1. Field of the Invention
The disclosed technology generally relates to single-carrier communication systems wherein adaptive feedback equalization is applied.
2. Description of the Related Technology
In outdoor wireless applications the multipath channel exhibits very long impulse responses, which can lead to delay spreads of several tens of microseconds. This can be of particular importance in e.g. cellular or broadcast applications. When transmitting at high rate over such channels, the introduced intersymbol interference (ISI) can span hundreds of symbols and, hence, severely distorts the received signals. To cope with such ISI, very long decision feedback equalizers (DFE) are required at the receiver to properly recover the transmitted signal. The design of such equalizer can be very complex and occupy a large chip area since certain performance requirements have to be met.
Decision feedback equalizers (DFE), consisting of a feedforward (FF) and a feedback (FB) filter, are well known in the art. They are preferred to linear equalizers because of their effectiveness at reducing the ISI. This stems mainly from their ability to cancel very efficiently the post-cursor portion of the ISI. This is done by optimally computing the feedforward and feedback coefficients, provided that the channel impulse response is known at the receiver. When this is not the case and the transmission extends over a large number of symbols (as e.g. in broadcasting systems), an adaptive DFE provides the means to compute adaptively the filter coefficients without any prior knowledge about the channel. According to the system definition, this adaptation can be carried out in a data-aided (trained) mode or non data-aided (blind) mode.
Although trained or blind DFEs may provide satisfactory performance, even in high delay spread channels, their implementation complexity grows linearly with the channel length expressed in number of symbol instants. When this length reaches several hundreds of symbol instants, the number of operations can easily exceed a thousand complex multiplications per symbol. For a symbol rate of 10 MHz, which is typical of broadband systems, this translates into 10 billion complex multiplications per seconds, which requires a very high power consumption and/or silicon area. To reduce the burden of the DFE implementation, frequency domain (FD) techniques have been proposed, which enable performing so-called fast convolutions (or correlations), thereby significantly reducing the implementation complexity.
Considering the use of FD processing for the linear equalizer is quite straightforward because it is a direct application of the fast convolution or correlation. On the contrary, FD processing for the feedback part of a DFE has been rarely and not always convincingly addressed. The FD DFE presented in the papers “Blind Decision Feedback Equalization for Terrestrial Television Receivers” (M. Ghosh, Proc. IEEE, vol. 86, no. 10, pp. 2070-2081, October 1998) and “Overlap and Save Frequency Domain DFE for Throughput Efficient Single Carrier Transmission” (S. Tomasin, IEEE 16th Int'l Symp. Personal, Indoor and Mobile Radio Communications, September 2005) only have the feedforward part in the FD. The main reason is that the FD block processing does not lend itself easily to the feedback process. The FD approach in “Frequency Domain Feedforward Filter Combined DFE Structure in Single Carrier Systems over Time-varying Channels” (B. Liu et al., IEEE Trans. Cons. Electronics, vol. 54, no. 4, November 2008) targets mainly single-carrier (SC) systems with cyclic extension and is also limited to the FF part. In “Frequency-domain and Multirate Adaptive Filtering” (J. Shynk, IEEE Signal Proc. Magazine, vol. 9, no. 1, pp. 14-37, January 1992) both the FF and FB part, including the weight update, are done in the FD, but the FB is less performant than in the TD case because the new decisions obtained after the DFTs are not fed back until the next DFT is computed.
Regarding notational conventions, normal Latin characters are used for time-domain signals (a) and tilde characters for frequency-domain signals (ã), vectors and matrices are denoted by a single and double under-bar, respectively, (a and A). The superscripts *, T and H denote the complex conjugate, the matrix transpose and complex conjugate transpose, respectively. The Hadamard (i.e. element-wise) product of vectors is denoted by the ⊙ operator. Any matrix in this description represented by a Q followed by a subscripted letter denotes a zero padding matrix. The operator F represents a FFT block operation.
Before a decision feedback equalization scheme as known in the art is described more in detail, a system model is introduced. The transmission is considered of a sequence of possibly complex symbols over a complex multipath channel and affected by additive white Gaussian noise. The discrete-time equivalent of the received signal x(t) can be expressed as:
                              x          ⁡                      (            k            )                          =                                            ∑                              l                =                0                                            L                -                1                                      ⁢                                          s                ⁡                                  (                                      k                    -                    l                                    )                                            ⁢                              h                ⁡                                  (                  l                  )                                                              +                      n            ⁡                          (              k              )                                                          (        1        )            where s(l) is the sequence of the transmitted symbols and h(k) denotes the channel impulse response of length L. The received signal x(k) is fed to a DFE (see FIG. 1) that satisfies the following relationship
                              y          ⁡                      (            k            )                          =                                            ∑                              i                =                0                            A                        ⁢                                                            a                  i                  *                                ⁡                                  (                  k                  )                                            ⁢                              x                ⁡                                  (                                      k                    +                                          d                      a                                        -                    i                                    )                                                              +                                    ∑                              i                =                1                            C                        ⁢                                                            c                  i                  *                                ⁡                                  (                  k                  )                                            ⁢                              d                ⁡                                  (                                      k                    -                    i                                    )                                                                                        (        2        )                                          d          ⁡                      (            k            )                          =                  f          ⁢                      {                          y              ⁡                              (                k                )                                      }                                              (        3        )            where ai(k) are the A coefficients of the feedforward filter, ci(k) are the C coefficients of the feedback filter, da is the delay of the feedforward filter and f{.} is the demodulation operator (slicer—in FIG. 1 named ‘decision device’). In many practical systems, the channel is not known at the receiver and it may be difficult to estimate the channel directly. Adaptive algorithms are then used to adapt the weights ai(k) and ci(k) so as to reduce the variance of the estimation error. Although many high performance algorithms exist, the least-mean square (LMS) algorithm is often used in practice for complexity reasons, especially when the filter lengths A and C are large. The LMS adaptation rule is as follows:e(k)=d(k)−y(k)  (4)ai(k+1)=ai(k)+2μax(k−i)e(k)  (5)ci(k+1)=ci(k)+2μcx(k−i)e(k)  (6)
The LMS algorithm used in combination with the proposed implementation of the DFE provides the simplest adaptation algorithm. However, its complexity increases as the channel length becomes larger. This is because the filtering operations in (2) involve convolutions of length A and C and the updates in (5) and (6) involve correlations of length A and C as well. Since the filter length must be approximately equal to or even greater than the channel length, the total complexity of the LMS-DFE is approximately equal to 4L possibly complex multiplications per output symbol, which may be challenging for channel lengths with hundreds of taps. It has been identified that the frequency domain (FD) processing is an interesting alternative to the time domain (TD) convolutions and correlations as in (2), (5) and (6). Indeed, it is well known that for a length Lx sequence x and length Ly sequence yconv(x,y)=IDFT(DFT(x)⊙DFT(y))  (7)corr(x,y)=IDFT(DFT(x)*⊙DFT(y))  (8)provided that the discrete Fourier transform (DFT) and inverse discrete Fourier transform (IDFT) are taken over a length larger than or equal to Lx+Ly−1. Zeros must be padded to x and y before taking the DFT to achieve this. This alternative implementation is attractive for large Lx and/or Ly if the fast direct and inverse Fourier transforms (FFT and IFFT) are used. When one of the sequences is infinite, it must be split into blocks and special techniques, known as overlap-and-save and overlap-and-add, must be used to recombine the blocks after (7) or (8) is applied.
Hence, there is a need for an adaptive feedback equalization scheme of reasonable complexity, which is suitable for applications where relatively long impulse responses are encountered.