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MSE-Optimal Memoryless Scalar Quantization

Module by: Phil Schniter. E-mail the author

Summary: The mean-squared error minimizing scalar quantizer (the Lloyd-Max quantizer) is derived here using Lagrange optimization. Background on Lagrange optimization is also provided. Finally, error variance is derived for the asymptotic case of many quantization levels.

  • Though uniform quantization is convenient for implementation and analysis, non-uniform quantization yeilds lower σq2 when px(•)px(•) is non-uniformly distributed. By decreasing |q(x)||q(x)| for frequently occuring x (at the expense of increasing |q(x)||q(x)| for infrequently occuring x), the average error power can be reduced.
  • Lloyd-Max Quantizer: MSE-optimal thresholds {xk}{xk} and outputs {yk}{yk} can be determined given an input distribution px(•)px(•), and the result is the Lloyd-Max quantizer. Necessary conditions on {xk}{xk} and {yk}{yk} are
    ∂σq2∂xk=0fork∈{2,⋯,L}and∂σq2∂yk=0fork∈{1,⋯,L}.∂σq2∂xk=0fork∈{2,⋯,L}and∂σq2∂yk=0fork∈{1,⋯,L}.
    (1)
    Using equation 2 from Memoryless Scalar Quantization (third equation), ∂/∂b∫abf(x)dx=f(b)∂/∂b∫abf(x)dx=f(b), ∂/∂a∫abf(x)dx=-f(a)∂/∂a∫abf(x)dx=-f(a), and above,
    ∂σq2∂xk=(xk-yk-1)2px(xk)-(xk-yk)2px(xk)=0⇒xk⋆=yk⋆+yk-1⋆2,k∈{2⋯L},x1⋆=-∞,xL+1⋆=∞,∂σq2∂yk=2∫xkxk+1(x-yk)px(x)dx=0⇒yk⋆=∫xk⋆xk+1⋆xpx(x)dx∫xk⋆xk+1⋆px(x)dx,k∈{1⋯L}∂σq2∂xk=(xk-yk-1)2px(xk)-(xk-yk)2px(xk)=0⇒xk⋆=yk⋆+yk-1⋆2,k∈{2⋯L},x1⋆=-∞,xL+1⋆=∞,∂σq2∂yk=2∫xkxk+1(x-yk)px(x)dx=0⇒yk⋆=∫xk⋆xk+1⋆xpx(x)dx∫xk⋆xk+1⋆px(x)dx,k∈{1⋯L}
    (2)
    It can be shown that above are sufficient for global MMSE when ∂2logpx(x)/∂x2≤0∂2logpx(x)/∂x2≤0, which holds for uniform, Gaussian, and Laplace pdfs, but not Gamma. Note:
    • – optimum decision thresholds are halfway between neighboring output values,
    • – optimum output values are centroids of the pdf within the appropriate interval, i.e., are given by the conditional means
      yk⋆=E{x|x∈Xk⋆}=∫xpx(x|x∈Xk⋆)dx=∫xpx(x,x∈Xk⋆)Pr(x∈Xk⋆)dx,=∫xk⋆xk+1⋆xpx(x)dx∫xk⋆xk+1⋆px(x)dx.yk⋆=E{x|x∈Xk⋆}=∫xpx(x|x∈Xk⋆)dx=∫xpx(x,x∈Xk⋆)Pr(x∈Xk⋆)dx,=∫xk⋆xk+1⋆xpx(x)dx∫xk⋆xk+1⋆px(x)dx.
      (3)
    Iterative Procedure to Find {xk⋆}{xk⋆} and {yk⋆}{yk⋆}:
    1. Choose y^1y^1.
    2. For k=1,⋯,L-1k=1,⋯,L-1,
      given y^ky^k and x^kx^k, solve Equation 2 (lower equation) for x^k+1x^k+1,
      given y^ky^k and x^k+1x^k+1, solve Equation 2 (upper equation) for y^k+1y^k+1.end;
    3. Compare y^Ly^L to yL calculated from Equation 2 (lower equation) based on x^Lx^L and xL+1=∞xL+1=∞. Adjust y^1y^1 accordingly, and go to step 1.
  • Lloyd-Max Performance for large L: As with the uniform quantizer, can analyze quantization error performance for large L. Here, we assume that
    • – the pdf px(x)px(x) is constant over x∈Xkx∈Xk for k∈{1,⋯,L}k∈{1,⋯,L},
    • – the pdf px(x)px(x) is symmetric about x=0x=0,
    • – the input is bounded, i.e., x∈(-xmax,xmax)x∈(-xmax,xmax) for some (potentially large) xmax.
    So with assumption
    p x ( x ) = p x ( y k ) for x , y k ∈ X k p x ( x ) = p x ( y k ) for x , y k ∈ X k
    (4)
    and definition
    Δ k : = x k + 1 - x k , Δ k : = x k + 1 - x k ,
    (5)
    we can write
    P k : = Pr { x ∈ X k } = p x ( y k ) Δ k where we require ∑ P k = 1 P k : = Pr { x ∈ X k } = p x ( y k ) Δ k where we require ∑ P k = 1
    (6)
    and thus, from equation 2 from Memoryless Scalar Quantization(lower equation), σq2 becomes
    σ q 2 = ∑ k = 1 L P k Δ k ∫ x k x k + 1 ( x - y k ) 2 d x . σ q 2 = ∑ k = 1 L P k Δ k ∫ x k x k + 1 ( x - y k ) 2 d x .
    (7)
    For MSE-optimal {yk}{yk}, know
    0 = ∂ σ q 2 ∂ y k = 2 P k Δ k ∫ x k x k + 1 ( x - y k ) d x ⇒ y k ⋆ = x k + x k + 1 2 , 0 = ∂ σ q 2 ∂ y k = 2 P k Δ k ∫ x k x k + 1 ( x - y k ) d x ⇒ y k ⋆ = x k + x k + 1 2 ,
    (8)
    which is expected since the centroid of a flat pdf over Xk is simply the midpoint of Xk. Plugging yk⋆yk⋆ into Equation 7,
    σ q 2 = ∑ k = 1 L P k 3 Δ k ( x - x k / 2 - x k + 1 / 2 ) 3 x k x k + 1 = ∑ k = 1 L P k 3 Δ k ( x k + 1 / 2 - x k / 2 ) 3 - ( x k / 2 - x k + 1 / 2 ) 3 = ∑ k = 1 L P k 3 Δ k • 2 Δ k 2 3 = 1 12 ∑ k = 1 L P k Δ k 2 . σ q 2 = ∑ k = 1 L P k 3 Δ k ( x - x k / 2 - x k + 1 / 2 ) 3 x k x k + 1 = ∑ k = 1 L P k 3 Δ k ( x k + 1 / 2 - x k / 2 ) 3 - ( x k / 2 - x k + 1 / 2 ) 3 = ∑ k = 1 L P k 3 Δ k • 2 Δ k 2 3 = 1 12 ∑ k = 1 L P k Δ k 2 .
    (9)
    Note that for uniform quantization (Δk=ΔΔk=Δ), the expression above reduces to the one derived earlier. Now we minimize σq2 w.r.t. {Δk}{Δk}. The trick here is to define
    α k : = p x ( y k ⋆ ) 3 Δ k so that σ q 2 = 1 12 ∑ k = 1 L p x ( y k ⋆ ) Δ k 3 = 1 12 ∑ k = 1 L α k 3 . α k : = p x ( y k ⋆ ) 3 Δ k so that σ q 2 = 1 12 ∑ k = 1 L p x ( y k ⋆ ) Δ k 3 = 1 12 ∑ k = 1 L α k 3 .
    (10)
    For px(x)px(x) constant over Xk and yk∈Xkyk∈Xk,
    ∑ k = 1 L α k = ∑ k = 1 L p x ( y k ⋆ ) 3 Δ k | y k ⋆ = x k + x k + 1 2 = ∫ - x max x max p x ( x ) 3 d x = C x (a known constant) , ∑ k = 1 L α k = ∑ k = 1 L p x ( y k ⋆ ) 3 Δ k | y k ⋆ = x k + x k + 1 2 = ∫ - x max x max p x ( x ) 3 d x = C x (a known constant) ,
    (11)
    we have the following constrained optimization problem:
    min { α k } ∑ k α k 3 s.t. ∑ k α k = C x . min { α k } ∑ k α k 3 s.t. ∑ k α k = C x .
    (12)
    This may be solved using Lagrange multipliers.

    Aside: Optimization via Lagrange Multipliers:

    Consider the problem of minimizing N-dimensional real-valued cost function J(x)J(x), where x=(x1,x2,⋯,xN)tx=(x1,x2,⋯,xN)t, subject to M<NM<N real-valued equality constraints fm(x)=amfm(x)=am, m=1,⋯,Mm=1,⋯,M. This may be converted into an unconstrained optimization of dimension N+MN+M by introducing additional variables λ=(λ1,⋯,λM)tλ=(λ1,⋯,λM)t known as Lagrange multipliers. The uncontrained cost function is
    J u ( x , λ ) = J ( x ) + ∑ m λ m f m ( x ) - a m , J u ( x , λ ) = J ( x ) + ∑ m λ m f m ( x ) - a m ,
    (13)
    and necessary conditions for its minimization are
    ∂ ∂ x J u ( x , λ ) = 0 ⇔ ∂ ∂ x J ( x ) + ∑ m λ m ∂ ∂ x f m ( x ) = 0 ∂ ∂ λ J u ( x , λ ) = 0 ⇔ f m ( x ) = a m for m = 1 , ⋯ , M . ∂ ∂ x J u ( x , λ ) = 0 ⇔ ∂ ∂ x J ( x ) + ∑ m λ m ∂ ∂ x f m ( x ) = 0 ∂ ∂ λ J u ( x , λ ) = 0 ⇔ f m ( x ) = a m for m = 1 , ⋯ , M .
    (14)
    The typical procedure used to solve for optimal x is the following:
    1. Equations for xn, n=1,⋯,Nn=1,⋯,N, in terms of {λm}{λm} are obtained from Equation 14 (upper equation).
    2. These N equations are used in Equation 14 (lower equation) to solve for the M optimal λm.
    3. The optimal {λm}{λm} are plugged back into the N equations for xn, yielding optimal {xn}{xn}.
    Necessary conditions are
    ∀ ℓ , ∂ ∂ α ℓ ∑ k α k 3 + λ ∑ k α k - C x = 0 ⇒ λ = - 3 α ℓ 2 ⇒ α ℓ = - λ / 3 ∂ ∂ λ ∑ k α k 3 + λ ∑ k α k - C x = 0 ⇒ ∑ k α k = C x , ∀ ℓ , ∂ ∂ α ℓ ∑ k α k 3 + λ ∑ k α k - C x = 0 ⇒ λ = - 3 α ℓ 2 ⇒ α ℓ = - λ / 3 ∂ ∂ λ ∑ k α k 3 + λ ∑ k α k - C x = 0 ⇒ ∑ k α k = C x ,
    (15)
    which can be combined to solve for λ:
    ∑ k = 1 L - λ 3 = C x ⇒ λ = - 3 C x L 2 . ∑ k = 1 L - λ 3 = C x ⇒ λ = - 3 C x L 2 .
    (16)
    Plugging λ back into the expression for αℓ, we find
    α ℓ ⋆ = C x / L , ∀ ℓ . α ℓ ⋆ = C x / L , ∀ ℓ .
    (17)
    Using the definition of αℓ, the optimal decision spacing is
    Δ k ⋆ = C x L p x ( y k ⋆ ) 3 = ∫ - x max x max p x ( x ) 3 d x L p x ( y k ⋆ ) 3 , Δ k ⋆ = C x L p x ( y k ⋆ ) 3 = ∫ - x max x max p x ( x ) 3 d x L p x ( y k ⋆ ) 3 ,
    (18)
    and the minimum quantization error variance is
    σ q 2 ∥ min = 1 12 ∑ k p x ( y k ⋆ ) Δ k ⋆ 3 = 1 12 ∑ k p x ( y k ⋆ ) ∫ - x max x max p x ( x ) 3 d x 3 L 3 p x ( y k ⋆ ) = 1 12 L 2 ∫ - x max x max p x ( x ) 3 d x 3 . σ q 2 ∥ min = 1 12 ∑ k p x ( y k ⋆ ) Δ k ⋆ 3 = 1 12 ∑ k p x ( y k ⋆ ) ∫ - x max x max p x ( x ) 3 d x 3 L 3 p x ( y k ⋆ ) = 1 12 L 2 ∫ - x max x max p x ( x ) 3 d x 3 .
    (19)
    An interesting observation is that αℓ⋆3αℓ⋆3, the ℓthℓth interval's optimal contribution to σq2, is constant over ℓ.

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