Analysis of frequency domain frame detection and synchronization in OQAMOFDM systems
 Christoph Thein^{1}Email author,
 Malte Schellmann^{2} and
 Jürgen Peissig^{1}
https://doi.org/10.1186/16876180201483
© Thein et al.; licensee Springer. 2014
Received: 30 October 2013
Accepted: 20 May 2014
Published: 4 June 2014
Abstract
For future communication systems, filter bank multicarrier schemes offer the flexibility to increase spectrum utilization in heterogeneous wireless environments by good separation of signals in the frequency domain. To fully exploit this property for frame detection and synchronization, the advantage of the filter bank should be taken at the receiver side. In this work, the concept of frequency domain processing for frame detection and synchronization is analyzed and a suitable preamble design as well as corresponding estimation algorithms is discussed. The theoretical performance of the detection and estimation schemes is derived and compared with simulationbased assessments. The results show that, even though the frequency domain algorithms are sensitive to carrier frequency offsets, satisfactory frame detection and synchronization can be achieved in the frequency domain. In comparison to time domain synchronization methods, the computational complexity increases; however, enhanced robustness in shared spectrum access scenarios is gained in case the described frequency domain approach is utilized.
Keywords
1 Introduction
The increasing practical interest in filter bank multicarrier systems for nextgeneration wireless communication systems raises the demand for efficient synchronization methods making use of the favorable frequency containment of the filter bank to improve robustness in spectrum sharing scenarios. The focus of this work is on offset quadrature amplitude modulation orthogonal frequency division multiplexing (OQAMOFDM), since this modulation scheme provides optimal bandwidth efficiency with respect to symbol density in the timefrequency grid [1]. Common dataaided synchronization schemes, e.g., [2], use the time domain signal for symbol timing offset (STO) and carrier frequency offset (CFO) estimation. In this case, the analysis filter bank at the receiver is not involved and the advantage of separation of subchannels offered by the filter bank multicarrier systems is disregarded. As a result, the time domain synchronization in spectrum sharing scenarios needs to be enhanced, for example, by introducing a prefiltering stage, and its complexity increases.
In [3], Stitz et al. make use of the analysis filter bank for synchronization and propose a method based on a training sequence embedded into each subchannel that enables a persubchannel evaluation of the received signal. The authors showed that the interference from neighboring subchannels limits the estimation range of the CFO. The task of STO correction is thereby left to a threetap equalizer, which lowers the demand for accurate STO estimation. For that reason, only frame detection is considered. The same authors describe a pilotbased approach for frequency domain synchronization in [4], proposing a closedform approach to STO and CFO estimation. The training sequence is embedded as pilots in the payload data, which results in a limited detection range when the STO and CFO are estimated directly. In [5], SaeediSourck and Sadri utilize a modified preamble structure, which has been originally proposed in [6], and that mitigates the effect of selfinterference by occupying only every second subchannel. Closedform CFO estimation is enabled by the repetition of identical preamble symbols. The STO is estimated based on the received energy in the frequency domain as also mentioned in [6]. The STO estimation demands for a samplewise demodulation of the received preamble as well as for a sufficiently large gap between preamble and payload to find the maximum of the metric. Based on a reduced version of this preamble, the same authors propose an iterative approach to CFO and STO estimation in [7] that comes close to the maximum likelihood estimator. Both contributions, however, face the drawback of samplewise demodulation that leads to a high complexity, for which reason they are not considered for comparison in this work. In [8], we proposed a synchronization scheme that utilizes a similar approach to preamble design compared with that of SaeediSourck and Sadri in [5] to minimize the effects of interference, yet only symbolwise demodulation of the received signal is necessary for CFO and STO estimation. This leads to a reduced complexity compared to samplewise processing. Additionally, our design reduces the gap of unused symbols between preamble and payload by utilizing auxiliary pilots to remove selfinterference [4]. It enables the estimation of offsets in a range that is comparable to common time domain synchronization methods while keeping the amount of training sequence overhead small.
In this work, the performance of the frame detection algorithm and the CramérRao vector bound (CRVB) of the STO and CFO estimation are derived analytically based on the proposed training sequence structure from [8]. The performance of the proposed STO and CFO estimation algorithms is evaluated against the CRVB followed by the assessment of two different synchronization concepts. The proposed concepts show an improved estimation range compared to previously known frequency domain methods and achieve a performance that is similar to the one of commonly used time domain solutions. The focus of this analysis is on preamblebased direct estimation algorithms due to their reasonable tradeoff between efficiency and complexity. For that reason, interference cancelation techniques are not considered.
The paper is structured as follows. In Section 2, the signal model including the preamble design is specified. The detection and estimation metrics are introduced in Section 3, followed by the analytical derivation of their performance. Section 4 shows and discusses the results. The conclusion from this work is provided in Section 5.
2 System model and preamble design
In OQAMOFDM, the real and imaginary parts of the complexvalued QAM modulated symbols are staggered in time by half a symbol period T/2. The resulting realvalued symbols d_{k,m} are multiplied by the factor θ_{k,m}=j^{m o d(m+k,2)}, depending on the symbol index m and subchannel index k, enabling the realfield orthogonality of the OQAMOFDM symbols.
with the filtered noise samples Ψ_{k,m}.
One of the critical issues related to frequency domain processing is the mismatch between synthesis and analysis filter bank in the case of STO and CFO. The degradation of the received amplitude of the signal and the introduction of interference from neighboring symbols put limits on the performance of the frequency domain processing [4]. By separation of the pilots in time and frequency direction within the preamble, the selfinterference is reduced.
This interference from the surrounding payload symbols is obtained prior to transmission with the help of the distortionfree system response of the synthesis and analysis filter banks. Their use reduces the amount of guard symbols needed between preamble and payload part to a minimum while limiting the interference from the payload part. It is worth to note here that sparsely occupied preambles are also used for channel estimation for similar reasons, as discussed in [11].
with λ_{ k }∈{−1,1}. The subset ${\mathbb{K}}_{2}$ of ${\mathbb{K}}_{u}$ only contains K_{2}=K_{ u }/2 subchannels with either even or oddnumbered indices out of the set of utilized subchannels ${\mathbb{K}}_{u}$. K_{ u } is equal to the number of subchannels used for transmission. The sequence b_{k,m} can be arbitrarily chosen and optimized, e.g., to achieve a low peaktoaverage power ratio.
For the pilots in the preamble, we assume a pilot boost factor of $\gamma =2\sqrt{K/{K}_{u}}\in R$, resulting in d_{k,m}=γ b_{k,m} and ${\stackrel{~}{b}}_{k,m}={\stackrel{~}{d}}_{k,m}$. As a result from the sparse preamble design, the factor θ_{k,m} is the same for each pilot symbol. Hence, without loss of generality, the factor θ is assumed to be 1 and neglected in the following. Given this, (1) and (3) lead to the signal model of the preamble, which is used in the following for the analytical derivation of the CRVB. ${\stackrel{~}{b}}_{k,m}$ describes the recovered but not synchronized preamble pilots at the receiver after the demodulation.
where W=2K_{2} is the number of demodulated preamble symbols, called observations in the following, that are taken into account. In this notation, each row of a vector or matrix is related to one subchannel, denoted by the index k. In (10), this index directly affects the value of the diagonal elements of the matrix E. Furthermore, the definitions ${\mathbf{\text{b}}}_{i}={[\dots ,{b}_{k1,i},{b}_{k,i},{b}_{k+1,i},\dots \phantom{\rule{0.3em}{0ex}}]}^{T}\forall k\in {\mathbb{K}}_{2}$ and ${\mathbf{\Psi}}_{i}={[\dots ,{\Psi}_{k1,i},{\Psi}_{k,i},{\Psi}_{k+1,i},\dots \phantom{\rule{0.3em}{0ex}}]}^{T}\forall k\in {\mathbb{K}}_{2}$ are used.
3 Detection and estimation
In this section, the metric for frame detection in the frequency domain is introduced and analytically evaluated. Furthermore, the maximum likelihood estimator (MLE) for the CFO is presented and the MLE for the STO is derived. The maximum likelihood estimation of the symbol timing leads to an estimator with an insufficient estimation range for which reason two alternative STO estimators are motivated and described. In addition, the CRVB is obtained as a lower bound on the variance of the estimators.
In time domain processing, the common metric is based on an autocorrelation window that is shifted samplewise. In frequency domain processing, a samplewise shift of the received signal requires a complete demodulation process per sample shift, leading to a high complexity. The focus of this work is on symbolwise demodulation to efficiently perform detection and estimation in the frequency domain. Therefore, the processing is based on the offsetafflicted received pilots ${\stackrel{~}{b}}_{k,m}$ obtained after demodulation.
3.1 Frame detection
where 0<ρ<1 defines the threshold value. It follows that the detected symbol index $\widehat{m}$ is equal to m, if (15) is true. Two measures are of importance to characterize the quality of the detection algorithm: probability of missed detection P_{md} and probability of false alarm P_{fa}. Both probabilities depend on the decision threshold ρ. The first one indicates the probability of a detection failure if a preamble is present but is not detected. The second one provides the probability that a preamble is detected if only noise is received.
For the analytical derivation of the P_{fa} for the case that payload symbols plus noise are present, the detection metric is performed on the demodulated symbols ${\stackrel{~}{b}}_{k,m}$. However, compared to the case with pure noise, the received symbols are no longer normal distributed but depend on a discrete modulation alphabet. A tractable analytical solution can be obtained only if we approximate the distributions with normal distributions, resulting in expressions for f(Q_{ Ψ }) and $f\left({C}_{\Psi}^{2}\right)$ that are equivalent to the case of pure noise. Hence, this P_{fa} case is only evaluated based on simulations in Section 4.
3.2 Frequency offset and symbol timing estimation
The corresponding metric applied to two identical signal parts in the time domain yields the maximum likelihood estimator for the frequency offset, as derived in [14] and also used in [5]. For small offsets, when the influence of interference and amplitude degradation can be neglected, the metric for the frequency domain yields the MLE for the frequency offset. In case of larger offsets, the estimator is influenced not only by noise but also by interference from neighboring subchannels and the misalignment of the transmit and receive filters. Hence, due to the simplicity and the optimality in case of small offsets, the MLE from [14] is considered a useful and practical solution for frequency domain estimation.
holds with $\mathcal{F}\{\xb7\}$ representing the discrete Fourier transform (DFT) operation and the frequency domain samples given by X_{ k }. It follows that each received preamble symbol ${\stackrel{~}{\mathbf{b}}}_{i}$ contains information about the parameter τ and that it can be evaluated individually.
For ${\mathbb{K}}_{2}$ being a sufficiently large set or at sufficiently high SNR, the MLE is unbiased [15]. The estimation of the MLE is limited by the phase ambiguity of the subchannel with the highest subchannel index k. Setting the maximum subchannel index k_{ m a x }=K/2−1 into (23) yields the phase of ${e}^{j\frac{2\pi}{T}(\frac{K}{2}1)\tau}$ to be smaller than π only for τ≤T_{ s }. Furthermore, the assumption that ϕ≈0 is not always justified, and therefore, the MLE is not considered a practical option for STO estimation. Nevertheless, the MLE will be used later in Section 4 for verification of the CRVB.
The expression in (33) is maximized if $\stackrel{~}{\tau}=\tau $ which indicates that the estimator is unbiased at high SNR.
3.3 CramérRao vector bound
Including a Rayleigh fading channel matrix in (8) leads to the same lower bound on the estimation performance as for the AWGN case.
4 Results and discussion

The number of subchannels K is set to 32 with ${\mathbb{K}}_{u}=\{1,2,\cdots \phantom{\rule{0.3em}{0ex}},K\}$ and ${\mathbb{K}}_{2}=\{2,4,6,\cdots \phantom{\rule{0.3em}{0ex}},K\}$. It follows that the number of all usable subchannels is K_{ u }=32 and the number of subchannels occupied by the preamble pilots is K_{2}=16. For the simulation results, presented in Figures 4, 5, 6, 7, and 8, the number of payload symbols and auxiliary pilots is set to zero. Otherwise, the number of OQAMOFDM payload symbols is set to four and 4QAM modulation is applied. The number of realizations used in simulations is 10^{5}.

The prototype filter p[n T_{ s }] is designed following the frequency sampling approach to filter design. We use the filter introduced in [16], which is defined by the overlapping factor β=3 and the corresponding design parameter equal to 0.91697069.

The STO τ and the normalized CFO ν are assumed to be uniformly distributed in the range of {−T/4+T_{ s },T/4−T_{ s }} and (−0.5,0.5), respectively, if not otherwise stated. For the case of Rayleigh fading, τ refers to the delay of the rounded mean path delay of each realization of the channel.

The Rayleigh fading channel is emulated with an exponentialdecaying power delay profile according to $E\left[\righth\left[{\mathit{\text{nT}}}_{s}\right]{}^{2}]\propto {e}^{\frac{1}{2}n}$ with n∈{0,⋯,K/4−1}. A normalization of the power delay profile with ${\sum}_{n=0}^{K/41}E\left[\righth\left[{\mathit{\text{nT}}}_{s}\right]{}^{2}]=1$ is applied. The channel is static for each run but is varied between runs.
4.1 Frame detection
Beginning with the previously derived analytical expressions for P_{md} and P_{fa}, Figure 4 shows the comparison of the analytical with the simulationbased results. The analytical derivation of P_{md} over the threshold value ρ is based on the assumption that the STO and CFO are small. To take the influence of the offsets into account, two scenarios are considered in the simulations. Firstly, the P_{md} for the ideal case with no offset is evaluated, shown as the lower solid curve in Figure 4. Secondly, the offsetafflicted P_{md} is plotted, represented by the upper solid curve. The curve of the analytically derived performance is lying in between these two. The analytical and zerooffset results well agree with the results for the corresponding time domain metric presented in [13]. Furthermore, it shows that in the presence of STO and CFO, the detection rate degrades due to the amplitude degradation of the received preamble symbols and the introduction of interference.
The analytically derived P_{fa}, which assumes the presence of pure noise, is given by the upper dashed curve in Figure 4. As observed from this figure, it provides a pessimistic performance prediction compared to the outcome of the simulations, given by the two lower dashed curves. The difference stems from the approximations made during the derivation of P_{fa}. For the simulation results, two different scenarios are differentiated here. The lower dashed curve states the detection performance in the presence of pure noise, whereas the dashed line in the middle specifies the case that offsetafflicted payload symbols are received. For frame detection in time domain, these two cases are considered equal, because the time domain multicarrier signal is assumed to be distributed according to a normal distribution, yielding similar characteristics as the noise [13]. For the frequency domain approach, however, this assumption is no longer valid, as indicated by the simulation results.
4.2 Offset estimation
Hence, the sparse preamble exhibits a loss of at least a factor of 2 in number of observations W, affecting the corresponding CRVB. For K_{ u }=K, however, it can be shown that the gains and losses compensate each other. It follows from this consideration that the frequency domain CRVB is close to its time domain counterpart, as confirmed by looking at the two lower solid curves in Figure 5. In Figure 5 and as well in Figure 7, the CRVB is shown for τ=0 and ν=0.
The MLE for the STO (29) can achieve the CRVB for high SNR values for the limited estimation range of −T_{ s }≤τ≤T_{ s } and in AWGN conditions. The results indicate that the given ${\mathbb{K}}_{2}$ and an SNR value of 21 dB are sufficient for the MLE to be unbiased and asymptotically optimal [15]. For verification of the derived CRVB, the MLE is evaluated only in AWGN conditions. The CFE approaches the CRVB but is suboptimal since a gap between the RMSE of the estimation and the theoretical bound persists even for high SNR values in case of AWGN. The CFE does not account for the subchannel index k in (30), as the MLE does in (28), and hence, it does not deploy the complete received information. The performance of the CFE is significantly lowered by the Rayleigh fading, which results from the spread of received power over multiple channel taps and the corresponding interaction between different paths at the pilot positions in the frequency domain. Additionally, the estimation is subject to rounding errors as the CFE estimates the mean delay of the channel, which is compared to the rounded mean delay of the channel. The CCE achieves a similar performance as the CFE for the given reasons but is not subject to rounding errors for the Rayleigh fading case. In the case of AWGN and for the given number of realizations, the CCE, based on the integer nature of the estimation $\stackrel{~}{\tau}$, produces no error. This complies with the observation that the RMSE values of the closedform estimators are well below the rounding threshold of 0.5, where rounding the residual error to the next integer would yield zero as well. Since Figures 2 and 3 clearly suggest that the CFO has the most dominant effect on the estimation performance, the influence of the CFO on the RMSE is investigated in Figure 6, where the performance of the CCE and the CFE is shown over fixed values of the ν while τ is spanning the complete range. In both cases, AWGN and Rayleigh fading, the estimation of τ only weakly depends on the CFO.
The proximity of CRVB and CRVB^{TD} can as well be observed in Figure 7 where the RMSE of the CFO estimation is given. The CFO estimation in the frequency domain shows a slightly higher bound which is assumed to result from the different approximations used during derivation. As a verification of the derived CRVB, the MLE of the CFO is simulated with zero frequency offset and a small timing offset of −T_{ s }≤τ≤T_{ s }. Figure 7 clearly shows that the MLE yields the derived CRVB, suggesting that the CRVB^{TD} is too optimistic. The results for the MLE with offsets spanning the complete range show a performance approaching the CRVB for low SNR, while for higher values of the SNR, the RMSE runs into a performance floor. In Rayleigh fading environments, the CFO estimator exhibits almost the same performance as in the AWGN case and is only slightly degraded by the effects of the channel. The performance floor is mainly due to the remaining intrinsic interference from intercarrier interference between pilot symbols in the presence of frequency offsets and is the dominant impairment for high SNR, as Figure 8 indicates. The position of the performance floor is calculated in the Appendix 5 to be at 1.82×10^{−2}. Even though the CRVB degrades only slightly with increasing CFO, the difference between the CRVB and the MLE is 1 order of magnitude higher for the maximum CFO close to 0.5 compared to the case of zero CFO as a result of the intrinsic interference. For the case that the CFO is below 0.1, the MLE achieves the CRVB. This leads to the conclusion, that given ν<0.1, the resulting interference is sufficiently small to obtain an estimate close to the optimum.
4.3 Synchronization concepts

Concept 1. The demodulation of the received samples is only performed once, and CFO estimation and STO estimation are performed on the same, unsynchronized demodulated signal.

Concept 2. After demodulation, the CFO is estimated, which will bring the residual CFO down to ±10% of the subchannel spacing as indicated in Figure 8. After correcting the CFO in the time domain based on the first estimate, the CFO can be estimated a second time, now yielding an error below 2% according to Figure 8, which will significantly lower the performance floor.The two concepts are assessed in terms of achievable RMSE in Figures 10 and 11 and by means of BER in Figure 12 for the two relevant STO estimators CFE and CCE. The frame structure used in the evaluation is the one described in Figure 1.
In contrast to the previous evaluation of the core algorithms, outliers showing an absolute error greater than T/4 and 0.5 for the estimation of STO and CFO, respectively, are considered as falsely detected and are not included in the following results. The number of discarded estimations of this kind is well below 1% in the case of AWGN and below 5% in the case of Rayleigh fading channel conditions. The results from Figure 6 show only a weak influence of the CFO on the STO estimation. Given this observation, the difference between concepts 1 and 2 regarding the STO estimation does not vary significantly, which is as well indicated in Figure 10. In the case of concept 2 with CCE, one exception can be observed. Obviously, the additional consideration of the CFO to improve the estimation leads in some rare cases to timing errors due to an erroneous compensation of the CFO in (32) which could not be observed in the results for concept 1. In all cases, the additional payload leads to a worse estimation due to the increased interference. Figure 11 clearly shows that, in contrast to the STO estimation, concept 2 offers a significantly lower interferenceinduced performance floor for the CFO estimation. Comparing the results for concept 1 and concept 2, it can be concluded that concept 2 benefits from the smaller CFO after the second demodulation, confirming the results in the previous section. The evaluation of the BER, plotted in Figure 12, assumes perfect channel knowledge in combination with a onetap zeroforcing equalizer to recover the data symbols, which are modulated using a 4QAM symbol constellation. It is assumed that the channel is obtained at the preamble position, and the common phase error is zero at this position. The residual CFO after synchronization leads to a linearly increasing phase per OQAMOFDM symbol, which affects the demodulation of the payload symbols. In general, it can be observed that the difference between the CFE and the CCE is not significant when it comes to BER. This can be explained by the ability of the channel equalizer to effectively reduce the distortion of the phase per subchannel caused by small timing offsets. On the contrary, residual frequency offsets result in a phase drift over time with a high impact on the constellation diagram at receiver side if they are not tracked. As a result, concept 1 approaches the BER floor at 2×10^{−2} in both AWGN and Rayleigh fading conditions. The results for concept 2 indicate that the gain in CFO estimation accuracy is sufficient to get close to the ideal BER performance for AWGN and to match it in Rayleigh fading environments.
As a result, it can be shown that frequency domain synchronization methods can cope with the offsetafflicted selfinterference and offer a system performance that reaches the ideal case. To achieve this, it is sufficient to estimate and compensate for the CFO in an initial stage to further improve the CFO estimation, while the STO only needs to be estimated in the second stage.
4.4 Computational complexity
5 Conclusions
In this contribution, we showed that frame detection and synchronization can efficiently and satisfactorily be achieved in the frequency domain, taking advantage of the analysis filter bank at the receiver side. Our analysis concludes that, in theory, frequency domain synchronization schemes achieve a similar performance as time domain approaches. This is indicated by the CramérRao bounds that have been derived as part of this work. In practice, the results reveal that the performance of the algorithms strongly depends on the interference introduced by the carrier frequency offset. This drawback is removed effectively by the introduction of a frequency correction stage, leading to a bit error rate that is close to the ideal one. Even though the complexity analysis demonstrates that the frequency domain approach calls for a significantly higher computational effort, its advantage lies in shared spectrum scenarios where frequency bands, which are assigned to individual users or systems, can be synchronized and processed separately.
Appendix 1
Definition of C_{ Ψ }
R has only nonzero entries on the first offdiagonals, which are occupied by some value α<1. Under the assumption that p[n T_{ s }] is, in good approximation, a perfect reconstruction pulse shape that overlaps only with the subchannels directly adjacent to itself, P P^{ H }=I+R holds.
To distinguish between time and frequency domain noise energy, ${\sigma}_{n}^{2}$ is renamed to ${\sigma}_{\Psi}^{2}$ for ${\mathbf{C}}_{\mathbf{\Psi}}^{K\times K}$ in (50). Considering the structure of the preamble with only every second subchannel occupied, the noise that adds to the received pilot symbols after the analysis filter bank can still be considered white Gaussian noise. Furthermore, it is independent of the offset that affects the received pilot symbols. This can also be understood intuitively by looking at the transfer function of the prototype filter which expands only over adjacent subchannels. Therefore, the relevant covariance matrix is C_{ Ψ }, as defined in (25).
Appendix 2
Derivations for P_{ md }and P_{ fa }
Taking the absolute value of C_{ b } according to ${C}_{b}^{A}=\sqrt{{\left({C}_{b}^{R}\right)}^{2}+{\left({C}_{b}^{I}\right)}^{2}}$ results in ${C}_{b}^{A}$ to be Riciandistributed. For large values of W, the approximation $f\left({C}_{b}^{A}\right)\approx f\left({C}_{b}^{R}\right)$ can be made due to the dominant influence of $f\left({C}_{b}^{R}\right)$ on the distribution.
and $f\left({C}_{b}^{A}{Q}_{b}\right)=\mathcal{N}\left({\mu}_{{C}_{b}^{R}{Q}_{b}},{\sigma}_{{C}_{b}^{R}{Q}_{b}}^{2}\right)$ can be calculated depending on W and the ratio $\frac{{\gamma}^{2}}{{\sigma}_{\Psi}^{2}}$. The ratio $\frac{{\gamma}^{2}}{{\sigma}_{\Psi}^{2}}$ corresponds to a signaltonoise ratio per subchannel in the frequency domain.
$f\left({C}_{\Psi}^{2}\right)$ and f(Q_{ Ψ }) can be used to calculate the P_{fa} from (21). It has been shown in [13] that ${\sigma}_{\Psi}^{2}$ is only a scaling factor for $f\left({C}_{\Psi}^{2}\right)$ and f(Q_{ Ψ }) resulting in P_{fa} to be independent of the noise power.
Appendix 3
Regularity condition
In the last calculation step, the expectation can be taken for each coefficient of the product ∂ Ψ_{ w }/∂ v_{ i }Ψ_{ w } independently because ∂ Ψ_{ w }/∂ v_{ i } and Ψ_{ w } are statistically independent. With $E\left[{\Psi}_{w}\right]=E\left[{\Psi}_{w}^{\ast}\right]=0$, the regularity condition is fulfilled and the CRVB can be calculated for the given problem.
Appendix 4
Fisher information matrix
with W the number of observations per symbol equal to the elements used in the set of subchannels ${\mathbb{K}}_{2}$.
Appendix 5
Performance floor calculation
In the considered case of K=32 and the pulse shape p[n T_{ s }], RMSE_{min}(ν) yields 1.82×10^{−2}, which is close to the RMSE value that the performance floor in Figure 7 approaches.
Declarations
Authors’ Affiliations
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