3.2 Gain over pcm

An introduction to source-coding Page 1 / 1

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
$σ_{r}^{2} |_{T C} = \frac{1}{N} \sum_{ℓ = 0}^{N - 1} σ_{q_{ℓ}}^{2} = γ_{y} 2^{- 2 R} {(\prod_{k = 0}^{N - 1}, σ_{y_{k}}^{2})}^{1 / N} .$
We can compare to uniformly quantized PCM, where $σ_{r}^{2} |_{P C M} = γ_{x} σ_{x}^{2} 2^{- 2 R}$ . Since an orthogonal transform implies
$σ_{x}^{2} = \frac{1}{N} \sum_{k = 0}^{N - 1} σ_{y_{k}}^{2},$
we have the following gain over PCM:
$G_{TC} = \frac{σ_{r}^{2} |_{P C M}}{σ_{r}^{2} |_{T C}} = \frac{γ_{x}}{γ_{y}} \frac{\frac{1}{N} \sum_{k = 0}^{N - 1} σ_{y_{k}}^{2}}{{(\prod_{k = 0}^{N - 1}, σ_{y_{k}}^{2})}^{1 / N}} .$
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 $γ_{y} / γ_{x}$ accounts for changes in distribution which affect uniform-quantizer efficiency.For example, if T caused uniformly distributed x to become Gaussian distributed y _k , $γ_{y} / γ_{x}$ would contribute a 7 dB loss in TC-to-PCM performance.If, on the other hand, x was Gaussian, then y _k would also be Gaussian and $γ_{y} / γ_{x} = 1$ .

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Source: OpenStax, An introduction to source-coding: quantization, dpcm, transform coding, and sub-band coding. OpenStax CNX. Sep 25, 2009 Download for free at http://cnx.org/content/col11121/1.2

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