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Sliding Window-Frequency Domain Equalization for Multi-mode Communication Systems
Research Paper / Feb 2014

978-1-4577-1343-9/12/$26.00 ©2013 IEEE Sliding Window-Frequency Domain Equalization for Multi-mode Communication Systems Jialing Li1, Erdem Bala2, Rui Yang3 InterDigital Communications Inc. Melville, NY Jialing.Li@interdigital.com1, Erdem.Bala@interdigital.com2, Rui.Yang@interdigital.com3 Abstract— Future wireless communications should be designed efficiently to support multi-mode operations for different types of applications, such as machine-to-machine, video streaming, web browsing and voice. In the physical layer multi-mode operation could include the coexistence of single carrier and multicarrier modulations. To reduce overall complexity and power consumption, it is highly desirable to design the receiver such that some of the components are shared or reused by multiple modulation schemes. In this paper we propose a receiver architecture which is composed of a common channel equalizer for all carrier modulation schemes. Furthermore, we investigate the feasibility of using a well-studied time domain equalizer designed for single carrier modulation, i.e., the sliding window- frequency domain equalizer (SW-FDE) for general multicarrier modulation (MCM) receivers. As a specific illustration of the concept, we present the application of this equalizer to an OFDM-offset QAM system. Keywords-multi-mode system; orthogonal frequency division multiplexing-offset quadrature amplitude modulation; OFDM- OQAM; equalization I. INTRODUCTION MCM techniques enable transmission of a set of data over multiple narrow band subcarriers simultaneously. With an advanced wideband modulation and coding scheme, a system with MCM can achieve much higher spectral efficiency in frequency selective channels compared to those using single carrier modulation techniques. Orthogonal Frequency Division Multiplexing (OFDM) may be the most well-known MCM scheme that has been used in many standardized systems [1]. To support future systems with high spectral agility, such as cognitive radio, Filter Bank Multicarrier, e.g., OFDM-offset QAM (OFDM-OQAM), and its application to various multiple access systems have been investigated by many researchers recently [2]. OFDM can be considered as a type of filter bank multicarrier (FBMC) modulation whose time domain prototype filter is a simple rectangular pulse. Such a pulse, however, introduces large sidelobes, which create several challenging issues in practical systems, including high sensitivity to frequency offsets and large out-of-band emissions (OOBE). In OFDM-OQAM, a carefully designed prototype filter is used so that OOBE is much less than that of OFDM [3], [4]. The signals of adjacent subcarriers overlap each other to achieve a high spectral efficiency. Different from OFDM, 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 side lobes small. In general, the channel equalization for OFDM-OQAM in fading channels could be grouped into two main categories: The first one is equalization before the OFDM-OQAM analysis filter bank (AFB), which may require using a cyclic prefix (CP) in the transmit signal and results in low spectral efficiency and large OOBE [2], [5]. The second one is equalization after the OFDM-OQAM AFB, e.g., the most popular subcarrier-based equalization, with single tap or multiple taps [6]-[18]. It has been shown that even in mildly selective channels, a per-subcarrier multi-tap equalizer is needed for decent BER performance. Therefore, it is evident that to achieve good performance in highly frequency selective channels with long delay spreads, a very complex equalizer is required. In this paper, we consider using the sliding window frequency domain equalizer (SW-FDE) [19] in multi-mode systems. The equalizer is applied to the received signals before demodulation, and therefore, can be applied to MCM as well as single-carrier (SC) signals. As a specific illustration of the concept, we present the application of this equalizer to an OFDM-OQAM system, where the equalization is performed before the AFB. The same idea has been used in single carrier systems [19]-[20] as well as OFDM systems [21]. The proposed equalizer uses a sliding window to perform equalization for the central portion of the samples of each window (block) and slides the window along the block to cover all received samples. In the equalization of each block, it approximates the system model as if the channel impairment is a circular convolution with the channel impulse response. The inherent interference due to this approximation is analyzed. Simulations show that the SW-FDE outperforms the simple and practical single-tap minimum mean square error (MMSE) equalizer after the OFDM-OQAM AFB in terms of raw BER performance in highly selective channels. The raw BER performance due to inherent interference is also compared for different block lengths. The rest of the paper is organized as follows: In Sec. II, the OFDM-OQAM system model and architecture is presented. Our proposed SW-FDE is described in Sec. III. Then, simulation results for practical channel models are given in Sec. IV, followed by conclusions in Sec. V. II. SYSTEM MODEL AND ARCHITECTURE Without loss of generality, in this paper, we consider a FBMC system based on OFDM-OQAM shown in Figure 1, in which the SW-FDE along with a channel estimator is set before the AFB. The detailed description of the SW-FDE will be given in the next section. Figure 1. The high level system architecture The synthesis filter bank (SFB) and AFB of the system based on OFDM-OQAM may be implemented in several ways; each has different complexity in terms of number of arithmetic calculations. The structures of the SFB and AFB in this paper are shown in Figure 2. This design requires that the prototype filter length, ܮ௣, is an odd number. In Figure 2, ܮ is the FFT size, ܯሺ൑ ܮሻ is the number of active subcarriers, and ܾ ൌ ௅೛ିଵଶ . ሼݏ௞ைሾ݊ሿ, ݇ ൌ 0, … , ܯ െ 1ሽ are the OQAM modulated, real valued, symbols. And, ሼܣ௞ሺݖሻ, ݇ ൌ0, … , ܮ െ 1ሽ is the polyphase filters (with respect to ܮ) of the corresponding prototype filter, in which ܣ௞ሺݖሻ ൌ ∑ ݌௘ሾݍ ൅ ݉ܮሿݖି௠ඃ௅೛/௅ඇିଵ௠ୀ଴ and ݌௘ሾ݊ሿ is the prototype filter coefficients. The permutation operation in Figure 2(a), ࡼగ௕ , for an integer ܾ is defined by the permutation matrix ۾గ௕ ൌ ൤ ૙௕ൈሺ௅ି௕ሻ ۷௕ ۷௅ି௕ ૙ሺ௅ି௕ሻൈ௕൨ (1) where ۷௫ is the rank-ݔ identity matrix and ૙௫ ൈ ௬ is an ݔ by ݕ zero matrix. And the permutation operation in Figure 2(b), is defined by the permutation matrix ۾గ௔ ൌ ۏێ ێێ ۍ1 00 ڮ ڮ ڮ 0 ڮ 0 1 0 ڮ ڭ گ 0 1 0 1 0 گ گ ڭ 0 ڮ 0 ے ۑۑ ۑ ې ۾గ௕் (2) It should be mentioned that the AFB can be implemented using either an FFT or an IFFT. Here, similar to [4], we choose to use the FFT based implementation of AFB since it can fall back to OFDM operation to enable a multi mode design, even though the IFFT based implementation could be less complex. On the other hand, comparing to the design in [4], the proposed SFB and AFB with permutation operation can provide a less complex implementation. Regarding the structure shown in Figure 1, channel estimation is performed before the AFB. In practice, this can be achieved by attaching a preamble with reference signals for each block of transmitted OFDM-OQAM modulated signals, similar to 802.11 standards [22]. As an alternative, channel may be estimated after the AFB by using scattered pilots [23]. However, this would result in a delay and higher complexity since the received signal has to be processed by the AFB twice. The use of a channel equalizer prior to the demodulation process does not depend on the type of waveform as long as the reference signal is designed properly to support the channel estimation. This means that the architecture shown in Figure 1 can be applied to any waveform. For a multi-mode system that supports multiple waveforms, this type of equalizer can be shared or reused so that the overall complexity and cost is reduced. (a) Synthesis Filter Band (SFB) L-point FFT A0(z2) A1(z2) A2(z2) AL-1(z2) ↓L/2 ↓L/2 ↓L/2 ↓L/2 Z-1 Z-1 Z-1 y[n] * * û0[n] û1[n] û2[n] ûL-1[n] Polyphase filtering S/P conversion ŝO0 [n] ŝO1 [n] ŝO2 [n] ŝOM-1 [n] Pπa ûb[n] ûb-1[n] û0[n] ûL-1[n] ûL-2[n] ûb+1[n] (b) Analysis Filter Bank (AFB) Figure 2. The SFB (a) and AFB (b) of OFDM-OQAM considered in this paper III. SLIDING WINDOW-FREQUENCY DOMAIN EQUALIZER (SW-FDE) In this section, we introduce our proposed sliding window- frequency domain equalizer (SW-FDE) for OFDM-OQAM. It does not use a CP and hence does not introduce spectral distortion to the transmit waveform. It uses a sliding window to perform equalization for the central portion of the samples of This work was supported by InterDigital Communications Inc. each window (block). In the equalization of each block, it approximates the system model as if the received signal block is a circular convolution of the transmit signal block and the channel impulse response. In such an approximation, the system still suffers from inter-block interference even in the absence of noise, especially at the edges of each equalized block. Therefore, only the central portion of the equalized block is passed to the next stage of the receiver so as to minimize interference, and a sliding window is used to slide the observing window to cover all received samples. A. SW-FDE Sliding Window: Assume the block length is ܮ௕ samples, and the sliding window step size is ܮ௦௦ , which satisfies 1 ൑ ܮ௦௦ ൑ ܮ௕ . The total number of blocks is ݊஻. Denote the ith length-ܮ௕ block of the SFB output ܠ௜ as ܠ௜ ൌ ሾݔሾሺ݅ െ 1ሻܮ௦௦ሿ ݔሾሺ݅ െ 1ሻܮ௦௦ ൅ 1ሿ ڮ ݔሾሺ݅ െ 1ሻܮ௦௦ ൅ ܮ௕ െ 1ሿሿ், ݅ ൌ 0,1, ڮ , ݊஻ െ 1. (3) Note that for the 0th block, the elements of ݔ with negative indices are background interference and noise only. Due to the definition of the sliding window, if ܮ௦௦ ൏ ܮ௕ , there is overlapping between adjacent windows (blocks). As in Figure 1, there is no CP insertion. The ith length-ܮ௕ block of the transmitted signal is equal to the ith length-ܮ௕ block of the SFB output, i.e., ܠത௜ ൌ ܠ௜, ݅ ൌ 0,1, ڮ , ݊஻ (4) Also denote the ith length-ܮ௕ interference block as ܑ௜ ൌ ሾݔሾሺ݅ െ 1ሻܮ௦௦ െ ܮ௕ሿ ݔሾሺ݅ െ 1ሻܮ௦௦ െ ܮ௕ ൅ 1ሿ ڮ ݔሾሺ݅ െ 1ሻܮ௦௦ െ 1ሿሿ், ݅ ൌ 0,1, ڮ , ݊஻ െ 1. (5) It has inter-block interference (IBI) to ܠത௜. The sliding window operation is illustrated in Figure 3. Channel: Then, the ith length-ܮ௕ block of the received signal is ܡത௜ ൌ ۶௅್ܠത௜ ൅ ۶T,௅್ܑ௜ ൅ ܖഥ௜ (6) where the ܮ௕ ൈ ܮ௕ Toeplitz channel matrix ۶௅್ ൌ ۏ ێ ێ ێ ێ ۍ ݄଴ 0 ڮڭ ڰ ڰ ݄௅೓ିଵ ڮ ݄଴ ڮ ڮ 0 ڰ ڰ ڭ 0 ڰ ڭ 0 ݄௅೓ିଵ ڮ ڭ ڰ ڰ 0 ڮ 0 ݄଴ ڰ ڭ ڭ ڰ 0 ݄௅೓ିଵ ڮ ݄଴ے ۑ ۑ ۑ ۑ ې (7) represents the channel seen by ܠത௜. The second ܮ௕ ൈ ܮ௕ matrix ۶T,௅್ ൌ ۏ ێ ێ ێ ێ ۍ 0 ڮ 0ڭ ڰ ڰ ڭ ڰ ڰ ݄௅೓ିଵ ڮ ݄ଵ ڰ ڰ ڭ ڰ 0 ݄௅೓ିଵ ڭ ڰ ڰ ڭ ڰ ڰ 0 ڮ ڮ ڰ ڰ 0 ڰ ڰ ڭ ڮ ڮ 0 ے ۑ ۑ ۑ ۑ ې (8) represents the tail end of the channel’s impulse response that generates IBI to the first ܮ௛ െ 1 samples in the succeeding block. These two matrices have the interesting property of ۶௔ ൅ ۶T,௔ ൌ ۶ୡ,௔ (9) where ۶ୡ,௔ is an ܽ ൈ ܽ circulant matrix with the first column being ൣܐ் ૙ଵൈሺ௔ି௅೓ሻ൧ ் . FDE: Equation (6) could be rewritten as ܡത௜ ൌ ۶ୡ,௅್ܠത௜ ൅ ܖෝ௜ (10) where the circulant matrix ۶ୡ,௅್ ൌ ۶௅್ ൅ ۶T,௅್ is defined in (9), and the interference and noise is ܖෝ௜ ൌ ۶T,௅್ሺܑ௜ െ ܠത௜ሻ ൅ ܖഥ௜ (11) Note that the first ܮ௛ െ 1 samples suffer from the interference that comes from the last ܮ௛ െ 1 samples in ܑ௜ and ܠത௜. Since the circulant matrix ۶ୡ,௅್ satisfies ۶ୡ,௅್ ൌ 1 ܮ௕ ۴௅್ ு ઩௛۴௅್ (12) where ۴௅್ is the ܮ௕ ൈ ܮ௕ DFT matrix ۴௅ ൌ ൦ 1 1 ڭ 1 1 ௅ܹ ڭ ௅ܹ௅ିଵ ڮ ڮڰ ڮ 1 ௅ܹ௅ିଵ ڭ ௅ܹ ሺ௅ିଵሻሺ௅ିଵሻ ൪ , (13) ۴௅್ு is the ܮ௕ ൈ ܮ௕ IDFT matrix, and ઩௛ ൌ ݀݅ܽ݃ሼૃ௛ሽ (14) with ૃ௛ ൌ ൣߣ௛,଴, ߣ௛,ଵ, … , ߣ௛,௅್ିଵ൧ ் ൌ ۴௅್ܐ (15) then, the following equalizer, which is also a circulant matrix, ۳௅್ ൌ 1 ܮ௕ ۴௅್ ு ઩௘۴௅್ (16) Figure 3. Sliding window operation where ઩௘ ൌ ݀݅ܽ݃ሼૃ௘ሽ (17) with ૃ௘ ൌ ൣߣ௘,଴, ߣ௘,ଵ, … , ߣ௘,௅್ିଵ൧ ் (18) could be applied to the ith block, ܡ௜ ൌ ۳௅್ܡത௜ (19) Note that the expression of ߣ௘,௟, for ݈ ൌ 0, … , ܮ௕ െ 1, depends on the type of equalizer. In a minimum mean square error (MMSE) approach, ۳ெெௌா,௅್ ൌ ߪ௫ଶ۶ୡ,௅್ு ൫ߪ௫ଶ۶ୡ,௅್۶ୡ,௅್ு ൅ ߪ௡ଶ۷௅್൯ ିଵ (20) where it is assumed that ܧሺܠ௜ܠ௜ுሻ ൌ ߪ௫ଶ۷௅್ and ܧሺܖෝ௜ܖෝ௜ுሻ ൌߪ௡ଶ۷௅್ . Substituting (12) into (20), after some math manipulations, we obtain ۳ெெௌா,௅್ ൌ 1 ܮ௕ ۴௅್ ு ઩௘,ெெௌா۴௅್ (21) where ઩௘,ெெௌா ൌ ߪ௫ଶ઩௛ு൫ߪ௫ଶ઩௛઩௛ு ൅ ߪ௡ଶ۷௅್൯ ିଵ (22) i.e., ߣ௘,ெெௌா,௤ ൌ ߣ௛,௤כ ߪ௫ଶ หߣ௛,௤หଶߪ௫ଶ ൅ ߪ௡ଶ , ݍ ൌ 0,1, ڮ , ܮ௕ െ 1 (23) The MMSE equalizer in (20) could be implemented by using an ܮ௕ -point DFT, a 1-tap FDE (i.e., with the ߣ௘,௤ - multipliers), and an ܮ௕-point IDFT. Note that due to the block- wise operation, the equalizer itself should have an S/P and P/S conversion pair. The P/S conversion in the equalizer could be combined with the S/P conversion in the OFDM-OQAM AFB to reduce latency. B. Inherent Interference Analysis After the FDE, all recovered samples suffer from interference from the last ܮ௛ െ 1 samples in ܑ௜ and ܠത௜ . Substituting (10) and (11) into (19), the recovered block becomes ܡ௜ ൌ ܠത௜ ൅ ۳௅್۶T,௅್ሺܑ௜ െ ܠത௜ሻ ൅ ۳௅್ܖഥ௜ (24) where ۳௅್۶T,௅್ሺܑ௜ െ ܠത௜ሻ is the residual interference or estimation error. To analyze the residual interference, we consider the case in the absence of noise. In this case, ۳௅್ is essentially ۶ୡ,௅್ିଵ , and the noise term ۳௅್ܖഥ௜ is zero. Examples of the interference powers to each sample are shown for ܮ௕ ൌ ܮ ൌ 1024 samples/block and ܮ௕ ൌ 2ܮ ൌ 2048 samples/block, respectively, in Figure 4 for a 600-channel OFDM-OQAM system with 1024 samples per symbol duration. The blocks are quartered using dotted lines. As can be seen from this figure, the edge quarters have much higher interference than the middle of the blocks. This is the edge effect in SW-FDE. It is due to the much larger coefficients in the first and last few rows in ۳௅್۶T,௅್ . The interference level at either edges is not identical because the coefficients in first few rows of ۳௅್۶T,௅್ have larger magnitudes than these in the last few rows. Comparing the interference powers for different window (block) sizes, we find that the interference powers of the middle half of blocks (i.e., samples ௅್ସ ൅ 1, ௅್ ସ , ڮ , ଷ௅್ ସ of each block) become smaller as the block size increases. The suggests an appropriate sliding window step size to be ܮ௦௦ ൌ ௅್ଶ . (a) SW-FDE with ܮ௕ ൌ ܮ ൌ 1024 (b) SW-FDE with ܮ௕ ൌ 2ܮ ൌ 2048 Figure 4. Interference edging effect in SW-FDE C. Complexity and Latency Discussion The complexity of the FDE is introduced by the ܮ௕-point DFT and IDFT, and the ߣ௘,௤ -multipliers. The total additional number of real multiplications per block is ߤௌௐିி஽ா ൌ 2ߤிி்஼ ሺܮ௕, ܮ௕ሻ ൅ 4ܮ௕ ൌ 2ܮ௕ logଶ ܮ௕ െ 2ܮ௕ ൅ 8. Note that the complexity of calculating the ߣ௘,௤ -multipliers is not included here since the ߣ௘,௤ -multipliers are assumed given by the channel estimation block. Assume the total number of sample of the frame is ܮ௧ ൌ ݊஻ܮ௕. If ܮ௦௦ ൌ ௅್ଶ , its total complexity is 2݊஻ሺ2ܮ௕ logଶ ܮ௕ െ 2ܮ௕ ൅ 8ሻ ൌ 4ܮ௧ ቂlogଶ ܮ௕ െ 1 ൅ ସ௅್ቃ ൌ 4ܮ௧ ቂ݀ െ 1 ൅ ସଶ೏ቃ , where ܮ௕ ൌ 2ௗ . Therefore, as ݀, and correspondingly ܮ௕, get larger, complexity increases. SW-FDE uses a 1: ܮ௕ S/P and ܮ௕: 1 P/S pair and replaces the 1: ܮ S/P conversion in the OFDM-OQAM AFB. Therefore, the additional inherent latency is ߬ௌௐିி஽ா ൌ ܮ௕ ௦ܶ െ ܮ ௦ܶ ൌ ௕ܶ െ ܶ. IV. SIMULATIONS In this section, we present the simulation setup and results. 0 100 200 300 400 500 600 700 800 900 1000 -25 -20 -15 -10 -5 0 5 sample index in the block in te rfe re nc e po w er (dB ) Interferece edging effect in SW-FDE (1024 samples/block) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 -30 -20 -10 0 sample index in the block in te rfe re nc e po w er (dB ) Interferece edging effect in SW-FDE (2048 samples/block) A. Simulation Setup The performance of the SW-FDE equalizer is evaluated with simulations and compared to the single-tap MMSE equalizer. Since the discussion of different equalization (e.g., multi-tap equalizers) after OFDM-OQAM AFB is not the focus of the paper, we only consider the single-tap MMSE equalizer here that is widely used in existing OFDM systems and recommended for OFDM-OQAM systems in mildly frequency selective channels. The point is to demonstrate that our proposed SW-FDE would provide satisfying BER performance for multi-mode systems, e.g., an OFDM-OQAM system, in mildly and highly selective fading channels. The simulation parameters are selected to match the widely used LTE parameters [24]. Specifically, 1024 subcarriers, of which 600 are loaded with data, are used. The subcarrier spacing is set to be 15 kHz and the block size used in the SW-FDE is chosen as ܮ௕ ൌ 4ܮ ൌ 4096. Two vehicular channel models, Vehicular-A (Veh-A) and Vehicular-B (Veh-B), specified by the ITU-R [25] are used. The Doppler shift is set to 5 Hz for both channel models. Veh-A is a moderately selective channel with rms delay spread of 0.37µs while Veh-B is a highly selective with rms delay spread of 4µs. The parameters for the channel models are given in Table I. TABLE I. CHANNEL MODELS USED IN THE SIMULATIONS Tap Vehicular A Vehicular B Delay (ns) Average power (dB) Delay (ns) Average power (dB) 1 0 0 0 -2.5 2 310 -1 300 0 3 710 -9 8900 -12.8 4 1090 -10 12900 -10 5 1730 -15 17100 -25.2 6 2510 -20 20000 -16 B. Simulation Results in Practical Channel Models Figure 5 illustrates the raw BER of the OFDM-OQAM system with the single-tap equalizer and SW-FDE, respectively, when the Veh-A channel model is used. We can see from this figure that the performance of the two equalizers are very close to each other with the SW-FDE showing a slightly better performance than the single-tap equalizer at high SNR. Figure 6 illustrates the raw BER of the OFDM-OQAM system with the single-tap equalizer and SW-FDE when the Veh-B channel model is used. We can see from this figure that the SW-FDE equalizer outperforms the single-tap equalizer at medium and high SNR range for all modulation types. Because Veh-B is a highly selective channel and the delay spread is very large, the performance of the single-tap equalizer degrades significantly. The SW-FDE equalizer, on the other hand, effectively covers the channel delay spread with a sufficiently large block size. At high SNR, however, SW-FDE experiences an error floor due to IBI. The effect of the block length ܮ௕ on the BER performance of the SW-FDE equalizer is also evaluated and the result is illustrated in Figure 7. The block length is varying and equal to ݇ܮ, ݇ ൌ 1, 2, … , 8 . Veh-B channel model is used in this simulation with infinite SNR. As we can see from the figure, the BER reduces with increasing block length. However, the rate of this reduction decreases with increasing block length. Comparing Figure 6 and Figure 7, we could see that with sufficiently large block size (i.e., ܮ௕ ൒ 3ܮ), SW-FDE outperforms the single-tap equalizer. Figure 5. Raw BER performance of OFDM-OQAM in Veh-A channel Figure 6. 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