Elsevier

Physical Communication

Volume 42, October 2020, 101144
Physical Communication

Full length article
Design of DGT-based linear and non-linear equalizers for GFDM transmission

https://doi.org/10.1016/j.phycom.2020.101144Get rights and content

Abstract

This paper exploits the parallelism between Discrete Gabor Transform (DGT) and Generalized Frequency-Division Multiplexing (GFDM) that exists when the synthesis function, i.e. the pulse shaping filter, and the analysis function, i.e. the receiving filter, satisfy the Wexler–Raz identity. Choosing functions that satisfy the Wexler–Raz condition allows optimal symbol-by-symbol detection for a DGT-based GFDM receiver in case of transmission over an additive white Gaussian noise channel. However, multipath fading is the major problem of the wireless communication channel, hence, when transmission takes place over frequency selective channel, symbol-by-symbol detection is no longer optimal due to interference generated among the transmitted symbols. In this work, we deal with the design of linear and non-linear receivers for DGT-based GFDM transmission over a frequency selective channel that allows a good trade-off between complexity and performance. Different equalization schemes to mitigate distortions, such as Maximum Likelihood, Zero-Forcing and Minimum Mean-Squared Error, are developed and analyzed. Monte Carlo simulations are used to evaluate the error rate performance achieved with the considered design. A comparison is done with other works in the literature.

Introduction

In the recent years, intensive successful testing, proof-of-concepts and trials have supported the launch of the fifth generation (5G) cellular networks. The development of 5G introduces a new paradigm. In fact, it is expected to be based on complete wireless communications without limitations [1], i.e. the new technology will be available for each user experience and each part of the access network. Thus, a lot of challenges will influence the design of communication networks and many open-ended research opportunities [1], [2].

In particular, 5G will provide: enhanced Mobile Broadband communication (eMBB), Ultra-Reliable Low-Latency Communications (URLLC), and Machine Type Communications (MTC) [3], [4]. Among these, eMBB is expected to allow theoretical user throughputs up to 10  Gbps in uplink and up to 20  Gbps in downlink by adopting technologies such as multi band carrier aggregation, enhanced channel modulation schemes, massive Multiple-Input/Multiple-Output (MIMO), and licensed assisted access [1]; URLLC implies a reduction of the time taken by a packet to go from the transmitter to the receiver with a low probability of error; MTC is an innovative form of data communication which involves one, or more, entities that do not necessarily need human interaction.

On the other hand, the concept of network slicing, which uses resources when, and where, needed and then releases them, will play a critical role in 5G networks because of the very wide gamut of expected use cases and services. Extending slicing to physical layer is still an open issue [2].

These new 5G benefits will allow for a more “connected world”, i.e. a single platform that enables a variety of different services, animated driving, Industry 4.0, Internet-of-Everything (smart home appliances) [1], [2].

To address the above challenges at the physical layer, the concept of Software Defined Artificial Intelligence and Air Interface (SD-AI2) has been recently proposed with the aim to dynamically adapt the numerology of one link based on the user environment [2]. To enable SD-AI2, link adaptation mechanisms for several fundamental building blocks, such as waveform, frame structure, multiple access scheme, modulation and coding, etc. need to be well designed [2].

A key requirement in the physical layer of future cellular networks is the flexibility to support mixed services with different waveform parameters within one carrier [5], [6], [7], [8]. From the perspective of waveforms design, many new solutions have been investigated. These need to be characterized by very high spectral efficiency, relaxed synchronization, low out-of-band (OOB) emission [4] and, additionally, to be able to support variable and customizable pulse shaping filters, achieving a better trade-off between time-domain and frequency-domain localization [3], [4].

Among the most discussed waveforms, there are Orthogonal Frequency-Division Multiplexing (OFDM) and Generalized Frequency Division Multiplexing (GFDM). Both of them are based on a Frequency-Division Multiplexing (FDM) approach where signals are transmitted in parallel using different sub-carriers. The two proposed systems are very similar but with different advantages and disadvantages. Since pros and cons of OFDM are well known, in what follows we will focus on GFDM. In particular, GFDM exploits both frequency and time domain for symbols transmission and relies on traditional filter bank multicarrier concept and on circular filtering at sub-carrier level [9], [10]. Compared to OFDM, the main advantages of GFDM consist in a reduction of the OOB emission [7], and in an increase of the spectral efficiency, obtained through the introduction of tail biting, which makes the length of the cyclic prefix (CP) independent from that of the pulse shaping filter [11], [12]. The low latency and malleability requirements, which are the major challenges in the tactile Internet scenario, can be fulfilled by GFDM due its flexible block structure, which allows to cover both CP-OFDM and single carrier transmission, such as Single Carrier FDM (SC-FDM) or DFT-spread-OFDM [11].

The high peak-to-average power ratio (PAPR) of the OFDM is a very well known limitation and can impede good downlink and uplink performance [13], [14]. In contrast, the additional degree of freedoms from the adjustable sub-carrier filters in GFDM allow further control of the PAPR [15] and, moreover, several advantages of GFDM have been already brought to MIMO application without increasing the system complexity [16].

However, these GFDM performance gains come at the cost of non-orthogonal (or semi-orthogonal) transmission, which leads to an increase of Bit Error Rate (BER) and requires a more complex receiver. In fact, when it comes to complexity comparison, GFDM requires higher complexity than OFDM. The main issue of GFDM compared to OFDM is the need of equalization, implemented by block-based processing in time or frequency domain, which is required even in the case of transmission over an ideal channel [10], [17].

For an efficient implementation of the GFDM receiver in time-domain, a relationship between GFDM and discrete Gabor transform (DGT) was proposed in [18]. It was shown that GFDM transmission and reception are equivalent to a finite discrete Gabor expansion and DGT in critical sampling, respectively. The author of [18] provided an efficient algorithm for calculation of specific GFDM receiver filters in time domain for a non-frequency selective channel. An equivalent interpretation of the DGT receiver in frequency-domain was given in [19], which allows for signal recovery with lower complexity compared to the time-domain approaches. It is worth observing that when transmission over a frequency-selective channel is considered, the DGT interpretation with critical sampling loses its validity. In this case, to restore the condition required for using DGT at the receiver, the effect of the channel must be taken into account in the equalization of the whole GFDM symbol. This aspect was considered in [19], where it was observed that the performance of the proposed low complexity frequency-domain equalization approach for the DGT-based GFDM system was close to that of OFDM only when the number of sub-symbols transmitted on each sub-carrier is low. When this number increases a rapid degradation in the performance is observed due to the inter-sub-symbol interference (ISSI) among the sub-symbols transmitted on the same sub-carrier [20], which is not properly considered in [19]. Thanks to this novel concept, we investigate the possibility to use the time-domain approach of [18], which takes into account the frequency selectivity of the channel, to achieve a good trade-off between error rate and complexity. Moreover, the use of the Dirichlet function, here proposed, allows the DGT based GFDM system to reach the lower-bound of the theoretical performance in case of non-frequency selective channels, which is an aspect not discussed in [18], [19]. Furthermore, the main contribution of this paper consists in the use of a mathematical model for the received signal to design time-domain equalizers that combat ISSI on a sub-carrier basis. This makes possible to evaluate the best strategy for detecting the transmitted symbols according to the desired performance and degree of complexity. In particular, we will focus on the design of linear equalization schemes, such as Zero-Forcing (ZF), Minimum Mean-Squared Error (MMSE) and Matched Filter (MF) and we will compare their performance and complexity with respect to the optimal non-linear Maximum Likelihood Detection (MLD). Instead, in case of a multi-path frequency selective channel, the superior performance achieved by different equalization approaches that exploit the proposed modeling will be shown with respect the other solutions present in the literature that do not take into account such a knowledge. The proposed GFDM design can also be beneficial to develop future non-linear and recursive detection algorithms.

The structure of the paper is as follows. In Section 2 we will introduce the DGT interpretation of GFDM, while the results in case of transmission over a non-frequency selective channel are shown in Section 3. The mathematical model in case of transmission over frequency selective channels is reported in Section 4. The design of the different types of receivers will be considered and discussed in Section 5. Section 6 will present the results of Monte Carlo simulations and Section 7 analyzes the complexity of the proposed receivers. Finally, conclusion will be drawn in Section 8.

Section snippets

DGT-based GFDM system model

Dennis Gabor introduced in 1946 in his “Theory of Communication” a method to represent a signal as a linear combination of time and frequency coordinates. According to the given representation, each coordinate is well concentrated in time and frequency domain. Therefore, it is possible to define a complete collection of building blocks to decompose complicated signals. Yet, despite the long history of the Gabor framework and a lot of work by mathematicians, physicists and engineers alike, there

Transmission over non-frequency selective channels

The Additive White Gaussian Noise (AWGN) model describes a channel whose effect consists in the addition of a white Gaussian noise process to the transmitted signal, as shown in Fig. 1.

The channel is mathematically described by y(n)=x(n)+w(n),n=0,1,,N1,where the nth sample y(n) of the received signal is given by the sum of the transmitted signal x(n) and a zero-mean complex white Gaussian process w(n) with variance N0. The receiver performs the DGT, as in (4), for each m,q pairs as: Xq(m)=n=

Transmission over frequency-selective channels

When transmission takes place over a classical wireless channel, i.e. characterized by delay spread and fading effect, different types of interference arise, requiring a higher complexity for the GFDM demodulation procedure with respect to the OFDM case. This is one of the main motivations behind this work. In fact, low complexity solutions must be looked for to reduce the complexity of the system and that are suitable for hardware implementation at the same time.

As in the case of OFDM, the

Design of linear and non-linear receivers

With reference to (24), we define the vector of the received signal Yq=Yq(0),Yq(1),,Yq(M1)T on the qth sub-carrier, which is given by Yq=H̄qMXq+Wq,q=1,,K1,where Wq = Wq(0),Wq(1),,Wq(M1)T and see the equation given in Box I.

It is worth noting that the model in (24) is the same as that used to describe the received vector in a MIMO system [24]. With this interpretation it is possible to design several types of receivers according to the desired trade-off between performance and complexity.

Simulation results

The performance of the proposed linear and non-linear designs is evaluated by means of Monte Carlo simulations. Now on, the metric used to evaluate the reliability of the proposed GFDM communication system is the Symbol Error Rate (SER) versus SNR, which is defined as the ratio between the energy of the transmitted symbol Es and the power spectral density of the noise N0. The SER is used as performance metric in this section, since the aim of the paper is to show the impact of the interference

Complexity analysis

In this section the implementation complexity of the operations performed at the receiver is evaluated by taking into account the number of complex multiplications (see Table 1.). The main contributions to the computational cost are:

  • MK×M multiplications to calculate γq,m[n]y(n);

  • MK-point FFT for computing Yq(m), as given in the first row of (14).

  • implementation of the different detection algorithms, as given in Table 2.

The first two are common to all the detection methods and, therefore,

Conclusion

In this work a mathematical framework for the description of the transmission and reception of DGT based GFDM signal over a frequency selective channel has been introduced. The description of the concept of Inter Sub Symbols Interference (ISSI) was modeled in closed form.

Thanks to the novel and suggested ISSI modeling, different types of linear and non-linear detection methods were designed and considered. Each one is characterized by pro and cons that were analyzed, investigated, and described

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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