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    By: Sidney BurrusAs a part of collection: "An Introduction to Source-Coding: Quantization, DPCM, Transform Coding, and Sub-band Coding"

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Gain over PCM

Module by: Phil Schniter. E-mail the author

Summary: The total reconstruction error of transform coding with optimal bit allocation is compared to that of uniformly quantized PCM, and the ratio is found to depend on a type of spectral flatness measure.

  • With an orthogonal transform and the optimal bit allocation equation 8 from Optimal Bit Allocation (lower equation), the total reconstruction error equals
    σr2|TC=1N=0N-1σq2=γy2-2Rk=0N-1σyk21/N.σr2|TC=1N=0N-1σq2=γy2-2Rk=0N-1σyk21/N.
    (1)
    We can compare to uniformly quantized PCM, where σ r 2 | P C M =γxσx22-2R σ r 2 | P C M =γxσx22-2R. Since an orthogonal transform implies
    σx2=1Nk=0N-1σyk2,σx2=1Nk=0N-1σyk2,
    (2)
    we have the following gain over PCM:
    GTC= σr2 | PCM σr2 | TC =γxγy1Nk=0N-1σyk2k=0N-1σyk21/N.GTC= σr2 | PCM σr2 | TC =γxγy1Nk=0N-1σyk2k=0N-1σyk21/N.
    (3)
    Note that the gain is proportional to the ratio between arithmetic and geometric means of the transform coefficient variances. (Note similarities to the spectral flatness measure.) The factor γy/γxγy/γx accounts for changes in distribution which affect uniform-quantizer efficiency. For example, if T caused uniformly distributed x to become Gaussian distributed yk, γy/γxγy/γx would contribute a 7 dB loss in TC-to-PCM performance. If, on the other hand, x was Gaussian, then yk would also be Gaussian and γy/γx=1γy/γx=1.

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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