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Performance

Module by: Phil Schniter. E-mail the author

Summary: Here we analyze the optimal reconstruction error for transform coding. As the number of channels grows to infinity, the performance gain over PCM is shown to depend on the spectral flatness measure. Meanwhile, the performance of transform coding with an infinite number of channels is shown to equal that of DPCM with an infinite-length predictor. However, when the DPCM predictor length is equal to the number of transform coding channels, we show that DPCM always yields better performance.

Asymptotic Performance Analysis

  • For an N×NN×N transform coder, Equation 1 from "Gain over PCM" presented an expression for the reconstruction error variance σr2|TCσr2|TC written in terms of the quantizer input variances {σyk2}{σyk2}. Noting the N-dependence on σ r 2 | T C σ r 2 | T C in Equation 1 from "Gain over PCM" and rewriting it as σr2|TC,Nσr2|TC,N, a reasonable question might be: What is σr2|TC,Nσr2|TC,N as N→∞N→∞?
  • When using the KLT, we know that σyk2=λkσyk2=λk where λk denotes the kthkth eigenvalue of R x R x . If we plug these σyk2σyk2 into Equation 1 from "Gain over PCM", we get
    σr2|TC,N=γy2-2R∏k=0N-1λk1/N.σr2|TC,N=γy2-2R∏k=0N-1λk1/N.
    (1)
    Writing (∏kλk)1/N=exp(1N∑klnλk)(∏kλk)1/N=exp(1N∑klnλk) and using the Toeplitz Distribution Theorem (see Grenander & Szego)
    Foranyf(·),limN→∞1N∑kf(λk)=12π∫-ππf(Sx(ejω))dωForanyf(·),limN→∞1N∑kf(λk)=12π∫-ππf(Sx(ejω))dω
    (2)
    with f(·)=ln(·)f(·)=ln(·), we find that
    limN→∞σr2|TC,N=γy2-2Rexp12π∫-ππlnSx(ejω)dω=γyσx22-2RSFMxlimN→∞σr2|TC,N=γy2-2Rexp12π∫-ππlnSx(ejω)dω=γyσx22-2RSFMx
    (3)
    where SFMxSFMx denotes the spectral flatness measure of x(n)x(n), redefined below for convenience:
    SFMx=exp12π∫-ππlnSx(ejω)dω12π∫-ππSx(ejω)dω.SFMx=exp12π∫-ππlnSx(ejω)dω12π∫-ππSx(ejω)dω.
    (4)
    Thus, with optimal transform and optimal bit allocation, asymptotic gain over uniformly quantized PCM is
    GTC,N→∞=σr2|PCMσr2|TC,N→∞=γxσx22-2Rγyσx22-2RSFMx=γxγySFMx-1.GTC,N→∞=σr2|PCMσr2|TC,N→∞=γxσx22-2Rγyσx22-2RSFMx=γxγySFMx-1.
    (5)
  • Recall that, for the optimal DPCM system,
    GDPCM,N→∞=σr2|PCMσr2|DPCM,N→∞=σx2σe2|min,GDPCM,N→∞=σr2|PCMσr2|DPCM,N→∞=σx2σe2|min,
    (6)
    where we assumed that the signal applied to DPCM quantizer is distributed similarly to the signal applied to PCM quantizer and where σ e 2 | min σ e 2 | min denotes the prediction error variance resulting from use of the optimal infinite-length linear predictor:
    σe2|min=exp12π∫-ππlnSx(ejω)dω.σe2|min=exp12π∫-ππlnSx(ejω)dω.
    (7)
    Making this latter assumption for the transform coder (implying γy=γxγy=γx) and plugging in σe2 | min σe2 | min yields the following asymptotic result:
    GTC,N→∞=GDPCM,N→∞=SFMx-1.GTC,N→∞=GDPCM,N→∞=SFMx-1.
    (8)
    In other words, transform coding with infinite-dimensional optimal transformation and optimal bit allocation performs equivalently to DPCM with infinite-length optimal linear prediction.

Finite-Dimensional Analysis: Comparison to DPCM

  • The fact that optimal transform coding performs as well as DPCM in the limiting case does not tell us the relative performance of these methods at practical levels of implementation, e.g., when transform dimension and predictor length are equal and ≪∞≪∞. Below we compare the reconstruction error variances of TC and DPCM when the transform dimension equals the predictor length. Recalling that
    GDPCM,N-1=σx2σe2|min,N-1GDPCM,N-1=σx2σe2|min,N-1
    (9)
    and
    σe2|min,N-1=|RN||RN-1|σe2|min,N-1=|RN||RN-1|
    (10)
    where R N R N denotes the N×NN×N autocorrelation matrix of x(n)x(n), we find
    GDPCM,N-1=σx2|RN-1||RN|,GDPCM,N-2=σx2|RN-2||RN-1|,GDPCM,N-3=σx2|RN-3||RN-2|,⋯GDPCM,N-1=σx2|RN-1||RN|,GDPCM,N-2=σx2|RN-2||RN-1|,GDPCM,N-3=σx2|RN-3||RN-2|,⋯
    (11)
    Recursively applying the equations above, we find
    ∏k=1N-1GDPCM,k=(σx2)N-1|R1||RN|=(σx2)N|RN|∏k=1N-1GDPCM,k=(σx2)N-1|R1||RN|=(σx2)N|RN|
    (12)
    which means that we can write
    |RN|=(σx2)N∏k=1N-1GDPCM,k-1.|RN|=(σx2)N∏k=1N-1GDPCM,k-1.
    (13)
    If in the previously derived TC reconstruction error variance expression
    σr2|TC,N=γy2-2R∏ℓ=0N-1λℓ1/Nσr2|TC,N=γy2-2R∏ℓ=0N-1λℓ1/N
    (14)
    we assume that γy=γxγy=γx and apply the eigenvalue property ∏ℓλℓ=|RN|∏ℓλℓ=|RN|, the TC gain over PCM becomes
    GTC,N=σr2|PCMσr2|TC,N=γxσx22-2Rγx2-2R·σx2∏k=1N-1GDPCM,k-1/N=∏k=1N-1GDPCM,k1/N<GDPCM,N.GTC,N=σr2|PCMσr2|TC,N=γxσx22-2Rγx2-2R·σx2∏k=1N-1GDPCM,k-1/N=∏k=1N-1GDPCM,k1/N<GDPCM,N.
    (15)
    The strict inequality follows from the fact that GDPCM,kGDPCM,k is monotonically increasing with k. To summarize, DPCM with optimal length-N prediction performs better than TC with optimal N×NN×N transformation and optimal bit allocation for any finite value of N. There is an intuitive explanation for this: the propagation of memory in the DPCM prediction loop makes the effective memory of DPCM greater than N, while in TC the effective memory is exactly N.

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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.

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