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Motivated by the cascade of memoryless quantization and entropy coding, the entropy-minimizing scalar memoryless quantizer is derived. Using a compander formulation and tools from the calculus of variations, it is shown that the entropy-minimizing quantizer is the simple uniform quantizer. The penalty associated with memoryless quantization is then analyzed in the asymptotic case of many quantization levels.
  • Say that we are designing a system with a memoryless quantizer followed by an entropy coder, and our goal is to minimize theaverage transmission rate for a given σ q 2 (or vice versa). Is it optimal to cascade a σ q 2 -minimizing (Lloyd-Max) quantizer with a rate-minimizing code?In other words, what is the optimal memoryless quantizer if the quantized outputs are to be entropy coded?
  • A Compander Formulation: To determine the optimal quantizer,
    1. consider a companding system: a memoryless nonlinearity c ( x ) followed by uniform quantizer,
    2. find c ( x ) minimizing entropy H y for a fixed error variance σ q 2 .
    Compander curve: nonuniform input regions mapped to uniform output regions (for subsequent uniform quantization)
  • First we must express σ q 2 and H y in terms of c ( x ) . [link] suggests that, for large L , the slope c ' ( x ) : = d c ( x ) / d x obeys
    c ' ( x ) | x X k = 2 x max / L Δ k ,
    so that we may write
    Δ k = 2 x max L c ' ( x ) | x X k .
    Assuming large L , the σ q 2 -approximation equation 9 from MSE-Optimal Memoryless Scalar Quantization (lower equation) can be transformed as follows.
    σ q 2 = 1 12 k = 1 L P k Δ k 2 = x max 2 3 L 2 k = 1 L P k c ' ( x ) 2 | x X k = x max 2 3 L 2 k = 1 L p x ( x ) c ' ( x ) 2 Δ k | x X k since P k = p x ( x ) | x X k Δ k for large L = x max 2 3 L 2 - x max x max p x ( x ) c ' ( x ) 2 d x .
    Similarly,
    H y = - k = 1 L P k log 2 P k = - k = 1 L p x ( x ) Δ k log 2 p x ( x ) Δ k | x X k = - k = 1 L p x ( x ) Δ k log 2 p x ( x ) | x X k - k = 1 L p x ( x ) Δ k log 2 Δ k | x X k = - - x max x max p x ( x ) log 2 p x ( x ) d x h x : ``differential entropy'' - - x max x max p x ( x ) log 2 2 x max L c ' ( x ) Δ k d x = h x - log 2 2 x max L - x max x max p x ( x ) d x = 1 + - x max x max p x ( x ) log 2 c ' ( x ) d x = constant + - x max x max p x ( x ) log 2 c ' ( x ) d x
  • Entropy-Minimizing Quantizer: Our goal is to choose c ( x ) which minimizes the entropy rate H y subject to fixed error variance σ q 2 . We employ a Lagrange technique again, minimizingthe cost - x max x max p x ( x ) log 2 c ' ( x ) d x under the constraint that the quantity - x max x max p x ( x ) c ' ( x ) - 2 d x equals a constant C . This yields the unconstrained cost function
    J u c ' ( x ) , λ = - x max x max p x ( x ) log 2 c ' ( x ) + λ p x ( x ) c ' ( x ) - 2 - C φ ( c ' ( x ) , λ ) d x ,
    with scalar λ , and the unconstrained optimization problem becomes
    min c ' ( x ) , λ J u ( c ' ( x ) , λ ) .
    The following technique is common in variational calculus (see, e.g., Optimal Systems Control by Sage&White). Say a ( x ) minimizes a (scalar) cost J a ( x ) . Then for any (well-behaved) variation η ( x ) from this optimal a ( x ) , we must have
    ϵ J a ( x ) + ϵ η ( x ) | ϵ = 0 = 0
    where ϵ is a scalar. Applying this principle to our optimization problem, we search for c ' ( x ) such that
    η ( x ) , ϵ J u c ' ( x ) + ϵ η ( x ) , λ | ϵ = 0 = 0 .
    From [link] we find (using log 2 a = log 2 e · log e a )
    J u ϵ | ϵ = 0 = - x max x max ϵ φ c ' ( x ) + ϵ η ( x ) , λ | ϵ = 0 d x = - x max x max ϵ p x ( x ) log 2 ( e ) log e c ' ( x ) + ϵ η ( x ) + λ p x ( x ) c ' ( x ) + ϵ η ( x ) - 2 - C | ϵ = 0 d x = - x max x max log 2 ( e ) p x ( x ) c ' ( x ) + ϵ η ( x ) - 1 η ( x ) - 2 λ p x ( x ) c ' ( x ) + ϵ η ( x ) - 3 η ( x ) | ϵ = 0 d x = - x max x max p x ( x ) c ' ( x ) - 1 log 2 ( e ) - 2 λ c ' ( x ) - 2 η ( x ) d x
    and to allow for any η ( x ) we require
    log 2 ( e ) - 2 λ c ' ( x ) - 2 = 0 c ' ( x ) = 2 λ log 2 e a constant! .
    Applying the boundary conditions,
    c ( x max ) = x max c ( - x max ) = - x max c ( x ) = x .
    Thus, for large- L , the quantizer that minimizes entropy rate H y for a given quantization error variance σ q 2 is the uniform quantizer. Plugging c ( x ) = x into [link] , the rightmost integral disappears and we have
    H y | uniform = h x - log 2 2 x max L Δ ,
    and using the large- L uniform quantizer error variance approximation equation 6 from Memoryless Scalar Quantization ,
    H y | uniform = h x - 1 2 log 2 12 σ q 2 .
    It is interesting to compare this result to the information-theoretic minimal average rate for transmission of a continuous-amplitudememoryless source x of differential entropy h x at average distortion σ q 2 (see Jayant&Noll or Berger):
    R min = h x - 1 2 log 2 2 π e σ q 2 .
    Comparing the previous two equations, we find that (for a continous-amplitude memoryless source) uniform quantizationprior to entropy coding requires
    1 2 log 2 π e 6 0 . 255 bits/sample
    more than the theoretically optimum transmission scheme, regardless of the distribution of x . Thus, 0.255 bits/sample (or 1 . 5 dB using the 6 . 02 R relationship) is the price paid for memoryless quantization .

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
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salma
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Abhi
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ninjadapaul
20/(×-6^2)
Salomon
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ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
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ninjadapaul
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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how did you get the value of 2000N.What calculations are needed to arrive at it
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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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