A. Mhes and K. Zeger, "Binary lattice vector quantization with linear block codes and affine index assignments," IEEE Trans. Inform. Theory, vol. 44, pp. 79--95, Jan. 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....search algorithm to vector quantization. For the uniform source and uniform quantizer on a binary symmetric channel, the optimality of the natural binary assignment was shown in [10] Later, this result was rederived and generalized by 3 McLanghin and Neuhoff [11] Recently, Mehes and Zeger [12] considered optimal index assignment for a binary asymmetric channel. They showed that the natural binary assignment is not optimal on the binary asymmetric channel for uniform scalar quantization. While channel optimized quantization and optimal index assignment are approaches to joint ....

A. Mehes and K. Zeger, "Binary Lattice Vector Quantization With Linear Block Codes and Affine Index Assignments," IEEE Trans. Inform. Theory, vol. 44, pp. 79-94, 1998. 120

....Gersho [568] who dubbed the approach pseudo Gray coding. Other index assignment algorithms include [210] 543] 287] For binary symmetric channels and certain special sources and quantizers, analytical results have been obtained [555] 556] 250] 501] 112] 351] 42] 232] 233] [352]. For example, it was shown by Crimmins et al. in 1969 [112] that the index assignment that minimizes mean squared error for a uniform scalar quantizer used on a binary symmetric channel is the natural binary assignment. However, this result remained relatively unknown until rederived and ....

A. Mehes and K. Zeger, "Binary lattice vector quantization with linear block codes and a#ne index assignments," IEEE Trans. Inform. Theory, vol. 44, pp. 79--94, Jan. 1998.

....distortion due to channel errors) are helpful tools. Most previous work regarding analytical expressions for the channel distortion (see, e.g. 9,10,12 14,16 20] requires the two assumptions that: i) the channel is binary and symmetric, and that; ii) the transmitted VQ indices are equiprobable [17,19], corresponding to full encoder entropy [10,12,14,20] Some results not relying on the full entropy assumption (ii) can be found in [10] and [19] and results for binary a symmetric channels can be found in [19] However, no previous results valid for non binary channels are known to the author. ....

.... requires the two assumptions that: i) the channel is binary and symmetric, and that; ii) the transmitted VQ indices are equiprobable [17,19] corresponding to full encoder entropy [10,12,14,20] Some results not relying on the full entropy assumption (ii) can be found in [10] and [19], and results for binary a symmetric channels can be found in [19] However, no previous results valid for non binary channels are known to the author. Motivated by this, the main purpose of the present paper is to provide a generalization of parts of the earlier theory to arbitrary VQ s (no ....

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A. Mehes and K. Zeger, \Binary lattice vector quantization with linear block codes and ane index assignments," IEEE Transactions on Information Theory, vol. 44, no. 1, pp. 79-94, Jan. 1998.

....Gersho [568] who dubbed the approach pseudo Gray coding. Other index assignment algorithms include [210] 543] 287] For binary symmetric channels and certain special sources and quantizers, analytical results have been obtained [555] 556] 250] 501] 112] 351] 42] 232] 233] [352]. For example, it was shown by Crimmins et al. in 1969 [112] that the index assignment that minimizes mean squared error for a uniform scalar quantizer used on a binary symmetric channel is the natural binary assignment. However, this result remained relatively unknown until rederived and ....

A. M'ehes and K. Zeger, "Binary lattice vector quantization with linear block codes and affine index assignments," IEEE Trans. Inform. Theory, vol. 44, pp. 79--94, Jan. 1998.

....to source data) This interpretation is illustrated in Figure 3. The Hadamard transform has previously been proven to give advantages in VQ analysis: It has been used for IA optimization [20, 26] for soft decoding over memoryless channels [22] and for general analysis of VQ for noisy channels [27, 28, 42]. Furthermore, the Hadamard transform is a helpful tool in finding robust low complexity codes [20, 27, 28] This latter work can be employed in our case as follows: The codebook, fz(i)g, can be represented as in [20] where only some of the basis vectors of the transform are used. In this case ....

A. Mehes and K. Zeger, "Binary lattice vector quantization with linear block codes and affine index assignments," IEEE Transactions on Information Theory, vol. 44, no. 1, pp. 79--94, Jan. 1998.

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A. Mhes and K. Zeger, "Binary lattice vector quantization with linear block codes and affine index assignments," IEEE Trans. Inform. Theory, vol. 44, pp. 79--95, Jan. 1998.

....sense for the binary symmetric channel. McLaughlin, Neuhoff, and Ashley [2] generalized this result for certain uniform vector quantizers and uniform vector sources. Other than these papers, there are no others presently known in the literature giving index assignment optimality results. In [3] explicit mean squared error formulas were computed for uniform sources on binary asymmetric channels with various structured classes of index assignments. In [4] it was shown that for the uniform source and uniform quantizer the mean squared error resulting from a randomly chosen index ....

....an arbitrarily large fraction of all index assignments have an arbitrarily large fraction of codepoints arbitrarily close to the source mean as n 1 (Theorem 6.1) Then we extend a result in [4] by showing that most index assignments are asymptotically bad (Theorem 7. 1) and we extend results in [3], 6] and [7] by computing the mean squared error resulting from the Natural Binary Code (Theorem 7.3) the Folded Binary Code (Theorem 7.5) the Gray Code (Theorem 7.7) and a randomly chosen index assignment (Theorem 7.9) As comparisons, we state previously known mean squared error formulas ....

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A. Mehes and K. Zeger, "Binary Lattice Vector Quantization with Linear Block Codes and Affine Index Assignments," IEEE Transactions on Information Theory, vol. 44, no. 1, pp. 79-94, January 1998.

....that the performance of such a system can be significantly affected by the choice of index assignment. The problem of algorithmically finding good index assignments has been previously studied in [1] 6] and analytic formulas have been found for binary symmetric channels and certain sources [7] [12]. The optimality of the natural binary code was conjectured in [8] and proved in [10] for uniform scalar quantization of a uniform source and later extended to binary lattice vector quantizers with equiprobable quantization points in [11] In this paper, we derive an index assignment which ....

....nonsingular n 2 n generator matrix, t is an n dimensional binary translation vector, and the arithmetic is performed in Z n 2 .Ift = 0, then is called linear. The family of affine index assignments is attractive due to its lowimplementation complexity and was first systematically studied in [12] and [14] 16] An unstructured index assignment requires a table of size O(n2 n ) bits to implement, whereas affine assignments can be described by O(n 2 ) bits. Many useful index assignments are known to be affine, including the natural binary code, folded binary code, gray code, and two s ....

[Article contains additional citation context not shown here]

A. Mehes and K. Zeger, "Binary lattice vector quantization with linear block codes and affine index assignments," IEEE Trans. Inform. Theory, vol. 44, pp. 79--95, Jan. 1998.

....the performance of such a system can be significantly affected by the choice of index assignment. The problem of algorithmically finding good index assignments has been previously studied in [1, 2, 3, 4, 5, 6] and analytic formulas have been found for binary symmetric channels and certain sources [7, 8, 9, 10, 11, 12]. The optimality of the Natural Binary Code was conjectured in [8] and proved in [10] for uniform scalar quantization of a uniform source, and later extended to binary lattice vector quantizers with equiprobable quantization points in [11] In this paper we derive an index assignment which ....

....n Theta n generator matrix, t is an n dimensional binary translation vector, and the arithmetic is performed in Z n 2 . If t = 0, then is called linear. The family of affine index assignments is attractive due to its low implementation complexity, and was first systematically studied in [14, 15, 16, 12]. An unstructured index assignment requires a table of size O(n2 n ) bits to implement, whereas affine assignments can be described by O(n 2 ) bits. Many useful index assignments are known to be affine, including the Natural Binary Code, Folded Binary Code, Gray Code, and Two s Complement Code ....

[Article contains additional citation context not shown here]

A. M'ehes and K. Zeger, "Binary Lattice Vector Quantization with Linear Block Codes and Affine Index Assignments," IEEE Trans. Info. Theory, vol. IT-44, pp. 79--95, January 1998.

....grows. Hence, we seek a decay schedule of the channel code rate as a function of the overall transmission rate which minimizes the overall distortion. The channel codes we examine are classical binary linear block codes including repetition codes, ReedMuller codes, and BCH codes. We call (as in [8]) the structured source coders in this paper Binary Lattice Vector Quantizers. Vector quantizers with essentially identical structure have been extensively studied under various different names in [8 11] The main results of this paper are collected into Theorem 1 in Section 3, which gives ....

....linear block codes including repetition codes, ReedMuller codes, and BCH codes. We call (as in [8] the structured source coders in this paper Binary Lattice Vector Quantizers. Vector quantizers with essentially identical structure have been extensively studied under various different names in [8 11]. The main results of this paper are collected into Theorem 1 in Section 3, which gives achievable bounds on the asymptotic mean squared error performance of binary lattice vector quantizers and several useful families of binary linear block channel codes on a binary symmetric channel. The bounds ....

[Article contains additional citation context not shown here]

A. M'ehes and K. Zeger, "Binary Lattice Vector Quantization with Linear Block Codes and Affine Index Assignments," IEEE Trans. Info. Theory, vol. IT-44, pp. 79--95, January 1998.

....grows. Hence, we seek a decay schedule of the channel code rate as a function of the overall transmission rate which minimizes the overall distortion. The channel codes we examine are classical binary linear block codes including repetition codes, Reed Muller codes, and BCH codes. We call (as in [10]) the structured source coders in this paper Binary Lattice Vector Quantizers. Vector quantizers with essentially identical structure have been extensively studied under various different names in [11, 12, 13, 14, 10, 15] The main results of this paper are collected into Theorem 1 in Section 4, ....

....linear block codes including repetition codes, Reed Muller codes, and BCH codes. We call (as in [10] the structured source coders in this paper Binary Lattice Vector Quantizers. Vector quantizers with essentially identical structure have been extensively studied under various different names in [11, 12, 13, 14, 10, 15]. The main results of this paper are collected into Theorem 1 in Section 4, which gives achievable bounds on the asymptotic mean squared error performance of binary lattice vector quantizers and several useful families of binary linear block channel codes on a binary symmetric channel. The bounds ....

[Article contains additional citation context not shown here]

A. M'ehes and K. Zeger, "Binary Lattice Vector Quantization with Linear Block Codes and Affine Index Assignments," IEEE Trans. Info. Theory, vol. IT-44, pp. 79--95, January 1998.

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