## Discrete Multitone Modulation With Principal Component Filter Banks (2002)

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Venue: | IEEE Trans. Circuits Syst. I |

Citations: | 7 - 3 self |

### BibTeX

@ARTICLE{Vaidyanathan02discretemultitone,

author = {P. P. Vaidyanathan and Yuan-pei Lin and Member Ieee and Sony Akkarakaran and Student Member and See-May Phoong},

title = {Discrete Multitone Modulation With Principal Component Filter Banks},

journal = {IEEE Trans. Circuits Syst. I},

year = {2002},

volume = {49},

pages = {1397--1412}

}

### OpenURL

### Abstract

this paper that principal component filter banks (PCFB) which are known to be optimal for data compression and denoising applications, are also optimal for a number of criteria in DMT communication. For example, the PCFB of the effective channel noise power spectrum (noise psd weighted by the inverse of the channel gain) is optimal for DMT modulation in the sense of maximizing bit rate for fixed power and error probabilities. We also establish an optimality property of the PCFB when scalar prefilters and postfilters are used around the channel. The difference between the PCFB and a traditional filter bank such as the brickwall filter bank or DFT filter bank is significant for effective power spectra which depart considerably from monotonicity. The twisted pair channel with its bridged taps, next and fext noises, and AM interference, therefore appears to be a good candidate for the application of a PCFB. This will be demonstrated with the help of numerical results for the case of the ADSL channel. Keywords'. Discrete multitone modulation (DMT), frequency division multiplexing (FDM), principal component filter banks (PCFB), digital subscriber loops (DSL), channel capacity

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Citation Context ...lent to the Nyquist(M)orzero-crossing constraint gk(Mn)=δ(n). Similarly for biorthogonal filter banks, the product Hk(z)Fk(z) isNyquist(M). Orthonormal filter banks have been thoroughly studied [36],=-= [42]. Here are som-=-e of their properties: (a) � 2π |Hk(e 0 jω )| 2dω/2π =1(unit energy property), (b) |Hk(ejω )| 2 ≤ M (boundedness), and (c) �M−1 k=0 |Hk(ejω )| 2 = M (power complementarity). Stated here ... |

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26 |
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Citation Context ...rization property. That is, assuming σ 2 qi are in decreasing order, the power spectra Si(e jω )ofqi(n) are ordered such that Si(e jω ) ≥ Si+1(e jω ) 13 (12)spointwise for all ω. It has been sh=-=own in [37]-=- that total decorrelation and spectral majorization are necessary and sufficient for the PCFB property in the class Cideal. For classes other than Cideal and the transform coder class, such conditions... |

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Citation Context ... signal processing [4,24,37,46]. The role of a specific class of PCFBs in the optimality of DMT systems was first observed in [22]. A related problem, namely the optimization of filter bank precoders =-=[13]-=-, [44] has been considered in great depth in a series of recent papers by Giannakis and his group [13, 28, 29]. The precoder typically introduces redundancy (like a non maximally decimated filter bank... |

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Citation Context ...CFB is not guaranteed for arbitrary classes of orthonormal filter banks (see [4] for counterexample). When a PCFB does exist, there is a sequential algorithm that can be used to construct the filters =-=[23], -=-[4], [5]. Closely associated with PCFBs is the notion of an optimal compaction filter H(e jω ) for a signal q(n) with power spectrum Sqq(e jω ). Such a filter has the property that its output in res... |

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Citation Context ...ce been proved to be optimal for a general class of objective functions in signal processing [4,24,37,46]. The role of a specific class of PCFBs in the optimality of DMT systems was first observed in =-=[22]-=-. A related problem, namely the optimization of filter bank precoders [13], [44] has been considered in great depth in a series of recent papers by Giannakis and his group [13, 28, 29]. The precoder t... |

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Citation Context ...thonormal filter bank {Fk(e jω )} and the prefilter D(e jω ). The next result shows how this power can be minimized. It is an extension to the DMT system, of a familar result in subband coding theor=-=y [12]-=-. Theorem 1. Optimum prefiltered orthonormal FB for DMT. Assume that the modulation symbols xk(n) are white, and uncorrelated for different k. For fixed probabilities of error Pe(k) and bit rates bk, ... |

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Citation Context ...e DFT at the expense of higher complexity of implementation. 1.1. Outline And Relation To Past Work The PCFB was introduced first in [32] and its optimality for a variety of problems was suggested in =-=[35]-=-. It has since been proved to be optimal for a general class of objective functions in signal processing [4,24,37,46]. The role of a specific class of PCFBs in the optimality of DMT systems was first ... |

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Citation Context ...ore entering the bank of filters {Hk(z)}. The path from xk(n) toyk(n) now has the effective form shown in Fig. 2(a). 2 In actual practice there are many ways to approximate this equalized system (see =-=[25]-=- and references therein). One approach would be to use a time domain equalizer in cascade with the channel and reduce the effective channel to be FIR with a short impulse response. This effective FIR ... |

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Citation Context ...e filter banks used for this purpose are DFT filter banks which can be implemented efficiently with the FFT. The filters in these DFT filter banks provide poor separation between adjacent subchannels =-=[27]-=-. It is known that the use of better filters improves performance (e.g., higher bit rate for fixed error probabilities and power). This was clearly demonstrated in [27] using cosine modulated filter b... |

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Citation Context ... of subchannels M is fixed. The channel transfer function C(z) and the noise power spectrum See(e jω ) are assumed to be fixed and known as well. A brief overview of these results will also appear in=-= [40]. 5.1.-=- Minimizing Transmitted Power Recall that the total transmitted power can be expressed in terms of the error probability and noise variances 15 ω ωsas in Eq. (10), reproduced below: P = M−1 � k=... |

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Citation Context ...r use in digital subscriber loops (DSL), and excellent descriptions of this can be found in [10] and [31], The DMT system can be regarded as a filter bank in transmultiplexer configuration [1], [36], =-=[41]-=-. Typically, the filter banks used for this purpose are Manuscript received December 11, 2001; revised April 3, 2002. This work was supported in part by the National Science Foundation under Grant MIP... |

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Citation Context ...ng where a signal x(n) to be quantized is first decomposed into subbands. Background material on the DMT system and more generally on the use of digital filter banks in communications can be found in =-=[1,10,15,17,20,33]. 2.-=-1. Perfect Reconstruction In absence of the channel noise e(n), the DMT system of Fig. 1(a) is LTI, with the transfer function from xm(.) toyk(.) given by � � Dkm(z)=Hk(z)C(z)Fm(z) In general the ... |

6 |
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Citation Context ...thonormal DMT systems (which are biorthogonal rather than orthonormal) are considered in Sec. 7 and it is shown again that the PCFB has a role in optimality. Some parts of this paper have appeared in =-=[38] a-=-nd [39]. 1.2. DSP And Multirate Notations Bold faced letters denote matrices and vectors. The transpose, conjugate, and transpose-conjugate of a matrix are denoted, respectively, as A T ,A ∗ , and A... |

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Citation Context ... addition, the PCFB for also induces the spectral majorization property. That is, assuming are in decreasing order, the power spectra of are ordered such that pointwise for all . It has been shown in =-=[37]-=- that total decorrelation and spectral majorization are necessary and sufficient for the PCFB property in the class .For classes other than and the transform coder class, such conditions for the PCFB ... |

5 |
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Citation Context ...was introduced first in [32] and its optimality for a variety of problems was suggested in [35]. It has since been proved to be optimal for a general class of objective functions in signal processing =-=[4,24,37,46]-=-. The role of a specific class of PCFBs in the optimality of DMT systems was first observed in [22]. A related problem, namely the optimization of filter bank precoders [13], [44] has been considered ... |

5 |
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Citation Context ...ng where a signal x(n) to be quantized is first decomposed into subbands. Background material on the DMT system and more generally on the use of digital filter banks in communications can be found in =-=[1,10,15,17,20,33]. 2.-=-1. Perfect Reconstruction In absence of the channel noise e(n), the DMT system of Fig. 1(a) is LTI, with the transfer function from xm(.) toyk(.) given by � � Dkm(z)=Hk(z)C(z)Fm(z) In general the ... |

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4 |
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Citation Context ...observed in [22]. A related problem, namely the optimization of filter bank precoders [13], [44] has been considered in great depth in a series of recent papers by Giannakis and his group [13], [28], =-=[29]-=-. The precoder typically introduces redundancy (like a non maximally decimated filter bank) to combat intersymbol interference. The precoder and receiver filters can be optimized according to several ... |

3 | Discrete multitone communication with principal component filter banks
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(Show Context)
Citation Context ... values from a rectangular constellation as demonstrated in Fig. 12(d). More efficient constellations exist [26]. APPENDIX B. PROOF OF CONVEXITY OF BIT RATE The following proof was first presented in =-=[6]. Consider Eqn-=-. (8) and delete all dependence on k for simplicity. Without using the approximation 1 − 2 −b ≈ 1wewill show that b is convex in σ 2 q. First notice that σ 2 q 3P = g(b)= � Q −1 � Pe 2(1... |

3 |
bank optimization with convex objectives, and the optimality of principal component forms
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Citation Context ...was introduced first in [32] and its optimality for a variety of problems was suggested in [35]. It has since been proved to be optimal for a general class of objective functions in signal processing =-=[4]-=-, [24], [37], [46]. The role of a specific class of PCFBs in the optimality of DMT systems was first observed in [22]. A related problem, namely the optimization of filter bank precoders [13], [44] ha... |

3 | on principal component filter banks: Colored noise suppression and existence issues - “Results |

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Citation Context ...the PCFB serves as an optimal theoretical benchmark in both cases. The complete duality between the optimization of subband coders and DMT systems can also be seen in a more basic way as explained in =-=[21]-=-. The use of nonuniform filter banks and PCFBs for DMT communication has not been addressed in this paper. Such an extension finds application in the so-called DWMT (discrete wavelet multitone) modula... |

2 |
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Citation Context ...eriously for use in digital subscriber loops (DSL), and excellent descriptions of this can be found in [10] and [31], The DMT system can be regarded as a filter bank in transmultiplexer configuration =-=[1]-=-, [36], [41]. Typically, the filter banks used for this purpose are Manuscript received December 11, 2001; revised April 3, 2002. This work was supported in part by the National Science Foundation und... |

2 |
extension of the KLT for wavelets and perfect reconstruction filter banks
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(Show Context)
Citation Context ...the DFT at the expense of higher complexity of implementation. A. Outline and Relation to Past Work The PCFB was introduced first in [32] and its optimality for a variety of problems was suggested in =-=[35]-=-. It has since been proved to be optimal for a general class of objective functions in signal processing [4], [24], [37], [46]. The role of a specific class of PCFBs in the optimality of DMT systems w... |

1 |
Optimizing the capacity of orthogonal and
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Citation Context ... DMT systems (which are biorthogonal rather than orthonormal) are considered in Sec. 7 and it is shown again that the PCFB has a role in optimality. Some parts of this paper have appeared in [38] and =-=[39]. 1.-=-2. DSP And Multirate Notations Bold faced letters denote matrices and vectors. The transpose, conjugate, and transpose-conjugate of a matrix are denoted, respectively, as A T ,A ∗ , and A † . We u... |