Analysis of an FBMC/OQAM scheme for asynchronous access in wireless communications
 Davide Mattera^{1},
 Mario Tanda^{1}Email author and
 Maurice Bellanger^{2}
https://doi.org/10.1186/s1363401501914
© Mattera et al.; licensee Springer. 2015
Received: 31 July 2014
Accepted: 31 December 2014
Published: 10 March 2015
Abstract
The OFDM/OQAM transceiver belongs to the filterbankbased multicarrier (FBMC) family and, unlike OFDM schemes, it is particularly able to meet the requirements of the physical layer of cognitive radio networks such as high level of adjacent channel leakage ratio and asynchronous communications. The paper proposes and analyzes a new implementation structure, named frequency spreading, for the OFDM/OQAM transceiver. On flat channels, it is equivalent to the standard one in terms of inputoutput relations, though more complex. On multipath channels, it offers a crucial advantage in terms of equalization, which is performed in the frequency domain, leading to high performance and no additional delay. With its flexibility and level of performance, the analyzed scheme has the potential to outperform OFDM in the asynchronous access context and in cognitive radio networks.
Keywords
1 Introduction
The cognitive radio transmission context exhibits a number of specific features which make it significantly different from the conventional transmission environment [1,2]. First, the available bandwidth is likely to be fragmented, i.e., it is made of nonadjacent spectrum chunks that have to be exploited jointly for high speed data communications. Then, the sections of the spectrum that are not available might be occupied by a primary user and a high level of protection must be provided. Specifically, the transmission system must guarantee a high level of adjacent channel leakage ratio (ACLR)^{a}. Next, the transmission band is likely to be changing on short notice or even without notice. On the exploitation side, the total bandwidth available might be dedicated to a single user requiring high bitrates or it can be dynamically shared by several users in proportion to their instantaneous capacity needs. If opportunistic operation is contemplated, these users have the freedom to show up and disappear as they wish. In such conditions, a rigid communication procedure, where each user must be aligned before the transmission can start, is inadequate. In fact, asynchronous operation is necessary to reach an acceptable level of spectral efficiency. Clearly, to cope with such a context, an appropriate physical layer is required.
The spectrum granularity offered by multicarrier transmission techniques has proven its efficiency for spectrum exploitation, and the most popular technique, orthogonal frequency division multiplexing (OFDM), has been widely used in communications for more than a decade now. However, for the cognitive radio context as described above, it lacks flexibility and it is likely to lead to poor spectral efficiency, even with the introduction of additional processing [3]. Therefore, an enhanced multicarrier technique is needed, as pointed out in [4], where it is shown that a filterbankbased multicarrier (FBMC) physical layer can meet the ACLR requirements [2,4]. In particular, FBMC/OQAM can overcome the limits of OFDM provided that we impose a constraint on the cutoff frequency of the prototype filter, which cannot exceed the subchannel spacing as pointed out in [5]. High stopband attenuation filters have been proposed that do not satisfy this constraint, such as isotropic orthogonal transform algorithm (IOTA) [6] and Hermite filters. These filters are associated with singletap equalizers as mentioned in [7]: such a scenario does not allow the exploitation of the potential advantages of FBMC/OQAM systems for cognitive and, therefore, they cannot compete with CPOFDM. On the contrary, the high performance equalization objective can be met if the prototype filter (employed in the FBMC/OQAM scheme) is designed using the frequency sampling technique introduced in [8] and developed in [5]. With this simple approach, the coefficients are derived from a few samples of the filter frequency response, which makes the implementation of the filter bank in the frequency domain practical.
When comparison between such FBMC/OQAM system and OFDM system is considered, it appears that a key advantage of OFDM with cyclic prefix (CP) is the capability to achieve perfect channel equalization, as long as the channel impulse response remains shorter than the guard time provided by the CP. Thus, in order to be accepted, the FBMC/OQAM approach must have a high performance equalization capability, particularly in the asynchronous context, characterized by the fact that the system must compensate simultaneously the timing offset, the frequency offset, and the channel distortion.
In the absence of CP, the equalization capability of the FBMC system rests on the subchannel equalizers, which cannot be singletap as in OFDM, but must be multitap to reach similar performance. However, the use of multitap equalizers implies an increase of the receiver latency; this motivates the search for an equalizer structure that does not introduce such a disadvantage. The main contribution of the paper lies in the proposal of a new transceiver structure for FBMC/OQAM systems that is able to provide satisfactory performance without increasing the transceiver delay and accepting the presence of significant timing and frequency offsets among the users that are performing the multiple access, as it is common in a cognitive radio scenario.
The concept has been presented in [9,10], along with preliminary performance results, under the name frequency spreading (FS)FBMC, but a rigorous analysis of the corresponding scheme is still missing. An objective of the present paper is to provide such an analysis and prove the equivalence of FSFBMC with the conventional polyphase network (PPN)FFT scheme in both transmitter and receiver. This equivalence is important because it opens the way to mixed implementations. For example, in uplink transmission, the distant user can be equipped with the conventional IFFTPPN transmitter, while the high performance but more complex FSFBMC receiver is implemented at the base station.
Many advances in the applications of FBMC to various scenarios will be able to take advantages from the proposed transceiver structure. In particular, the capability to use multiple antennas at the transmitter and/or at the receiver, which significantly increases bandwidth efficiency, can be easily carried out along the lines introduced in [1113], which however do not take into account the frequencydespreading structure at each receiver; further works is needed to define the details of the MIMO extension of the proposed structure. Alternative structures are also under consideration for achieving the same goal of operating on multipath channel with a minimum implementation complexity. For example, the fastconvolution structure [14], which is currently under study for its extension on the multipath channel, is superior to the proposed structure in terms of computational complexity. It is equivalent in terms of flexibility (e.g., it shows a similar capability to easily compensate a timeoffset in the frequencydomain as suggested in [15]), while the fastconvolution transceiver latency is larger than that achieved by the frequencydespreading system [14].
The organization of the paper is as follows. In Section 2, the FSFBMC scheme for the transmitter is described and the proof of the equivalence with the standard FBMC, namely the IFFTPPN cascade, is provided. Section 3 is dedicated to the receiver structures and, again, the proof of the equivalence between FSFBMC and standard FBMC, namely the cascade of PPN and FFT, is provided; moreover, in Section 3, it is shown that the FSFBMC structure is computationally more complex while in Section 4, it is shown that on a multipath channel, it offers a crucial advantage in terms of equalization, which is performed in the frequency domain, just like OFDM, leading to high performance and no additional delay. In Section 5, the performance of the proposed scheme is illustrated and contrasted with the results obtained for OFDM and standard FBMC when the subchannel equalizer has a singletap. In Section 6, the main aspects of FSFBMC are summarized and the potential impact is discussed.
Notation: \(j \stackrel {{\triangle }}{=} \sqrt {1}\), superscripts (·)^{∗}, (·)^{ T }, and (·)^{ H } denote the complex conjugation, the transpose, and the conjugate transpose, respectively, ℜ[ ·] is the real part, log is the base2 logarithm, ⊗ is the linear convolution, δ[ k] is the Kronecker delta, ceil[x] is the smallest integer larger than or equal to x, and 〈·〉 denotes the time average, i.e., \(\left \langle x [\!n] \right \rangle \stackrel {{\triangle }}{=} {\lim }_{N \rightarrow + \infty } \frac {1}{2N+1} \sum _{n=N}^{N} x[\!n]\) and mod_{ M }(ℓ)=△ℓ−q M with q such that mod_{ M }(ℓ)∈{0,1,…,M−1}. Moreover, we denote with DFT[ x] the vector \(\widetilde {\mathbf {x}}\) whose kth component can be written as \(\widetilde {x}_{k} = \frac {1}{N} \sum _{i=0}^{N1} x_{i} e^{j \frac {2 \pi }{N} k i }\) and with IDFT[x] the vector \(\widehat {\mathbf x}\) whose kth component can be written as \({\hat {x}}_{k} = \sum _{i=0}^{N1} x_{i} e^{j \frac {2 \pi }{N} k i }\) where x _{ i } is the ith component of the N×1 input vector x. Finally, lowercase boldface letters denote column vectors, × the componentwise product between two vectors and, finally, 0 denotes the null vector.
2 The transmitter with standard and frequencyspreading structures
where T is the multicarrier symbol interval, \(\mathcal A \subset \{0, 1, \dots, M1 \}\) is the set of active subcarriers whose size is M _{ u }, the sequences \(a_{n,k}^{R}\) and \(a_{n,k}^{I}\) indicate the real and imaginary parts of the complex data symbols transmitted on the kth subcarrier during the nth QAM symbol, N _{ b } is the number of training symbols, N _{ s } is the number of payload symbols, while g(t) is the prototype filter. It is assumed that the data symbols \(a_{n,k}^{R}\) and \(a_{n,k}^{I}\) are statistically independent with zeromean and variance \({\sigma _{a}^{2}}\).
The generation of the sequence s ^{ R }[ i] is equivalent to the generation of the sequence of M×1 vectors \(\mathbf {d}_{n}^{(R)}\) whose kth component \(d_{n,k}^{(R)}\) is equal to s ^{ R }[ n M+k] for k∈{0,1,…,M−1}. In the following, we consider two implementation structures and their implementation complexities: though the standard implementation structure based on an IFFT over M points exhibits a reduced computational complexity, the frequencyspreading structure based on an IFFT over a larger number of points provides useful insights into the structure of the transmitted signal.
2.1 Standard transmitter structure
2.2 Frequencyspreading structure
Note that the 2K−1 nonnull values \(\{G_{k} \}_{k=(K1)}^{K1}\) are the free parameters of the prototype filter when it is designed according to the frequencysampling procedure used in [5].
where p∈{0,1,…,M} and k∈{0,1,…,K−1}.
 1.
to use the input symbols \(a_{n,k}^{R}\) to calculate (for ℓ∈{0,1,…,K M−1}) the components z _{ n,ℓ } of z _{ n } according to Equation 35. Note that the symbol \(a_{n,p}^{R}\) is spread over 2K−1 components of the vector z _{ n } and for this reason, the structure is named FSFBMC; in fact, each component of z _{ n } is dependent on two adjacent symbols and each symbol a _{ n,p }, according to Equation 35, not only determines the component pK of the frequencydomain vector z _{ n } but also spreads its effect, weighted by the frequency response of the prototype filter, on the different components of the same vector ranging from p K−(K−1) up to p K+K−1;
 2.
to determine h _{ n } starting from z _{ n } by performing the IDFT over KM points in the righthand side of Equation 31;
 3.
to evaluate \(\mathbf {d}_{n}^{(R)}\) by the overlapandadd processing defined in Equations 24 and 25.
2.3 Complexity comparison of the two structures
The standard transmitter structure requires to calculate (a) the IFFT over M samples according to the definition in Equation 9 then to calculate the vector \(\mathbf {d}_{n}^{(R)}\) according to the PPN (Equation 15). The frequencyspreading structure requirements have been just summarized.
In a structure with a single processor, the complexity comparison is equivalent to the count of the number of flops required by the two structures. The number of complex flops for calculating the IFFT over M samples can be written as 1.5M log(M) while the number of real flops can be written^{b} as 4M log(M)−6M+8 as in the splitradix.
The number of complex multiplications for calculating IFFT over M samples can be written as 0.5M×(logM−1) while the number of real multiplications can be written as M log(M)−3M+4 [19] by removing most of the trivial operations and using three real multiplications per complex multiplication.
To calculate the vector \(\mathbf {d}_{n}^{(R)}\) according to the PPN (Equation 15), the following number of realvalued flops are necessary: 2M realvalued multiplications for each of the K terms and M complexvalued additions for each of the K−1 couples of vectors to be summed.
With the frequencyspreading structure, we need 2K M−M realvalued multiplications for calculating z _{ n } according to Equation 35 while the IFFT for calculating h _{ n } by using Equation 31 requires 4K M log(K M)−6K M+8 realvalued flops. Finally, (K−1)M complexvalued additions are needed by the overlapandsum structure in Equation 25.
The first approximation is obvious while the second approximation holds provided that M is sufficiently large and K sufficiently small (e.g., for K=4 and M=1024, we have a normalized approximation error of 6.1%). Moreover, \(C_{m}^{(ST)}/C_{m}^{(FT)}\) is around 58% for K=2, 35% for K=4, and 24% for K=8, mainly independently of M∈{512,1024,2048,4096}. Therefore, the frequencyspreading structure is about K times more complex when implemented using a structure with a single processor.
3 The receiver for standard and frequencyspreading structures
In fact, according to Equation 39, o ^{(R)}[ i] denotes the additive signals present in the received signal r[ i] that do not depend on the useful symbol \(a_{n,k}^{R}\); the condition in Equation 44 therefore implies that such additive signals do not interfere with the useful signal when the matchedfilter projection (designed according to the useful term) is performed (i.e., the result of the matchedfilter projection is independent of the interfering signals); on the other hand, the matchedfilter projection is optimum (in the maximumlikelihood sense) when the interference signals are not present and only the noisy version of the useful term is taken into account. Consequently, the decision variable in Equation 41 operating on the flat channel without synchronization error is optimum (in the maximumlikelihood sense) for estimating statistically independent information symbols; analogously, the same optimality holds for the decision variable in Equation 42. Obviously, the receiver implemented according to Equation 41 has to be modified in order to operate on a multipath channel. However, before discussing such modifications, we first need to describe the two structures implementing Equation 41. Consequently, we recall the standard structure for implementing Equation 41 in Subsection 3.1 and we introduce an alternative structure in Subsection 3.2; moreover, we compare their complexities in Subsection 3.3.
3.1 The standard receiver structure
3.2 The frequencydespreading receiver structure
where we have taken into account the properties of the prototype filter (see Equations 14, 28, and 29) and the fact that it is real and, consequently, \(G_{k} = G_{k}^{*}\).
 1.
 2.
Calculate the vector R _{ n } in Equation 56 by FFT over KM points (and half period later the vector \(\mathbf {r}_{n}^{(I)}\) in Equation 63);
 3.
Perform M different projections of the vector R _{ n } according to Equations 60 and 58 in order to obtain each decision variable \(d_{n,k}^{(R)}\) for k∈{0,1,…,M−1} (and half period later according to Equations 61 and 62 to obtain \(D_{n,k}^{(I)}\)). Thus, to obtain the datum \(a_{n,k}^{(R)}\) that in the transmitter has been spread over 2K−1 components of the vector z _{ n } (see the first point of the sentence after Equation 35), the same components of the vector R _{ n } are exploited by using as weights the (conjugate) Fourier coefficients \(G_{k'}^{*}\) with k ^{′}∈{−(K−1),…,K−1}. For this reason, such a structure is called frequencydespreading receiver: in fact, it collects all the components of the frequencydomain vector R _{ n } dependent on the useful symbol \(a_{n,k}^{R}\), due to the spreading performed at the transmitter, and weights them according to the frequency response of the prototype filter, achieving the despreading of the useful symbol.
The importance of the proposed structure is not limited by the assumption in Equation 28 because, when it is necessary to introduce a possible mismatch (i.e., to use at the transmitter a prototype filter that does not satisfy Equation 28), it can be managed with very marginal performance loss (i.e., the frequency despreading receiver can be still employed at the receiver, with its advantages considered in the paper and without appreciable disadvantages due to the presence of a mismatch).
3.3 Complexity comparison of the two structures
where the first and the second approximations are obvious while the third one holds provided that M is sufficiently large and K is sufficiently small (e.g, for M=1024 and K=4, 4 log(M)+4 log(K)+2=50 while 4 log(M)+4K−8=48). Therefore, under the same assumptions used at the transmitter side, we can obtain the following approximation: \(C^{FR}_{f} = K C^{SR}_{f}\); moreover, \(C^{SR}_{m}/C^{FR}_{m}\) is about 50% for K=2, 30% for K=4, and 21% for K=8, independently of M∈{512,1024,2048,4096}. Thus, also for the receiver case, the frequencyspreading structure has a computational complexity about K times larger.
Structure complexity
Structure  Number of flops  Number of multiplications 

Standard transmitter  8+M[4 log(M)+4K−8]  M[log(M)+2K−3]+4 
Standard receiver  8+M[4 log(M)+4K−8]  M[log(M)+2K−3]+4 
FS transmitter  8+M{4K[log(K M)−0.5]−3}  M{K[log(K)+log(M)−1]−1}+4 
FS receiver  8+M[4K log(K M)+2K−6]  4+M[K log(K M)+K−2] 
4 Adapting the frequencydespreading structure to the multipath channel
In the present section, we first define the adaptation of the frequencydespreading structure to the multipath channel, then we recall a standard approach to adapt the standard structure to the multipath channel and we finally compare their performance.
4.1 Frequencydespreading structure operating on multipath channel
where the complexvalued sequence h[ i] of length L _{ h }+1 models the multipath channel.
where the complexvalued coefficients \(F_{k'}^{(k)}\) in Equation 69 replace the coefficients \(G_{k^{\prime }}\) in Equation 54; therefore, the coefficients \(F_{k'}^{(k)}\) can be set to \(G_{k^{\prime }}\) obtaining the structure for a flat channel; on a multipath channel, we can set them in order to equalize the channel improving the receiver performance. The value of α to be used in Equation 43 is given by \(\alpha \stackrel {{\triangle }}{=} \sqrt {P_{r}/P_{s}}\) where \(P_{r} \stackrel {{\triangle }}{=} \left \langle E \left [\left  \rule {0mm}{3mm} h [\!i] \otimes s[\!i] \right ^{2} \right ] \right \rangle \) represents the average power of the useful component of the received signal r[ i] and \(P_{s} \stackrel {{\triangle }}{=} \left \langle E\left [s[i] ^{2}\right ] \right \rangle \) represents the average power of the transmitted signal s[ i].
where \(I_{0,0}^{(\mathrm {R},k)}\) represents the coefficient of the useful term \(a_{n,k}^{R}\), η _{ n,k } describes the effect of the background noise, and I _{ n,k } describes the intersymbol and intercarrier interferences of the symbols \(a_{n m,kq}^{R}\) ((m,q)≠(0,0)) and \(a_{nm,kq}^{I}\) on the useful symbol \(a_{n,k}^{R}\). Such interferences would be negligible on a flat channel but they become significant on the multipath channel (see Equation 67).
influence the useful coefficient and the interference power; we can, therefore, set the vector f _{ k } in order to equalize the effects of the multipath channel in Equation 67.
The noise term η _{ n,k } in Equation 70 is a zeromean complexvalued Gaussian random variable with variance \(\frac {2 \\mathbf {f}_{k}\^{2} {\sigma _{a}^{2}}}{\gamma \\mathbf {G}^{(K)}\^{2} }\) where γ is defined as the signaltonoise ratio per subcarrier, i.e., \(\gamma \stackrel {{\triangle }}{=} \frac {E_{s}}{N_{0}}\) where E _{ s } is defined as the energy of the useful term of the received signal in a multicarrier symbol period that is dedicated to each active subcarrier.
The satisfaction of the condition in Equation 74 is equivalent to the condition in Equation 44 and it concerns the design of the prototype filter. Moreover, it guarantees the optimality (in the maximumlikelihood sense) of the receiver structure on the flat channel.
When \(v_{n,q,k'}^{(\text {FLAT},k)}=0\), then the denominator of Equation 76 is null and Equation 75 is still valid provided that we replace with unit both the denominator of Equation 76 and the same quantity in Equation 73.
guarantees to the structure the same performance achieved on flat channel provided that L _{ h }≪M (see Figure 1 for a scheme of the FS structure in Equation 69 when Equation 82 is chosen).
4.2 Recalling the standard approach
The standard approach consists in including an equalizer stage in cascade with the structure described in Subsection 3.1. The effects of the multipath channel can be equalized by using a singletap structure [7]. In this case, the kth entry of the DFT output (see Equations 50 or 51) is multiplied by 1/H(F _{ k }) with \(F_{k}= \frac {k}{M}\) in the standard structure for k∈{0,1,…,M−1}.
More sophisticated multitap structures could be used and have also been proposed with reference to the standard structure [22]. Since they operate in the frequency domain and at twice the multicarrier symbol rate, they introduce an additional delay proportional to the number of taps. We consider the singletap equalizer in both structures since it maintains limited the overall latency of the transceiver. It may appear that the FS equalizer be equivalent to a multitap subchannel equalizer following the standard structure and therefore that the considered comparison be unfair. However, they are not equivalent for two reasons: (a) because the delay introduced by the two structures is different and obtaining the minimum delay is important in a transceiver, like the OFDM/OQAM one, with an already larger delay in comparison with the OFDM system; (b) the PPNFFT scheme performs equalization after sampling rate reduction which introduces an interpolation operation. The distinction of the two structures in terms of sampling rate reduction lies in the fact that the FS structure performs equalization in its internal behavior and therefore before sampling rate reduction while the PPN structure performs equalization after sampling rate reduction and consequently needs to use the singletap equalizer to not increase the transceiver delay.
4.3 Comparing the signaltointerferenceandnoise ratios of the two structures
In other terms, the choice in Equation 83 makes the FS receiver equivalent to the PPN structure equipped with the singletap equalizer of coefficient 1/H _{ kK }. Therefore, by comparing Equations 82 and 83, we can obviously note that the advantage of the frequencydespreading equalizer lies in its capability of using the coefficient \(H_{kK+k^{\prime }}\) instead of the constant term H _{ kK }. Since the FS structure in Equation 69 first extracts the DFT of the input vector \(\mathbf {r}^{(K)}_{n}\) according to Equation 56 and subsequently uses the coefficient \(F_{k'}^{(k)}\) to equalize the channel effect at frequency (k K+k ^{′})/(K M)=k/M+k ^{′}/(K M), the FS structure with the choice in Equation 82 uses the right coefficient (i.e., \(H_{kK+k^{\prime }}\)) to equalize the channel response at frequency (k K+k ^{′})/(K M) while the FS structure with the choice in Equation 83, which is equivalent to the PPN structure equipped with the singletap equalizer, always uses the same coefficient H _{ kK } to equalize the channel responses at the different frequencies (k K+k ^{′})/(K M) for k ^{′}∈{−(K−1),…,−1,0,1,…,K−1}. In other terms, differently from the PPN structure equipped with the singletap equalizer, the FS structure is able to equalize with different coefficients the different parts of the subcarrier band. Since the effect of an offset n _{ τ } in timing synchronization, perfectly compensated however in the frequency domain, can be obtained by setting h[ ℓ]=δ[ ℓ−n _{ τ }], our analysis shows that a performance improvement of the FS equalizer, which for n _{ τ }=0 (flat channel) is equivalent to the singletap equalizer, appears when larger values of n _{ τ } determine faster variations of the channel frequency response so that nonnegligible variations appear within the subcarrier band; in such a case, the FS structure is able to use different coefficients to equalize each part of the subcarrier band and can therefore achieve improved performance in comparison with the PPN structure equipped with singletap equalizer. Such a superior capability of the FS structure is irrelevant in the presence of a flat channel; therefore, the two structures show the same performance on the flat channel or on channels where the variations on the subcarrier band (of length 1/M) are marginal.
where the SINR of the frequencydespreading receiver is determined by the use of the vector f _{ k } described in Equation 82 whereas the singletap equalizer is determined by the use of the vector f _{ k } in Equation 83. Note that the signaltointerference ratio SIR_{ k } can be obtained by employing Equation 84 without the first term at the denominator.
5 Performance comparison of the two structures
In the present section, we assess via computer simulations the equalization performance achieved by using the frequencyspreading structure and compare it with that achieved by the standard structure. We have also included in the performance comparison the classical OFDM system that is often considered for opportunistic transmissions because it is a classical scheme employing the multicarrier approach where many practical difficulties have already been resolved; this has a strong impact on the overall cost. However, for the cognitive radio context, it lacks flexibility and it is likely to lead to poor spectral efficiency. Since the latency^{d} of the FBMC receiver is K times larger than that of the OFDM receiver with the same number of subcarriers, we have set the number of subcarriers in the OFDM transceiver K times larger in order to compare two structures with the same latency. Moreover, with such a choice, the OFDM receiver and the FSFBMC receiver perform the FFT procedure on the same size, though FSFBMC has still to perform it to a rate 2K times larger. In particular, we have used 2,048 subcarriers for OFDM while we have used only 512 subcarriers for FBMC transceiver and we have used K∈{2,3,4} in order to verify the effect of the overlap factor.
 1.
The considered FBMC and OFDM systems have a bandwidth \(\frac {1}{T_{\mathrm {s}}}= 11.2 \ \text { MHz}\);
 2.
The transmitted symbols are the real and imaginary parts of 64QAM symbols;
 3.
The considered multipath fading channel model is the ITUR Vehicular B [23];
 4.
The used prototype filter is that proposed in [5]. Actually, any type of prototype filter can be implemented with an extended FFT, due to the equivalence between time and frequency domains. However, in order to be practical, the number of frequency domain filter coefficients must be the smallest possible, which is the case of the used filter;
 5.
The channel is fixed in each run but it is independent from one run to another;
 6.
The residual timing offset (RTO) and the normalized residual carrier frequency offset (RCFO) are controlled as simulation parameters;
 7.
Both systems exploit a onetap subcarrier equalizer with perfect knowledge of the channel and of the residual timing error, i.e., when simulating the presence of the timing offset n _{ τ }, we have used exp(−j2π n _{ τ } k/M)/H(k/M) instead of 1/H(k/M) as coefficient of the singletap equalizer in the standard structure and exp(−j2π n _{ τ } k/K M)/H(k/K M) as coefficient of the singletap equalizer for the frequencydespreading structure;
 8.
The effect of the RCFO on the phase of each decision variable in the frequency domain, which increases [20] linearly with time, is not compensated; therefore, the BER is dependent on the specific multicarrier symbol interval considered for equalization. In order to maintain sufficiently limited the effects of such nonideal receiver behavior, we evaluate the BER on the data transmitted in one of the first multicarrier symbol intervals, the eighth one;
 9.
In order to use the same bandwidth in both FBMC and OFDM, which exhibits a larger spectral leakage, we have set the percentage of active subcarriers in OFDM transceiver as 82% of the overall number of subcarriers while we have set to 89% this percentage in OFDM/OQAM transceiver;
 10.
The length of the cyclic prefix is 1/8 of the OFDM multicarrier symbol period (note that since in the FBMC system the cyclic prefix is not used, in the considered case an increase of the bitrate nearly equal to 11.1% with respect to the OFDM system is obtained).
Note that, in consequence of the choices reported at the points 9 and 10, the data rate of the OFDM system is about 82% of the data rate of the FBMC system.
We have also performed other simulation experiments to verify the performance on the less hostile ITUR Vehicular A channel: here, the condition L _{ h }≪M is better satisfied. In fact, the corresponding results, shown in Figure 7, report that the two structures and the OFDM system are practically equivalent on channel A. Only for larger values of E _{ b }/N _{0} we can observe some difference; in particular, we note that the singletap equalizer for K=2 provides the worst performance; moreover, the three dashedline curves, corresponding to the singletap equalizer for K=3 and K=4 as well as the frequency spreading structure for K=2. Only for K=3 and K=4 the frequency despreading structure behaves practically equivalent to the OFDM system. Therefore, we can conclude that similar performance is achieved by OFDM and FSFBMC transceivers also if the latter uses only 512 subcarriers while the former uses 2,048 subcarriers.
In order to better simulate the working conditions of a cognitive radio scenario, we next consider an uplink scenario where all active subcarriers are blockwise shared among four users whose delays are mutually independent and uniformly distributed within {−M/2,−M/2+1,…,M/2−1}. Only a single subcarrier is left as guard between adjacent sets of active subcarriers whereas a maximum value of the normalized CFO equal to 0.1 is admitted on each user. The value of K is set to 4 as, in the previous experiments, it resulted to be the best choice for K.
The considered uplink scenario is quite general and can be encountered in a number of cognitive network architectures, in particular when the cognitive cellular networks are taken into account [24]. For example, consider the case where the four terminals are located in an area where a large number of cells have been deployed by different operators for the local coverage enhancement of a potentially crowded zone interested in opportunistic multimedia downloads at low cost. Each relay is provided of a proper backhaul connection and cooperates with the other relays (and with the terminals) by means of a control channel, designed in dedicated, common, or underlay fashion [25]. The cooperation is aimed at determining according to an optimization procedure (e.g., that proposed in [26]), the subcarriers available for the transmission of each active terminal, and its transmitted power (to be minimized). Such a cooperation obviously does not include a timing alignment procedure not only because it would make much more complex the access control but mainly because it is impossible, i.e., the distances among relays imply that the transmission of each terminal is received by different relays. Therefore, in absence of a timing alignment procedure, the delay and the CFO of the user of interest are perfectly compensated and, then, all the other users remain asynchronous; moreover, we first consider the case where each signal, coming from the other users, at the receiver arrives with the same power of the useful signal.
It is interesting to note that, differently from what happens on channel ITUR Vehicular A, on channel ITUR Vehicular B the performance of the frequency despreading receiver is not equivalent to its performance on the flat channel. This is due to the fact that the powers of the interference terms are increased since the conditions that guarantee the optimality of the frequency despreading structure are not satisfied. Therefore, the interferences limit the performance for larger values of E _{ b }/N _{0}. When the performance achieved by the frequency despreading structure is not satisfactory, transceiver performance can be improved by using another procedure for the design the coefficients \(F_{k'}^{(k)^{*}}\), more sophisticated than that in Equation 82. An alternative approach would require to introduce a timedomain filtering of each component of the DFT output, which would however also increase the receiver latency (however, the introduced latency would not be worse than that required by a lighter timedomain filtering approach following the PPN structure).
6 Conclusions

smaller guard bands in the frequency domain, which means improved performance of the multiplexing scheme. Such a performance advantage can be quantified in terms of the overall bitrate resulting from the use of OFDM/OQAM or equivalently can be quantified by comparing the complexity of the two transceivers since OFDM may need a heavy digital filtering of the signal to be transmitted in order to maintain similar overall bitrate, as pointed out in [2];

robustness to residual timing and carrier frequency offsets;

capability to fully exploit fragmented spectrum;

more flexibility in multiuser exploitation since asynchronous users can be accommodated.
The main disadvantage of the FSFBMC receiver is related to the higher rate at which the FFT has to be performed when M/K subcarriers are employed in the frequencyspreading structure in comparison with the classical OFDM transceiver employing M carriers. Here, however, a pipelining approach to the FFT implementation may significantly reduce the importance of such a disadvantage.
7 Endnotes
^{a} The adjacent channel leakage ratio (ACLR) is the ratio of the filtered mean power centered on the assigned channel frequency to the filtered mean power on the adjacent channel frequency.
^{b} We use the approximate expression 4M log(M)−6M+8 as a first approximation and to better appreciate the behavior of the complexity; however, the exact calculation of the number of flops for IFFT is given [27] by \(\frac {34}{9} M \log (M)  \frac {124}{27} M 2 \log (M) + \frac {16}{27} (1)^{\log (M)} (1  3/8 \log (M)) +8\).
^{c} Also if a component of the vector \(\mathbf {w}_{m,q}^{(k)}\) is null the condition existing on a flat channel can be still restored provided that we accept that the definition (77), which does not apply to such a case, is obviously meant in the sense that the corresponding component of the vector f _{ k } is null.
^{d} The delay of each symbol transmitted with OFDM/OQAM system is equal to the length of the prototype filter while the delay of each symbol transmitted with OFDM system is equal to the length of the multicarrier symbol period. Thus, the FBMC and the OFDM transceivers have the same delay if the OFDM one has K times more subcarriers than OFDM/OQAM.
8 Appendix A
9 Appendix B
From Equation B.4, Equation 35 directly follows.
In the derivation, we have used the property that \(j^{\text {mod}_{M} (p+1)} = j^{p+1}\) that holds for p∈{0,1,…,M−1} under the condition, always satisfied, that M is multiple of 4. In fact, the property is trivial for p∈{0,1,…,M−2} since mod_{ M }(p+1)=p+1 while, for p=M−1, it is equivalent to the property j ^{ M }=1, which holds only for M integer multiple of 4.
10 Appendix C
The last equality can be equivalently rewritten as in Equation 51.
11 Appendix D
Taking into account Equations 62 and 59, Equation D.2 can be written as in Equation 61.
Declarations
Authors’ Affiliations
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