Optimum low complexity filter bank for generalized orthogonal frequency division multiplexing

As seen in Fig. 2, PR-QMF bank consists of synthesis filter bank as multiplexing part and analysis filter bank as de-multiplexing 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 PR-QMF 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, PR-QMF banks can also be classified as cyclic orthogonal filter banks [15]. Two-band PR-QMF banks are the simplest form of PR-QMF banks which are easily designed by using a prototype filter. Assume h(n) to be L-tap prototype filter-derived such that L is an even number and its Z transform satisfies the following condition:

Perfect reconstruction property yields [16]andwhere h₁(n) and h₂(n) are low-pass and high-pass filters at the synthesis part and g₁(n) and g₂(n) are low-pass and high-pass filters at the analysis part of 2-band PR-QMF bank, respectively. Based on (8), considering R(z) = H(z)H(z⁻¹), the L-tap prototype filter of 2-band PR-QMF bank can be obtained as the causal part of 2L-tap half-band filter R(z). The half-band filter R(z) is a linear phase FIR filter, and thus, we may use the Parks-McClellan algorithm to derive R(z) so as to provide the desired features for prototype filter H(z) [17].

Using a uniform tree structure of log₂K levels, 2-band filter banks can be used for building K-band filter banks for the case where K is a power of two. By using the same prototype filter for generating 2-band filter banks in all levels of the tree, we may obtain the conventional tree-structured filter bank. However, 2-band 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 4-band tree-structured filter bank in which the filters with superscripts 1 and 2 belong to the first and second levels of the tree, respectively.

In this part, we introduce the tree-structured PR-QMF filter bank with localized frequency response sub-band filters. Our approach is the design of non-conventional tree structure wherein different levels of tree have different 2-band filter banks but 2-band filter banks of the same level are similar. Let M be the size of input sequences at the synthesis part of K-band tree-structured filter bank and h ⁱ (n) be the impulse response of the prototype filter employed to derive 2-band 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 : log₂K such that the following optimization problem is satisfiedwhere 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 K-band tree-structured filter will be established when the 2-band 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 : log₂K.

In our approach, first, the Parks-McClellan algorithm is used to design R ⁱ (n) where the designing parameters are length, cut-off 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 − 1M which is changed to 2 ⁱ M after up-sampling 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 havewhere L _i is the length of h ⁱ (n). Since more spread in the time domain permits designing of more localized sub-band 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 half-band filter R ⁱ (n) will be 2 × 2 ⁱ M − 1. In addition, cut-off 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 Parks-McClellan algorithm and on the conditions that the order and cut-off 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 32-tap optimum prototype filter (case 2) versus two non-optimum 32-tap prototype filters of cases 1 and 3 with larger and smaller maximum-ripples, respectively. The filters are derived as the causal parts of 64-tap half-band filters with the order and cut-off frequency equal to 61 and 0.5, respectively. For case 2, the maximum-ripple of respective half-band filter is numerically derived as 0.0121 while the maximum-ripples 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 maximum-ripple compared to non-optimum prototype filters.

In this section, we propose a practical implementation algorithm in order to reduce the computational complexity of the optimum cyclic tree-structured filter bank. In order to reach this goal, the cyclic tree-structured filter bank is implemented in the frequency domain using Discrete Fourier Transform (DFT) and Inverse DFT (IDFT), where M-point DFT and IDFT are defined as:where \( {W}_M={e}^{-j\left(\frac{2\pi }{M}\right)} \). Time domain operations for a 2-band cyclic filter bank are cyclic convolution, up-sampling, and down-sampling by the order of two. Throughout the paper, we will denote the frequency domain operations equivalent to the time domain factor 2 up-sampling and down-sampling as 2-Copy and 2-Fold, respectively, where the definitions of 2-Copy and 2-Fold are derived in the following lemmas.

Lemma 1 If the M-symbol sequence x(n) is up-sampled by the factor of 2 to generate x ^e (n), then X ^e (k)=2-Copy{X(k)}, where X(k) is a sequence of M-point DFT coefficients of x(n), X ^e (k) is a sequence of 2M-point DFT coefficients of x ^e (n) and \( {X}^{\mathrm{e}}(k)=2-\mathrm{Copy}\left\{\boldsymbol{X}(k)\right\}=\Big\{{\displaystyle \begin{array}{c}\frac{1}{\sqrt{2}}X(k)\kern.5em for\kern.5em k=0:M-1\\ {}\frac{1}{\sqrt{2}}X\left(k-M\right)\kern.5em for\kern.5em k=M:2M-1\end{array}}\operatorname{} \) _.

Lemma 2 If 2M-symbol sequence x(n) is down-sampled by the factor of 2 to generate x ^d (n), then X ^d (k)= 2-Fold {X(k)}, where X(k) is a sequence of 2M-point DFT coefficients of x(n) and X ^d (k) is a sequence of M-point 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 down-sampling 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 M-symbol sequence x(n) with the M-coefficient filter g(n) is equivalent to the multiplication of DFT coefficients of x(n) and frequency domain coefficients of g(n) given by

Substituting up-sampling, down-sampling, and cyclic convolution operations of K-band tree-structured filter bank with 2-Copy, 2-Fold, 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 M-point DFT are required to transfer input blocks to the frequency domain. The K-band FDTS synthesis filter bank is the subsequent stage, and then, N-point 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 N-point DFT. The K-band FDTS analysis filter bank and K blocks of M-point DFT are the subsequent stages.

Figure 5 shows how to implement a 4-band tree-structured filter bank in frequency domain where \( {\boldsymbol{x}}_i(n)\ \mathrm{and}\ {\hat{\boldsymbol{x}}}_i(n) \) for i = 1 : 4 are M-symbol sequences and \( \mathbf{y}(n)\ \mathrm{and}\ \hat{\mathbf{y}}(n) \) are 4M-symbol 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 M-symbol 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 M-coefficient 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 PR-QMF 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 : log₂K as

In this part, the complexities of frequency and time domain implementations of tree-structured 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 K-band tree-structured 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.

Deploying the optimum low-complexity tree-structured PR-QMF 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 tree-structured PR-QMF 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.

In what follows, we compare GOFDM with GFDM in terms of spectrum localization, error performance, numerical complexity, and peak to average power ratio. Optimum tree-structured PR-QMF bank is employed for GOFDM. In the GFDM structure, the prototype filter is root-raised cosine (RRC) filter with roll-off 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 PR-QMF 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.

In order to reduce OOB emission, the normal approach is to use empty subcarriers at the corner sides of assigned bandwidth. For block-based 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 sub-band filters of the K-band tree-structured filter bank are disorderly arranged in the spectrum for the case where K is higher than 2. The 4-band frequency split of the spectrum is illustrated in Fig. 8 for ideal band-pass filters. As explained, 4-band tree-structured filter bank has two stages, stage 1 consists of two 2-band filter banks and stage 2 consists of one 2-band filter bank. It is seen in Fig. 8 that filter bank of stage 2 reverses the arrangement of two higher sub-bands and the frequency response of sub-band 3 comes before sub-band 4. Therefore, the order of 4-band tree-structured filter bank in the spectrum may be shown as {1, 2, 4, 3}. Similarly, 8-band tree-structured filter bank may be considered as a stage of two 4-band tree-structured filter banks prior to one 2-band filter bank. Combining two 4-band tree-structured filter banks using next 2-band filter bank reverses the order of higher 4-band filter bank, and thus, the sub-bands are ordered in the spectrum as {1,2,4,3,7,8,6,5}. Following such approach, one can find the order of sub-band filters of K-band tree-structured filter bank in the spectrum is matched to a sequence of gray numbers. In our case study, the 256-number 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 in-band and out of band, respectively. The OOB radiations of G-OFDM, G-GFDM, and G-GOFDM 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 GW-OFDM, GW-GFDM, and GW-GOFDM 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 PR-QMF 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.

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 one-tap MMSE-FDE and analysis PR-QMF 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 ZF-FDE, matched filters and DSIC algorithm. One-tap coefficients of MMSE-FDE and ZF-FDE for k = 1:N are defined as \( \frac{1}{C(k)} \) and \( \frac{C^{\ast }(k)}{{\left|C(k)\right|}^2+{\sigma}_w^2} \), respectively, where C(k) and \( {\sigma}_w^2 \) are N-point 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 self-interference 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 (PC-CC) 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.

Among the methods introduced to implement GFDM receivers, authors in [18] proposed the lowest complexity modem for GFDM zero-forcing 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 towhere I is the number of adjacent sub-bands 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 K-band 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.

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 low-cost 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, single-carrier FDM system is considered. Then, a pulse-shaping raised cosine filter with roll-off = 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 single-carrier 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.