Multi-carrier modulation analysis and WCP-COQAM proposal
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Abstract
In the vision towards future radio systems, where access to information and sharing of data is to be available anywhere and anytime to anyone for anything, a wide variety of applications and services are therefore envisioned. This naturally calls for a more flexible system to support. Moreover, the demand for drastically increased data traffic, as well as the fact of spectrum scarcity, would eventually force future spectrum access to a more dynamic fashion. For addressing the challenges, a powerful and flexible physical layer technology must be prepared, which naturally brings us to the question whether the legacy of the OFDM system can still fit in this context. In fact, during the past years, extensive research effort has been made in this area and several enhanced alternatives have been reported in the literature. Nevertheless, up to date, all of the proposed schemes have advantages and disadvantages. In this paper, we give a detailed analysis on these well-known schemes from different aspects and point out their open issues. Then, we propose a new scheme that aims to maximally overcome the identified drawbacks of its predecessors while still trying to keep their advantages. Simulation results illustrate the improvement achieved by our proposal.
Keywords
FBMC FMT GFDM OFDM OQAM1 Introduction
In today’s mobile communication systems, cyclic-prefix-based orthogonal frequency division multiplexing (CP-OFDM) is widely adopted. It shows that the concept of multi-carrier modulation (MCM) is well recognized as an efficient mode for broadband transmission. However, for moving towards future radio systems beyond 2020, it is time to ask whether the traditional fashion for MCM can effectively meet particular demands from some emerging scenarios, e.g., ultra dense networks (UDN) and heterogenous networks (HetNet). To be more specific, future radio systems aim to offer drastically increased data volume. To address this challenge, an envisioned approach is to increase the network densification, together with dynamic spectrum access (DSA) techniques based on the cognitive radio (CR) concept, leading to the so-called UDN [1]. In order to effectively implement DSA solutions, it relies on the physical layer modulations. Indeed, an important requirement is that the radio signal must provide a good spectrum shape, i.e., low out-of-band radiation. However, an identified drawback of the OFDM systems is that the employed rectangular pulse yields an unsatisfactory sidelobe attenuation which may cause severe consequences in UDN. For instance, the high out-of-band radiation may severely pollute the neighbors in the adjacent bands. The second drawback of OFDM is the lack of waveform flexibility which could be problematic in the HetNet. Considering the heterogeneity of future networks, where cells with different sizes are integrated, doubly dispersive, i.e., time/frequency-dispersive, channel appears in the signal transmission environment, while the dominant parts can be varying from time to time [2]. Research analysis has shown that the waveform localization property in time and frequency domains both play an important role in the design criteria for addressing the two-dimensional fading [3]. Moreover, in a system with imperfect synchronization conditions, a waveform that is designed with different weights towards time and frequency localization will provide different degree of robustness against the time and frequency residual synchronization errors as well [4, 5]. Therefore, a suitable system should be able to employ flexible waveforms, depending on different transmission circumstances in future HetNet scenarios.
To overcome these two drawbacks, during the past decades, people tried to improve the OFDM with some other pulse shapes. However, the mathematicians proved that for a MCM system it cannot simultaneously employ a flexible pulse, remain orthogonality, and transmit at the Nyquist rate. This statement was later known as the Balian-Low Theorem (BLT) [6]. Thus, novel MCM transmission fashion should be used to solve this bottleneck. In the literature, two main MCM alternatives have been proposed. The first scheme is called Filter Bank Multi-carrier/Offset Quadrature Amplitude Modulation (FBMC/OQAM), whose pioneering work of Saltzberg [7] can be also analyzed in the Gabor perspective. A key idea of his multi-carrier system is to split the complex data symbols into their real and imaginary parts. By this way, the orthogonality condition is relaxed and only applies to the real field, which eventually allows us to escape from the requirements defined by the BLT. The resulting degree of freedom can then be used to improve the signal spectrum shape and further to address the waveform flexibility. On the other hand, the second scheme, which is called Filter Multi-Tone (FMT), consists in relaxing the Nyquist rate transmission by employing an over Nyquist sampling, which, in practice for a given symbol duration, leads to an increased frequency spacing. This means that if the pulse spectrum is band-limited to the extended subcarrier spacing, no interference exists between the subcarriers. Therefore, the waveform design only needs to satisfy a one-dimensional Nyquist condition, e.g., square root raised cosine (SRRC), leaving more room for the signal spectrum improvement as well as the waveform flexibility. The first proposal in this direction dates back from 1971 [8] and it was later known under different acronyms, such as fraction-spaced multi-carrier modulation [9], over-sampled OFDM [10], over-sampled DFT [11], and more recently, it is often denoted as FMT [12]. Aside from the above-mentioned advantages, both FBMC/OQAM and FMT have some weak points. Since there is no CP in the FBMC/OQAM and FMT, the system orthogonality is ruined when the transmission passes through a multi-path channel. Therefore, more equalizers with higher complexity are needed on the receiver side for maintaining an acceptable interference level [13, 14, 15]. Moreover, due to this orthogonality loss, even more complex receiver is requested when further considering these schemes in combination with space-time encoded multiple-input-single-output (MISO) transmission, e.g., Alamouti coding [16]. In addition, for the FMT scheme, due to its over-sampled nature, it inherits a spectral efficiency (SE) loss.
Besides these two schemes, a new concept of MCM has recently appeared in the literature. It replaces the linear filtering with a circular filtering for pulse shaping. This idea was originally raised with the introduction of generalized frequency division multiplexing (GFDM) [17]. By using a circular filtering, the overall MCM system can maintain a block transform processing so that a CP can be easily inserted. Moreover, an SRRC pulse is used in the GFDM scheme to further improve the signal power spectrum density (PSD). Due to this advantage, the GFDM was first presented for communications over TV white space. Later, its good out-of-band energy attenuation rapidly captured a lot of attention in the CR field [18, 19, 20, 21] and further towards next generation of cellular systems [22]. However, a primary obstacle is that the GFDM is not an orthogonal system, again, restricted by the BLT [23]. This non-orthogonality nature could be a vital drawback in a practical usage. The in-band interference severely causes a performance degradation, which further was proved to be pulse shape dependent [23]. To mitigate the performance degradation, one possible solution is to use an iterative interference cancelation [20]. However, the receiver complexity gets largely increased and might even get exploded when considering it together with MIMO transmission. After GFDM, the circular filtering concept is further adopted for the FMT, which gives the birth for the cyclic block FMT (CB-FMT) [24]. It can successfully overcome the orthogonality issue. Nevertheless, as the modulation kernel remains FMT, the SE loss as well as the trade-offs between orthogonality and waveform flexibility still exist.
To the best of our knowledge, a comparative analysis among all of the above-mentioned MCM schemes has never been reported in the literature. In this paper, we first provide such analysis among these MCM schemes following four main criteria: spectral efficiency, power spectral density, orthogonality for distortion-free/multi-path channel, and waveform flexibility. Although the selected four criteria do not fully cover all of the research aspects, they can already be used to identify some drawbacks of each of these schemes. Then, motivated by the analysis, we further introduce a new type of FBMC/OQAM, which also adopts the circular filtering to the FBMC/OQAM and we call it FBMC/circular OQAM (COQAM)^{a}. The investigation of this new scheme aims to overcome the identified drawbacks of the State-of-the-Art (SoTA) MCM schemes. The rest of the paper is organized as follows: in Section 2, we give a brief recall of the SoTA MCM schemes and provide an analysis on their pros and cons with regard to four criteria. In Section 3, we present the concept of the FBMC/COQAM modulation and its motivation behind. In Section 4, the efficient implementation algorithm is presented. In Section 5, we illustrate a detailed transceiver design by considering the FBMC/COQAM in a multi-path transmission context. The efficiency evaluations of the proposed scheme are given in Section 6. In Section 7, some discussions and remarks are addressed to further clarify the proposal interest. Finally, in Section 8, we draw some conclusions and perspectives. For simplicity, in the following, we omit the term FBMC for FBMC/OQAM and FBMC/COQAM.
2 SoTA MCM schemes analysis
In this section, we first give a brief description of the SoTA schemes and their mathematical expressions. Then we provide an analysis on the pros and cons from different aspects, which will motivate our research proposal.
2.1 SoTA MCM description
In what follows, the five MCM schemes, i.e., CP-OFDM, OQAM, FMT, GFDM, and CB-FMT, are briefly recalled, including their general concepts and baseband expressions.
2.1.1 CP-OFDM modulation
with M the number of subcarriers and c_{ m } the complex-valued data symbols, e.g., QAM constellations. The overall CP-OFDM system can be efficiently realized by fast Fourier transform (FFT/IFFTs), and one important advantage is that the CP-OFDM can maintain a full orthogonality, which requires only a simple equalizer at the receiver. However, it employs a rectangular pulse with several disadvantages that will be analyzed later on. It is worth noting that since the rectangular pulse is used in OFDM system, the pulse shaping is implicitly realized by the Fourier transform.
2.1.2 OQAM modulation
where ^{∗} denotes the complex conjugation, δ_{m,p}=1 if m=p and δ_{m,p}=0 if m≠p. The OQAM system only holds a perfect orthogonality in the distortion-free case.
2.1.3 FMT modulation
If the employed pulse has a spectrum that is restrictively limited within the increased subcarrier spacing, no spectra-crossing happens between adjacent subcarriers, which confirms an inter-carrier interference (ICI) free transmission. Thus, the orthogonality condition (5) is changed to a one-dimension only, i.e., any traditional Nyquist pulse can be used. This can be interpreted as an alternative way to relax the orthogonality condition. Moreover, like OQAM, the FMT only holds a perfect reconstruction in the distortion-free case as well.
2.1.4 GFDM modulation
The periodic filter is used to realize the circular convolution at the transmitter, which is equivalent to the tail-biting process [17]. Note that there does not exist any orthogonality condition for the filter design because the GFDM itself is a non-orthogonal system.
2.1.5 CB-FMT modulation
Note that if we set N_{2}=M, then the CB-FMT turns to be GFDM. Thus, the best filter that can guarantee (5) is only rectangular filter.
2.2 Pros and cons analysis
In this section, we provide a detailed analysis on the advantages and drawbacks of the above schemes. This analysis is conducted from four aspects, namely SE, PSD, orthogonality, and waveform flexibility.
2.2.1 SE analysis
Indeed, the measure of SE is inversely proportional to the product (8). If the product value gets greater, it means that there exists a SE loss in either time domain, i.e., taking longer time to transmit one symbol or in frequency domain, i.e., using more frequency bands for the transmission or the combination of both causes. Denoting by SEI, the spectral efficiency indicator, we have 0≤SEI≤1 with SEI =1 being the optimum.
In a special case where K=1, its SEI is identical to that of CP-OFDM.
2.2.2 PSD analysis
2.2.3 Orthogonality analysis
The orthogonality is analyzed in two cases, i.e., end-to-end (E2E) distortion-free case and multi-path (MP) channel case. The former shows whether the MCM can form a perfect reconstruction system. While the latter reveals the robustness against a frequency selective fading, meaning whether a simple one-tap equalizer can completely remove the multi-path interference, as such in the CP-OFDM case. It hints further on the complexity of the receiver design. This feature is particularly important for the case of CP-OFDM-based space-time encoded MISO transmission, since the space-time decoding can therefore be simply realized by a one-tap equalization, resulting in a low-complexity receiver. Nevertheless, there still remain open issues when combining space-time encoded MISO with some MCM schemes other than CP-OFDM. A well-known example is the possibility of enabling a low-complexity Alamouti transmission with advanced MCM schemes [16]. Thus, in general, more weights are given to the orthogonality than to the SE for the MCM evaluation. In the following, we assume that the inserted CP is always sufficiently longer than the MP channel length for CP-OFDM, GFDM, and CB-FMT. Note that the Doppler effect is not considered in our orthogonality analysis, as it ruins the orthogonality for all the MCM schemes in general. The robustness against Doppler effect will be investigated in future publications.
The CP-OFDM system is a well-known orthogonal system. It holds the perfect reconstruction in the E2E case. In a MP case, with the aid of CP, neither the ICI nor the inter-symbol interference (ISI) is resulted. Hence, the orthogonality is fully addressed in CP-OFDM systems. Thanks to this advantage, the CP-OFDM is recognized as a low-complexity system because the channel compensation can be easily accomplished with a simple one-tap equalization^{d}.
Beyond this threshold, the spectra of the adjacent subcarriers start to reach across with each other, preventing from the ICI-free assumption. Indeed, in line with [12], the wider the pulse spectrum is, the severer the ICI will be resulted. That is why the EVM curves for FMT and CB-FMT turn around at the threshold RO point and continuously go up as the RO factor increases. The GFDM scheme provides the worst orthogonality performance in the E2E case due to its non-orthogonal nature.
2.2.4 Waveform flexibility analysis
3 COQAM: motivation and concept
As our analysis showed, the CP-OFDM cannot address any flexibility and its PSD has a high out-of-band radiation. The drawbacks of OQAM and FMT are lack of the orthogonality under multi-path channel, which requires more complicated receiver design and limits the feasibility with MIMO transmission. Even more severe orthogonality issue is inherited in the GFDM scheme, which further increases the system complexity and prevents waveform flexibility. The CB-FMT can completely solve the orthogonality issue under some constraints on the pulse design and it manages to improve the PSD shape compared with CP-OFDM. However, a compromise is laid among waveform flexibility, orthogonality, and SE loss. In addition, the CB-FMT inherits a two-dimensional SE loss.
with K^{′} the number of real symbol slots per each block. Note that the real symbols are obtained from taking the real and imaginary parts of QAM constellations. Thus, its relation to the symbol slot K introduced in the GFDM system is that K^{′}=2K. This also implies that the COQAM and GFDM have a same block length, i.e., K^{′}N_{1}=K M. The rest of the parameters are in line with those presented in the OQAM scheme. To implement a circular convolution with a prototype filter g of length L=K M=D+1, we introduce a pulse shaping filter denoted $\stackrel{~}{g}$, obtained by the periodic repetition of duration KM of the prototype filter g, i.e., $\stackrel{~}{g}\left[\phantom{\rule{0.3em}{0ex}}k\right]=g\left[\text{mod}\right(k,\mathit{\text{MK}}\left)\right]$.
4 Practical implementation schemes
A direct implementation of (11) cannot be envisioned in practice. In this section two efficient methods are proposed for this implementation. The first one is based on a conventional inverse fast Fourier transform (IFFT), while the second one takes advantage of a pruned IFFT scheme.
4.1 IFFT-based algorithm
with g^{k,n}=g[mod(k−n N_{1},M K)], for k∈ [0,M K−1] and n∈ [0,K^{′}−1].
4.2 Pruned IFFT-based algorithm
In (16), the former part reflects the number of multiplications needed for OFDM modulation, processing the same amount of the data symbols. Thus, M K^{2} represents the additional arithmetic complexity. In fact, the exact complexity calculation depends on the implementation algorithm. For example, the algorithm presented in this section is one of the possible ways to implement COQAM system. One may also consider using a frequency domain implementation concept which is introduced for GFDM and CB-FMT. Thus, the resulting complexity differs as a function of implementation algorithms. Moreover, the algorithm preference must be analyzed in the context of concrete system parameter setting, e.g., frame structure and concrete M and K values.
5 COQAM transceiver design in a radio system
where L_{GI} is the CP part used to fight against the multi-path channel interference and L_{RI} is the portion devoted to windowing.
Note that a lot of computation savings can be envisioned here, because a large number of the coefficients of G[ q] are trivial values. Finally, a K^{′}-point IFFT is performed for each subcarrier and the post-phase processing is given by
6 Performance evaluation
Our research objective is to find an enhanced MCM scheme to maximally address the considered four aspects. Therefore, we repeat the same analysis that has been reported in Section 2.2. This time, we take the WCP-COQAM into the comparison. In addition, the Bit Error Rate (BER) vs. Signal-to-Noise Ratio (SNR) comparison is also given for further illustrating the efficiency of the proposal in the presence of the background noise.
6.1 Spectral efficiency evaluation
6.2 Power spectral density evaluation
6.3 Why is circular convolution necessary?
WCP-COQAM transforms the continuous-processing to the block-processing such that with the aid of CP, the one-tap frequency domain equalization can become effective. Nevertheless, one may wonder if the circular filtering is really necessary. For the classical OQAM scheme, it can also introduce a CP then applies the frequency domain equalization at the receiver. Such scheme is named CP-OQAM and it was proposed in [37]. In this section, we explain the necessity of using circular filtering instead of linear filtering and the difference between WCP-COQAM and CP-OQAM.
6.4 Orthogonality evaluation
Power loss due to CP and over-sampling
Power loss (dB) | |
---|---|
CP-OFDM | 0.97 |
OQAM | 0 |
FMT | 0.97 |
GFDM | 0.03 |
CB-FMT | 1 |
WCP-COQAM | 0.03 |
7 Discussions
7.1 WCP-COQAM receiver complexity
Similar to the transmitter side, the receiver complexity is also of paramount importance to be evaluated. Here, we provide a simple complexity analysis in terms of the number of arithmetical computations, i.e., complex multiplications (CM). Then we provide a complexity and performance comparison of WCP-COQAM vs. the classical OQAM receiver employing non-trivial equalizers. This comparison reveals the interest of using WCP-COQAM scheme.
For WCP-COQAM receiver as shown in Figure 9, the arithmetical computation for processing one block of the received data (L=M K samples) can be divided into the computation of one FFT of size L, FDE for one block, polyphase filtering (product of FDE outputs and G^{q,m}), and IFFTs of size K^{′} for M subcarriers, noting that the cyclic shift operation does not consume any arithmetical computations. Here we do not count the post-phase processing as it does not represent additional complexity with regard to the classical OQAM receiver. In more detail, the L-point FFT needs (L/2) log2L CMs. The FDE, which represents one-tap equalizer, yields L CMs. The product between FDE outputs and G^{q,m} is a special case. As G^{q,m} represents the shifted version of the frequency coefficients of the prototype filter, it turns out that only a small portion of the frequency coefficients are non-zero-valued. If one uses the SRRC filter, the number of the non-zero coefficients can be analytically calculated. Assuming a SRRC prototype filter with RO factor r, its bandwidth thus yields $\frac{1+r}{{\mathit{\text{MT}}}_{\text{s}}}$ (Hz), where T_{s} stands for the sample duration. From (19) it shows that frequency resolution for calculating the frequency coefficients is 1/M K T_{s} (Hz). Therefore, the number of non-zero coefficients can be obtained with the division between the bandwidth and the frequency resolution, i.e., ⌈(1+r)K⌉ with ⌈·⌉ being the ceiling operation. Thus, the polyphase filtering consumes M⌈(1+r)K⌉ CMs. At last, each of the K^{′}-point IFFT results in (K^{′}/2) log2K^{′}=K log22K CMs which lead to a total of M K log22K CMs.
Receiver complexity analysis
Schemes | Number of complex multiplications |
---|---|
WCP-COQAM | $\mathit{\text{MK}}\left(\frac{\underset{2}{log}\mathit{\text{MK}}}{2}+\underset{2}{log}2K+1\right)+M\lceil (1+r)K\rceil $ |
OQAM-EIC | M K log2M+M K^{2}+(8κ^{2}+8κ+2)M K |
OQAM-MTE | M K log2M+M K^{2}+2η M K |
As explained in Section 2.2.2, the SRRC filter was chosen in our previous performance evaluation to ensure a fair comparison among different MCM schemes. However, the drawback of SRRC filter comes from the necessity of using a long filter length which results in a high receiver complexity. In practice, for the WCP-COQAM and classical OQAM schemes, the orthogonality condition (3) can be perfectly held with a moderate filter length, e.g., overlapping factor ≤4. In what follows, we show the performance comparison among WCP-COQAM, OQAM-EIC, and OQAM-MTE.
Power loss due to CP
WCP-COQAM | OQAM-ZF | OQAM-EIC | OQAM-MTE | |
---|---|---|---|---|
Power loss | 0.26 dB | 0 dB | 0 dB | 0 dB |
Complexity comparison in a specific context
Schemes | Number of complex multiplications |
---|---|
WCP-COQAM | 2,560 |
OQAM-EIC | 7,168 |
OQAM-MTE | 4,096 |
7.2 How can WCP-COQAM combine with Alamouti?
As discussed in the previous section, the WCP-COQAM has similar robustness of the multi-path interference immunity as for the CP-OQAM scheme. Therefore, the solutions proposed for CP-OQAM scheme can be easily tailored to WCP-COQAM system. Among them, one solution called pseudo-Alamouti encoding was proposed to CP-OQAM [38] which allows for a simple low-complexity decoding algorithm on the receiver side. To extend this scheme for WCP-COQAM, the encoding process can remain the same as [38], while the receiver structure should be modified according to Figure 9. The details of the WCP-COQAM-based Alamouti transmission will be given in our future publication. Moreover, in the literature, there are proposals on the Alamouti-FBMC/OQAM, e.g., [39, 40]. In [39], a block-wise Alamouti encoding is proposed, where the notion of block is defined by inserting zero-symbols, leading to a compromise between the spectral efficiency and receiver memory buffer. In [40], an FFT-FBMC scheme, together with a special data transmission strategy has been proposed. This scheme can be seen as a subcarrier-wise precoded FBMC system. The ISI at each subcarrier can be easily removed by a simple equalization, thanks to the subcarrier-wise IFFT precoding and CP insertion, while the ICI level can be controlled by a special data transmission strategy and good frequency-localized prototype filter. It is claimed that this scheme can well enable the Alamouti transmission with a reasonable added complexity. The performance evaluation as well as the complexity comparison among these schemes for Alamouti transmission will be envisioned in our future work.
7.3 WCP-COQAM receiver flexibility
7.4 Differentiation of WCP-COQAM and Abdoli et al
In the end, we would like to spend some space to discuss the commonality and the difference between our contributions and that of in [41]. First off, [41] has a different motivation that aims at the reduction of the overhead of the OQAM system in a burst transmission, which ends up with a similar concept that uses a circular convolution filtering instead of the linear convolution, and a time-windowing is also proposed to improve the PSD. Nevertheless, the differences between the proposal of [41] and ours are that our proposal is based on a regular circular convolution, while in [41] it is claimed that the regular circular convolution is avoided and instead, a weighted circular convolution version is used. Second, the system realization is totally different and the windowing process is also different. Moreover, the orthogonality enhancement in a multi-path channel context is not in the investigation scope of [41]. Finally, comparing the classical OQAM scheme, with the same SRRC prototype filter the proposal in ([41], Tab. I), leads to an orthogonality loss, while this is not the case in our proposal.
8 Conclusions
In this paper, we analyzed five SoTA MCM schemes from different aspects which are deemed as the important factors for future radio systems. The analyzed results showed that none of the SoTA schemes can simultaneously address these aspects. Then, we proposed a novel MCM scheme, which is a combination version of the classical OQAM with the circular convolution concept. Its concept and efficient implementation as well as its transceiver algorithm were reported. Finally, the efficiency of the proposed scheme was analyzed in terms of the evaluation criteria, and it turned out that our proposed scheme can indeed address these aspects to a maximum degree. Nevertheless, the selected evaluative criteria do not cover all of the research aspects that naturally triggers further investigations. To continue the work, our future research will cover the performance and complexity analysis for Alamouti-encoded transmission, in which different proposals in the literature will be compared together. Moreover, the sensitivity of WCP-COQAM to the synchronization error will also be addressed. Furthermore, effort will also be made to address the remaining issues such as how to adapt the pilot structure to the WCP-COQAM system and how to take advantage of the WCP-COQAM receiver flexibility.
Endnotes
^{a} During the paper preparation, we spotted that a similar idea was independently investigated in [41] A discussion will be given at the end of this paper to differentiate the different contributions between our proposal and [41].
^{b} In this paper, the PSD is simulated with Welch’s method, where the segment length is set to M for CP-OFDM, OQAM, and GFDM and N_{2} for FMT and CB-FMT. The initial shift between segments is taken equal to half of the segment length.
^{c} In the literature, the only reported filter for the GFDM and CB-FMT systems is the SRRC. Therefore, in this paper, we stick to the SRRC to ensure the filter alignment, even though it is not the most favorable choice for the OQAM and FMT schemes as it cannot exactly hold an orthogonal condition with a finite length.
^{d} A condition for this statement is that the CP-OFDM symbol duration is smaller than the channel coherence time.
^{e} On-line arithmetical computation means the calculation needed for each received sample. Its counterpart is the periodic mathematical computation, meaning that the calculation is only needed punctually.
Notes
Acknowledgements
This work has been performed in the framework of the FP7 project ICT-317669 METIS. The authors would like to acknowledge the contributions of their colleagues.
Supplementary material
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