compact multidimensional broadband wireless: the convergence of wireless mobile and access. IEEE Commun. Mag, 119–123 (2000)

ETSI Normalization Committee, in Digital Broadcasting Systems for Television, Sound and Data Services; Framing Structure, Channel Coding and Modulation for Digital Terrestrial Television, ed. by . Norme ETSI, document pr ETS 300 744 (European Telecommunications Standards Institute, SophiaAntipolis, Valbonne, 1996)

ANSI T1E1.4 Committee Contribution, The DWMT: A Multicarrier Transceiver for ADSL using Mband Wavelets (Technical Report, ANSI, 1993)

Broadband Radio Access Networks (BRAN), HIPERLAN Type 2 Technical Specification Part 1Physical Layer. Eur. Telecommun. Standards Inst. (ETSI), DTS/BRAN030 0031

DP Palomar, JM Cioffi, MA Lagunas, Joint TxRx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization. IEEE Trans. Signal Process 51(9) (2003)

W Zou, Y Wu, COFDM: an overview. IEEE Trans. Broadcast 41, 1–8 (1995). Publisher Full Text

D Divsalar, MK Simon, Multiple trellis coded modulation (MTCM). IEEE Trans. Commun 36, 410–419 (1988). Publisher Full Text

AJ Viterbi, Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Trans. Inform. Theory IT13, 260–269 (1967)

I Lee, Spacetime bitinterleaved coded modulation for OFDM systems. IEEE Trans. Signal Process 52, 820–825 (2004). Publisher Full Text

Z Liu, Y Xin, GB Giannakis, Linear constellation precoding for OFDM with maximum multipath diversity and coding gains. IEEE Trans. Commun 51, 416–427 (2003). Publisher Full Text

J Boutros, E Viterbo, Signal space diversity: a power and bandwidthefficient diversity technique for the Rayleigh fading channel. IEEE Trans. Inform. Theory 44, 1453–1467 (1998). Publisher Full Text

AO Hero, TL Marzetta, Cutoff rate and signal design for the quasistatic Rayleigh fading spacetime channel. IEEE Trans. Inform. Theory 47, 2400–2416 (2001). Publisher Full Text

Y Rong, SA Vorobyov, AB Gershman, Linear block precoding for OFDMsystems based on maximization of mean cutoff rate. IEEE Trans. Signal Process 53(12), 4691–4696 (2005)

V Le Nir, M Helard, Reducedcomplexity spacetime block coding and decoding schemes with block linear precoding. IEE Electron. Lett 39(14), 1066–1068 (2003). Publisher Full Text

V Le Nir, M Helard, R Le Gouable, Spacetime block codes with maximal diversity in OFDMbased systems. Proceedings of the 5th World Wireless Congress (WWC'2004) San Francisco, USA (2004)

B Muquet, Z Wang, GB Giannakis, M de Courville, P Duhamel, Cyclic prefixing or zero padding for wireless multicarrier transmissions? IEEE Trans. Commun 50, 2136–2148 (2002). Publisher Full Text

MS Alouini, A Goldsmith, Adaptive modulation over Nakagami channels. J. Wirel. Commun 13(12), 119–143 (2000)

A Goldsmith, S Chua, Variable rate variable power MQAM for fading channels. IEEE Trans. Commun. 45, 1218–1230 (1997). Publisher Full Text

T Keller, L Hanzo, Adaptive multicarrier modulation: a convenient framework for timefrequency processing in wireless communications. Proc. IEEE 88, 611–640 (2000). Publisher Full Text

GG Raleigh, JM Cioffi, Spatiotemporal coding for wireless communication. IEEE Trans. Commun 46, 357–366 (1998). Publisher Full Text

T Keller, L Hanzo, Adaptive multicarrier modulation: a convenient framework for timefrequency processing in wireless communications. Proc. IEEE 88, 611–640 (2000). Publisher Full Text

K Wong, R Cheng, K Letaief, R Murch, Adaptive antennas at the mobile and base stations in an OFDM/TDMA system. IEEE Trans. Commun 49, 195–206 (2001). Publisher Full Text

O Edfors, M Sandell, JJ van de Beek, SK Wilson, PO Brjesson, OFDM channel estimation by singular value decomposition. IEEE Comm. Lett 46(7), 931–939 (1998)

A Gusmo, R Dinis, N Esteves, On frequencydomain equalization and diversity combining for broadband wireless communications. IEEE Commun. Lett 51(7), 1029–1033 (2003)

MT Heideman, C Sidney Burrus, On the number of multiplications necessary to compute a length2n DFT. IEEE Trans. Acoust. Speech. Sig. Proc 34, 91–95 (1986). Publisher Full Text

P Duhamel, M Vetterli, Fast Fourier transforms: a tutorial review and a state of the art. Signal Process, 259–299 (1990)