In this paper, the performance analysis of the adaptive line enhancer when the input signal consists of multiple sinusoids embedded in noise is investigated. The performance is evaluated in terms of the signal-to-noise ratio gain at the filter's output. It is shown that, for multiple sinusoids, this gain is not only a function of the filter length, but also of three additional factors — the number of sinusoids, the noise power, and the amplitude of each sinusoid. Simulation results for a dual noisy sinusoidal input are presented to illustrate the validity of this analysis.
In general, the physical telecommunications infrastructure in the United States is primarily copper between local telecommunication companies and their users. Although the late-90's ideologues forecasted a timely transition from old copper telephone infrastructure to fiber, where there was talk of fiber-to-the-curb or fiber to each person's home, it appears now that there is little to no push by the local telephone exchanges to change the distribution medium to their customers. It is copper and will remain copper for sometime to come. This places the local exchange in a precarious situation in terms of competition with its natural nemesis; the cable television provider [1]. Aside from power cables, telephone wires and coaxial cable television are commonly distributed to the majority of homes. However, a coaxial cable has a distinct advantage over a two-wire telephone connection, since a coaxial cable is naturally shielded [8] and [9]. Therefore, the standard telephone connection faces greater noise and will pose a particular design problem to those telephone engineers that are working to bring Internet access to densely populated areas [1].
In competition with this service, local telephone companies provide a service known as Asynchronous Digital Subscriber Line (ADSL) over their non-coaxial infrastructure [7]. ADSL has been touted as the solution for an end-user that plans to use telephone wires to gain access, since it provides a high speed uplink and a much higher speed downlink, which is ideal for web browsing [1]. Since ADSL occupies the 0– band, a broadband noise management approach is in order [8]. To provide this service reliably, they must combat a number of problems including line attenuation, group delay, and noise, with the noise being the unique obstacle.
An interesting application for performing noise reduction on a telephone wire connection is the use of adaptive filtering, where the adaptive line enhancer (ALE) is chosen in this investigation due to its simplicity and ease of implementation [11]. Although the ALE is known to be applicable to narrowband signals in broadband noise [3] and [5], it is not clear as to why the ALE does not extend well to a problem like this one, since the ALE provides enhancement on a sinusoid by sinusoid basis. In other terms, it is not evident as to why its performance does not extrapolate well to a host of sinusoids (i.e. a band).
Adaptive line enhancement techniques for tracking sinusoids in noise have been widely used and their performance for improving the signal-to-noise ratio (SNR) has been also studied [4], [10], [2] and [12]. The main goal of this paper is to explore the limitations of the adaptive line enhancer by analytically describing its performance to predict the algorithm's SNR gain. For multiple sinusoids, it is shown that this gain is not only a function of the filter length L, but also of three other additional factors — the number of sinusoids, the noise power, and the amplitude of each sinusoid. Furthermore, it is shown that the gain can be exactly predicted for all L corresponding to any integer multiple of the noise-free signal's fundamental frequency. At other values of L, the gain is approximated. Simulation results for a noisy sinusoidal input are presented to illustrate the validity of this analysis. Note that the model of a fundamental and its harmonics are used for signaling in this simulation. Furthermore, since a set of harmonic sinusoids is a subset of the entire domain of individual sinusoids, the extrapolation of individual sinusoids into a broadband is somehow similar to the extrapolation of groups of harmonic sinusoids into a broadband.
A quantitative analysis of the ALE method has been performed in details. For multiple sinusoids in noisy environment, it is shown that the output SNR gain is a function of not only the filter length, but also is a function of three additional factors — the number of sinusoids, the noise power, and the amplitude of each sinusoid. Simulation results reveal the performance trends in the ALE implementation and the effect of these factors. Accordingly, careful consideration has to be taken before using the ALE in telecommunication type applications.
