Precoded GFDM transceiver with low complexity time domain processing
- Maximilian Matthé^{1}Email author,
- Luciano Mendes^{2},
- Ivan Gaspar^{1},
- Nicola Michailow^{1},
- Dan Zhang^{1} and
- Gerhard Fettweis^{1}
https://doi.org/10.1186/s13638-016-0633-1
© Matthéet al. 2016
Received: 19 February 2016
Accepted: 8 May 2016
Published: 25 May 2016
Abstract
Future wireless communication systems are demanding a more flexible physical layer. GFDM is a block filtered multicarrier modulation scheme proposed to add multiple degrees of freedom and to cover other waveforms in a single framework. In this paper, GFDM modulation and demodulation is presented as a frequency-domain circular convolution, allowing for a reduction of the implementation complexity when MF, ZF and MMSE filters are employed as linear demodulators. The frequency-domain circular convolution shows that the DFT used in the GFDM signal generation can be seen as a precoding operation. This new point-of-view opens the possibility to use other unitary transforms, further increasing the GFDM flexibility and covering a wider set of applications. The following three precoding transforms are considered in this paper to illustrate the benefits of precoded GFDM: (i) Walsh Hadamard Transform; (ii) CAZAC transform and; (iii) Discrete Hartley Transform. The PAPR and symbol error rate of these three unitary transform combined with GFDM are analyzed as well.
Keywords
1 Introduction
Fifth generation (5G) networks will face new challenges that will require a higher level of flexibility from the physical layer (PHY). 3D and 4k video will push the throughput and spectral efficiency. Tactile Internet [1] will demand latencies at least one order of magnitude smaller than what fourth generation (4G) can achieve. Machine type communication (MTC) [2] will require loose synchronization, low power consumption and a massive number of connections. Wireless Regional Area Network (WRAN) [3] needs to cover wide areas to provide Internet access in low populated regions. The future mobile networks must have a flexible PHY to address several different scenarios.
Generalized Frequency Division Multiplexing (GFDM) [4] is a recent waveform that can be engineered to address various use cases. GFDM arranges the data symbols in a time-frequency grid, consisting of M subsymbols and K subcarriers, and applies a circular prototype filter for each subcarrier. GFDM can be easily configured to cover other waveforms, such as Orthogonal Frequency Division Multiplexing (OFDM) [5] and Single Carrier Frequency Domain Multiple Access (SC-FDMA) [6] as corner cases. The subcarrier filtering can reduce the out-of-band (OOB) emissions, control peak to average power ratio (PAPR) and allows dynamic spectrum allocation. GFDM, with its block-based structure, can reuse several solutions developed for OFDM, for instance, the concept of a cyclic prefix (CP) to avoid inter-frame interference. Hence, frequency-domain equalization can be efficiently employed to combat the effects of multipath channels prior to the demodulation process. With these features, GFDM can address the requirements of 5G networks.
The main disadvantage of a flexible waveform is the complexity required for its implementation. Reducing the GFDM complexity is essential to bring its flexibility to 5G networks. The understanding of GFDM as a filtered multicarrier scheme leads to a modem implementation based on time-domain circular convolution for each subcarrier, performed in the frequency domain [4]. This paper follows the well-known polyphase implementation of filter bank modulation [5–7] and applies it to GFDM.
As one main contribution, the investigation in this paper reveals that the Poisson summation formula [8] can be utilized in the demodulation process, considering a per-subsymbol circular convolution and decimation in the frequency domain. This operation can be performed as an element-wise multiplication with subsequent simple M-fold accumulation in the time domain. This simple re-orientation of the data symbol processing allows for a considerable reduction in complexity, which makes GFDM applicable in a wide range of scenarios.
This paper also contributes to reformulate the modulation and demodulation as matrix-vector operations based on M (K)-point discrete Fourier transforms (DFTs). As an outcome, it becomes evident that DFT can actually be considered as a precoding operation. Moreover, they can be replaced by other transformations as well. This observation facilitates arbitrary precoding of the GFDM data, which adds a new level of flexibility to the system, which will be exemplified by the use of three different transforms: (i) Walsh-Hadamard transform (WHT) [9]; (ii) constant amplitude zero auto-correlation (CAZAC) [10] and; (iii) discrete Hartley transform (DHT) [11].
It is worth mentioning that the proposed low complexity signal processing complements the work in [4] because the requirement for block alignment is loosened. It can be usable for supporting pipeline inner receiver implementations, particularly when building synchronization and channel estimation circuits for embedded training sequences [12, 13]. It can also be beneficial to develop future non-linear and recursive detection algorithms.
The remainder of this paper is organized as follows: Section 2 describes the classical approach for GFDM modulation and demodulation and introduce reformulated expressions for a less complex implementation. Section 3 presents the matrix notation to describe the modulation and demodulation processes. Section 4 introduces precoding to the GFDM communication chain. Section 5 shows the complexity analysis of the precoded schemes. Section 6 brings three examples of precoding using different unitary transforms and Section 7 concludes the paper.
2 Classical GFDM description and low-complexity reformulation
where \(n=0,\dots,N-1\), \(\circledast \) describes circular convolution carried out with period N and g[ n] denotes the impulse response of the transmit prototype filter.
The subcarrier filtering does not guarantee orthogonality between subcarriers, unless specific prototype filters are used, such as the Dirichlet pulse [15]. Offset Quadrature Amplitude Modulation (OQAM) mapping can also be employed to achieve real-orthogonality [16] and avoid self-interference. In a more general case, the demodulator needs to deal with self-interference. Matched filter (MF) can be combined with successive interference cancellation (SIC) [17] or, alternatively, the self-interference can be mitigated with zero-forcing (ZF) or minimum mean square error (MMSE) filters, for which a simple design based on the Zak transform is available [18].
The sequence obtained from the Inverse Discrete Fourier Transform (IDFT) is concatenated M times in time-domain due to \(n=0,\dots,MK-1\). Such multiplication in the time-domain expresses the convolution of the subcarrier filter with the data in the frequency domain for each subsymbol.
Equation 7 describes a sampled short-time Fourier transform (STFT). This is clear since the GFDM demodulator is actually performing a Gabor transform [18], which can be understood as a sampled STFT. For this reason, (7) does not allow to employ different prototype filters per subcarrier. This constraint is circumvented by performing equalization before demodulation or the presented scheme can be used to initially access the channel state information by demodulation of training data. Also, the proposed scheme facilitates the implementation of odd number of subsymbols [18] avoiding the use of M-point DFT and IDFT required in [4].
Notice that (7) holds when MF, ZF or MMSE filters are used. ZF or MMSE filters can deliver good symbol error rate (SER) performance, while MF needs SIC to remove the SER floor caused by the self-interference. Hence, (7) shall be used to implement ZF and MMSE demodulators, while MF can be considered when the transmit filters are orthogonal or OQAM is employed.
3 GFDM matrix model
where \(\mathbf {d}^{\text {c}}_{m} = \left (d_{0,m} \,\,\, d_{1,m} \,\,\, \dots \,\,\, d_{K-1,m}\right)^{\mathrm {T}}\) denotes the data symbols transmitted in the mth subsymbol.
where y is the equalized vector at the input of the demodulator, Γ=diag(W _{ N } γ) with γ being the demodulation filter impulse response, e.g., MF, ZF, or MMSE filters.
Observe that (13) is similar to a polyphase filter structure, but with the difference that cyclic time-shifts are used here and frequency domain convolution is performed in time domain as element-wise vector multiplication. With this approach, the first step is to obtain a time domain version of \( \mathbf {d}^{\text {c}}_{m} \) by multiplying it with an inverse DFT (IDFT) matrix \(\mathbf {W}_{K}^{\mathrm {H}}\). M times upsampling in frequency domain is performed in time by duplicating the transformed data symbols with a repetition matrix R ^{(M,K)}. Each subsymbol is then pulse-shaped with g. P ^{(m)} shifts the mth subsymbol to its respective position in time. The GFDM signal is obtained by summing all the pulse-shaped subsymbols, with no need of the N-IDFT domain conversion in (10).
While the new approach describing GFDM modulation and demodulation as convolution in the frequency domain reduces the implementation complexity; in the next section, this principle will be expanded to a more general structure, where domain conversions will be understood as a precoding process.
4 DFT-based precoding for GFDM
is the information coefficient vector to be transmitted and T is a transformation matrix. For classical GFDM, where a time-frequency grid is used to transmit the information, \(\mathbf {T}=\mathbf {W}_{K}^{H}\). However, (16) can be seen as precoding the information vector by a generic matrix T. Different matrices can be used to achieve different requirements. Hence, by transmitting precoded data, GFDM can be even more flexible to address requirements of future wireless networks.
Precoding in time (T) and frequency (F) domains
Domain | Operation | Number of multiplications |
---|---|---|
FT | \({\vphantom {{\sum _{1}^{1}}}}\boldsymbol {\Delta } = \mathbf {W}_{K}^{\mathrm {H}} \mathbf {D} \mathbf {I}_{M}\) | \(MK\log _{2}(K)\) |
TT | \({\vphantom {{\sum _{1}^{1}}}}\boldsymbol {\Delta } = \mathbf {I}_{K} \mathbf {D} \mathbf {I}_{M}\) | 0 |
FF | \({\vphantom {{\sum _{1}^{1}}}}\boldsymbol {\Delta } = \mathbf {W}_{K}^{\mathrm {H}} \mathbf {D} \mathbf {W}_{M}^{\mathrm {H}}\) | \(MK\log _{2}(K)+KM\log _{2}(M)\) |
TF | \({\vphantom {{\sum _{1}^{1}}}}\boldsymbol {\Delta } = \mathbf {I}_{K} \mathbf {D} \mathbf {W}_{M}^{\mathrm {H}}\) | \(KM\log _{2}(M)\) |
The choice of the data domain has impact on the robustness of the system regarding time- and frequency- selective fading. The concept is not limited to DFT and, in general, any meaningful transform can be applied to the rows and columns, as exemplified in Section 5.
can be seen as an even more general way of precoding that is not restricted to subcarriers or subsymbols but allows arbitrary coupling between any elements of d= vec(D). The precoding is then applied leading to \(\boldsymbol {\delta } = \mathbf {\mathcal {T}} \mathbf {d}\).
5 Out of band, PAPR and complexity analysis
Number of multiplications of DFT models. Modulation and demodulation complexity are the same with this criterion
Type | Matrix model | Complexity | Resource | # op. |
---|---|---|---|---|
OFDM | \( \mathbf {x} = \frac {1}{N}\mathbf {W}^{\mathrm {H}}_{N} \mathbf {d} \), \(\, \hat {\mathbf {\!d}} = \mathbf {W}_{N} \mathbf {y} \) | \( C_{\text {OFDM}} = C_{\text {DFT,N}}=N\log _{2}(N)\) | K=2048, M=1 | 22,528 |
GFDM | \( \mathbf {x} = \frac {1}{N}\mathbf {W}_{N}^{\mathrm {H}} \sum \limits _{k=0}^{K-1} \mathbf {P}^{(k)} \mathbf {G} \mathbf {R}^{(K,M)} \mathbf {W}_{M} \mathbf {d}^{\text {r}}_{k} \) | C _{GFDM,ZF}=C _{DFT,N}+L(N+C _{DFT,M}) | K=128, M=15 | 274,202 |
Time conv. | \( \,\hat {\mathbf {d}}^{\text {r}}_{k} = \frac {1}{M}\mathbf {W}_{\!\!M\,\,}^{\mathrm {H}} \left (\mathbf {R}^{(K,M)}\right)^{\mathrm {T}} \mathbf {\Gamma } \left (\mathbf {P}^{(k)} \right)^{\mathrm {T}} \mathbf {W}_{N} \mathbf {y}, \forall k\) | L=K | ||
GFDM | \(\mathbf {x} = \sum \limits _{m=0}^{M-1} \mathbf {P}^{(m)} \text {diag}(\boldsymbol {g}) \mathbf {R}^{(M,K)} \mathbf {W}_{K}^{H} \mathbf {d}^{\text {c}}_{m} \) | C _{GFDM}=M(N+C _{DFT,K}) | K=128, M=15 | 42,240 |
Freq. conv. | \( \,\hat {\mathbf {d}}^{\text {c}}_{m} = {\mathbf {W}_{K}}\left (\mathbf {R}^{(M,K)}\right)^{\mathrm {T}} \text {diag}(\boldsymbol {\gamma }) \left (\mathbf {P}^{(m)} \right)^{\mathrm {T}} \mathbf {y}, \forall m\) |
The GFDM modulator and demodulator proposed in [4] require the complexity of the M-point DFT algorithm with additional K times complex multiplications over repeated chunks of M complex samples, resulting from a DFT operation. Even when the frequency response of the demodulation filter is assumed to be sparse, e.g., when roll-off is small and a ZF filter span L can be smaller than the total number of subcarriers, the modem scheme presented in this paper is less complex than the solution proposed in [4]. Another obstacle for the implementation in [4] is that it is based M-point DFT operations and M is optimally chosen to be odd as shown in [18]. Hence, DFT of odd length would need to be implemented which cannot be achieved by radix-2 FFT algorithms.
Recently, an alternative formulation for low-complexity implementation of GFDM modulators and demodulators was shown in [20]. The presented algorithm relies on a block-diagonalization of the transmitter and receiver matrix by using a block-DFT matrix, which can be easily implemented by means of the FFT. The achieved complexity statistics in [20] are comparable to the ones presented in the present paper. However, [20] relies on a specific structure of the transmitter and receiver filters, such that the diagonalization can be employed. Fortunately, at the receiver, this structure is obeyed for MF, ZF and MMSE receiver filters. However, the solution is not general in the sense that any filter at the receiver can be employed with the algorithm in [20]. For example, different filters might be employed when the channel noise is correlated, polluted by interference or not all subcarriers are allocated. In contrast, the algorithm presented in this paper does not rely on any structure of the receiver filter and hence equally works with any time-domain function γ[ n] in (5). This is achieved by smartly rearranging the terms in (5) without making any assumptions on γ[ n].
6 Expanding GFDM features with unitary transform precoding
Several different unitary transforms can be associated with GFDM to achieve specific goals. Three examples based on WHT, CAZAC, and DHT will be explored in the following.
6.1 WHT-GFDM
and (15) can be used to generate the WHT-GFDM symbol.
6.2 CAZAC-GFDM
with \(i=0,\,1,\,\dots,\,K-1\), \( l=0,\,1,\,\dots,\,K-1\) and \(\mathbf {C}_{K}^{\text {H}}\mathbf {C}_{K}=\mathbf {I}_{K}\).
6.3 DHT-GFDM
Notice that \(\boldsymbol {\Theta }_{K}^{\text {T}}\boldsymbol {\Theta }_{K}=\mathbf {I}_{K}\).
6.4 Precoding performance analysis
Simulations parameters
Parameter | Value |
---|---|
Modulation order | 16-QAM |
Number of subcarriers | 512 |
Number of subsymbols | 5 |
CP length | 16 samples |
Transmit filter | RC with α=0.25 |
Channel impulse response | h _{dB}[ i]=−2i/3 for \(i=0, 1, \dots, 15\) |
with H[ k] and G[ k] being the channel frequency response and the prototype filter frequency response, respectively.
7 Conclusions
This paper introduces a GFDM modulator and demodulator based on a low-complexity multiplication in the time domain. The new matrix model allows to consider the coefficients transmitted by the subcarriers as the precoding of the data symbols. This approach broadens the GFDM flexibility, allowing to use different transforms to achieve different goals. Precoding can be used to improve the system SER performance over frequency-selective channels and to reduce the PAPR. The three unitary transforms considered in this paper presented the same SER performance. The CAZAC-GFDM achieved the lowest PAPR among the considered transforms. The computational complexity of the new GFDM modulator and demodulator have been compared with the solution proposed in [4], showing that the modem solution proposed in this paper can reduce the number of complex valued multiplications, even when sparse filters in the frequency domain are employed.
Declarations
Acknowledgements
This work has been performed in the framework of the FP7 ICT-619555 project RESCUE, which is partly funded by the European Union. The work presented in this paper was sponsored by the Federal Ministry of Education and Research within the progarmme “Twenty20 - Partnership for Innovation” under contract 03ZZ0505B - “fast wireless” and within in the framework of the SATURN project with contract no. 100235995 which is funded by the Europaischer Fonds Für regionale Entwicklung (EFRE). This work was partially supported by Finep/Funttel Grant No. 01.14.0231.00, under the Radiocommunication Reference Center (Centro de Referência em Radicomunicações - CRR) project of the National Institute of Telecommunications - Inatel, Brazil. The computations were performed at the Center for Information Services and High Performance Computing (ZIH) at TU Dresden.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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