Comparisons of Filter Bank Multicarrier Systems

Transcription

1 Comparisons of Filter Bank Multicarrier Systems Juan Fang 1, Zihao You 2, I-Tai Lu 5 ECE Department Polytechnic Institute of NYU Brooklyn, NY, USA jfang1985@gmail.com 1, zyou1@students.poly.edu 2, itailu@poly.edu 5 Abstract In this paper, we compare three filter bank multicarrier (FBMC) techniques, i.e., Orthogonal Frequency Division Multiplexing (), -Offset Quadrature Amplitude Modulation (-OQAM) and wavelet packet modulation (WPM), based on multiple criteria, including bandwidth efficiency, computational complexity, latency, out-ofband emission, peak-to-average power ratio, and sensitivity to timing and frequency offsets. An efficient implementation of -OQAM is also proposed for a special case. Keywords-Multicarrier; ; -OQAM; Wavelet Packet Modulation I. INTRODUCTION Multicarrier modulation (MCM) techniques enable transmission of a set of data over multiple narrow band subcarriers simultaneously. With an advanced wideband modulation and coding scheme (MCS), a system with MCM can achieve much higher spectral efficiency in frequency selective channels compared to those using single carrier modulation techniques. Filter bank multicarrier (FBMC) modulation is a family of MCM techniques in which a prototype filter is designed to achieve a certain goal, such as minimizing inter-symbol interference (ISI), inter-carrier interference (ICI) and/or stop band energy. The well-known Orthogonal Frequency Division Multiplexing () can be considered a type of FBMC whose time domain prototype filter is a simple rectangular pulse. From the complexity perspective, the main advantage of over other FBMC techniques is that it is very easy to be implemented. However, large sidelobes of the rectangular pulse in create challenging issues in practical systems. For example, the performance of the systems at the physical (PHY) layer is very sensitive to frequency offset. In addition, in systems such as those in TV white space and heterogeneous networks with small cells, multiple radio links coexist in congested spectral bands, but are loosely controlled or coordinated in resource usage (frequency, timing and power). In such a network, strong adjacent channel interference may be generated from large out-of-band emissions partially contributed from large sidelobes at the baseband. Another drawback of is that the modulated signal exhibits large peak-to-average power ratio (PAPR) which usually leads to low efficiency power amplifiers (PAs). To overcome the aforementioned drawbacks, researchers have been looking into different FBMC techniques, including OQAM- [1]-[4] and Wavelet Packet Transform (WPT) [5]. In OQAM-, subcarriers of the signal overlap each other to achieve a high spectral efficiency. Jialing Li 3, Rui Yang 4 InterDigital Communications, Inc. Melville, NY, USA Jialing.Li@interdigital.com 3, Rui.Yang@interdigital.com 4 Different from, the real and imaginary parts of the QAM symbols are processed separately with 2 symbol rate. A prototype filter needs to be carefully designed to minimize or zero out ISI and ICI while keeping the sidelobes small. On the other hand, wavelet packet modulation (WPM), which uses the wavelet packet transform (WPT) instead of the discrete Fourier transform (DFT), is receiving growing attention because of some of its interesting features in terms of spectrum usage and signal construction flexibilities [5]. Some studies have also shown that the WPM signal exhibits smaller PAPR than [6]. Little about the out-of-band leakage from WPM waveforms is found in the literature. In this paper, we compare with other FBMC techniques including -OQAM and WPM in terms of several metrics that are commonly considered when designing a communication system. The metrics include bandwidth efficiency, computational complexity, latency, out-of-band emission, PAPR, and bit error rate (BER). Sensitivity to frequency and timing offsets is also measured and compared. An efficient implementation of the -OQAM synthesis and analysis filter banks when the prototype filter is of odd length is also proposed. II. FILTER BANK MULTICARRIER SYSTEMS Fig.1 shows a block diagram for a general FBMC system.the transmitted signal, x[n], can be expressed as, where M is the number of subcarriers, s, is the l th symbol in the k th subcarrier, L s is the number of samples per transmit symbol spacing, and is the synthesis filter for the k th subcarrier. At the receiver, the estimated l th symbol s, in the k th subcarrier is, (2) where y[n] is the received signal and is the impulse response of the analysis filter for the k th subcarrier. Define the number of samples per symbol duration as L, where. Note that L s and L are not necessarily the same. For an ideal channel where, a QAM symbol s, in and WPM will be the same as the input symbol s, if the filter satisfies the orthogonal condition:, (3) where is the Kronecker delta. Define the l th input symbol vector along all subcarriers as,,,, (4) (1) /12/$ IEEE

2 the corresponding output signal vector from the transmitter as , (5) the corresponding received signal vector at the receiver as , (6) and the estimated l th symbol vector as,,,. (7) In the following subsections, we give the synthesis and analysis filters for three different MCM schemes in details. s, l s kl, s M 1, l g gk gm 1 x[ n ] yn [ ] f fk fm 1 sˆ, l ŝ kl, ŝ M 1, l Figure 1. General structure for an FBMC system. A. In, the input symbols, are QAM symbols and. The synthesis and analysis filters are defined as (8) where /. The prototype filter 1/ for,1,, 1 and elsewhere. Substituting (8) into (1), after some math manipulations, the transmitted signal for the l th symbol is obtained as 1 (9) where the (i,j) th element of the DFT matrix is,, 1,, (1) and the inverse DFT (IDFT) matrix is given by /. stands for the identity matrix. Similarly, substituting (8) into (2), after some math manipulations, the estimated l th symbol is obtained as 1 (11) The transmitter can be implemented by using zeropadding, inverse fast Fourier transform (IFFT) and parallel-toserial (P/S) conversion; the receiver can be implemented by using serial-to-parallel conversion (S/P) conversion and fast Fourier transform (FFT). For multipath channels, cyclic prefix (CP) is usually used in. B. -OQAM In -OQAM, the input symbols, are OQAM symbols and /2. The OQAM symbols are defined as,, (12) where, is the real input sequence:,,, 2 (13),, 2 1 where, and, are the real and imaginary parts of the th input QAM symbol,, respectively. In (12), the factor introduces the π/2 phase shift between any pair of adjacent OQAM symbols (in both the frequency and time) to ensure orthogonality. In -OQAM, (3) is not satisfied. According to (12), the estimated real sequence is,, (14) And, according to (12), the estimated th QAM symbol is,, (15), As in [3], the synthesis and analysis filters are (16) where 1/2, and the length- prototype filter is a square-root Nyquist filter with roll-off factor not greater than 1. The prototype filter should be designed to strike a balance between minimizing structure based interference, and channel/radio impairment mitigation. The prototype filter design is out of the scope of this paper and we will report it in the future. In this paper we use the frequency sampling based prototype filters [3] in our simulation. We further assume, where the integer K is the overlapping factor (the number of symbol durations in the prototype filter length), and is an odd integer. So, is an integer in this case. The synthesis and analysis filters can be implemented in different ways. An efficient implementation of the -OQAM will be presented in Sec. III. C. WPT WPM is a multicarrier modulation based on WPT. In WPM, the input symbols, are QAM symbols and. The synthesis filter in (1) is derived by having the input data symbols go through several sub-stages where the input of each sub-stage first gets upsampled by 2, and then goes through a pair of filters and. A 2 sub-stage (4 subcarriers) example is shown in Fig.2. Let, and denote the Z-transform of, and, respectively. Then,,,,,,,,. Figure 2. Block diagram of a WPM transmitter example with M = 4. (17) If an additive white Gaussian noise (AWGN) channel is assumed, the structure of the receiver is simply the reverse of the transmitter, where the analysis filter in (2) is determined by another pair of half-band low-pass filter and half-band high-pass filter, respectively, as shown in Fig.3 (4 sub-carriers, i.e., 2 sub-stages, are drawn in this figure). These two pairs of filters are jointly designed so that (3) is /12/$ IEEE

3 satisfied. In this paper, we use Daubechies (Db) wavelets [7] of different lengths as,, and. Figure 3. Block diagram of a WPM receiver example with M = 4 III. EFFICIENT IMPLEMENTATION OF -OQAM Efficient polyphase implementations of -OQAM synthesis filter bank (SFB) and analysis filter bank (AFB) have been investigated in [1]-[4] (IFFT-based SFB and AFB for 1 [1] and for general [4], IFFT-based SFB and FFT-based AFB [2]-[3]). To make a system operable using with either or -OQAM, the IFFT-based SFB and FFT-based AFB should be chosen. It has been shown in [2]-[3] that the complexity of such implementation for the special case of 1 has lowest complexity. In this section, we extend it to a novel efficient implementation for the case when is an arbitrary odd integer. A. Synthesis Filter Bank (SFB) The Z-transform s of the synthesis filters are 1 (18) where (19) 1. (2) The polyphase filters in (19) are defined as,,1,,1. (21) Define the Z-transform of the input symbols (the Z- transform is performed along the symbol index l) as. Similar to [3], the Z-transform of can be written as. (22) where 1 (23) Then, can be further rewritten as (24) where. (25) and the permutation matrix 1 (26) From (24), the -OQAM SFB can be implemented by using zero-padding, IFFT, permutation, polyphase filters, and P/S conversion (via upsampling by /2 and delay chain). Its polyphase structure is given in Fig.4. B. Analysis Filter Bank (AFB) Define the Z-transform of the received signal as and that of the estimated symbols (the Z-transform is,,,, performed along the symbol index l) as. Since the analysis filters are the same as the synthesis filters, in an AWGN channel, the IFFT-based AFB could be written as. (27) Note that the DFT and IDFT matrix could be related through (28) where is the permutation matrix (29) 1 Substituting (28) into (27), we obtain, (3) where the permutation matrix. (31) Based on (3), the FFT-based -OQAM AFB can be implemented by using S/P conversion (via delay chain and downsampling by L/2), polyphase filters, permutation, FFT, and discarding irrelevant outputs. Its polyphase structure is given in Fig.5. s,l s 1,l s 2,l s M-1,l L-point IFFT b b+1 L-1 1 b-1 P πb 1 2 L-1 A (z 2 ) A 1(z 2 ) A 2(z 2 ) A L-1(z 2 ) Polyphase filtering + x[n] Z -1 + Z -1 + Z -1 P/S conversion Figure 4. Polyphase implementation of the -OQAM SFB Figure 5. Polyphase implementation of the -OQAM AFB Note that the inputs to the SFB (resp. the outputs from the AFB) are purely real or imaginary, and so, a L/2-point cosine modulated filter bank and a L/2-point sine modulated filter bank could be used to replace the IFFT at the SFB (resp. the FFT at the AFB), as in the critically sampled exponentially modulated filter bank [8]. As such, the AFB s complexity could be reduced almost by half. However, if the system design strategy is to make it operable using either or /12/$ IEEE

4 -OQAM, the IFFT-based SFB and FFT-based AFB structure should be chosen. IV. SYMBOL DENSITY, COMPLEXITY AND LATENCY A. QAM Symbol Density Define the time and frequency spacings for transmitting one QAM symbol (equivalently, two PAM symbols) as T QAM and F QAM, respectively. The bandwidth efficiency [9] in terms of QAM symbol/sec/hz, is defined as 1 (32) Denote one symbol duration as (in seconds). In, assuming using CP, 1/, and where is the CP duration. In -OQAM, 1/, and. In WPM [1], for each sub-stage of the transmitter, the time resolution is doubled and the frequency resolution is halved. Therefore the product of these two values remains 1. The QAM symbol densities of the three FBMC systems are summarized in Table 1. It is clear that, with CP, is less bandwidth efficient than -OQAM and WPM. B. Computational Complexity Based on the FBMCs efficient implementations described in the previous sections, we discuss their computational complexities in the critically sampled scenario, i.e.,. We evaluate the complexity in terms of the number of real multiplications (but multiplications with 1 and are not included since they are flip of sign and/or flip of real and imaginary parts) per M QAM input symbols. For, note that the number of real multiplications of an -point FFT/IFFT (via Split Radix FFT [11]) with L complex inputs is log 34. Therefore, the total number of real multiplications of per M QAM symbols is 2 2log 68. For -OQAM, note that, via Split Radix FFT [11], the number of real multiplications of an -point IFFT with L purely real or imaginary inputs is log 3 4. Therefore, the total number of real multiplications of - OQAM per M QAM symbols (equivalently, 2M PAM symbols) is log According to [5], the total number of real multiplications of WPM per M QAM symbols is 4 1, where denotes the length of filters and. For example, the Dbx wavelet is of length 2x, where x is a positive integer. The complexities in terms of numbers of real multiplications in, -OQAM with typical K values, and WPM with various filters (Db1 and Db7 [7]) are shown in Table 2. We see that the computational complexities of the -OQAM and WPM could be up to 6 and 7 times higher than the, respectively. This more or less reflects the difference in implementation cost (e.g., gate and memory counts if implemented in ASIC) although they don t map to the cost directly. However, due to the decoder complexity, the contribution of the MCM processor to the overall receiver complexity is not as significant. Therefore the increased complexity of the -OQAM or WPM processors relative to the processor may not be of concern when considered in the context of the entire baseband processor. TABLE 1 QAM SYMBOL DENSITY (SYMBOL/SEC/HZ) -OQAM WPM / 1 1 TABLE 2 COMPUTATIONAL COMPLEXITY ( 1) K = 3 K = 4 K = 5 Db1 Db TABLE 3 LATENCIES T CP 5.2μs 4.7μs 16.7μs.667ms.719ms.714ms.834ms TABLE 4 -OQAM LATENCIES K ms.3667 ms.4334 ms TABLE 5 WPM LATENCIES (ASSUMING Dbx Db1 Db4 Db ms.5333 ms.9333 ms C. Latency The latency discussed in this sub-section is the inherent latency introduced in FBMC s structure from the input of the SFB to the output of the AFB. The latency due to the arithmetic operations (e.g., multiplication) is ignored since they depend on hardware or DSP implementation. Let the sample duration be /. In, latency comes from the P/S and S/P conversion pair and CP (of duration ). The latency due to an L:1 P/S and 1:L S/P conversion pair is. The total latency is. In -OQAM, the prototype filter support is 1. Latency comes from the OQAM modulation (which is T/2), P/S and S/P conversion pair, and filtering. The total latency is. In WPM, latency comes from the P/S and S/P conversion pair and filtering. The latency caused by filtering is 1 1 1, then the total latency is 1 1. Assuming the symbol duration T = 66.7μs, the latencies of for three different CP durations used in 3GPP LTE, of -OQAM for typical K values, and of WPM for various filters (Db1, Db4 and Db7 [7]) are shown in Table 3, Table 4 and Table 5, respectively. Note that Table 5 shows the latency upper bounds by assuming. It is clear that the latency from MCM for -OQAM and WPM is much higher than that for with CP. However, comparing to the latency requirement in IMT-Advanced (1ms for control plane and 1ms for user plan), such an increase in latency should still be manageable, especially for -OQAM. V. EVALUATION OF TRANSMITTED SIGNAL Simulations are carried out for a system with 64 subcarriers. Assuming subcarrier spacing 15 or the symbol duration T = 66.7μs, the total bandwidth of the system is 96 khz. Let the number of samples per symbol, L, be 128 unless /12/$ IEEE

5 specified. Then, the sample duration is.52μs. QPSK symbol modulation is assumed unless specified. In, CP is not used. In -OQAM, the frequency sampling prototype filters (with three typical K values) that minimize the total structure based interference [3] are used. In WPM, two Daubechies wavelets, Db1 and Db7, are used. A. PAPR The PAPR of the l th transmitted symbol is defined as (33) PAPR 1 where is defined in (1). Fig.6 shows the complementary cumulative distribution function (CCDF) of PAPRs for different MCM signals when L = 64. The PAPR performance of -OQAM does not depend on the K value so that only the results of K = 3 is shown for simplicity. It is observed that the PAPR performance depends primarily on the number of subcarriers and is insensitive to the type of FBMC schemes considered in this paper. CCDF WPM(db1) -OQAM(K=3) WPM(db7) PAPR reference value (db) Figure 6. CCDF of the three FBMC systems. B. Power Spectral Density (PSD) Fig.7 shows the PSDs of the three types of FBMC signals. The -OQAM has the largest stopband attenuation because its prototype filter has small sidelobes. In addition, the stopband attenuation for K = 4 is about 2dB larger than that for K = 3. The WPM has the worst out-of-band leakage because its overall synthesis filters have large sidelobes. The WPM with Db7 has smaller out-of-band leakage than the WPM with Db1 since the former uses longer filter than that of latter. Note that in this simulation, the impact from nonlinear PA was not considered. With nonlinear PA, the out-of-band emission for all MCM schemes will increase and difference among them could be reduced. The investigation of the performance using nonlinear PA will be left as future work. VI. PERFORMANCE EVALUATION AT RECEIVER In this section, the simulation parameters are the same as those in Section V. A. Bit Error Rate (BER) Fig.8 shows the BER performance comparison using QPSK and 16QAM modulation in AWGN channel. The unit of the abscissa, Es/No, stands for the symbol signal-to-noise ratio. The three FBMC systems obtain almost the same BER performances for each modulation scheme and they are consistent with the theoretical results. PSD(dB) f(khz) Figure 7. PSDs of, -OQAM (K = 3 & 4), and WPM (Db1 & Db7). BER Dashed line: 16QAM Solid line:qpsk WPM-Db7 WPM-Db1 -OQAM(K=3) -OQAM(K=4) Theory -OQAM(K=3) WPM-Db1 WPM-Db Es/N(dB) Figure 8. BER performance of, -OQAM and WPM. B. Sensitivity to Timing Offset (TO) and Carrier Frequency Offset (CFO) As and other multicarrier schemes, -OQAM is sensitive to TO and CFO. Table 6 shows the real part of the estimated input symbol for -OQAM. In Table 6, is the TO in seconds, and,, (34) represents the interference of the signal modulated at the m th subcarrier to that at the k th subcarrier due to CFO with and (35) 2 where is the Fourier transform of. It is shown that both ICI and ISI are generated when there is TO or CFO. Since the stopband attenuation is large for OQAM-, the TO-generated-ICI only exists between adjacent subcarriers. This feature can be used in designing TO estimation. Fig.9 and Fig.1 display the BER performance degradation of the three FBMC systems due to TO and CFO, respectively. For, CP was not included so that all MCM schemes have the same bandwidth efficiency. Here we assume the channel is noise free so that the only distortion to the signal is from the ISI and ICI. For the case with CFO, pilots are /12/$ IEEE

6 inserted every 1ms for the receiver to correct the phase error caused by the CFO. From Fig.9 and Fig.1, we see that WPM (Db1) is least sensitive to TO and CFO, compared to others. Compared to, -OQAM is more sensitive to TO (or CFO) when TO (or CFO) is small but less sensitive to TO (or CFO) when TO (or CFO) is large. The break point depends on the constellation size. Besides, in asynchronous uplink or cognitive radio -OQAM systems are more robust than systems to misalignments among users because of its sufficient stopband attenuation. VII. CONCLUSION In this paper we reviewed three different MCM techniques, i.e.,, -OQAM and WPM, and modeled them under the same FBMC framework. For -OQAM, we proposed an efficient implementation of the IFFT-based SFB and FFT-based AFB by using simple permutations to avoid complex multipliers. We compared these MCM techniques using multiple metrics. In general, -OQAM and WPM are more bandwidth efficient than since they don t need CP. -OQAM has much lower out-of-band emission comparing to and WPM, which may make it more suitable to cognitive radio systems. On the other hand, for -OQAM the computational complexity is about 6 times greater and the latency is 5 times longer than, but still in the manageable range for practical systems. For WPM, those numbers could be even larger. In terms of PAPR, the three MCM techniques with their nominal corresponding parameters have very similar characteristics. Regarding the timing and frequency offsets, WPM is least sensitive. - OQAM is less sensitive to those offsets for large offset values, but more sensitive for small values. So, for -OQAM, to achieve better in-band performance, tighter time and frequency error control is needed. Timing and frequency error correction and the investigation into performance with a nonlinear PA is left as future work. REFERENCES [1] P. Siohan, C. Siclet and N. Lacaille, "Analysis and design of /OQAM systems based on filterbank theory," Signal Processing, IEEE Transactions on, vol.5, no.5, pp , May 22. [2] A. Viholainen, T. Ihalainen, T. Hidalgo, M. Renfors and M. Bellanger. Prototype filter design for filter bank based multicarrier transmission, in Proc. of 17th European Signal Processing Conference, Aug. 29. [3] A. Viholainen, M. Bellanger and M. Huchard, Prototype filter and structure optimization, Physical Layer for Dynamic Spectrum Access and Cognitive Radio, Jan. 29. [4] B. Farhang-Boroujeny, "Tutorial: Filter Bank Multicarrier for Next Generation of Communication Systems," 21 Wireless Symposium and Summer School at Virgina Tech, June 3, 21. [5] A. Jamin and P. Mahonen, Wavelet packet Modulation for Wireless Communications, Wireless Communications & Mobile Computing Journal, John Wiley and Sons Ltd., vol. 5, no. 2, p , Mar. 25. [6] M. Gautier, C. Lereau, M. Arndt, and J. Lienard, PAPR analysis in wavelet packet modulation, in Proc. of 3rd International Symposium on Communications, Control and Signal Processing (ISCCSP), Mar. 28. [7] I. Daubechies, Ten Lectures on Wavelets, SIAM Publications, [8] A. Viholainen, J. Alhava, and M. Renfors, Efficient implementation of complex modulated filter banks using cosine and sine modulated filter banks, EURASIP Journal on Advances in Signal Processing, vol. 26, Article ID 58564, 1 pages, doi: /ASP/26/58564, 26. [9] B. Farhang-Boroujeny and C. H. G. Yuen, Cosine modulated and offset QAM filter bank multicarrier techniques: A continuous-time prospect, EURASIP Journal on Advances in Signal Processing, vol. 21, Article ID , 16 pages, doi: /21/165654, 21. [1] M. K. Lakshmanan and H. Nikookar, A Review of Wavelets for Digital Wireless Communication, Wireless Personal Communications: An International Journal, vol.37 no.3-4, p , May 26. [11] H. S. Malvar, Signal Processing with Lapped Transforms, Artech House, Boston, Mass, USA, CFO : Hz TO: / BER TABLE 6 ESTIMATED DATA AT THE RECEIVER, 1 2,,, , 1 2,, 2, 3 2, 1 2 Figure 9. BER performance as function of TO. -OQAM (K=3) WPM (Db1) Dashed line: QPSK Solid line:16qam Frequency Offset(Hz) Figure 1. BER performance as function of CFO /12/$ IEEE