On the Information Rate of Single-Carrier FDMA Using Linear Frequency Domain Equalization and Its Application for 3GPP-LTE Uplink
© HanguangWu et al 2009
Received: 31 January 2009
Accepted: 19 July 2009
Published: 9 September 2009
This paper compares the information rate achieved by SC-FDMA (single-carrier frequency-division multiple access) and OFDMA (orthogonal frequency-division multiple access), where a linear frequency-domain equalizer is assumed to combat frequency selective channels in both systems. Both the single user case and the multiple user case are considered. We prove analytically that there exists a rate loss in SC-FDMA compared to OFDMA if decoding is performed independently among the received data blocks for frequency selective channels. We also provide a geometrical interpretation of the achievable information rate in SC-FDMA systems and point out explicitly the relation to the well-known waterfilling procedure in OFDMA systems. The geometrical interpretation gives an insight into the cause of the rate loss and its impact on the achievable rate performance. Furthermore, motivated by this interpretation we point out and show that such a loss can be mitigated by exploiting multiuser diversity and spatial diversity in multi-user systems with multiple receive antennas. In particular, the performance is evaluated in 3GPP-LTE uplink scenarios.
In high data rate wideband wireless communication systems, OFDM (orthogonal frequency-division multiplexing) and SC-FDE (single-carrier system with frequency domain equalization), are recognized as two popular techniques to combat the frequency selectivity of the channel. Both techniques use block transmission and employ a cyclic prefix at the transmitter which ensures orthogonality and enables efficient implementation of the system using the fast Fourier transform (FFT) and one tap scalar equalization per subcarrier at the receiver. There has been a long discussion on a comparison between OFDM and SC-FDE concerning different aspects [1–3]. In order to accommodate multiple users in the system, OFDM can be straightforward extended to a multiaccess scheme called OFDMA, where each user is assigned a different set of subcarriers. However, an extension to an SC-FDE based multiaccess scheme is not obvious and it has been developed only recently, called single-carrier FDMA (SC-FDMA) . (A single-carrier waveform can only be obtained for some specific sub-carrier mapping constraints. In this paper we do not restrict ourself to these constraints but refer SC-FDMA to as DFT-precoded OFDMA with arbitrary sub-carrier mapping.) SC-FDMA can be viewed as a special OFDMA system with the user's signal pre-encoded by discrete Fourier transform (DFT), hence also known as DFT-precoded OFDMA or DFT-spread OFDMA. One prominent advantage of SC-FDMA over OFDMA is the lower PAPR (peak-to-average power ratio) of the transmit waveform for low-order modulations like QPSK and BPSK, which benefits the mobile users in terms of power efficiency . Due to this advantage, recently SC-FDMA has been agreed on to be used for 3GPP LTE uplink transmission . (LTE (Long Term Evolution) is the evolution of the 3G mobile network standard UMTS (Universal Mobile Telecommunications System) defined by the 3rd Generation Partnership Project (3GPP).) In order to obtain a PAPR comparable to the conventional single carrier waveform in the SC-FDMA transmitter, sub-carriers assigned to a specific user should be adjacent to each other  or equidistantly distributed over the entire bandwidth , where the former is usually referred to as localized mapping and the latter distributed mapping.
This paper investigates the achievable information rate using SC-FDMA in the uplink. We present a framework for analytical comparison between the achievable rate in SC-FDMA and that in OFDMA. In particular, we compare the rate based on a widely used transmission structure in both systems, where equal power allocation (meaning a flat power spectral density mask) is used for the transmitted signal of each user, and linear frequency domain equalization is employed at the receiver.
The fact that OFDMA decomposes the frequency-selective channel into parallel AWGN sub-channels suggests a separate coding for each sub-channel without losing channel capacity, where independent near-capacity-achieving AWGN codes can be used for each sub-channel and accordingly the received signal is decoded independently among the sub-channels. This communication structure is of high interest both in communication theory and in practice, since near-capacity-achieving codes (e.g., LDPC and Turbo codes) have been well studied for the AWGN channel. We show that although SC-FDMA can be viewed as a collection of virtual Gaussian sub-channels, these sub-channels are correlated; hence separate coding and decoding for each of them is not sufficient to achieve channel capacity. We further investigate the achievable rate in SC-FDMA if a separate capacity-achieving AWGN code for each sub-channel is used subject to equal power allocation of the transmitted signal. The special case that all the sub-carriers are exclusively utilized by a single user, that is, SC-FDE, is investigated in , and it is shown that the SC-FDE rate is always lower than the OFDM rate in frequency selective channels. However, an insight into the cause of the rate loss and its impact on the performance was not given. Such an insight is of interest and importance to design appropriate transmission strategies in SC-FDMA systems, where a number of sub-carriers and multi-users or possibly multiple antennas are involved. In this paper, based on the property of the circular matrix we derive a framework of rate analysis for SC-FDMA and OFDMA, which is a generalization of the result in , and it allows for the calculation of the achievable rate using arbitrary sub-carrier assignment methods in both the single user system and the multi-user system subject to individual power constraints of the users. We analyze the cause of the rate loss and its impact on the achievable rate as well as provide the geometrical interpretation of the achievable rate in SC-FDMA. Moreover, we reveal an interesting relation between the geometrical interpretation and the well-known waterfilling procedure in OFDMA systems. More importantly, motivated by this geometrical interpretation we show that such a loss can be mitigated by exploiting multi-user diversity and spatial diversity in the multi-user system with multiple receive antennas, which is usually available in mobile systems nowadays.
The paper is organized as follows. In Section 2 we introduce the system model and the information rate for OFDMA and SC-FDMA. In Section 3 we derive the SC-FDMA rate result and provide its geometrical interpretation assuming equal power allocation without joint decoding. Then we extend and discuss the SC-FDMA rate result for the multi-user case and for multi-antenna systems in Section 4. Simulation results are given in Section 5, and conclusions are drawn in Section 6.
2. System Model and Information Rate
In the following, we first briefly review the achievable sum rate in the OFDMA system and then show the sum rate relationship between OFDMA and SC-FDMA. We assume in the uplink that the users' channels are perfectly measured by the base station (BS), where the resource allocation algorithm takes place and its decision is then sent to the users via a signalling channel in the downlink. For simplicity, we start with the single-user single-input single-output system and then extend it to the multi-user case with multiple antennas at the BS. For convenience, the following notations are employed throughout the paper. is the Fourier matrix with the th entry , and denotes the inverse Fourier matrix. Further on, the assignment of data symbols to specific sub-carriers is described by the sub-carrier mapping matrix with the entry
2.1. OFDMA Rate
After CP removal at the receiver, the received block can be written as
where is the transmitted block of the OFDMA system, and is a circulant matrix with the first column . The following discussion makes use of the important properties of circulant matrices given in the appendices (Facts 1 and 2). Performing multi-carrier demodulation using FFT and sub-carrier demapping using , we obtain the received block
where is an diagonal matrix with its diagonal entries being the channel frequency response at the selected sub-carriers of the user. This relationship can be readily verified since has only a single nonzero unity entry per column, and this structure of also leads to
where the step (9) to (10) follows from Fact 2 (see Appendix ), (10) to (11) follows from (7) since is also a diagonal matrix, and the step (11) to (12) results from (8). Therefore, is a vector consisting of uncorrelated Gaussian noise samples. The frequency domain ZF equalizer is given by the inverse of the diagonal matrix which essentially preserves the mutual information provided that is invertible. Here we assume that is always invertible since the BS can avoid assigning sub-carriers with zero channel frequency response to the user. Due to the diagonal structure of and independent noise samples of (uncorrelated Gaussian samples are also independent), (6) can be viewed as the transmit signal components or the data symbols on the assigned sub-carriers propagating through independent Gaussian sub-channels with different gains. This structure suggests that coding can be done independently for each sub-channel to asymptotically achieve the channel capacity. The only loss is due to the cyclic prefix overhead relative to the transmit signal block length. The achievable sum rate of an OFDMA system can be calculated as the sum of the rates of the assigned sub-carriers, which is given by
where is the power allocated to the th sub-carrier. Note that the employment of a zero forcing (ZF) equalizer performing channel inversion for each sub-carrier preserves the capacity since the resulting signal-to-noise ratio (SNR) for each sub-carrier remains unchanged. To maximize the OFDMA rate subject to the total transmit power constraint , the assignment of the transmit power to the independent Gaussian sub-channels should follow the waterfilling principle, and so the optimal power of the th sub-carrier is given by
where stands for the trace of the argument. It should be noted that the waterfilling procedure implicitly selects the optimal sub-carriers out of the available sub-carriers in the system and assigns optimal transmit power to each of them. Therefore, it is possible that some sub-carriers are not used. In our model, the waterfilling procedure amounts to mapping to the desired sub-carriers and at the same time constructing having diagonal covariance matrix with entries equal to the optimal power allocated to the desired sub-carriers.
2.2. SC-FDMA Rate
where we denote by the residual noise vector after ZF equalizer and IDFT. With (16) the transmit data components in SC-FDMA system can be viewed as propagating through virtual sub-channels distorted by the amount of noise given by . Note that is a Gaussian vector due to the linear transformation but it is entries are generally correlated which we show in the following:
where is applied to elementwise, and the step from (17) to (18) follows from the fact that is a diagonal matrix. The matrix is hence also diagonal with the diagonal entries being the reciprocal of channel power gains of the assigned sub-carriers of the user, which are usually not equal in frequency selective channels. Hence is a circulant matrix according to Fact 2 (see Appendix ) with nonzero values on the off diagonal entries. Therefore, the residual noise on the virtual sub-channels is correlated and hence SC-FDMA does not have the same parallel AWGN sub-channel representation as OFDMA. However, note that the DFT at the SC-FDMA transmitter does not change the total transmit power due to the property of the Fourier matrix , that is,
The property of power conservation of the DFT precoder at the transmitter and invertibility of IDFT at the receiver leads to the conclusion that the mutual information is preserved. Hence, the mutual information between the transmit vector and post-detection vector is equal to that of OFDMA . In other words, for any sub-carrier mapping and power allocation methods in OFDMA system, there exists a corresponding configuration in SC-FDMA which achieves the same rate as OFDMA. For example, suppose, for a given time invariant frequency selective channel, that is the optimal covariance matrix given by the waterfilling solution in an OFDMA system. To obtain the same rate in an SC-FDMA system, the covariance matrix of the transmitted signal can be designed as
where in the last step we use Fact 2 (see Appendix ). Hence, is a circulant matrix with the first column . Since both the covariance matrix of the transmitted signal and residual noise exhibit a circulant structure in an SC-FDMA system, correlation exists in both the transmitted symbols before DFT and the received symbols after IDFT. Such correlation complicates the code design problem in order to achieve the same rate as in OFDMA. This paper makes no attempt to design a proper coding scheme for SC-FDMA but we mention that SC-FDMA is not inferior to OFDMA regarding the achievable information rate from an information theoretical point of view. Instead, it can achieve the same rate as OFDMA if proper coding is employed. Note that the above statement implies using the same sub-carriers to convey information in both systems. Therefore, SC-FDMA and OFDMA are the same regarding the rate if they both use the same sub-carrier and the same corresponding power for each sub-carrier to convey information. However, in SC-FDMA coding and decoding should be applied across the transmitted and received signal components, respectively.
3. SC-FDMA Rate Using Equal Power Allocation without Joint Decoding
The waterfilling procedure discussed above is computationally complex which requires iterative sub-carrier and power allocation in the system. An efficient sub-optimal approach with reduced complexity is to use equal power allocation across a properly chosen subset of sub-carriers , which is shown to have very close performance to the waterfilling solution. In other words, this approach assumes and designs a proper sub-carrier mapping matrix to approximate the waterfilling solution, where the number of used sub-channels is also a design parameter. This approach can also be applied to an SC-FDMA system to approximate the waterfilling solution since DFT precoding and decoding are information lossless according to our discussion in Section 2. Note that DFT precoding does not change the equal power allocation property of the transmitted signal according to Fact 2 (see Appendix ), that is, ( ). Therefore, to obtain the same rate as in OFDMA, coding does not need to be applied across transmitted signal components, and only correlation among the received signal components needs to be taken into account for decoding.
3.1. SC-FDMA Rate without Joint Decoding
We are interested to see what the achievable rate in SC-FDMA is if a capacity-achieving AWGN code is used for each transmitted component, which is decoded independently at the receiver. Under the above given condition, the achievable rate in SC-FDMA is the sum of the rate of each virtual subchannel for which we need to calculate the post-detection SNR, that is, the post-detection SNR of the th virtual subchannel can be expressed as
In step (21) we denote which is the harmonic mean of by definition. In the last step we let since the post-detection SNR is equal for all the virtual subchannels. Using Shannon's formula the achievable rate in SC-FDMA can be obtained as
which is a function of the harmonic mean of the power gains at the assigned sub-carriers. Note that the result in  is a special case of (23) where all the available sub-carriers in the system are used by the user. It is perceivable that because noise correlation between the received components is not exploited to recover the signal. In the following, we will prove this inequality analytically. In order to prove
it is equivalent to prove
The Hoehn-Niven theorem  states the following: Let be the harmonic mean and let be the positive numbers, where the 's are not all equal, then
where the last step follows from the fact that is a constant value so that it can be factored out of the operation. Therefore, by applying the transitive property of inequality to (26) and (28) it follows that
holds, which corresponds to the case of frequency flat fading. Therefore, (24) holds in general.
The harmonic mean is sensitive to a single small value. tends to be small if one of the values is small. Therefore, the achievable sum rate in SC-FDMA depending on the harmonic mean of the power gain of the assigned sub-carriers would be sensitive to one single deep fade whose sub-carrier power gain is small. To give an intuitive impression how sensitive it is, we make use of the geometrical interpretation of the harmonic mean by Pappus of Alexandria  which is provided in Appendix .
3.2. Relation to OFDMA
In the following, we will show that the achievable sum rate of SC-FDMA using equal power allocation without joint decoding is equivalent to that achieved by nonprecoded OFDMA system with equal gain power (EGP) allocation among the assigned sub-carriers. This conclusion will lead to our geometrical interpretation of the SC-FDMA system.
In an OFDMA system, the EGP allocation strategy pre-equalizes the transmitted signal so that all gains of the assigned sub-carriers are equal, that is,
Upon insertion of (33) into (13), the achievable sum rate using EGP can be calculated as
which is equal to in (23), provided that both the SC-FDMA and OFDMA systems use the same assigned sub-carriers. This result leads to the conclusion that ZF equalized SC-FDMA with equal power allocation can be viewed as a nonprecoded OFDMA system performing EGP allocation among the assigned sub-carriers. It is worthy to point out that we find that EGP allocation shares a similar geometrical interpretation with waterfilling. This statement can be proven by applying logarithmic operation at both sides of the objective function of (32), which becomes
4. Extension to Multiuser Case and Multiantenna Systems
The information rate analysis in Sections 2 and 3 assumes only one user in the system. However, the principle also holds for the multi-user case where each user's signal will be first individually precoded by DFT and then mapped to a different set of sub-carriers. It is known that in the multi-user OFDMA system, the maximum sum rate of all the users can be obtained by the multi-user waterfilling solution  where each user subject to an individual power constraint is assigned a different set of sub-carriers associated with a given power. Therefore, the information rate achieved in the system can be calculated as a sum of rate of each user, which can again be calculated similarly as in the single-user system. As a result, a multi-user SC-FDMA system can achieve the same rate as a multi-user OFDMA system since DFT and IDFT essentially preserve the mutual information of each user if the same resource allocation is assumed. If equal power allocation of the transmitted signal without joint decoding is assumed for each user, the system sum rate of users can be straightforward extended from (23), that is,
where is the length of the transmitted signal block of the th user whose post-detection SNR is denoted as , is the power of the th transmitted symbol of the th user, and is the channel frequency response at the th assigned subcarrier of the th user. The geometrical interpretation of the achievable sum rate in the multiuser SC-FDMA system can be straightforward interpreted as performing multiuser EGP allocation in the system, where each user, subject to a given transmit power constraint, performs EGP allocation in the assigned set of subcarriers. It can be proven that by summing up the rate of all the users, each of which obeys (24), where the equality occurs when the channel frequency response at the assigned sub-carriers of each user is equal; that is, each user experiences flat fading among the assigned subcarriers for communication but the channel power gains can be different for different users. Note that the optimal multi-user waterfilling solution tends to exploit multi-user diversity and schedule at any time and any subcarrier of the user with the highest sub-carrier power gain-to-noise ratio to transmit to the BS. Consequently, from the system point of view, only the relatively strong sub-carriers, possibly from different users, are selected and the relative weak ones are avoided. In other words, each user is only assigned a set of relative strong sub-carriers. It will be a good choice if the above sub-carrier allocation scheme is applied for each user in SC-FDMA systems, because it is essentially equivalent to performing EGP among the relative strong sub-carriers for each user. As the number of users increases, the weak sub-carriers can be more effectively avoided due to the multi-user diversity. As a result, the effective channel for each user becomes less frequency selective, and the rate loss in SC-FDMA compared to OFDMA becomes smaller. The same effect happens if the BS is equipped with multiple antennas to exploit the spatial diversity to harden the channels. For SC-FDMA with the localized mapping constraint or the equidistantly distributed mapping constraint, multi-user diversity may help to reduce the rate loss with respect to an OFDMA system but with less degrees of freedom because multi-user diversity cannot guarantee that good sub-carriers assigned to each user are adjacent to each other or equidistantly distributed in the entire bandwidth. In this case, spatial diversity is much more important because it can always reduce frequency selectivity of each user's channel by using, for example, a maximum ratio combiner (MRC) at the receiver. As a result, the user specific resource allocation has less influence on the achievable rate no matter which sub-carriers are selected by the users but only the number of sub-carriers assigned to each user is needed to be considered. Consequently, not only is the rate loss mitigated but also the multi-user resource scheduler is greatly simplified. As an additional advantage, SC-FDMA can offer lower PAPR than OFDMA with negligible rate loss.
5. Simulation Results
Parameter assumptions for simulation
1.25 MHz, 2.5 MHz, 5 MHz, 10 MHz, 15 MHz and 20 MHz
Number of subcarriers in the system
75, 150, 300, 600, 900 and 1200
Number of subcarriers per RB
3GPP SCME urban macro 
Number of UEs
up to 6
Number of BSs
Antennas per UE
Antennas per BS
1, 2, 3
BS antenna spacing
We have presented a framework for an analytical comparison between the achievable information rate in SC-FDMA and that in OFDMA. Ideally, SC-FDMA can achieve the same information rate as in OFDMA since DFT and IDFT are information lossless; however, proper coding across the transmitted signal components and decoding across the received signal components have to be used. We further investigated the achievable rate if independent capacity achieving AWGN codes is used and accordingly decoding is performed independently among the received components for SC-FDMA, assuming equal power allocation of the transmitted signal. A rate loss compared to OFDMA was analytically proven in the case of frequency selective channels, and the impact of the weak sub-carriers on the achievable rate was discussed. We also showed that the achievable rate in SC-FDMA can be interpreted as performing EGP allocation among the assigned sub-carriers in the nonprecoded OFDMA systems which has a similar geometrical interpretation with waterfilling. More importantly, it was pointed out and shown in 3GPP-LTE uplink scenario that the rate loss could be mitigated by exploiting multi-user diversity and spatial diversity. In particular, with spatial diversity we showed that while being able to achieve a system sum rate very close to that in OFDMA, SC-FDMA provides an additional lower PAPR advantage.
A. Properties of the Circulant Matrix
Fact 1 (, Diagonalization of a circulant matrix).
Denote by the first column of a circulant matrix and by the diagonal matrix with the argument on the diagonal entries, then can be diagonalized by pre- and postmultiplication with a -point FFT and IFFT matrices, that is, , where is a diagonal matrix with diagonal entries being a scale version of the Fourier transform of .
where circulant denotes a circulant matrix with the first column . Equation (A.1) means that a circulant matrix can be written as a multiplication of IFFT matrix, diagonal matrix, and FFT matrix. In particular, if and only if all entries of are equal, then holds which is also a diagonal matrix with equal entries.
B. Geometrical Interpretation of the Harmonic Mean, the Arithmetic Mean, and the Geometric Mean
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