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Optimum low complexity filter bank for generalized orthogonal frequency division multiplexing
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 12 (2018)
Abstract
Generalized frequency division multiplexing (GFDM) is one of the multicarrier modulation candidates proposed for the 5th generation of wireless networks. Among GFDM linear receivers, GFDM MMSE receiver achieves the best error performance for multipath fading channels at the cost of high numerical complexity. Hence, the combination of GFDM match filter (MF) receiver and doubleside successive interference cancellation (DSIC) method is used instead. However, there is a significant gap between the error performance of GFDM MMSE and DSIC/MF receivers for the case of employing modern channel coding. Recently, we have proposed a new multicarrier scheme based on GFDM architecture called generalized orthogonal frequency division multiplexing (GOFDM). This study derives an optimized cyclic treestructured perfect reconstructionquadrature mirror filter (PRQMF) bank for GOFDM transceiver and then introduces a novel method for implementation of the optimum filter bank in the frequency domain. Employing such a fast and optimum filter bank provides several advantages for GOFDM transceiver. GOFDM transmitter mitigates outofband spectrum leak to the level of that of GFDM. In addition, choosing an appropriate configuration of filter bank yields lower peak to average power ratio in transmit signal of GOFDM compared to that of OFDM. On the other hand, while GOFDM MMSE receiver has lower numerical complexity compared with GFDM DSIC/MF receiver, its coded bit error rate curve is close to that of GFDM MMSE receiver. The aforementioned advantages envision GOFDM as a competitive candidate to be employed in the physical layer of new wireless applications.
Introduction
Spectrum is a limited media for growing demand of radio communications, and hence, spectrum efficiency is one of the important parameters to be taken into account for any modulation proposed as the air interface of the 5th generation (5G) wireless networks. Low computational complexity of the transceiver algorithm is another important parameter especially for use cases that require low battery consumption. OFDM has been widely used in wireless transmission due to its low computational complexity and good performance for multipath channels [1, 2]. In order to provide the possibility of using a simple onetap frequency domain equalizer (FDE) at the receiver side of OFDM, cyclic prefix (CP) is generally added to OFDM symbols. Since CP length should, at least, be equal to the channel length, CP overhead ratio is high for large channel spreads. Moreover, OFDM suffers from high outofband spectrum leak [3] which forces many subcarriers to be left vacant in order to fit the OFDM spectrum in an emission spectrum mask. Therefore, OFDM achieves low spectral efficiency due to large guard time and guard band and thus could not be recommended as an appropriate choice for new wireless applications.
Among the alternative filtered multicarrier schemes [4, 5], general frequency division multiplexing (GFDM) is known as a versatile multicarrier modulation being able to address many of the requirements of new wireless applications [6]. GFDM was first introduced in [7] and then has been widely investigated in the 5GNOW project [8, 9]. Three common linear receivers, i.e., zeroforcing (ZF), matched filter (MF), and minimum mean square error (MMSE) receivers, are evaluated for the receiver side of GFDM. Among GFDM linear receivers, GFDM MMSE receiver yields the lowest error rate at the cost of the highest complexity which makes it unusable for many applications [10]. Therefore, matched filter (MF) receiver in combination with the doubleside successive interference cancellation (DSIC) algorithm is used instead to provide a tradeoff between computational complexity and bit error rate (BER) performance [11].
On the basis of GFDM architecture, we have recently introduced a generic cyclic filtered multicarrier scheme called extendedGFDM. Employing perfect reconstructionquadrature mirror filter (PRQMF) bank in the generic scheme yields generalized orthogonal frequency division multiplexing (GOFDM). GOFDM benefits from architectural advantages of GFDM including flexibility of the structure, high spectral efficiency, and low latency. We have shown GOFDM MMSE outperforms GFDM ZF receiver in terms of error performance on the assumption of perfect synchronization. In addition, we have proved GOFDM MMSE receiver has a simple structure that consists of onetap MMSEFDE prior to PRQMF receiver filter bank [12].
The first contribution of this paper is to propose an optimized treestructured PRQMF bank in order to achieve low outofband (OOB) spectrum emission. Simulation results show that employing proposed optimized PRQMF bank in GOFDM structure results in OOB spectrum leak as low as that of GFDM transmitter. In addition, it is shown GFDM and GOFDM transmitters which employ an appropriate configuration of filter bank can improve peak to average power ratio (PAPR) compared with that of OFDM transmit signal. As the second contribution, a novel method for implementing the cyclic treestructured filter bank in the frequency domain is introduced. Such implementation results in a lower computational complexity for GOFDM MMSE receiver compared with existing implementation techniques proposed for GFDM DSIC/MF receiver under multipath fading channels. Further, it will be seen GOFDM MMSE receiver yields better error performance compared with OFDM for multipath fading channel where modern error control coding is presented in the setup. In the similar conditions, GFDM encounters complexity and error performance challenges for MMSE and DSIC/MF receivers, respectively. The remaining parts of the paper are organized as follows. In Section 2, the extendedGFDM scheme is described. In Section 3, the first optimum lowcomplexity PRQMF bank is derived and then the GOFDM transceiver block diagram based on such filter bank is introduced. Simulation results presented in Section 4 compare GOFDM and GFDM in terms of spectrum localization, computational complexity, error performance, and PAPR considering OFDM as a benchmark. The paper is concluded in Section 5.
ExtendedGFDM scheme
Figure 1 illustrates a general form of baseband multicarrier modulation inspired from GFDM architecture; therefore, it is called extendedGFDM scheme. At the transmitter side, x_{ ti }(n) for i = 1 : K are K blocks of M complex symbols generated by a QAM mapper. Upsampling x_{ ti }(n) sequences by the factor of K generates \( {\boldsymbol{x}}_{ti}^e(n) \) sequences of N = MK symbols whose symbols are given as
Assume g_{ i }(n) and h_{ i }(n) for i = 1 : K to be impulse responses of transmitter and receiver filters with length less than or equal to N, we have
where \( \overset{N}{\circledast } \) denotes Npoint circular convolution. Substituting for \( {x}_{ti}^e(n) \) symbols from (1) yields y(n) symbols as
where \( {\tilde{g}}_i\left(n jK\right) \) is achieved through jK cyclic shift of g_{ i }(n) after extending to N symbols through zero padding. Next, inserting CP at the beginning of y(n) sequence generates y_{ t }(n) sequence. At the receiver side, removing CP symbols yields y_{ r }(n) sequence that may be expressed as the cyclic convolution of y(n) and the impulse response of multipath fading channel. Employing FDE for y_{ r }(n) yields y_{ Eq }(n), where FDE needs an Npoint DFT to transfer symbols to frequency domain and an Npoint IDFT to return them back to time domain. Subsequently, \( {\boldsymbol{x}}_{ri}^e(n) \) for i = 1 : K are Nsymbol sequences obtained as
and then Kfactor downsampling operations yield x_{ ri }(n) symbols as
Since each g_{ k }(n) is a bandpassed filter located at a separate part of frequency domain, it can be considered as a subcarrier waveform. Assuming the sampling time to be T_{s}, (3) shows that it will take KT_{ s } seconds to transmit one symbol of each subcarrier, and thus, M symbols of each branch are carried within M time slots taking the total duration of MKT_{ s } seconds. Therefore, the scheme is Ksubcarrier and M time slot modulation with block based output of N = MK symbols. This result is in contrast with OFDM block (OFDM symbol) in which each subcarrier transmits one symbol during one time slot. The generic scheme gains some architectural advantages that are described as follows:

Low latency: Since equalization is processed within one block, extendedGFDM has the advantage of low delay. In addition, cyclic filtering used in extendedGFDM scheme prevents to generate tail symbols, and thus, it has lower latency compared to linear filtered multicarrier modulations.

Spectral efficiency: ExtendedGFDM improves guard time overhead ratio compared with OFDM through sharing one CP among multiple time slots.

Flexible structure: In wireless cellular networks, smallM, largeK configurations can be employed for downlink to provide multiple access and largeM, smallK configurations may be used for uplink to mitigate PAPR.
Assume p(n) to be the impulse response of a prototype filter. Substituting \( {g}_i(n)=p(n)\mathit{\exp}\left(j2\pi \frac{\left(i1\right)n}{K}\right) \) for i = 1 : K in Fig. 1 creates a modulated filter bank; y(n) is thus obtained as
In GFDM transmitter, data of i branch is filtered by use of a prototype filter and shifted in the frequency domain through multiplying by \( \mathit{\exp}\left(j2\pi \frac{\left(i1\right)n}{K}\right) \) factor [13]; y(n) is thus given as
It is straightforward to show (6) and (7) are equal, and therefore, employing \( {g}_i(n)=p(n)\exp \left(j2\pi \frac{\left(i1\right)n}{K}\right) \) in the extendedGFDM scheme leads to GFDM structure.
Filter bank design for GOFDM transceiver
GOFDM emerges through employing synthesis and analysis PRQMF banks at the transmitter and receiver sides of extendedGFDM scheme, respectively. Since GOFDM follows extendedGFDM structure, it takes the common architectural advantages of extendedGFDM described in previous section. In this section, we will address the design of PRQMF bank to achieve low OOB spectrum leak and low order of computational complexity when employed in the GOFDM structure.
Background
As seen in Fig. 2, PRQMF bank consists of synthesis filter bank as multiplexing part and analysis filter bank as demultiplexing part. Connecting the output of the synthesis part directly to the input of the analysis part will completely reconstruct the original inputs of the filter bank. Furthermore, in PRQMF bank, synthesis filters are matched to corresponding analysis filters [14]. In addition, perfect reconstruction property is preserved even when the convolutions are performed circularly, and thus, PRQMF banks can also be classified as cyclic orthogonal filter banks [15]. Twoband PRQMF banks are the simplest form of PRQMF banks which are easily designed by using a prototype filter. Assume h(n) to be Ltap prototype filterderived such that L is an even number and its Z transform satisfies the following condition:
Perfect reconstruction property yields [16]
and
where h_{1}(n) and h_{2}(n) are lowpass and highpass filters at the synthesis part and g_{1}(n) and g_{2}(n) are lowpass and highpass filters at the analysis part of 2band PRQMF bank, respectively. Based on (8), considering R(z) = H(z)H(z^{−1}), the Ltap prototype filter of 2band PRQMF bank can be obtained as the causal part of 2Ltap halfband filter R(z). The halfband filter R(z) is a linear phase FIR filter, and thus, we may use the ParksMcClellan algorithm to derive R(z) so as to provide the desired features for prototype filter H(z) [17].
Using a uniform tree structure of log_{2}K levels, 2band filter banks can be used for building Kband filter banks for the case where K is a power of two. By using the same prototype filter for generating 2band filter banks in all levels of the tree, we may obtain the conventional treestructured filter bank. However, 2band filter banks used in different levels of a tree, even at each node of the same tree level, can be different for the most general case. Figure 3 shows 4band treestructured filter bank in which the filters with superscripts 1 and 2 belong to the first and second levels of the tree, respectively.
Optimized treestructured PRQMF bank
In this part, we introduce the treestructured PRQMF filter bank with localized frequency response subband filters. Our approach is the design of nonconventional tree structure wherein different levels of tree have different 2band filter banks but 2band filter banks of the same level are similar. Let M be the size of input sequences at the synthesis part of Kband treestructured filter bank and h^{i}(n) be the impulse response of the prototype filter employed to derive 2band filter banks used in ith level of the tree. The OOB spectrum leak depends on the behaviour of the filters within the passband and stopband. The optimization is thus aimed in designing h^{i}(n) for i = 1 : log_{2}K such that the following optimization problem is satisfied
where H^{i}(f) and H^{i}(z) are frequency response and Z transform of h^{i}(n), respectively. Substituting h^{i}(n) for h(n) in (9–12) yields \( {g}_1^i\left(\mathrm{n}\right),{g}_2^i\left(\mathrm{n}\right),{h}_1^i\left(\mathrm{n}\right) \) and \( {h}_2^i\left(\mathrm{n}\right) \) and, subsequently, the optimized Kband treestructured filter will be established when the 2band filter banks used in the ith level of tree are obtained by use of the derived impulse responses \( {g}_1^i\left(\mathrm{n}\right),{g}_2^i\left(\mathrm{n}\right),{h}_1^i\left(\mathrm{n}\right) \) and \( {h}_2^i\left(\mathrm{n}\right) \) for i = 1 : log_{2}K.
In our approach, first, the ParksMcClellan algorithm is used to design R^{i}(n) where the designing parameters are length, cutoff frequency and maximum ripple of the filter and then, h^{i}(n) is derived as the causal part of R^{i}(n). We determine the designing parameters of R^{i}(n) so that E_{ i } is minimized for h^{i}(n). Noting that going from each level to the successive higher one, the sequence size is duplicated, and hence, the size of input sequences at the ith level of tree is 2^{i − 1}M which is changed to 2^{i}M after upsampling by the order of two. In order to preserve the size of filtered sequences through cyclic filtering, the filters length has to be less than or equal to the size of filtered sequences, and thus, we have
where L_{ i } is the length of h^{i}(n). Since more spread in the time domain permits designing of more localized subband filters in the frequency domain, the length of h^{i}(n) is chosen to have the maximum possible value, i.e., L_{ i } = 2^{i}M. As a consequence, the order of respective halfband filter R^{i}(n) will be 2 × 2^{i}M − 1. In addition, cutoff frequency is set at the middle of frequency space (i.e., 0.5 for normalized frequency space). R^{i}(n) is then derived by use of ParksMcClellan algorithm and on the conditions that the order and cutoff frequency are fixed and the maximum ripple is numerically determined such that the achieved h^{i}(n) minimizes E_{ i }. Figure 4 compares frequency response of 32tap optimum prototype filter (case 2) versus two nonoptimum 32tap prototype filters of cases 1 and 3 with larger and smaller maximumripples, respectively. The filters are derived as the causal parts of 64tap halfband filters with the order and cutoff frequency equal to 61 and 0.5, respectively. For case 2, the maximumripple of respective halfband filter is numerically derived as 0.0121 while the maximumripples of cases 1 and 3 are chosen as 0.1 and 0.001, respectively. The values of E_{ i } for cases 1, 2, and 3 are 0.0944, 0.064, and 0.0712, respectively. The results show optimum filter with lowest E_{ i } value may have larger maximumripple compared to nonoptimum prototype filters.
Lowcomplexity treestructured PRQMF bank
In this section, we propose a practical implementation algorithm in order to reduce the computational complexity of the optimum cyclic treestructured filter bank. In order to reach this goal, the cyclic treestructured filter bank is implemented in the frequency domain using Discrete Fourier Transform (DFT) and Inverse DFT (IDFT), where Mpoint DFT and IDFT are defined as:
where \( {W}_M={e}^{j\left(\frac{2\pi }{M}\right)} \). Time domain operations for a 2band cyclic filter bank are cyclic convolution, upsampling, and downsampling by the order of two. Throughout the paper, we will denote the frequency domain operations equivalent to the time domain factor 2 upsampling and downsampling as 2Copy and 2Fold, respectively, where the definitions of 2Copy and 2Fold are derived in the following lemmas.
Lemma 1 If the Msymbol sequence x(n) is upsampled by the factor of 2 to generate x^{e}(n), then X^{e}(k)=2Copy{X(k)}, where X(k) is a sequence of Mpoint DFT coefficients of x(n), X^{e}(k) is a sequence of 2Mpoint DFT coefficients of x^{e}(n) and \( {X}^{\mathrm{e}}(k)=2\mathrm{Copy}\left\{\boldsymbol{X}(k)\right\}=\Big\{{\displaystyle \begin{array}{c}\frac{1}{\sqrt{2}}X(k)\kern.5em for\kern.5em k=0:M1\\ {}\frac{1}{\sqrt{2}}X\left(kM\right)\kern.5em for\kern.5em k=M:2M1\end{array}}\operatorname{} \)_{ . }
Lemma 2 If 2Msymbol sequence x(n) is downsampled by the factor of 2 to generate x^{d}(n), then X^{d}(k)= 2Fold {X(k)}, where X(k) is a sequence of 2Mpoint DFT coefficients of x(n) and X^{d}(k) is a sequence of Mpoint DFT coefficients of x^{d}(n) and \( 2\mathrm{Fold}\left\{\boldsymbol{X}(k)\right\}=\frac{1}{\sqrt{2}}\left[X(k)+X\left(k+M\right)\right]\ \mathrm{for}\ k=0,1,\dots, M \)_{ . }
Proof In the earlier study, we have expressed up and downsampling by the factor K in the frequency domain by use of DFT coefficients [11]. Substituting K with 2 in the derived results yields the results of Lemmas 1 and 2.
Further, we know that cyclic convolution of the Msymbol sequence x(n) with the Mcoefficient filter g(n) is equivalent to the multiplication of DFT coefficients of x(n) and frequency domain coefficients of g(n) given by
Substituting upsampling, downsampling, and cyclic convolution operations of Kband treestructured filter bank with 2Copy, 2Fold, and multiplication operations yields a new structure that we call it as frequency domain tree structure (FDTS) filter bank. Consequently, in order to completely implement the filter bank in the frequency domain, at the first stage of synthesis part, a set of K blocks of Mpoint DFT are required to transfer input blocks to the frequency domain. The Kband FDTS synthesis filter bank is the subsequent stage, and then, Npoint IDFT is required to return these symbols from the frequency domain back to the time domain. At the analysis part, the first stage is an Npoint DFT. The Kband FDTS analysis filter bank and K blocks of Mpoint DFT are the subsequent stages.
Figure 5 shows how to implement a 4band treestructured filter bank in frequency domain where \( {\boldsymbol{x}}_i(n)\ \mathrm{and}\ {\hat{\boldsymbol{x}}}_i(n) \) for i = 1 : 4 are Msymbol sequences and \( \mathbf{y}(n)\ \mathrm{and}\ \hat{\mathbf{y}}(n) \) are 4Msymbol sequences, respectively. In general case, based on the results of the previous section derived for optimum filter bank, when the inputs at the synthesis part are Msymbol sequences, \( {g}_1^i\left(\mathrm{n}\right),{g}_2^i\left(\mathrm{n}\right),{h}_1^i\left(\mathrm{n}\right) \) and \( {h}_2^i\left(\mathrm{n}\right) \) are 2^{i}Mcoefficient filters. Substituting \( {g}_1^i\left(\mathrm{n}\right),{g}_2^i\left(\mathrm{n}\right),{h}_1^i\left(\mathrm{n}\right) \) and \( {h}_2^i\left(\mathrm{n}\right) \) for g(n) in (17) yields the frequency domain coefficients of the filters to be employed at the ith level of FDTS filter bank, i.e., \( {G}_1^i(k),{G}_2^i(k),{H}_1^i(k) \) and \( {H}_2^i(k) \). Finally, it should be noted that since cyclic filtering creates a clockwise circular shift, outputs of the analysis part of PRQMF bank are not exactly equal to the corresponding inputs of synthesis part. In order to compensate such shifts and obtain full reconstruction cyclic filter bank, it is enough to change frequency domain coefficients of the synthesis filters for i = 1 : log_{2}K as
Complexity analysis
In this part, the complexities of frequency and time domain implementations of treestructured filter bank are compared. The complexity is calculated in terms of the number of needed basic operations of additions and multiplications to generate one output block. The filter bank is the optimal Kband treestructured filter bank where the inputs at the synthesis part are sequences of M symbols. In general, the coefficients of the filters may be complex numbers, and so, all multiplications and additions are generally complex instructions. We assume that the filter coefficients and the DFT coefficients of the filters are known at the receiver side and they are registered in the buffer in the initialization step.
Frequency domain implementation
Figure 6a, b shows a typical 2band frequency domain synthesis and analysis filter banks used at the ith level of the synthesis and analysis parts of optimum FDTS filter bank, respectively. Let V_{1}(k), V_{2}(k) be 2^{i − 1}Msymbol input sequences of 2band synthesis filter bank. Based on Lemma 1, the next 2Copy operations generate 2^{i}Msymbol sequences whose symbols are given as
and, subsequently, the output symbols of synthesis part are obtained as
For 2band analysis filter bank, letting \( \widehat{W}(k) \) be 2^{i}Msymbol input sequence yields
subsequently, based on Lemma 2, output symbols of 2band analysis filter bank are obtained as
Based on (22), the 2band frequency domain synthesis filter banks used at the ith level of FDTS synthesis filter bank require 2 × 2^{i}M multiplications and 2^{i}M additions. The Kband FDTS synthesis filter bank consists of log_{2}K levels, and there are \( \frac{K}{2^i} \) 2band filter banks at the ith level. Therefore, the number of multiplications and additions performed in the synthesis part of FDTS is
In order to implement Kband synthesis filter bank in the frequency domain, the total basic operations needed consists of basic operations for the calculation of K blocks of Mpoint DFT, Kband FDTS, and Npoint IDFT. K is always a power of two and N=MK , thus assuming that M to be a power of two yields N to be also a power of two. Therefore based on the radix2 FFT algorithm, the overall number of basic operations is obtained as
On the other hand, referring to Eqs. (23–26), 2band analysis filter banks used at ith level of analysis part need 2 × 2^{i}M multiplications for two multiplication operators and 2 × 2^{i − 1}M additions for two 2Fold operators. Therefore, the 2band analysis and synthesis filter banks used at the ith level of FDTS filter bank need equal number of basic mathematical calculations, and consequently, the total number of basic mathematical operations of synthesis and analysis parts of frequency domain implementation are similar.
Time domain implementation
The inputs of the ith level of synthesis part are blocks of 2^{i − 1}M symbols, and thus, after upsampling, the size of input blocks is changed to 2^{i}M where half of them are zeroes. Twoband filter banks used at the ith level of analysis part require two 2^{i}Mpoint cyclic convolutions and one 2^{i}Mpoint addition at the end. Therefore, number of basic operations performed for the 2band synthesis filter banks used at ith level of synthesis part is
The tree is built of log_{2}K levels where the ith level at the synthesis part consists of \( \frac{K}{2^i} \) 2band synthesis filter bank. Therefore, the total number of basic operations of the synthesis part yields as
In a similar approach, it can be shown that the numbers of basic operations at the analysis part is equal to those of the synthesis part. The results are presented in Table 1. It is seen that the complexity of frequency and time domain implementations are of the order of Nlog_{2}N and N^{2}, respectively.
GOFDM system model based on optimum lowcomplex filter bank
Deploying the optimum lowcomplexity treestructured PRQMF bank in the generic scheme of Fig. 1 yields the block diagram of Fig. 7 wherein lower case letters denote time domain symbols and higher case letters denote frequency domain symbols. As stated before, the advantage of implementing treestructured PRQMF filter bank in frequency domain instead of time domain is reducing the computational complexity of the structure. This advantage is more highlighted at the receiver side wherein the existing FDE implies that the received symbols are transferred to the frequency domain even in case of time domain implementation.
Simulation results
Simulation setup
In what follows, we compare GOFDM with GFDM in terms of spectrum localization, error performance, numerical complexity, and peak to average power ratio. Optimum treestructured PRQMF bank is employed for GOFDM. In the GFDM structure, the prototype filter is rootraised cosine (RRC) filter with rolloff factor of 0.3. Setting up the configuration parameters of filter banks needs some consideration. Provided that M and K being even numbers GFDM transmit matrix becomes singular [13] but PRQMF bank does not face such a problem and we are permitted to use M and K both as a power of 2. In the simulations, we evaluate two configurations where the number of subcarriers in GFDM and GOFDM schemes for configuration cases 1 and 2 are 256 and 16, respectively. Therefore, we encounter the GFDM limitation of using odd number for M values. Hence, M is chosen as 9 and 129 for GFDM scheme and the nearest power 2 numbers, i.e., 8 and 128 are chosen for GOFDM scheme in the configuration cases 1 and 2, respectively. In two GOFDM configurations, M × K = 2048 and the number of CP symbols is equal to 64; thus, both configurations have similar spectral efficiency. When OOB emission is analyzed, configuration with large number of subcarriers, i.e., case 1 is taken into account while PAPR is analyzed for configuration with small number of subcarriers, i.e., case 2. Both configurations are evaluated in terms of error performance and numerical complexity. The channel profile and other simulation parameters are presented in Table 2.
OOB leakage
In order to reduce OOB emission, the normal approach is to use empty subcarriers at the corner sides of assigned bandwidth. For blockbased modulations, abrupt changes of signal between successive blocks result in high OOB emission. Windowing is thus exploited at the transmitter side by multiplying output blocks with a window function to smooth abrupt changes and combat the OOB emission. For the case of windowed waveforms in addition to CP, cyclic suffix (CS) is also inserted. Through the simulations, G prefix denotes windowed waveform and GW prefix denotes windowing and guard subcarriers are simultaneously applied for a waveform. According to LTE standard for the case of 256 subcarriers, 150 subcarriers are occupied as allocated bandwidth. We consider six empty subcarriers of each side as guard band and, consequently, power spectral densities (PSDs) leak to other empty subcarriers may be considered as OOB emission. For the case of GOFDM, despite what might be expected, frequency responses of subband filters of the Kband treestructured filter bank are disorderly arranged in the spectrum for the case where K is higher than 2. The 4band frequency split of the spectrum is illustrated in Fig. 8 for ideal bandpass filters. As explained, 4band treestructured filter bank has two stages, stage 1 consists of two 2band filter banks and stage 2 consists of one 2band filter bank. It is seen in Fig. 8 that filter bank of stage 2 reverses the arrangement of two higher subbands and the frequency response of subband 3 comes before subband 4. Therefore, the order of 4band treestructured filter bank in the spectrum may be shown as {1, 2, 4, 3}. Similarly, 8band treestructured filter bank may be considered as a stage of two 4band treestructured filter banks prior to one 2band filter bank. Combining two 4band treestructured filter banks using next 2band filter bank reverses the order of higher 4band filter bank, and thus, the subbands are ordered in the spectrum as {1,2,4,3,7,8,6,5}. Following such approach, one can find the order of subband filters of Kband treestructured filter bank in the spectrum is matched to a sequence of gray numbers. In our case study, the 256number gray sequence starting from 1 shows the order of subcarriers in the spectrum.
Figure 9 compares PSDs of OFDM, GFDM, and GOFDM. In order to evaluate OOB emission, OOB radiation parameter is defined as the ratio of the energy that is emitted into the OOB range and the amount of energy within the allocated bandwidth, i.e.,
where BW and OOB are the set of frequencies that are considered as inband and out of band, respectively. The OOB radiations of GOFDM, GGFDM, and GGOFDM are calculated as − 25.7, − 34.2, and − 33.9 dB, respectively. In case of applying both windowing and guard subcarriers, the OOB radiations of GWOFDM, GWGFDM, and GWGOFDM are reduced to − 35.6, − 50.5, and −49.8 dB, respectively. As expected, the OOB emissions of cyclic multicarrier modulations are significantly less than that of OFDM. Furthermore, the optimized PRQMF bank causes GOFDM OOB radiations to be only 0.3 and 0.7 dB more than that of GFDM for the cases of inserting guard subcarriers and employing window and inserting guard subcarriers together, respectively. We now compare the bandwidth efficiency for three modulations. Noting that the overhead symbols are CP and CS guard symbols in the time domain and empty guard subcarriers in the frequency domain, we define the spectral efficiency as the ratio of useful symbols to the total transmitted symbols during each output block, and then, the spectral efficiency of OFDM, GOFDM, and GFDM are \( \frac{256}{256+64+16}=0.76,\frac{256\times 9}{256\times 9+64+32}=0.96 \) and \( \frac{256\times 8}{256\times 8+64+32}=0.95 \), respectively. The results show that GOFDM transmitter yields almost the same level of OOB leakage and spectral efficiency as those of GFDM transmitter.
Error performance
In the case of GFDM MMSE receiver, an inverse matrix has to be calculated according to changes in noise power. Therefore, GFDM MMSE receiver suffers from high complexity, and DSIC/MF receiver is suggested to use instead. In the earlier study, we have proved, on the assumption of perfect synchronization, GOFDM MMSE receiver which simply consists of onetap MMSEFDE and analysis PRQMF bank is the best GOFDM linear receiver in terms of error performance [12]. On the other hand, GFDM DSIC/MF receiver is the combination of ZFFDE, matched filters and DSIC algorithm. Onetap coefficients of MMSEFDE and ZFFDE for k = 1:N are defined as \( \frac{1}{C(k)} \) and \( \frac{C^{\ast }(k)}{{\leftC(k)\right}^2+{\sigma}_w^2} \), respectively, where C(k) and \( {\sigma}_w^2 \) are Npoint DFT coefficients of channel impulse response and the power of white Gaussian noise, respectively.
In this study, OFDM, GFDM DSIC/MF with four iterations of DISC algorithm, and GFDM MMSE and GOFDM MMSE receivers are compared in terms of error performance for transmitting windowed signals through multipath fading channel when complete synchronization is assumed. At the first step, uncoded BER is analyzed. As seen in Fig. 10a for the case of configuration 1, while GOFDM MMSE slightly outperforms OFDM, the BER curves of GFDM MMSE and DSIC/MF receivers are tightly close to that OFDM. For the configuration case 2 shown in Fig. 10b, GOFDM and GFDM MMSE receivers outperform OFDM in terms of error performance for signal to noise ratio (SNR) larger than 12 and 15 dB, respectively, and the situations are reversed for SNR smaller than these thresholds. However, there is a considerable gap between GFDM DSIC/MF and OFDM error performance curves which means the negative effect of selfinterference at the receiver side of GFDM MF receiver is more serious for configuration case 2 compared with configuration case 1. At the second step, the parallel concatenated convolutional code (PCCC) with code rate R = 1/3 is employed for transmitter side, and a soft demapper in combination with a turbo decoder with 10 iterations is employed at the receiver side. It is known that the performance of employed advanced receiver depends on the mean square error in the received symbols. Therefore, as seen in Fig. 10c, d, while GOFDM MMSE outperforms OFDM, there are significant gaps between coded BER curves of DSIC/MF GFDM and OFDM in both configurations. On the other hand, the coded error performance of GOFDM MMSE receiver compared to GFDM MMSE receiver depends on the configuration of filter bank and SNR value. It can be verified that BER curve of GOFDM MMSE is slightly lower/higher than that of GFDM MMSE for SNR smaller/larger than 9.7 and 7.5 dB for the configuration cases 1 and 2, respectively.
Computational complexity
Among the methods introduced to implement GFDM receivers, authors in [18] proposed the lowest complexity modem for GFDM zeroforcing and matched filter receivers. However, the proposed method does not support DSIC/MF receiver for the case of multipath fading channel. Therefore, sparse frequency domain processing method introduced in [19] still leads to lowest complexity for our case study. Based on this method, the number of multiplications in GFDM DSIC/MF receiver with the configuration of K subcarriers and M time slots ignoring the equalizer complexity is reduced to
where I is the number of adjacent subbands wherein the frequency response of prototype filter has significant values and J is the number of iterations for DSIC algorithm. I depends on the frequency response of prototype filter and usually is small; however, J depends on the order of QAM constellation and for the case of large QAM constellation, large J is required to achieve good error performance [19].
Ignoring the equalizer part, the complexity of GOFDM receiver with the configuration of K subcarriers and M time slots is equivalent to the complexity of Kband analysis filter bank with an input block of N = MK symbols. When M and K are a power of two, numerical complexity for such configuration is derived in (28).
We compare numerical complexity of GFDM DSIC/MF and GOFDM MMSE receivers in terms of the number of multiplications. In the case of GOFDM, K and M are power of 2 numbers where K is ranged from 16 to 2048 and M is ranged from 4 to 128. For the case of GFDM, K has the same range but M values increased by 1, i.e., M = 5, 9, 17, 33, 65, 129. The values of M in GFDM and GOFDM configurations are not equal; thus, we consider GOFDM configuration parameters as the reference and calculate its complexity relative to GFDM complexity with equal K but M values increased by 1. As seen in Fig. 11a, b, GOFDM MMSE receiver achieves less numerical complexity compared to GFDM DSIC/MF receiver especially for the case of large M and large J values. For example, in configuration case 1, GOFDM to GFDM complexity ratio for J = 4 and J = 8 are 0.4474 and 0.2724, respectively, and in configuration case 2, this parameter equals 0.2106 and 0.1203 for J = 4 and J = 8, respectively.
PAPR
The PAPR of y_{ t }(n) sequence is defined as
Larger PAPR values demand a higher linear range amplifier which is a challenge for designing lowcost terminals. Complementary cumulative density function (CCDF) of PAPR, i.e., the probability that the PAPR exceeds a certain value, is the typical measure to compare different systems in terms of PAPR. PAPR value of multicarrier schemes depends on the number of subcarriers, and the schemes with a few number of subcarriers perform better than the schemes with large number of subcarriers [20] in terms of PAPR. Authors in [21] showed considering GFDM multicarrier scheme for the uplink of a cellular network wherein only one subcarrier assigned to each user yields smaller PAPR compared to OFDM scheme. Therefore, we consider GFDM and GOFDM with the configuration case 2 for an uplink scenario where one subcarrier assigned to each of 16th users. For the case of OFDM, 16 contiguous subcarriers are allocated to each user. As the benchmark, singlecarrier FDM system is considered. Then, a pulseshaping raised cosine filter with rolloff = 0.5 is applied in all schemes. The average values of CCDF of PAPR of transmit signals of the waveforms are shown in Fig. 12. It is seen, OFDM is significantly outperformed by GFDM and GOFDM in terms of PAPR. It can also be verified that GOFDM is superior to singlecarrier FDM but it is outperformed by GFDM in terms of PAPR. According to results of previous parts, GFDM transmitter achieves such superiority at the costs of more computational complexity or lower error performance of the receiver side.
Conclusions
GOFDM encompasses the architectural advantages of GFDM including high spectral efficiency, low latency, and flexible structure. The novelty of current work relies on exploiting a fast and optimized PRQMF bank to be employed in the GOFDM structure. As a consequence, GOFDM transmitter yields the same level of OOB leakage as that of GFDM transmitter and GOFDM receiver yields lower computational complexity compared with GFDM DSIC/MF receiver. Further, it was shown that GOFDM MMSE receiver is superior to OFDM in terms of error performance when modern channel coding is used in the setup. In the similar conditions, GFDM DSIC/MF receiver could not perform adequately, and thus, it was required to use high complexity MMSE receiver for GFDM to achieve acceptable error performance. Furthermore, it was shown employing the configurations of small numbers of subcarriers, large numbers of time slots in GOFDM architecture preserves good spectral efficiency and, besides, yields PAPR even lower than singlecarrier FDM scheme. For such configurations, although GFDM transmitter is superior to GOFDM transmitter in terms of PAPR, GFDM MMSE and DSIC/MF receivers suffer from significant cost of numerical complexity and error performance, respectively. In spite of aforementioned advantages, GOFDM has to be precisely evaluated for the case of asynchronous transmitters which is a challenging issue for new 5G applications.
Abbreviations
 5G:

5th generation
 CCDF:

Complementary cumulative density function
 CP:

Cyclic prefix
 CS:

Cyclic suffix
 DSIC:

Doubleside successive interference cancellation
 FDE:

Frequency domain equalizer
 FDTS:

Frequency domain tree structure
 GFDM:

Generalized frequency division multiplexing
 GOFDM:

Generalized orthogonal frequency division multiplexing
 MF:

Matched filter
 MMSE:

Minimum mean square error
 OOB:

Outofband
 PAPR:

Peak to average power ratio
 PRQMF:

Perfect reconstructionquadrature mirror filter
 PSD:

Power spectral density
 RRC:

Rootraised cosine
 ZF:

Zeroforcing
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Acknowledgements
This work was supported in part by a grant from IPM and in part by a grant from the Iran National Science Foundation under grant 95824827.
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MHA and BHK have been responsible for providing the idea of the paper. MHA has also been responsible for implementation of ideas and performing simulations. MHA, BHK, and AH have been involved in writing the manuscript and proofreading it.
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Mohammad Hadi Abbaszadeh received his B.Sc. and M.Sc. degrees from Shiraz University and Iran University of Science and Technology, respectively. He is currently pursuing the Ph.D. degree in Tehran Science and Research Branch, Islamic Azad University, Tehran, Iran. His research interests include MIMO wireless communications and multicarrier modulations.
Babak Hossein Khalaj received his M.Sc. and Ph.D. degrees from Stanford University. At Stanford, he has been among the pioneering team working on adoption of multiantenna arrays in mobile networks. He has also been a senior member of Advanced Communications Research Institute (ACRI) at Sharif University, Tehran, Iran, and the recipient of Alexander von Humboldt Fellowship in 2007–2008.
Afrooz Haghbin received her M.Sc. and PhD degrees from Tehran University and Tarbiat Modares University, respectively. She is currently with the electrical and computer department of Science and Research Branch in Islamic Azad University, Tehran, Iran, as an assistant professor. Her research interests include MIMO wireless communications, channel coding, precoding, multicarrier modulation, and estimation theory.
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Abbaszadeh, M.H., Khalaj, B.H. & Haghbin, A. Optimum low complexity filter bank for generalized orthogonal frequency division multiplexing. J Wireless Com Network 2018, 12 (2018). https://doi.org/10.1186/s136380171017x
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DOI: https://doi.org/10.1186/s136380171017x
Keywords
 Generalized Frequency Division Multiplexing (GFDM)
 Treestructured Filter Bank
 Peak To Average Power Ratio (PAPR)
 MMSE Receiver
 Minimum Mean Square Error (MMSE)