Frequency domain concurrent channel equalization for multicarrier systems

Receivers for wireless Orthogonal Frequency Division Multiplexing (OFDM) systems usually perform the channel estimation based on pilot carriers in known positions of the channel spectrum. Interpolation between pilot carriers is applied to determine the channel transfer function in all carrier frequencies. Channel variations along time are compensated by means of interpolation between successive channel estimates on the same carrier frequency. However, not rarely, the fast channel variations exceed the time interpolator capability, as is the case for mobile operation. In this article we present a new channel compensation technique based on the concurrent operation of two stochastic gradient timedomain algorithms, one which minimizes a cost function that measures the received signal energy dispersion and other which minimizes the Euclidean distance between the received digital modulation symbols and the ones in the reference constellation assigned to each OFDM sub-channel. Results show that the new technique advantageously improves the system robustness to fast channel variations since, with a low computational cost, it dramatically reduces the demodulator symbol error rate even when the receiver is operating in an intense dynamic multipath scenario.

,1752'8&7,21 The Concurrent Equalizer (CE) is a blind deconvolution algorithm based on the principle of the concurrent operation between the direct decision (DD) equalizer and the constant modulus (CMA) equalizer [2]. The minimization of the Euclidean distance based DD cost function only takes place when the minimization of the energy dispersion based CMA cost function is judged to have achieved a successful adjustment with high certainty. Certainty is measured as the closeness of the output to the same IQ symbol in the reference constellation before and after a perturbation is imposed to the equalizer. Due to the high performance improvement achieved with this approach, the CE has been considered as a state of the art technique in the context of blind channel deconvolution [3].

)iELR ' ¶$JRVWLQL 6LUOHVLR &DUERQL -~QLRU )UHTXHQF\ 'RPDLQ &RQFXUUHQW 0DULD & ) 'H &DVWUR DQG )HUQDQGR & & 'H &DVWUR &KDQQHO (TXDOL]DWLRQ IRU 0XOWLFDUULHU 6\VWHPV
The CE is a time domain adaptive algorithm which was originally proposed for single carrier systems [2]. In this work we extend the use of the CE to multicarrier OFDM systems by applying the same principle in the frequency domain. Figure 1 shows the simplified model for a typical OFDM demodulator and the usual symbol aided channel estimation scheme [5]. Reference symbolsso-called pilot symbols -are applied to pilot sub-channels (carriers) before the IFFT at the transmitter. At the receiver, the pilot sub-channels in the FFT output are estimated by the received known pilot symbols at its respective frequencies of the channel spectrum.  Figure 2 shows an example of the grid resulting after the FFT for a received OFDM frame. Notice the rectangular arrangement of the symbol pilots, which denotes it as a rectangular pilot grid. The distance between two pilot symbols in frequency direction is I 1 carriers.
The distance between two pilot symbols in time direction is W 1 OFDM symbols.
The implementation of a control system as a program with concurrent objects is based on concurrency models of multithreading, which allows the representation of a system as a network of cooperative elements, in this case the Agents. The use of the same base class for deriving all classes guarantees the uniformity of interface for all system elements, thus making the program easier and minimizing implementation errors.
Let Q and L respectively be the frequency and time indexes in the grid for which the OFDM transmitter assigns a pilot symbol L Q 6 . These indexes are given by 1 is the spacing between two pilot symbols in time direction, and ª º ⋅ is the operator which returns the smallest integer larger than or equal to its argument. Notice we are assuming here that the first pilot symbol in the grid is located at the first carrier of the first OFDM symbol in an OFDM frame. Thus, the number of pilot symbols in an OFDM frame is given by However, in order to properly minimize the multipath effects in frequency domain, an OFDM receiver has to estimate the channel transfer function in all carrier frequencies between two pilot symbols spaced by I 1 in frequency direction and not only in those carrier frequencies for which the OFDM transmitter assigns a pilot symbol. Further, the channel usually presents a dynamic behavior so that its transfer function changes with time.
Thus, the OFDM receiver has to estimate the channel transfer function in all OFDM symbols between two pilot symbols spaced by W 1 OFDM symbols in time direction. Therefore, it is clear the need for some interpolation procedure between adjacent pilot symbols in time and in frequency. Indeed, the estimate of the complete channel transfer function for the OFDM frame represented in the grid is obtained by interpolating L Q + samples so that we can obtain L Q + Ö not only in grid positions which contain pilot symbols L Q 6 but also between them: Notice that the number of filter coefficients WDS 1 is given where {} ⋅ is the operator which returns the cardinality of its argument set. In Figure 2, for example, In practice, however, due to the high computational cost of a two-dimensional interpolating filter, the two-dimensional interpolation process is decomposed in two cascaded one-dimensional interpolation filters working sequentially [8].
In general, interpolation is first performed in time direction and then subsequently in frequency direction. Firstly, for a given frequency index Q′ to which a pilot symbol is assigned, the estimation of L Q + Ö ′ for a particular time index L is performed by a time direction interpolation filter with impulse response given by coefficients

)iELR ' ¶$JRVWLQL 6LUOHVLR &DUERQL -~QLRU )UHTXHQF\ 'RPDLQ &RQFXUUHQW 0DULD & ) 'H &DVWUR DQG )HUQDQGR & & 'H &DVWUR &KDQQHO (TXDOL]DWLRQ IRU 0XOWLFDUULHU 6\VWHPV
Subsequently, for a given time index L , the estimation of L Q + Ö for a particular frequency index Q is performed by a frequency direction interpolation filter with impulse response given by coefficients (8) are obtained for all frequencies Q′ to which a pilot symbol is assigned, they can be used with any given time index L as pilot symbol references in the subsequent frequency direction interpolation: The time direction filter coefficients  (11)  For the frequency direction interpolation we have where I ω is the vector whose WDS 1 components are the frequency direction filter coefficients which results from the frequency direction cross-correlation between the channel transfer function values and its estimates at frequency positions corresponding to pilot symbols, and whose WDS 1 components Q Q ′ θ are given by where 6 ) is the carrier frequency spacing and τ is the maximum expected delay spread in the channel [5].
The new channel compensation technique presented in this article is based on the CE algorithm, which is a concurrent time direction adaptive algorithm briefly described in the introduction section of this article. In the present work, the CE performs a further adjustment on the channel transfer function estimates obtained from a linear interpolation procedure between two adjacent pilot symbols in frequency direction. The reason for the adoption of linear interpolation instead of Wiener filter interpolation is twofold. First, the linear interpolator complexity is quite lower than the Wiener filter complexity. Second, the concurrent time domain channel compensation perform a local channel estimation, local in the sense that it ignores dependencies among pilot symbols far from each other. Thus, if initialized by estimates that result from Wiener filter interpolation, the concurrent time direction local compensation is "perturbed" by non-local pilot symbols, which might start the gradient in a less optimal descent path.
In order to keep spectral uniformity [1], this work adopts a scattered pilot symbol grid as shown in Figure 3, instead of the fixed pilot symbol grid of Figure 2. When using linear interpolation between two adjacent pilot symbols on the scattered pilot grid shown in Figure 3, the interpolation procedure is similar to the two cascaded one-dimensional interpolation Wiener filters. Firstly, for a given frequency index Q′ to which a pilot symbol is assigned, the estimation of Notice that L Q + Ö given by (17) (16) and (17)

&21&855(17 )5(48(1&< '20$,1 (48$/,=$7,21
The new technique presented in this article proposes a further update in the compensated grid samples , L Q < aiming to minimize the dynamic multipath effects, so that the further compensated grid yields L Q ,, L Q 6 < ≈ . Since all channel information stored in the pilot symbol grid has been used as reference to determine , L Q < , in order to perform any further update it is necessary to rely on references which doesn't depend on the pilot symbol grid. The CE, briefly described in the introduction section of this article is a high performance and low complexity blind adaptive algorithm which is perfectly suited for the task of performing a further update on , L Q < with no dependency on the pilot symbol grid. Figure 4 shows the block diagram for the CE-based channel compensation technique adopted for the deconvolution of each OFDM sub-channel in the OFDM frame, which we called Frequency Domain ConcurrentChannel Equalizer (FDCCE).
With reference to Figure 4, the CE-based channel compensation technique can be described as follows: the receiver samples the channel at twice the IFFT rate at the transmitter [4], so that after discarding the guard interval, the received & 1 7 −spaced complex baseband sequence is stored in two input buffers of size & 1 at the FFT input. It is assumed that the receiver clock recovery system is such that even-index samples of the received baseband sequence are synchronized to the samples of the IFFT output sequence at the transmitter. Thus each input buffer stores a 1 is the number of OFDM symbols per OFDM frame. Since the CE is a gradient based blind deconvolution algorithm, each block Φ corrects a possible multiple $ rotation of each CE output \ with respect to the reference IQ symbol constellation.

6,08/$7,21 5(68/76
In this section we evaluate the performance of the Frequency Domain Concurrent Channel Equalizer (FDCCE) proposed in this article when it is applied to the ISDB-T digital television broadcast system. Specifically, we compare the symbol error rate (SER) [1] of the IQ baseband output sequence in Figure 4, with and without the aid of the FDCCE, for the ISDB-T system operating in Mode I: 2048 total carriers, 64-QAM mapping, OFDM symbol duration 252μs and ¼ guard interval [6]. For the performance evaluation, we adopt the Brazil A-E channel multipath profiles [7] shown in Tables 2-5 with a channel bandwidth of 6 MHz. In order to evaluate the FDCCE performance under dynamic multipath, we apply Doppler rotation to the echo with highest amplitude level [10]. Since the purpose of the simulation is only to compare performances between channel estimation methods, it is assumed that there is no synchronization error.

&21&/86,21
The proposed Frequency Domain Concurrent Channel Equalizer (FDCCE) outperforms the classical channel estimation and compensation approaches for OFDM systems, such as Wiener interpolation and linear interpolation.
The adoption of the FDCCE after the linear interpolation procedure results in a lower symbol error rate when compared with the Wiener interpolator, while keeping the computational cost in a low level.
Actually, for a moderate multipath scenario such the Brazil A profile, and for a SNR above 27.5 dB, we obtained a zero symbol error rate, even when the mobile receiver is operating in an intense dynamic multipath scenario (Fdoppler = 100Hz).