Multicarrier waveforms with I/Q staggering: uniform and nonuniform FBMC formats
© Jošilo et al.; licensee Springer. 2014
Received: 2 December 2013
Accepted: 6 November 2014
Published: 26 November 2014
Starting from the well-known uniform filter bank multicarrier (FBMC) format with Nyquist spectral shaping of frequency-limited orthogonal (FLO) subchannels and its dual time-frequency, time-limited orthogonal (TLO) form, we introduce an orthogonal frequency division multiplex of nonuniformly spaced subchannels with unequal width.
The orthogonality conditions for nonuniform FBMC (Nu-FBMC) were evaluated through simulations using the ‘extended orthogonal frequency division multiplexing (OFDM)’ framework, as well as through analytical derivation in a rather pragmatic manner. The referent filter bank impulse responses were defined in frequency domain by straightforward aggregations of the pertaining uniform filter bank subchannels spectral shapes for FLO and by transforming the frequency domain of appropriately aggregated time-limited referent impulse responses of uniform TLO configuration. Nonsymmetrical spectral shaping of subchannels in the FLO case format has also been proposed. The spectral efficiency has been evaluated in the presence of nonlinear distortions caused by high power amplifier (HPA), and results have been given in accordance with power spectral density (PSD) constraints defined by IEEE 802.11a standard.
With the ever increasing demand for higher wireless network throughput and number of supported users, efficient utilization of spectrum and development of effective means to overcome the co-channel interference have become a very important and challenging area of research. The cyclic prefix-based orthogonal frequency division multiplexing (CP-OFDM) as the most common and widely used multicarrier technique has many disadvantages in terms of the above mentioned requirements, which necessitates further improvement of the existing multicarrier techniques. One of the possible and quite promising solution is the usage of staggered modulation formats that sequentially, at T/2 separated instants, transmit the in-phase (I or Re) and the quadrature (Q or Im) components of the M-ary quadrature amplitude modulation (M-QAM) data symbols of duration T, prevalently known as offset QAM (OQAM). This modulation format creates the basis for efficient utilization of spectrally shaped subchannels of the filter bank multicarrier (FBMC) aggregations, also known as OFDM/OQAM that can reduce the adjacent channel interference, at least without the presence of nonlinear distortions, whose significance has been discussed in (The cosine-modulated multitone (CMT) therein is actually a form of OFDM/OQAM with odd-spaced subchannels). On the other side, this modulation format offers an additional degree of freedom, enabling suppression of the kindred co-channel interference based on the quasi-rectilinear constellation and the intrinsically real domain orthogonality by means of application of the minimum mean square error widely linear filtering (MMSE WLF) equalization[2–4]. Starting from the time-frequency duality between standard OFDM/OQAM formats and TLO multicarrier formats, we introduce uniform (even- and odd-spaced) FBMC-FLO and FBMC-TLO configurations. The frequency-limited orthogonal (FLO) and the time-limited orthogonal (TLO) denote the FBMC options with the nominal frequency domain and the time-domain limited subchannels' signaling elements, respectively, that are the orthogonal multicarrier systems where there is respective overlapping between adjacent subchannels in each of the subchannels. Although the standard OFDM/OQAM can support prototype functions derived without such conceptualization, as is so called IOTA with identically shaped prototype impulse response and its Fourier transform, the paper considers more practical FLO prototype functions and its dual form named TLO becomes an extension of the OFDM/MSK approach therein.
In order to harmonize the advantages of using wider subchannels in terms of reduction of peak-to-average power ratio (PAPR) and the increase of spectral efficiency in situations when predetermined power spectral density (PSD) masks have to be obeyed, the need for a modification arises which would enable the utilization of subchannels with differing widths within scattered frequency bands (white zones). At the same time, it is important to be able to separate the adjacent channels by relatively narrow frequency guard intervals. This, in particular, relates to the available frequency gaps in the targeted private mobile radio (PMR) applications, the TV white spaces, and the wireless system bandwidths unoccupied by the primary users. This goal could be achieved, in principle, by adopting widely explored and quite well-studied nonuniform filter bank configurations (Nu-FBMC) known and used for source coding applications. In particular, channelization approach proposed in could be used; but due to specific orthogonality conditions for the data transmission, we have chosen a more pragmatic approach. Although the latter relies on spectral aggregation of uniformly spaced subchannels within the frequency-disjoined subbands, with the goal of their near-perfect reconstruction, its specialization to nonuniform data transmission related structures, in particular those of the TLO-type, turned out to be not that straightforward.
With appropriate modification of the uniform FLO and TLO configurations, we formed the nonuniform filter banks (Nu-FBMC) with non-equal subchannel width and with non-equidistant central frequencies by using the method of aggregation of the respective frequency- or time-domain shapes. The orthogonality conditions are analytically derived and confirmed through computer simulations for both the Nu-FBMC-FLO and the Nu-FBMC-TLO configurations. For this purpose, we have used the framework of extended OFDM. We also experimented and came up with the idea and confirmation of orthogonality conditions for the asymmetrical roll-off factors (in frequency domain) primarily to better control the latency inherent to the FBMC format. This will also allow for the multiplexing of unequally wide subchannels of FBMC waveforms belonging to different users without the need for an introduction of the frequency guard bands in the downlink (DL) direction of transmission.
After highlighting the time-frequency duality between uniformly spaced FBMC FLO and TLO formats in Section 2, the use of the extended OFDM format is reproduced in Section 3, along with hints to some critical aspects regarding the need for oversampling and equalization. The method of aggregation, the uniformly spaced subchannels, and the related derivation of nonuniform filter bank orthogonality conditions are dealt in Section 4. In Section 5, the asymmetrical subchannel spectral shaping is introduced. The similar aggregation procedure is applied to the nonuniform TLO format in Section 6, but in time domain by means of the adequate positioning of their spectral representations along the frequency. In Section 7, we presented the comparison of CP-OFDM and FBMC (FLO/TLO) formats in context of spectral and power efficiency and PAPR reduction. A brief comparison with conventional approaches of the nonuniform filter bank design in Section 8 is followed by the conclusions in the last Section 9.
2 Uniformly spaced TLO-MC and FBMC
The designations of TLO and FLO are meant to be applied to each subchannel individually; but, as seen from the waveforms' time-frequency representation and their positioning, the absence of their mutual overlapping in both domains is not implied.
The implementation of TLO and FLO-FBMC formats, both in the uniform and nonuniform configurations, is considered in Section 3 through the extended OFDM concept that is essentially a method for designing the signaling pulse-shape in frequency domain.
3 The extended OFDM framework
Figure 3a,b shows two nonadjacent and nonoverlapped subchannels with indexes k and k +2. Subchannel with index k +1 overlaps with both; orthogonality is provided by the usage of real inputs of the IFFT for k and k +2 and imaginary inputs for k +1, or vice versa.
The transposed configuration of the receiver part is shown in the block diagram of Figure 3b, whereby the 2K - 1 samples of the K•M-long FFT outputs are weighted with the same subchannel referent filter transfer function coefficients as in transmitter and summed to produce the subchannel outputs at T/2 intervals.
The separation between the peaks of the channel SRRC transfer function with 100% roll-off factor is 1/T, corresponding to the inverse of the QAM symbol rate. The frequency shift between even and odd subchannels' central frequencies is 1/2T. Each of the individual spectral shapes Ĝ n,k (ω), shown by triangles in Figure 4, contents 2K - 1 equidistant samples, as shown in Figure 3a,b.
The conditions for achieving (near) perfect orthogonality are arrangement dependant. For the even filter banks and the particular subchannel, the real and imaginary (including phase shift j) parts of QAM symbols are fed to the K•M-point IFFT inputs sequentially at even and odd T/2 instants. The order of the real and imaginary parts staggering is reversed on their adjacent subchannels. The I/Q staggering pattern shown in Figure 2 applies for this case. For the odd filter banks, the Re and j•Im parts of QAM symbol are brought to the K•M-point IFFT inputs (sent in all subchannels) at T/2 instants as purely real or purely imaginary values.
4 Aggregating subchannels for the Nu-FBMC-FLO format
Direct extension of the uniform filter bank towards the nonuniform filter bank configuration, for the FLO case, is to simply aggregate the subchannels of the even, odd, or combined uniform filter banks. Next, we propose the construction of nonuniform configurations and afterwards, we study the orthogonality conditions in terms of the Re and j•Im signaling instants and their relationship in the I/Q staggering pattern. For that purpose, we retain the roll-off region spectral symmetry from the uniform filter bank arrangements shown in Figure 4, with the frequency axis italicized by sampling frequency fs = M/T, where M is the number of subchannels, and T is the duration of QAM interval.
4.1 Concept of aggregation and selected configurations
where x and y define the range of subsequent (non-zero) uniform channels. For example, from the odd-spaced uniform filter bank with 32 uniform subchannels, the following arrangement of the non-equidistant subchannels can be made:
If the original subchannels in uniform arrangements have roll-off factor of 100% in the wider subchannels, it becomes progressively reduced to 50%, 25%, and 12.5% by the effective doubling of subchannels' bandwidth. For aggregated channels, the signaling intervals are correspondingly halved as can be inferred from the illustration in Figure 5. The orthogonality conditions between the subchannels formed in this way are reduced to the ones of the uniform FBMC as shown in Section 4.2. The sum of the vectors pertinent to the nonuniform subchannels is fed to the input of a K•M-size IFFT in the same way as in Figure 3a, and similarly reproduced at the output of demodulator by weighting/summing them at the K•M-size FFT output, as it was the case in Figure 3b.
Sequencing of the real and complex QAM symbol parts takes place at the beginning and the half of the corresponding QAM symbol intervals. The sequencing strategy generally differs from the ones used for uniform structures and has been empirically determined to be as follows: if particular subchannel, by its replication along the frequency axis, produces an odd uniform filter bank, its signaling always uses either Re or j•Im. If, however, the aggregated subchannel forms an even uniform filter bank, then interchangeably, Re and j•Im are brought to K•M-IFFT block, so that the adjacent (nonuniformly spaced) subchannels at the same time instant are purely real and purely imaginary. For the odd nonuniform filter bank arrangement of Figure 5a, the signaling of QAM parts is shown in Figure 5b. Alternatively, sCh1p, 3p, 4m, and sCh2m can transmit j•Im part, while remaining subchannels transmit the Re part. In all the figures, the Re part is marked by a square and the j•Im part by a diamond.
4.2 Analytical derivation of orthogonality conditions
The rather pragmatic approach to formal verification of orthogonality conditions in the case of nonuniform FLO and TLO will rely on orthogonality conditions for the related uniform filter banks.
T is QAM symbol period and f(t) is the impulse response of the square-root Nyquist filter with transfer function F(ω).
5 Using asymmetrical subchannels for the Nu-FBMC-FLO format
In all analyzed examples evaluated by the computer simulation, we essentially had a zero frequency gap between the positive and negative frequencies (spectral parts), made by omitting only one subchannel. For easier separation and filtering-out of the spectral images in the process of low-pass filtering at the transmitter and suppression of spectral images aliasing in the receiver (before modulation to the RF carrier frequency and after demodulation), a wider gap can be obtained by only appropriately increasing the base-band processing sampling frequency.
While there is no noticeable difference in the data reconstruction accuracy (MSE ~8.1·10-4 and MSE ~8.3·10-4 for symmetrical and asymmetrical cases, respectively), by comparing the subchannels impulse responses in Figure 10b and Figure 11b, it can be seen that they are much shorter in the latter case, in particular for the widest subchannels with the largest relative increase of roll-off factors.
6 Aggregating the time-domain signaling pulses for the Nu-FBMC-TLO format
It can be concluded that the MSE values in all above observed cases are very small, and the scattering around the nominal values on constellation diagrams is insignificant.
6.1 Concept of aggregation and selected configurations
Similar to Section 4.2 in which orthogonality for nonuniform FLO formats was derived based on the uniform FLO orthogonality conditions; the orthogonality of the nonuniform TLO formats can be reduced to the uniform TLO case.
Due to the absence of overlapping among the uniform TLO subchannels' impulse responses separated by (at least) one - the shortest - impulse response, the product is equals 0, so that the orthogonality criterion is reduced to the one corresponding to the uniform FBMC-TLO case.
7 Spectral efficiency with HPA
Considering the new emerging wireless technology, spectral and power efficiency are essential in the context of efficient utilization of spectrum under the adjacent channel interference avoidance constraints. The conventional CP-OFDM does not fulfill these demands for future wireless technologies. On the other hand, FBMC formats are expected to be more suitable in that sense.
Comparison of different FBMC formats and conventional CP-OFDM through the IEEE 802.11a standard PSD mask fitting are shown below. The LTE-related analysis that covers a wider frequency range around the nominal bandwidth are provided in deliverable D3.3, Section 4, http://www.ict-emphatic.eu.
8 Comparison with conventional approaches
The field of the nonuniform filter banks analysis for the source coding applications is fairly well developed, and two basic configurations have emerged - modifications of the uniform filter banks implementation by combination of (I)FFT and PNs by appropriately optimizing the referent low-pass filter impulse response[12, 13] and the application of quadrature-mirror filters based branching or tree architectures. While the first case the roll-off factors remain relatively high, the tree-branching method keeps the same roll-off factor for all subchannels, with maximal value inversely proportional to the number of nonuniform subchannels. While these two structures may have similar complexity of implementation, their flexibility appears to be much smaller.
We have presented a rather pragmatic approach for an extension of the uniformly spaced FLO and TLO FBMC formats towards the nonuniform ones. The paper provides the analysis and elaboration of nonuniform filter banks, with derivation of their orthogonality conditions.
The nonuniform FBMC framework can be of practical use as an element of a flexible channelization and, in particular, for the realization of communication channels for individual users. In the downlink, the asymmetrical spectral shaping removes the necessity for separation of adjacent users by frequency guard bands even if they individually use subchannels of different bandwidth. For the uplink directions, by deploying the nonuniform FBMC, the channels of individual users can be separated by a reduced loss in spectral efficiency, e.g., to accommodate PSD mask constraints, as demonstrated in Section 7.
The advantages of the relatively wide subchannels compared to those in the CP-OFDM regarding the robustness against frequency offset and phase jitter are implicit. The same applies to the influence of Doppler spread, provided that the channel variations can be tracked.
The potential resilience of the staggered I/Q formats and the communication (multicarrier) waveforms based on them against the (kindred) co-channel interference has yet to be fully explored in terms of the per subchannel equalization in the WLF context[15, 16]. The linear and nonlinear (DFE) MMSE equalization is directly applicable to the nonuniformly spaced subchannels of the FLO kind, while for the uniform and nonuniform TLO formats, the adequate oversampling at the output of the analysis filter bank might be needed. The alternative successive interference cancelation (SIC) framework proposed in for the FLO case is straightforwardly applicable by just increasing the number of adjacent subchannels to be taken into account.
aFrom implementation and applicability point of view, the fast-convolution filter bank framework will likely be a better alternative.
This work was supported by an FP7grant, project EMPhAtiC, and ICT-318362 (http://www.ict-emphatic.eu/). The authors thank the reviewers for their numerous useful comments and suggestions, which very much improved the paper.
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