Single-carrier and multicarrier transceivers based on discrete cosine ...

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 12, DECEMBER 2013

Single-Carrier and Multicarrier Transceivers Based on Discrete Cosine Transform Type-IV Fernando Cruz-Roldán, Senior Member, IEEE, M. Elena Dom´ınguez-Jiménez, Gabriela Sansigre Vidal, José Piñeiro-Ave, and Manuel Blanco-Velasco, Senior Member, IEEE

Abstract—Multicarrier and single-carrier communications can be performed by using different orthogonal transforms. In this work, we derive the conditions to use discrete cosine transform type-IV even (DCT4e) in these contexts. Basically, these conditions are the two following: 1) The channel impulse response has to be symmetric. 2) Redundancy (a prefix and also a suffix guard sequence) must be appended into each data symbol to be transmitted. To satisfy the former, a front-end prefilter must be incorporated at the receiver. With regard to the second condition, we use a matrix formulation to demonstrate how the global transmission channel matrix can be diagonalized when symmetric extension or zero padding are used as redundancy. The use of zero padding requires an additional block at the receiver, whose structure is also shown. Moreover, a new expression based on the DCT4e to obtain the value of the per-subcarrier coefficients of the frequency-domain equalizer is derived. Finally, the proposed transceivers are compared to the DFT-based standardized systems in different communication scenarios under the presence of carrier frequency offset. Index Terms—Multicarrier transceiver, multicarrier modulation (MCM), discrete Fourier transform (DFT), discrete cosine transform (DCT), single-carrier frequency domain equalization (SC-FDE), single-carrier frequency domain multiple access (SC-FDMA), orthogonal frequency-division multiplexing (OFDM), carrier-frequency offset (CFO).

I. I NTRODUCTION

M

ULTICARRIER MODULATION (MCM) based on the Discrete Fourier Transform (DFT) has been the dominant medium-access technique in the fixed and nomadic broadband communications (see e.g. [1], [2]). Since release 8, it is the medium access technique chosen for the downlink in next generation mobile communications (LTE and LTE-Advanced) [3], [4]. MCM has significant advantages such as its effectiveness against multipath or frequency selective fading, or the use of bit loading algorithms to improve system performance Manuscript received April 29, 2013; revised September 6, 2013; accepted September 25, 2013. The associate editor coordinating the review of this paper and approving it for publication was Luc Deneire. This work was partially supported by the Spanish Ministry of Economy and Competitiveness through project TEC2012-38058-C03-01, and by the Spanish Ministry of Education, Culture and Sports through project PRX12/00064. F. Cruz-Roldán, J. Piñeiro-Ave, and M. Blanco-Velasco are with the Department of Teor´ıa de la Señal y Comunicaciones, Escuela Politécnica Superior de la Universidad de Alcalá, 28871 Alcalá de Henares (Madrid), Spain (e-mail: {fernando.cruz, jose.pineiro, manuel.blanco}@uah.es). M. E. Dom´ınguez-Jiménez and G. Sansigre Vidal are with the Department of Matemática Aplicada a la Ingenier´ıa Industrial, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, 28006 Madrid, Spain (e-mail: {elena.dominguez, gabriela.sansigre}@upm.es). Digital Object Identifier 10.1109/TWC.2013.111313.130751

with a significant increase in data rate per subcarrier [5]. However, the DFT-based MCM also presents some drawbacks: sensitivity to time and, especially, to frequency synchronization, high ratio between peak-to-average power ratio (PAPR), and reduced frequency discrimination. This leads to poor system performance in fixed and mobile noisy environments. To avoid some of the above constraints, some standards include the use of other medium-access techniques. This is the case of LTE, which proposes the use of single-carrier frequency division multiple access (SC-FDMA) for the uplink [3]. Many different solutions have been proposed as alternatives to DFT-based systems, such as the use of filter banks [6]–[9] or discrete trigonometric transforms (DTTs) to carry out the block transforms included in the single-carrier or the multicarrier transceivers [10]–[17]. In [12], it is shown that DCT Type-II even1-based multicarrier modulators (DCT2e-MCM) can perfectly diagonalize frequency-selective channels without requiring channel knowledge at the transmitter, with DCT2e-MCM being more robust to carrier frequency offset (CFO) than DFT-MCM. An intercarrier interference (ICI) analysis for a DCT2e-MCM system operating in the presence of CFO over additive white Gaussian noise (AWGN) channels is presented in [13]. On the other side, a new single-carrier frequency division multiple access system based on the DCT2e has been proposed in [17].

A. Properties of DCT-based Systems The interest of using DCT2e instead of DFT is well documented in several works, e.g., [12], [13]: •

•

The DCT bases have excellent spectral compaction and energy concentration properties, and generate much more frequency selective filter banks than the DFT filter bank, that is, a better separation among subchannels is obtained and the spectral efficiency is improved. As a result, the channel estimation and also the system performance (BEP) can be improved in noisy environments. In the presence of frequency offset, the ICI coefficients in DCT-MCM are more concentrated around the main coefficient. As a result, DCT-MCM systems offer better robustness against CFO. As it is known, this effect results in severe OFDM performance degradation in wireless mobile communications.

1 Eight

different types of DCTs are shown in [20].

c 2013 IEEE 1536-1276/13$31.00

´ et al.: SINGLE-CARRIER AND MULTICARRIER TRANSCEIVERS BASED ON DISCRETE COSINE TRANSFORM TYPE-IV CRUZ-ROLDAN

•

•

•

The DCT uses only real arithmetic, which reduces the signal processing complexity/power consumption, especially for real pulse-amplitude modulation signalling. Direct-conversion architecture (DCA) is replacing the superheterodyne architecture in OFDM systems, simplifying the design of the radio frequency (RF) front-end at the expense of in-phase/quadrature (I/Q) imbalance, among other problems [18], [19]. On the contrary, for real-valued modulations (BPSK or PAM), in the absence of a quadrature modulator, DCT-based systems do not suffer from the in-phase/quadrature imbalance. DSL standards based on DFT specify block sizes of N subchannels with conjugate symmetry. The reason is that data to be transmitted must be a real number, and therefore, the frequency-domain input data must have conjugate symmetry. This means that there are N/2 effective subchannels, not N . In DCT-MCM with real-valued modulations (BPSK or PAM), all N subchannels could be used to obtain a real-valued signal to be transmitted. Conjugate symmetry is not needed in the input data to obtain real values after performing the DCT, and as a result, the transmission can be full-rate (one symbol per subcarrier).

B. DCT4e Compared to DCT2e Discrete Cosine Transform Type-IV even (DCT4e) has been successfully applied to spectrum estimation, adaptive filtering, and it has played a key role in the efficient implementation of orthogonal lapped transforms and modulated filter banks. DCT4e is closely related to modified discrete cosine transform (MDCT), which is widely used in audio compression standards (e.g. MPEG-1, MPEG-2, or Dolby LabsAC-3). In fact, it is widely accepted that the DCT4e-based fast MDCT algorithms are the most efficient both in terms of computational complexity and structural simplicity [21]. In this paper, the use of DCT4e for single-carrier and multicarrier transceivers is proposed as an alternative solution to DCT2e. Of concern is the fact that both DCT2e-based and DCT4e-based transceivers bring the attractive features described in subsection I-A. Furthermore, DCT4e presents as additional feature that the direct and the inverse transforms are defined by exactly the same expression, except for the scaling factor. This characteristic simplifies the implementation of the system, since the same hardware can be used to carry out the transform blocks at both the transmitter and the receiver. Regarding the computational complexity of DCT4e and DCT2e, several efficient algorithms have been proposed in recent years to implement them [22]–[25]. The lower flop counts are provided in [24], [25], in which the operation counts are reduced to 17 9 N log2 N + O (N ) for DCT2e and DCT4e, considering a power-of-two transform size. C. Main Contributions and Organization The main goal of this work is to design the new multicarrier and single-carrier transceivers of Fig. 1 based on the DCT4e as an alternative solution to DFT-based or DCT2e-based systems. For this purpose, we obtain the kind of redundancy (symmetric

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extension or zero padding) to be introduced into each data symbol x to be transmitted. As it is known, the introduction of redundancy helps to equalize the transmission channel more easily. Moreover, a new expression based on the DCT4e for the coefficients di that perform the frequency domain equalization by means of a one-tap equalizer, is also obtained. The rest of this article is organized as follows. First, a brief description of single-carrier and multicarrier transceivers is provided in Section II, focusing our attention on DFT-based transceivers, which are recommended in most of standards. In Section III, the conditions for designing DCT4-based single-carrier and multicarrier transceivers are derived by using the matrix formulation. The study is presented for two kinds of redundancy: Symmetric extension and zero padding. Section IV provides performance evaluation of the proposed transceivers and its comparison to DFT-based systems, and finally, concluding remarks are given in Section V. II. P RELIMINARY BACKGROUND A. General Block Diagrams Figure 1 shows the general block diagrams representing multicarrier and single-carrier transceivers. First, let us focus our attention on Figure 1(a). At the transmitter, the incoming data are processed by an N -point inverse transform (T−1 a ), with N being the number of subchannels or subcarriers. At the receiver, a discrete transform (Tc ) is performed. Next, the frequency domain equalization (FEQ) is carried out by means of a set of coefficients 1/di , which are obtained from a third transformation not indicated in the figure (Tb ). On the other side, the block diagram of single-carrier modulation with frequency division multiple access (SC-FDMA) is shown in Figure 1(b). The main difference between SC-FDMA and OFDMA lies in the sequential transmission of the subcarriers in the first one versus the parallel transmission in the second one [26]. SC-FDMA arises from the single-carrier modulation combined with frequency domain equalization (SC-FDE) (Figure 1(c)) [27]. In this work, we address the design of multicarrier and single-carrier transceivers shown in Fig. 1 by employing DCT4e to implement the T−1 a and the Tc matrix blocks, and additionally Tb , which allow us to obtain the di coefficients. This problem can be formulated in two different ways. The first one is based on the interpretation of the symmetric convolution. That is, if DCTs are used as block transforms in the system of Fig. 1(a), the idea consists of some way forcing the linear convolution performed by the channel to become a symmetric convolution in the time-domain, or equivalently, an element-by-element operation in the corresponding DCT domain. The above formulation is used in [28] to obtain the conditions for using the eight different DCTs for multicarrier data transmission and using symmetric extension as redundancy. However, it would be useful to formulate the above problem so that zero padding strategy could also be used as redundancy. When the frequency-selective finite-impulse response of the channel is known, the zero-padded (ZP) solution guarantees symbol recovery irrespective of channel null locations in the absence of noise (coherent modulations are assumed) [29],

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(a)

(b)

(c) Fig. 1. Block diagram of single-carrier and multicarrier transceivers over a channel with additive noise. (a) Multicarrier transceiver. (b) Single-carrier frequency division multiple access transceiver. (c) Single-carrier frequency domain equalization transceiver.

[30]. Moreover, in different setups, ZP systems are optimal solutions in the mean-square error sense [30], [31]. Also, ZP provides advantages such as accurate time synchronisation and transmit power saving. To this end, we use a second way of formulating this problem, consisting of deriving the conditions for using DCT4e for single-carrier and multicarrier data transmission using matrices. For the rest of this work, only symmetric channel filters h are considered. Different solutions to meet the symmetry condition are proposed in [12, p. 915] by means of the front-end filter w. Furthermore, although we focus our attention on MCM (Fig. 1(a)) and SC-FDE (Fig. 1(c)), the proposed solutions apply directly for multiple users systems, such as OFDMA or SC-FDMA.

B. DFT-based Systems DFT is recommended in most standardized systems to perform Ta , Tc , and Td transforms in Fig. 1. In the above, the insertion of redundancy, e.g., cyclic prefix, allows the channel matrix H to be modelled as a right-circulant matrix, which

can be expressed as HDF T = W−1 · D · W,

(1)

−1

where W and W are, respectively, the DFT and the IDFT matrices, and D is a diagonal matrix, in which the diagonal elements di , 0 ≤ i ≤ (N − 1), are obtained as the N -point DFT of the channel impulse response. The FEQ block of each system of Fig. 1 basically consists of a 1/di bank of scalars, which efficiently equalizes the frequency-selective channel in the frequency domain2. Particularizing Fig. 1(a) for DMT/OFDM: Transforms Ta , Tb and Tc are DFTs; the redundancy added in the transmitter is usually a cyclic prefix or a zero padding [29]; the front-end filter in the receiver (w) is only required if the order of the discrete-time equivalent channel is larger than the number of redundant samples added in the transmitter to each OFDM symbol [5]. On the other side, the SC-FDMA modulation proposed in LTE (and LTE-A) is based on the use of DFTs, meaning that the Td transform in Figure 1(b) is a DFT, and 2 Using this definition, the equalizer is designed with the zero-forcing criterion, which perfectly reverses the effect of the transmission channel.


x

hch

0 Fig. 2.

N-1

transform the channel matrix H into an equivalent channel matrix Hequiv perfectly diagonalizable by DCT4e: Hequiv = C−1 4 · D · C4 ,

0 n

Example of a sequence x and a channel impulse response hch .

the remaining blocks are configured as previously described for DMT/OFDM. Finally, SC-FDE has also usually been implemented based on efficient FFT/IFFT operations, i.e., with Ta , Tb and Tc being DFTs in Figure 1(c). III. DCT4e FOR S INGLE -C ARRIER AND M ULTICARRIER M ODULATIONS

B. Symmetric Extension In order to derive the conditions to use symmetric-extension (left and right prefixes), we follow the same notation and reasoning provided in [12]. First, the channel matrix given in (2) is split as H = [Hlp Ho Hrs ] where the N × ν matrix Hlp contains the first ν columns of H, the N × ν matrix Hrs its last ν columns, and Ho the N remaining central ones: ⎡

Let us consider the extended block xTe = xTlp xT xTrs

Hlp

of size (N + 2ν) × 1. The receiving data can be obtained as y = H · xe + z, Toeplitz matrix of size N × (N + 2ν) defined ··· hν .. . ···

h0 .. . ..

. 0

· · · h−ν .. . h0 .. .. . . hν · · ·

0 h−ν .. . h0

··· 0 .. .. . . .. . 0 · · · h−ν

⎤ ⎥ ⎥ ⎥ , (2) ⎥ ⎦

(4)

i.e, in order to transform the Toeplitz matrix (2) in an equivalent matrix as (3).

A. Channel Matrix

where H is a as ⎡ hν ⎢ ⎢ 0 H=⎢ ⎢ . ⎣ .. 0

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⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

hν 0 .. . 0 .. . 0

··· .. . .. . ··· ··· ⎡

⎤ h1 .. ⎥ . ⎥ ⎥ ⎥ hν ⎥ ⎥, 0 ⎥ ⎥ .. ⎥ . ⎦

⎡

0

··· .. .

··· ··· .. . ···

0

0

Hrs

..

. ···

h−ν .. .

..

.

··· .. . .. .

..

. 0

···

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ h−ν ⎢ ⎢ .. ⎣ . h−1

0

h0 ⎢ .. ⎢ . ⎢ ⎢ ⎢ Ho = ⎢ hν ⎢ 0 ⎢ ⎢ . ⎣ ..

0 .. . 0

hν

···

0 h−ν .. . h0

0 .. . 0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

h−ν

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

Using the above, the receiving data can be expressed as and z is a column vector related to the additive noise. The goal follows: ⎤ ⎡ is to transform the channel matrix H into a new N ×N matrix xlp Hequiv , which can be diagonalized via the DCT4e matrix, y = H · xe + z = [Hlp Ho Hrs ] ⎣ x ⎦ + z denoted by C4e , whose (k, j) −entry is given by xrs

(2k + 1) (2j + 1) π = H · x + H · x + H · x + z, lp lp o rs rs (C4e )k,j = 2 cos , k, j = 0, . . . , N −1. 4N or In this sense, it is well known from [20] that any N × N y = (Hlp · Flp + Ho + Hrs · Frs ) · x + z. matrix Hequiv can be diagonalized via DCT4e if and only if it can be written as a sum of a Toeplitz matrix G and a Hankel Notice that Ho is a Toeplitz matrix, and if we assume that the channel filter h is symmetric (h−k = hk ), then Ho is matrix M4e : also symmetric, and of the same kind as G in (3), by taking Hequiv = G + M4e tk = hk for k = 0, . . . , ν < N and zero otherwise. Therefore, ⎤ ⎡ it suffices to build Flp and Frs such that t0 t1 · · · tN −2 tN −1 ⎥ ⎢ . . . .. .. .. ⎢ t1 Hlp Flp + Hrs Frs = M4e tN −2 ⎥ ⎥ ⎢ ⎡ ⎤ ⎥ ⎢ .. . 0 ··· 0 h1 · · · hν .. ⎥ =⎢ . ⎥ ⎢ ⎢ .. .. ⎥ . . . ⎥ ⎢ .. .. .. ⎢ . .. .. .. . ⎥ ⎣ tN −2 . . . ⎢ ⎥ t1 ⎦ . . ⎢ (3) 0 ⎥ hν . ⎢ ⎥. tN −1 tN −2 · · · t1 t0 . =⎢ . . −hν ⎥ ⎡ ⎤ 0 ⎢ ⎥ t1 t2 · · .· tN −1 0 ⎢ .. .. ⎥ . . . . . . . . . ⎣ ⎢ t2 ⎥ . . . . . . ⎦ . −tN −1 ⎥ ⎢ ⎢ .. ⎥ .. 0 · · · 0 −hν · · · −h1 . . . +⎢ . ⎥. .. .. .. . ⎢ ⎥ . . . . ⎣ tN −1 This is possible if and only if . . −t2 ⎦

0 −tN −1 · · · −t2 −t1 (5) Flp = Jν 0ν×(N −ν) ,

Therefore, our main goal in this section is to show what Frs = 0ν×(N −ν) − Jν , kind of redundancy should be appended in x in order to

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Fig. 3. Symmetries in x and h to be used in DCT4e-based systems (N (2ν + 1)). HS, HA, and WS stand for Half-sample Symmetry, Half-sample Antisymmetry, and Whole-sample Symmetry, respectively [28], [32].

where Jν is the antidiagonal permutation matrix of order ν. Therefore, the prefix and suffix must be defined as T T xlp = Jν x0 · · · xν−1 = xν−1 · · · x0 , ⎤ ⎡ −xN −1 T ⎢ ⎥ .. xrs = −Jν xN −ν · · · xN −1 =⎣ ⎦. . −xN −ν This result is consistent with the one obtained in [28] for DCT4e. Fig. 3 illustrates an example of the symmetry to be imposed on x and h for the proposed DCT4e-based system. C. Zero Padding In this subsection, we show how to transform the channel matrix H into Hequiv using zeros as redundancy, i.e., zeros are padded into each transmitted block. To this end, let xTzp = (0, ...0, xT , 0, ...0) be the (N + 2ν) × 1 zero-padded version of the original symbol x of length N . Specifically, the total amount of redundant elements (zeros) appended into each transmitted symbol is 2ν: ν zeros as prefix, and ν zeros as suffix. Let xTr−zp = (0, ...0, xTzp , 0, ...0) be the (N + 4ν) × 1 vector that includes ν zeros from the previous symbol, xTzp , and ν zeros from the next coming symbol under consideration (see Fig. 4). This vector xr−zp is transformed by the (N + 2ν) × (N + 4ν) matrix H as in (2) to obtain the receiving data: yrT zp = y−ν · · ·

y−1

y0

···

yN−1

yN

···

yN+ν−1

.

It is easy to see that yr zp = H · xr−zp + zzp = Hm · x + zzp , where zzp is a term related to the noise and ⎤ ⎡ 0 ··· 0 hν .. ⎥ .. ⎢ .. . . . . ⎢ . . ⎥ ⎥ ⎢ ⎥ ⎢ .. .. ⎢ h0 . . 0 ⎥ ⎥ ⎢ ⎥ ⎢ .. . . .. ⎢ . . . h ν ⎥ ⎥ ⎢ ⎢ .. ⎥ Hm = ⎢ ... . . . . . . . ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ hν . . . . . . h0 ⎥ ⎥ ⎢ ⎢ .. ⎥ ⎢ 0 ... ... . ⎥ ⎥ ⎢ ⎢ . . .. ⎥ ⎣ .. . . . . . . ⎦ 0

···

0 hν

is an (N + 2ν) × N matrix whose columns are the N central columns of H. Our aim is to obtain Hequiv , which is a square

Fig. 4. Illustration of the zero-padding system. Only ν zeros must be inserted before and after each information symbol.

matrix derived from Hm that can be diagonalized by DCT4e. As it has been shown, such matrices should be split in the sum of a Toeplitz matrix and a Hankel one as in (3). Let us split Hm as T , Hm = HT1 HT2 HTc HT4 HT5 where H1 contains the rows from 1 to ν, H2 from ν +1 to 2ν, Hc from 2ν + 1 to N , H4 from N + 1 to N + ν, and finally H5 the remaining ν last rows. Observe that all the above are ν × N matrices, except for Hc , which is an (N − 2ν) × N matrix. In particular, the submatrices at both extremes are ⎤ ⎡ hν 0 ··· ··· ··· 0 ⎢ .. . . .. ⎥ .. . . . ⎥ H1 = ⎢ ⎦ ⎣ . .. . hν 0 ··· 0 h1 ⎤ ⎡ 0 ··· 0 hν · · · h1 ⎢ .. ⎥ . .. .. H5 = ⎣ ... . . . ⎦ 0

··· ··· ···

0 hν

Next, Hequiv is built from Hm just folding its first (or last) ν rows (through the permutation Jν ) as in a mirror, and adding (or substracting) them to the adjacent (or previous) ν rows: ⎤ ⎡ H2 + Jν H1 ⎦ Hc Hequiv = ⎣ ν H4 − J H5 We call this procedure ”mirror and add/substract” (MIAS). To reconstruct the original symbol x from the receiving data yr zp , we split yr zp and the noise zzp in the same way as Hm : T yr zp = y1T y2T ycT y4T y5T . where y1T = y2T = ycT = y4T = y5T =

y−ν

· · · yν−1

y0

yν

· · · yN −ν−1

yN −ν yN

· · · y−1

· · · yN −1 · · · yN +ν−1

, ,

, , .


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where DCT3e stands for discrete cosine transform type-III even, and hrZP is the N −length vector defined as hrZP = [h0 , h1 , · · · , hν , 0, · · · , 0] . The result in (6) can be expressed by means of the normalized DCT4e of the first column of Hequiv . This follows from (4) by observing the i-th element of the first column of D·C4e = C4e · Hequiv , from which we obtain di =

DCT4e (hequiv )i , 2 cos (2i+1)π 4N

(7)

where hequiv is the first column of Hequiv . To sum up, the coefficients of the FEQ can be computed using the DCT3e as (6), or using the DCT4e as in (7). In this last case, all the transforms Ta , Tb , and Tc are carried out by means of the DCT4e, which makes the transceiver implementation easier . IV. E XAMPLE D ESIGN Fig. 5.

Mirror and add/substract (MIAS) block processing at the receiver.

Then, the same transformations on the rows of yr zp must be performed to get the N × 1 receiving data: ⎤ ⎡ ν ⎤ ⎡ ν J H1 + H2 J y1 + y2 ⎦+z=⎣ ⎦x + z yc Hc y=⎣ y4 − Jν y5 H4 − Jν H5 ⎤ ⎡ ν Iν 0 0 0 J = ⎣ 0 0 I(N −2v) 0 0 ⎦ Hm · x + z, ν 0 0 0 I −Jν where Iν denotes the diagonal matrix of order ν. Therefore: y = Hequiv · x + z, since the equivalent channel matrix can be expressed as Hequiv = Υ · Hm , being

⎡

Jν ⎣ Υ= 0 0

Iν 0 0

0

I(N −2v) 0

0 0 Iν

⎤ 0 0 ⎦. −Jν

As a result, the channel matrix can be diagonalized via the DCT4e. Notice that the MIAS block processing, included in Υ, is required at the receiver before the Tc block transform (see Fig. 5). D. FEQ coefficients Now, we obtain the value of each di coefficient to efficiently equalize the frequency-selective transmission channel under the zero-forcing criterion. We have previously shown how Hequiv can be diagonalized by the DCT4e. In (4), D is a diagonal matrix with the main diagonal defined as d = [d0 , · · · , dN −1 ]. In [20], it is demonstrated that these diagonal entries can be obtained as d = DCT3e(hrZP ),

(6)

In this section, the capabilities of the proposed systems with respect to DCT2e-based and DFT-based transceivers are examined through computer simulations. The impact of the normalized CFO Δf T on the symbol error rate is also considered. CFO is mainly caused by mismatch or ill-stability of the local oscillators in the transmitter and the receiver, and by the Doppler effect in mobile systems. A. Setup As for the simulation configurations, the parameters used in our experiments are summarized in Table I. Almost all the parameters and conditions used in our simulations are assumed for both the DCT-based and the conventional DFT-based transceivers, but there are some exceptions that are clarified below. Symmetric extension (SE) and ZP are appended as redundancy for DCT-based transceivers, and CP or ZP for the DFT-based counterpart. The amount of redundant elements required to implement the proposed and the DCT2e-based systems is 32, whereas this length is 16 for the DFT-based system. For each SNR, defined by the ratio of the power of the transmitted signal and the variance of channel noise, at least 2·104 independent trials are performed. We consider that the channel remains unchanged within one symbol. However, it changes independently among different symbols. As can be seen in Table I, two wireless fading channels according to the ITU Pedestrian A and Vehicular A channels [33], are used as multipath channels. The Pedestrian A channel has a relatively short delay profile, and 5 Km per hour has been taken into account as the pedestrian velocity; it corresponds to a Doppler spread of 9.3 Hz for a carrier frequency of 2 GHz. On the other hand, 120 Km per hour is considered as the mobile speed for the Vehicular A channel. This speed leads to 222.4 Hz of Doppler spread for the carrier frequency used in the experiments. Finally, a perfect channel estimation is also considered at the receiver and the front-end prefilter for the DCT-based transceivers is implemented as the time-reversed (matched) filter to the estimated channel. For the case of DFT-based systems, the front-end prefilter is not considered.

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TABLE I S IMULATION PARAMETERS . Value 5 MHz 200 ns 2 GHz QPSK 512 16 symbols 16 32 (DCT4e and DCT2e) or 16 (DFT) ITU Pedestrian A and Vehicular A [33] Perfect Zero Forcing iid AWGN Hard Decision > 2 · 104

CP−DFT NCF CP−DFT CFO=0.08 CP−DFT CFO=0.1 CP−DFT CFO=0.12 SE−DCT4e NCF SE−DCT4e CFO=0.08 SE−DCT4e CFO=0.1 SE−DCT4e CFO=0.12 SE−DCT2e NCF SE−DCT2e CFO=0.08 SE−DCT2e CFO=0.1 SE−DCT2e CFO=0.12

−1

10

SER

Parameters System Bandwidth Sampling Period Carrier Frequency Modulation and Demodulation Total Subcarrier Number (N ) SC-FDMA Input Block Size SC-FDMA Input FFT Size (M ) Length of Redundancy Channel Models Channel Estimation Channel Equalization Noise Model Detection Number of Iterations

MCM (Pedestrian A Channel)

0

10

−2

10

−3

10

−4

10

0

5

10

15

20 SNR (dB)

25

30

35

40

B. Multicarrier transceivers

C. SC-FDE transceivers Figures 10–13 illustrate the SER performances of the proposed DCT4e-based SC-FDE transceivers compared to the DCT2e-based and the conventional DFT-based ones for three different values of the CFO. For both the Pedestrian A and the Vehicular A channels, the following can be noted. a) In the absence of CFO, the DFT-based transceivers outperform the proposed SE and CP DCT-SC-FDE (approximately 1 dB for the SER value of 10−3 ). b) ZP-DFT-SC-FDE shows better performance than the proposed transceiver and CP-DFT-SC-FDE in the presence of CFO. c) On the contrary, the DCT-based SC-FDE outperforms the CP-DFT-based 3 The simulation results are almost identical for both DCT2e-based and DCT4e-based transceivers in all the communication scenarios herein considered.

Fig. 6. SER performances of symmetrical extended DCT4e-MCM, DCT2e-MCM, and the conventional cyclic-prefixed DFT-MCM, with the pedestrian A channel.

ZP−MCM (Pedestrian A Channel)

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DFT NCF DFT CFO=0.08 DFT CFO=0.1 DFT CFO=0.12 DCT4e NCF DCT4e CFO=0.08 DCT4e CFO=0.1 DCT4e CFO=0.12 DCT2e NCF DCT2e CFO=0.08 DCT2e CFO=0.1 DCT2e CFO=0.12

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We investigate the effect of CFO on the performance of the proposed MCM systems. In Figures 6 and 7 (Pedestrian A channel), it can be seen that similar symbol error rates are obtained when SE and ZP are included as redundancies in DCT-MCM transceivers3 . In addition, both kind of systems have shown similar results to DFT-MCM (OFDM) for low frequency offset, whereas the performance of the CP-DFT-based MCM system suffers from higher degradation when the frequency offset reaches the maximum values. On the contrary, the ZP-DFT-MCM shows a gain (e.g., 0.5 dB for the symbol error rate (SER) value of 10−3 ) compared to both SE and ZP DCT4e-MCM without offset (CFO= 0) and for CFO= 0.08. Finally, both CP and ZP DFT-based systems suffer from an extreme degradation for CFO=0.12, whereas the SER equals 10−3 at SNR=35 dB for the proposed SE and ZP DCT4e-MCM. Similar results are obtained for the Vehicular A channel (Figs. 8 and 9), with the exception of the CP-DFT-MCM transceiver also reaching an error floor at SNR=35 dB for CFO=0.1. To sum up, since DCT2e-MCM and DCT4e-MCM involve an additional block (front-end prefilter) and the double of redundant elements, DFT-MCM is recommended in the absence of CFO. However, in the presence of frequency offset, the proposed DCT4e and also the DCT2e-based systems outperform the conventional DFT transceivers, especially with strong CFO.

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Fig. 7. SER performances of zero-padded DCT4e-MCM, DCT2e-MCM, and the conventional DFT-MCM, with the pedestrian A channel.

transceiver, for CFO=0.1. d) In the worst scenario, i.e., CFO=0.12, CP-DFT-SC-FDE and the proposed transceivers reach error floors, whereas the ZP-DFT-SC-FDE transceiver exhibits better performance. As an illustrative example, the SER equals 10−3 at SNR=35 dB when the Pedestrian A channel is considered. In summary, we observe from the above simulations that the proposed DCT4e-based system is more robust than CP-DFT-SC-FDE to CFO variation, but the best performance has been obtained with ZP-DFT-SC-FDE. V. C ONCLUSIONS Different DCT4e-based transceivers for multicarrier and single-carrier communications have been proposed. Using a matrix formulation, the conditions to perfectly diagonalize the global channel matrix into decoupled and memoryless subchannels, are derived. The problem is formulated using


MCM (Vehicular A Channel)

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CP−DFT NCF CP−DFT CFO = 0.08 CP−DFT CFO = 0.1 CP−DFT CFO = 0.12 SE−DCT4e NCF SE−DCT4e CFO = 0.08 SE−DCT4e CFO = 0.1 SE−DCT4e CFO = 0.12 SE−DCT2e NCF SE−DCT2e CFO = 0.08 SE−DCT2e CFO = 0.1 SE−DCT2e CFO = 0.12

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CP−DFT NCF CP−DFT CFO=0.1 CP−DFT CFO=0.12 SE−DCT4e NCF SE−DCT4e CFO=0.1 SE−DCT4e CFO=0.12 SE−DCT2e NCF SE−DCT4e CFO=0.1 SE−DCT2e CFO=0.12

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Fig. 8. SER performances of symmetrical extended DCT4e-MCM, DCT2e-MCM, and the conventional cyclic-prefixed DFT-MCM, with the vehicular A channel.

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Fig. 10. SER performances of symmetrical extended DCT4e-SC-FDE, DCT2e-SC-FDE, and the conventional cyclic-prefixed DFT-SC-FDE, with the pedestrian A channel.

ZP SC−FDE (Pedestrian A Channel)

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DFT NCF DFT CFO=0.08 DFT CFO=0.1 DFT CFO=0.12 DCT4e NCF DCT4e CFO=0.08 DCT4e CFO=0.1 DCT4e CFO=0.12 DCT2e NCF DCT2e CFO=0.08 DCT2e CFO=0.1 DCT2e CFO=0.12

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DFT NCF DFT CFO=0.1 DFT CFO=0.12 DCT4e NCF DCT4e CFO=0.1 DCT4e CFO=0.12 DCT2e NCF DCT2e CFO=0.1 DCT2e CFO=0.12

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Fig. 9. SER performances of zero-padded DCT4e-MCM, DCT2e-MCM, and the conventional DFT-MCM, with the vehicular A channel.

matrices so that symmetric extension or zero padding can be used as redundancy. The main complexity disadvantage of the DCT-based transceivers is that the receiver needs a front-end prefilter to guarantee that the channel impulse response is symmetric. Additionally, the receiver has to include a mirror and add/substract (MIAS) block processing when zero padding is used. However, in any case, the transmission channel is equalized by means of simple one-tap frequency domain equalizers. We have shown the expression based on the DCT3e to obtain the corresponding FEQ coefficients, and from it, we have derived a new expression that allows the calculation of these coefficients using the DCT4e. Finally, considering different scenarios, simulation results have shown that the proposed transceivers are a good alternative for multicarrier communications and, compared to cyclic-prefixed DFT-based systems, can also be considered for single-carrier

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Fig. 11. SER performances of zero-padded DCT4e-SC-FDE, DCT2e-SC-FDE, and the conventional DFT-SC-FDE, with the pedestrian A channel.

communications. However, the proposed DCT-SC-FDE does not outperform SC-DFT-FDE systems with zero padding, not even with strong CFO. ACKNOWLEDGEMENTS The authors would like to thank the Associate Editor and the anonymous Reviewers for their insightful recommendations, which have significantly contributed to the improvement of this paper. R EFERENCES [1] IEEE Std 802.11n, Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: Enhancement for Higher Throughput, 2009. [2] IEEE Std 802.16e, IEEE Standard for local and metropolitan area networks, part 16: Air interface for broadband wireless access systems, May 2009.

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SC−FDE (Vehicular A Channel)

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CP−DFT NCF CP−DFT CFO=0.1 CP−DFT CFO=0.12 SE−DCT4e NCF SE−DCT4e CF0=0.1 SE−DCT4e CF0=0.12 SE−DCT2e NCF SE−DCT2e CF0=0.1 SE−DCT2e CF0=0.12

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[11] [12] [13] [14]

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[17]

Fig. 12. SER performances of symmetrical extended DCT4e-SC-FDE, DCT2e-SC-FDE, and the conventional cyclic-prefixed DFT-SC-FDE, with the vehicular A channel.

[19]

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[18]

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[20]

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[21]

[22]

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[24] [25]

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Fig. 13. SER performances of zero-padded DCT4e-SC-FDE, DCT2e-SC-FDE, and the conventional DFT-SC-FDE, with the vehicular A channel.

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[3] 3GPP Technical Specification, TS 36.211, Evolved Universal Terrestrial Radio Access (E-UTRA); Physical channels and modulation, 3GPP, v 8.4.0 September 2008. [4] 3GPP Technical Specification, TS 36.212, Evolved Universal Terrestrial Radio Access (E-UTRA); Multiplexing and channel coding, 3GPP, v 8.4.0 September 2008. [5] A. Goldsmith, Wireless Communications. Cambridge University Press, 2006. [6] T. Fusco, A. Petrella, and M. Tanda, “Data-aided symbol timing and CFO synchronization for filter bank multicarrier systems,” IEEE Trans. Wireless Commun., vol. 8, no. 5, pp. 2705–2715, May 2009. [7] F. Cruz-Roldán and M. Blanco-Velasco, “Joint use of DFT filter banks and modulated transmultiplexers for multicarrier communications,” Signal Process., vol. 91, no. 7, pp. 1622–1635, July 2011. [8] N. Moret and A. M. Tonello, “Performance of filter bank modulation with phase noise,” IEEE Trans. Wireless Commun., vol. 10, no. 10, pp. 3121–3126, Oct. 2011. [9] R. Zakaria and D. Le Ruyet, “A novel filter-bank multicarrier scheme to mitigate the intrinsic interference: application to MIMO systems,” IEEE Trans. Wireless Commun., vol. 11, no. 3, pp. 1112–1123, Mar. 2012. [10] S. Attallah and J. E. M. Nilsson, “Sequences leading to minimum

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Fernando Cruz-Roldán (M’98-SM’06) was born in Baena, Spain, in 1968. He received his Technical Telecommunication Engineer degree from the Universidad de Alcalá (UAH), Spain, in 1990, his Telecommunication Engineer degree from the Universidad Politécnica de Madrid (UPM), Spain, in 1996, and Ph. D. in Electrical Engineering from the UAH, in 2000. Dr. Cruz-Roldán received the Universidad de Alcalá Prize for the most outstanding doctoral dissertation in the engineering discipline. He joined the Department of Ingenier´ıa de Circuitos y Sistemas (UPM), in 1990, where from 1993 to 2003, he was an Assistant Professor. From 1998 to February 2003, he was a Visiting Lecturer at Universidad de Alcalá. In March 2003, he joined the Universidad de Alcalá, Spain, as an Associate Professor, and since November 2009, he has been a Professor with the Department of Teor´ıa de la Señal y Comunicaciones, Universidad de Alcalá. His teaching and research interests are in digital signal processing, filter design, and multirate systems applied to subband coding and broadband digital communications. M. Elena Dom´ınguez-Jiménez was born in Madrid, Spain. She received the degree in Mathematical Sciences from the Universidad Complutense de Madrid in 1992 and the Ph.D. degree from the Universidad Politécnica de Madrid in 2001. Dr. Dom´ınguez received an Extraordinary Award from the Universidad Politécnica de Madrid for the best doctoral dissertation in 2002. Since 1992, she has been working at the Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Politécnica de Madrid. She became an Associate Professor in 2012. Her research interests include signal processing for communications, wavelets, filterbank theory, and compressive sampling.

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Gabriela Sansigre-Vidal was born in Madrid, Spain. She received the Ph.D. degree in Mathematical Sciences from the Universidad de Zaragoza in 1990. She is Associate Professor at the Departamento de Matemtica Aplicada, E.T.S.I. Industriales, Universidad Politécnica de Madrid (Spain) since 1992. Her research interests include Orthogonal Polynomials, Linear Algebra, Signal Processing for Communications and Filterbank Theory. José Piñeiro-Ave received the M.S. degree from the Universidad de Vigo in 1997 in telecommunication engineering. In 1999, he joined the Department of Teor´ıa de la Señal y Comunicaciones, Universidad de Alcalá (UAH), Madrid, Spain, where he was an Assistant Professor from 1999 to 2002. In January 2002, he joined the UAH as an Associate Professor. His current research interests are digital signal processing, video processing and infrared systems.

Manuel Blanco-Velasco (S’00-M’05-SM’10) received the engineering degree from the Universidad de Alcalá, Madrid, Spain in 1990, the MSc in communications engineering from the Universidad Politécnica de Madrid, Spain, in 1999, and the PhD degree from the Universidad de Alcalá in 2004. From 1992 to 2002, he has been with the Circuits and Systems Department at the Universidad Politécnica de Madrid as Assistant Professor. In April 2002, he joined the Signal Theory and Communications Department of the Universidad de Alcalá where he is now working as Associate Professor. His main research interests are biomedical signal processing and communications.