## On Lattice Quantization Noise (1996)

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Venue: | IEEE Trans. Inform. Theory |

Citations: | 72 - 20 self |

### BibTeX

@ARTICLE{Zamir96onlattice,

author = {Ram Zamir and Meir Feder},

title = {On Lattice Quantization Noise},

journal = {IEEE Trans. Inform. Theory},

year = {1996},

volume = {42},

pages = {1152--1159}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present several results regarding the properties of a random vector, uniformly distributed over a lattice cell. This random vector is the quantization noise of a lattice quantizer at high resolution, or the noise of a dithered lattice quantizer at all distortion levels. We find that for the optimal lattice quantizers this noise is wide-sense stationary and white. Any desirable noise spectra may be realized by an appropriate linear transformation ("shaping") of a lattice quantizer. As the dimension increases, the normalized second moment of the optimal lattice quantizer goes to 1=2ße, and consequently the quantization noise approaches a white Gaussian process in the divergence sense. In entropy coded dithered quantization, which can be modeled accurately as passing the source through an additive noise channel, this limit behavior implies that for large lattice dimension both the error and the bit rate approach the error and the information rate of an Additive White Gaussian Noise (AW...

### Citations

8563 |
Elements of Information Theory
- Cover, Thomas
- 1991
(Show Context)
Citation Context ...l measure for the distance of the distribution of the quantization noise from a Gaussian distribution is the information divergence, also called "relative entropy" or "Kullback-Leibler =-=distance"; see [5]-=-. Definition 3: The divergence from Gaussianity of a vector U 2 R n with a density f U , is D(U ; U ) = D(fU jjf Us) = Z R n f U log f U f U = h(U ) \Gamma h(U) (18) where h(\Delta) denotes differenti... |

1650 |
Vector Quantization and Signal Compression
- Gersho, Gray
- 1991
(Show Context)
Citation Context ... In high resolution quantization theory, it is common to assume that the quantization error of a uniform or lattice quantizer has a uniform distribution over the basic cell of the quantizer [1], [8], =-=[9]-=-. This approximation is completely accurate for all resolution levels in (subtractive) dithered quantization, where a uniformly distributed noise, the dither, is added intentionally to the source befo... |

1213 |
An algorithm for vector quantizer design
- Linde, Buzo, et al.
- 1980
(Show Context)
Citation Context ...g one monotonically reduces (i.e., improves) the normalized second moment of a given lattice quan6 tizer. Actually, Theorem 1 adds another necessary condition to the well known Lloyd conditions [15], =-=[13]-=- for the optimal quantizer. Second, the shaping procedure implies that by an appropriate linear transformation of a (nondegenerate) lattice quantizer any desired quantization noise spectra may be obta... |

842 | Least squares quantization in pcm
- Lloyd
- 1982
(Show Context)
Citation Context ...tioning one monotonically reduces (i.e., improves) the normalized second moment of a given lattice quan6 tizer. Actually, Theorem 1 adds another necessary condition to the well known Lloyd conditions =-=[15]-=-, [13] for the optimal quantizer. Second, the shaping procedure implies that by an appropriate linear transformation of a (nondegenerate) lattice quantizer any desired quantization noise spectra may b... |

645 | Vector Quantization
- Gray
- 1984
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Citation Context ... (subtractive) dithered quantization, where a uniformly distributed noise, the dither, is added intentionally to the source before quantization and then subtracted from the quantizer output; see e.g. =-=[10]-=-, [11], [21] and [22]. In any case, the (additive) uniform quantization noise model provides a convenient tool in analyzing schemes incorporating uniform, lattice or linear trellis quantizers. In ligh... |

397 |
Sphere packings, lattices, and groups
- Conway, Sloane
- 1998
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Citation Context ...on their limit properties as the lattice dimension increases. To be precise let us begin with the definition of a lattice quantizer, which is somewhat broader than the usual definition used, e.g., in =-=[4]-=-. A quantizer is defined by a set of code points and a partition which is associated with it. The code points of a K-dimensional lattice quantizer form a K-dimensional lattice L = fl i g, i.e., l i 2 ... |

314 |
Rate Distortion Theory: A Mathematical Basis For Data Compression
- Berger
- 1971
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Citation Context ...A Gaussian quantization noise is desirable, at least for Gaussian sources and MSE criterion, since it resembles the form of the noise in the forwardchannel realization of the rate-distortion function =-=[2]-=-. Furthermore, Gaussian quantization noise corresponds to an efficient covering of high dimensional spaces. In our context, the natural measure for the distance of the distribution of the quantization... |

199 |
Asymptotically optimal block quantization
- Gersho
- 1979
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Citation Context ...ction In high resolution quantization theory, it is common to assume that the quantization error of a uniform or lattice quantizer has a uniform distribution over the basic cell of the quantizer [1], =-=[8]-=-, [9]. This approximation is completely accurate for all resolution levels in (subtractive) dithered quantization, where a uniformly distributed noise, the dither, is added intentionally to the source... |

141 |
Information and Information Stability of Random Variables and Processes. San-Francisco: Holden-Day
- Pinsker
- 1964
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Citation Context ... as n ! 1. Note that all types of convergence in n which are used throughout this proof are with respect to the random initializations \Theta. Thus, by the semi-continuity of the divergence (see e.g. =-=[17]-=-) lim inf n!1 D(X + Z (Kn ) \Theta ; X + W )sD(X + W ; X + W ) a.s. ; (40) where, of course, (40) holds also in probability. Since the divergence is non-negative, (40) must hold also on the average ov... |

120 |
Asymptotic quantization error of continuous signals and the quantization dimension
- Zador
- 1982
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Citation Context ...2��e is also the limit of G opt K : Lemma 1 (G. Poltyrev) G opt K ! 1 2��e ; as K !1 ; (24) at a rate log i 2��eG opt K j = O ` log K K ' : (25) Lemma 1 was originally inferred from the wo=-=rk of Zador [20]-=- and a conjecture made by Gersho in [8]. The bounds obtained by Zador implied that the average normalized second moment of the cells of the optimal K-dimensional quantizer (for uniformly distributed d... |

101 |
Trellis coded quantization of memoryless and Gauss-Markov sources
- Marcellin, Fischer
- 1990
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Citation Context ...see e.g. [23] or any recent article on the CELP technique for speech coding). Finally, this theorem can be extended to infinite dimensional lattice quantizers, i.e., to trellis coded quantizers (TCQ) =-=[16]-=-, [3] having a linear structure. We denote by Q1 the trellis quantizer, and we replace the matrix A by the discrete time invariant shaping filter a = fa n g, n = 0; \Sigma1; \Sigma2; : : :, whose freq... |

71 |
Spectra of quantized signals
- Bennett
- 1948
(Show Context)
Citation Context ...troduction In high resolution quantization theory, it is common to assume that the quantization error of a uniform or lattice quantizer has a uniform distribution over the basic cell of the quantizer =-=[1]-=-, [8], [9]. This approximation is completely accurate for all resolution levels in (subtractive) dithered quantization, where a uniformly distributed noise, the dither, is added intentionally to the s... |

49 |
A dozen de Finetti-style results in search of a theory
- Diaconis, Freedman
- 1987
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Citation Context ...mal lattice quantizers become closer to balls. This observation provides a simple geometric explanation of the Gaussian limit behavior of the quantization noise, since, by Poincar'e theorem (see e.g. =-=[6]-=-), the projection on any finite dimensional hyperspace of a uniform distribution over a ball, becomes, in the limit as the ball dimension increases, a distribution of an i.i.d. Gaussian vector. By uti... |

48 | On universal quantization by randomized uniform/lattice quantizers
- Zamir, Feder
- 1992
(Show Context)
Citation Context ...e) dithered quantization, where a uniformly distributed noise, the dither, is added intentionally to the source before quantization and then subtracted from the quantizer output; see e.g. [10], [11], =-=[21]-=- and [22]. In any case, the (additive) uniform quantization noise model provides a convenient tool in analyzing schemes incorporating uniform, lattice or linear trellis quantizers. In light of this mo... |

19 | Information rates of pre/post filtered dithered quantizers
- Zamir, Feder
- 1996
(Show Context)
Citation Context ...oi partition (5) is optimal in many cases, sometimes the more general definition of the partition and the mapping rule (3) is needed. One such example occurs when we incorporate pre- and post-filters =-=[23]-=- in the quantization process. Other examples are noisy source quantization, and entropy constrained quantization. An important structure figure of lattice quantizers which is used extensively in the l... |

16 | Rate-distortion performance in coding bandlimited sources by sampling and dithered quantization
- Zamir, Feder
- 1995
(Show Context)
Citation Context ...ed quantization, where a uniformly distributed noise, the dither, is added intentionally to the source before quantization and then subtracted from the quantizer output; see e.g. [10], [11], [21] and =-=[22]-=-. In any case, the (additive) uniform quantization noise model provides a convenient tool in analyzing schemes incorporating uniform, lattice or linear trellis quantizers. In light of this model and t... |

12 | Asymptotic distribution of the errors in scalar and vector quantizers
- Lee, Neuhoff
- 1996
(Show Context)
Citation Context ...nd the wide use of lattices in signal coding, it is interesting to characterize the statistical properties of a random vector which is uniformly distributed over the basic cell of a lattice; see e.g. =-=[12]-=-. Thus, we analyze in this paper the spectral properties and the divergence from Gaussianity of this random vector, referred to as lattice quantization noise. We mainly focus on optimal lattice quanti... |

11 |
Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization
- Linder, Zeger
- 1994
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Citation Context ...imal) lattice dimension. A slight generalization of the lattice quantizer is the tessellating quantizer, in which the basic cell P 0 may be rotated, and not only translated, to get the i-th cell [8], =-=[14]-=-. For example, an equilateral triangle cell generates a tessellating quantizer which is not a lattice quantizer. Despite their slight generality, tessellating quantizers are not considered in this wor... |

5 |
Covering properties of convolutional codes and associated lattices
- Calderbank, Fishburn, et al.
- 1995
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Citation Context ...g. [23] or any recent article on the CELP technique for speech coding). Finally, this theorem can be extended to infinite dimensional lattice quantizers, i.e., to trellis coded quantizers (TCQ) [16], =-=[3]-=- having a linear structure. We denote by Q1 the trellis quantizer, and we replace the matrix A by the discrete time invariant shaping filter a = fa n g, n = 0; \Sigma1; \Sigma2; : : :, whose frequency... |

3 |
A matrix form of the Brunn-Minkowski inequality
- Zamir, Feder
- 1995
(Show Context)
Citation Context ... 2 = 1 2��e ffl G(Q1 ) : 1 This property is analogous to the behavior of the entropy rate of blocks of a stationary process, and it is shown easily using the notion of "conditional volume&quo=-=t; defined in [24]-=-. When a K dimensional lattice quantizer is shaped by a matrix A, its second moment is multiplied by tracefAA t g=K, and the volume of its basic cell is multiplied by jAj = jAA t j 1=2 , or equivalent... |

2 |
Private communications
- Pinsker
- 1995
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Citation Context ...ve for general sources (36). In addition, the proof in the Gaussian case is much simpler and follows directly from Lemma 1. It is conjectured, though, that (35) holds for non-Gaussian sources as well =-=[18]-=-. Proof: We start with the proof of the Gaussian source case, which is straightforward. As was shown in Section I, the optimal lattice quantizers are white, i.e., EfZ i g = 0, and EfZ i Z j g = ffl \D... |

2 |
Private communications
- Poltyrev
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Citation Context ... V 2=K : (7) For the uniform scalar quantizer G 1 = 1=12 ' 0:08333, while G 2 ' 0:080188 for the hexagonal lattice quantizer with a Voronoi partition (see Figure 1). It was recently shown by Poltyrev =-=[19] (se-=-e also section III below) that as K !1, the minimum value of GK ! 1=2��e ' 0:058550. The goal of this work is to analyze the quantization noise, given either by QK (x) \Gamma x, or by QK (x + z) \... |

1 | Probability Teory and Examples - Durrett - 1989 |

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Results on lattice quantization with dithering
- Kirac, Vaidyanathan
(Show Context)
Citation Context ...ractive) dithered quantization, where a uniformly distributed noise, the dither, is added intentionally to the source before quantization and then subtracted from the quantizer output; see e.g. [10], =-=[11]-=-, [21] and [22]. In any case, the (additive) uniform quantization noise model provides a convenient tool in analyzing schemes incorporating uniform, lattice or linear trellis quantizers. In light of t... |

1 | Sphere Packings, Lattices and Groups - Theory - 1995 |

1 | Vector Quantization and Signal Compression - Theory - 1979 |

1 | originally presented at the Inst - IT-28 - 1982 |

1 | 1221 -, “Rate distortion performance in coding band-limited sources by sampling and dithered quantization - Mar - 1992 |