In many digital communication systems, a source generates digital information, such as data, audio, or video, which is to be transmitted to multiple receivers. The digital information bits are divided into blocks that define a discrete alphabet of symbols. These symbols are used to modulate a radio frequency (RF) carrier's frequency, amplitude and/or phase. For example, a quadrature oscillator can be used to modulate the symbols onto the amplitude and phase of the RF carrier, and the signaling is referred to as Quadrature Amplitude Modulation (QAM). The time interval between symbols is referred to as the symbol or baud interval, and the inverse of this interval is referred to as the symbol or baud rate.
Most modern digital communication systems use a symbol rate that sends thousands or millions of symbols per second, over propagation media including satellite links through the earth's atmosphere, terrestrial links from towers to fixed or mobile land-based receivers, or wired links using ancient twisted-pair copper connections or more sophisticated fiber optic connections. Such media are dispersive, causing reflections and multiple path delays arriving coincidently at the receiver. Such behavior is known as multipath, and causes symbols to smear across multiple symbol boundaries, which is referred to as inter-symbol interference (ISI). Moreover, mismatches in transmitter and receiver filtering induce ISI. Noise is added to the received signal from transmitter and receiver component imperfections, and from sources through the propagation path. At the receiver, an equalizer is used to mitigate the effects of ISI and noise induced in the entire channel, including transmitter, propagation medium, and front-end receiver processing. Since the exact channel characteristics are not known apriori at the receiver, the equalizer is usually implemented with adaptive methods.
A common type of equalizer uses adaptive, linear filters, and the adjustment of filter coefficients can be done in a variety of ways. Trained equalization methods rely on the embedding of a pre-determined sequence in the transmitted data, referred to as a training or reference sequence. The receiver stores or generates a replica of the training sequence, and to the extent that the received sequence differs from the training sequence, an error measure is derived to adjust equalizer coefficients. Usually, equalizer coefficient convergence relies on multiple transmissions of the training sequence, and the channel characteristics are also time varying. Hence, periodic re-training is necessary.
A common method of trained coefficient adaptation uses the Least Mean Squares (LMS) algorithm, which minimizes a Mean Squared Error (MSE) cost function with a stochastic gradient descent update rule. The LMS algorithm was originally proposed by Widrow to distinguish a fetus' heartbeat from a mother's heartbeat, and is further and concisely described in a paper entitled “The complex LMS algorithm,” by Widrow, McCool, and Ball, in The Proceedings of the IEEE, vol. 63, no. 4, pp. 719-720, April 1975.
Unfortunately, the training sequence needed for LMS takes up valuable bandwidth that could be used for data transmissions. Hence, methods that do not rely on a reference signal, or derive a reference signal from the data itself, are desirable. Such methods are referred to as blind equalization methods. A common blind equalization method replaces the reference signal in the LMS algorithm with the receiver's best guess at the data, and is therefore referred to as Decision Directed LMS (DD-LMS), as proposed in a paper entitled “Techniques for adaptive equalization of digital communication systems,” by R. W. Lucky, in the Bell Systems Technical Journal, vol. 45, no. 2, pp. 255-286, February 1966. DD-LMS needs a reasonably low percentage of incorrect decisions to prevent algorithm divergence, and is therefore impractical from a cold-start initialization. Other blind algorithms are usually used from a cold-start.
The Constant Modulus Algorithm (CMA) was originally proposed by Godard to decouple equalization from carrier tracking for QAM signals, and further developed by Treichler and Agee for constant envelope Frequency Modulated (FM) signals. Godard's work can be found in a paper entitled “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” by. D. N. Godard, in IEEE Transactions on Communications, vol. 28, no. 11, pp. 1867-1875, October 1980. Treichler and Agee's later work can be found in a paper entitled “A new approach to multipath correction of constant modulus signals,” by J. R. Treichler, and B. G. Agee, in IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-31, no. 2, pp. 459-472, April 1983. CMA is likely the most popular blind equalization algorithm in practice, and is well studied in the archival literature, due to its robustness to realistic signaling environments, and LMS-like computational complexity and asymptotic performance. Instead of minimizing a MSE cost function, CMA minimizes a quartic Constant Modulus (CM) cost function that penalizes dispersion at the equalizer output.
Both LMS and CMA were originally used to adjust the coefficients of a linear, transversal filter processing baud-spaced data. In “Fractional tap-spacing equalizer and consequences for clock recovery in data modems,” IEEE Transactions on Communications, vol. COM-24, no. 8, pp. 856-864, August 1976, G. Ungerboeck shows that when over-sampled data is input to the equalizer, so that the distance between adjacent equalizer coefficients is less than the baud interval, the equalizer can significantly aid in timing recovery by performing interpolation. Thus, the equalizer is referred to as fractionally spaced. Not until the mid-1990's (see, for example, “Fast blind equalization via antenna arrays,” Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Minneapolis, Minn., vol. 4, pp. 272-275, April 1993, by L. Tong et al.) was it recognized that the over-sampling introduces redundancy which allows a finite-length, fractionally-spaced equalizer to perfectly recover the source sequence that has passed through a finite-length channel, providing mild restrictions on the channel's frequency response. Such a feat is not possible with a baud-spaced equalizer—in general, an infinite number of baud-spaced equalizer coefficients are required to perfectly recover the source sequence that has passed through a finite-length channel. Almost all modern linear equalizers for broadband communication systems are implemented as fractionally-spaced equalizers, or exploit some type of redundancy other than temporal, such as spatial or phase.
A Decision Feedback Equalizer (DFE) is generally believed to provide superior ISI cancellation with less noise gain than a finite impulse-response (FIR) equalizer structure. Austin was perhaps the first to propose a DFE, in a report entitled “Decision feedback equalization for digital communication over dispersive channels,” MIT Lincoln Labs Technical Report No. 437, Lexington, Mass., August 1967. A DFE acts to additively cancel ISI by subtracting filtered decisions (or best guesses, also known as hard decisions) from the received waveform. The feedback structure embeds a baud-spaced FIR filter in a feedback loop, fed by symbol estimates, and therefore has infinite impulse response (IIR).
Like DD-LMS, the DFE architecture requires a low percentage of incorrect decisions to prevent algorithm divergence and error propagation, a phenomenon whereby an incorrect decision causes more incorrect decisions due to the feedback loop of the DFE. Therefore, a DFE requires alternative methods from a cold-start. A summary of such techniques using linear IIR filters is presented in a chapter entitled “Current approaches to blind decision feedback equalization,” by R. A. Casas et al., in the textbook, “Signal processing advances in wireless and mobile communications: trends in channel estimation and equalization,” edited by G. Giannakis, et al., Prentice Hall, Upper Saddle River, N.J., 2000.
The headings provided herein are for convenience only and do not necessarily affect the scope or meaning of the claimed invention.
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