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Memoryless Scalar Quantization

Module by: Phil Schniter. E-mail the author

Summary: Memoryless scalar quantization is discussed, with a focus on the uniform quantizer. Uniform quantizer error variance is derived under the assumption of many quantization levels, and several examples are provided.

  • Memoryless scalar quantization of continuous-amplitude variable x is the mapping of x to output yk when x lies within interval
    Xk:={xk<x≤xk+1},k=1,2,⋯,L.Xk:={xk<x≤xk+1},k=1,2,⋯,L.
    (1)
    The xk are called decision thresholds, and the number of quantization levels is L. The quantization operation is written y=Q(x)y=Q(x).
  • When 0∈{y1,⋯,yL}0∈{y1,⋯,yL}, quantizer is called midtread, else midrise.
  • Quantization error defined q:=x-Q(x)q:=x-Q(x)
    Figure 1: (a) Uniform and (b) non-uniform quantization Q(x) and quantization error q(x)
    Figure 1 (img007.png)
  • If x is a r.v. with pdf px(·)px(·) and likewise for q, then quantization error variance is
    σq2=E{q2}=∫-∞∞q2pq(q)dq=∫-∞∞x-Q(x)2px(x)dx=∑k=1L∫xkxk+1(x-yk)2px(x)dxσq2=E{q2}=∫-∞∞q2pq(q)dq=∫-∞∞x-Q(x)2px(x)dx=∑k=1L∫xkxk+1(x-yk)2px(x)dx
    (2)
  • A special quantizer is the uniform quantizer:
    yk+1-yk=Δ,fork=1,2,⋯,L-1,xk+1-xk=Δ,forfinitexk,xk+1,-x1=xL+1=∞.yk+1-yk=Δ,fork=1,2,⋯,L-1,xk+1-xk=Δ,forfinitexk,xk+1,-x1=xL+1=∞.
    (3)
  • Uniform Quantizer Performance for large L: For bounded input x∈(-xmax,xmax)x∈(-xmax,xmax), uniform quantization with x2=-xmax+Δx2=-xmax+Δ and xL=xmax-ΔxL=xmax-Δ, and with y1=x2-Δ/2y1=x2-Δ/2 and yk=xk+Δ/2yk=xk+Δ/2 (for k>1k>1), the quantization error is well approximated by a uniform distribution for large L:
    pq(q)=1/Δ|q|≤Δ/2,0else.pq(q)=1/Δ|q|≤Δ/2,0else.
    (4)
    Why?
    • – As L→∞L→∞, px(x)px(x) is constant over Xk for any k. Since q = x - y k | x ∈ X k q = x - y k | x ∈ X k , it follows that pq(q|x∈Xk)pq(q|x∈Xk) will have uniform distribution for any k.
    • – With x∈(-xmax,xmax)x∈(-xmax,xmax) and with xk and yk as specified, q∈(-Δ/2,Δ/2]q∈(-Δ/2,Δ/2] for all x (see Figure 2). Hence, for any k,
      pq(q|x∈Xk)=1/Δq∈(-Δ/2,Δ/2],0else.pq(q|x∈Xk)=1/Δq∈(-Δ/2,Δ/2],0else.
      (5)
    Figure 2: Quantization error for bounded input and midpoint yk
    This figure is a cartesian graph, with horizontal axis x and vertical axis q. The graph shows a series of seven zig-zags, centered at the origin as in figure one. The first zig-zag on the right begins at a horizontal value of -x_max, and the final zig-zag ends at the horizontal value x_max. The height or amplitude of the zig-zags ranges from -Δ/2 to Δ/2, and the width between peaks is measured to be Δ.
    In this case, from Equation 2 (upper equation),
    σq2=∫-Δ/2Δ/2q21Δdq=1Δq33-Δ/2Δ/2=1ΔΔ33·8+Δ33·8=Δ212.σq2=∫-Δ/2Δ/2q21Δdq=1Δq33-Δ/2Δ/2=1ΔΔ33·8+Δ33·8=Δ212.
    (6)
    If we use R bits to represent each discrete output y and choose L=2RL=2R, then
    σq2=Δ212=1122xmaxL2=13xmax22-2Rσq2=Δ212=1122xmaxL2=13xmax22-2R
    (7)
    and
    SNR [dB]=10log10σx2σq2=10log103σx2xmax222R=6.02R-10log103xmax2σx2. SNR [dB]=10log10σx2σq2=10log103σx2xmax222R=6.02R-10log103xmax2σx2.
    (8)
    Recall that the expression above is only valid for σx2 small enough to ensure x∈(-xmax,xmax)x∈(-xmax,xmax). For larger σx2, the quantizer overloads and the SNR decreases rapidly.

    Example 1: SNR for Uniform Quantization of Uniformly-Distributed Input

    For uniformly distributed x, can show xmax/σx=3xmax/σx=3, so that SNR =6.02R SNR =6.02R.

    Example 2: SNR for Uniform Quantization of Sinusoidal Input)

    For a sinusoidal x, can show xmax/σx=2xmax/σx=2, so that SNR =6.02R+1.76 SNR =6.02R+1.76. (Interesting since sine waves are often used as test signals).

    Example 3: SNR for Uniform Quantization of Gaussian Input

    Though not truly bounded, Gaussian x might be considered as approximately bounded if we choose xmax=4σxxmax=4σx and ignore residual clipping. In this case SNR =6.02R-7.27 SNR =6.02R-7.27.

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