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1.1 Memoryless scalar quantization

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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 y k when x lies within interval
    X k : = { x k < x ≤ x k + 1 } , k = 1 , 2 , ⋯ , L .
    The x k are called decision thresholds , and the number of quantization levels is L . The quantization operation is written y = Q ( x ) .
  • When 0 ∈ { y 1 , ⋯ , y L } , quantizer is called midtread , else midrise .
  • Quantization error defined q := x - Q ( x )
    (a) Uniform and (b) non-uniform quantization Q(x) and quantization error q(x)
  • If x is a r.v. with pdf p x ( · ) and likewise for q , then quantization error variance is
    σ q 2 = E { q 2 } = ∫ - ∞ ∞ q 2 p q ( q ) d q = ∫ - ∞ ∞ x - Q ( x ) 2 p x ( x ) d x = ∑ k = 1 L ∫ x k x k + 1 ( x - y k ) 2 p x ( x ) d x
  • A special quantizer is the uniform quantizer :
    y k + 1 - y k = Δ , for k = 1 , 2 , ⋯ , L - 1 , x k + 1 - x k = Δ , for finite x k , x k + 1 , - x 1 = x L + 1 = ∞ .
  • Uniform Quantizer Performance for large L : For bounded input x ∈ ( - x max , x max ) , uniform quantization with x 2 = - x max + Δ and x L = x max - Δ , and with y 1 = x 2 - Δ / 2 and y k = x k + Δ / 2 (for k > 1 ), the quantizationerror is well approximated by a uniform distribution for large L :
    p q ( q ) = 1 / Δ | q | ≤ Δ / 2 , 0 else .
    Why?
    • As L → ∞ , p x ( x ) is constant over X k for any k . Since q = x - y k | x ∈ X k , it follows that p q ( q | x ∈ X k ) will have uniform distribution for any k .
    • With x ∈ ( - x max , x max ) and with x k and y k as specified, q ∈ ( - Δ / 2 , Δ / 2 ] for all x (see [link] ). Hence, for any k ,
      p q ( q | x ∈ X k ) = 1 / Δ q ∈ ( - Δ / 2 , Δ / 2 ] , 0 else .
    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. 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.
    Quantization error for bounded input and midpoint y k
    In this case, from [link] (upper equation),
    σ q 2 = ∫ - Δ / 2 Δ / 2 q 2 1 Δ d q = 1 Δ q 3 3 - Δ / 2 Δ / 2 = 1 Δ Δ 3 3 · 8 + Δ 3 3 · 8 = Δ 2 12 .
    If we use R bits to represent each discrete output y and choose L = 2 R , then
    σ q 2 = Δ 2 12 = 1 12 2 x max L 2 = 1 3 x max 2 2 - 2 R
    and
    SNR [dB] = 10 log 10 σ x 2 σ q 2 = 10 log 10 3 σ x 2 x max 2 2 2 R = 6 . 02 R - 10 log 10 3 x max 2 σ x 2 .
    Recall that the expression above is only valid for σ x 2 small enough to ensure x ∈ ( - x max , x max ) . For larger σ x 2 , the quantizer overloads and the SNR decreases rapidly.

    Snr for uniform quantization of uniformly-distributed input

    For uniformly distributed x , can show x max / σ x = 3 , so that SNR = 6 . 02 R .

    Snr for uniform quantization of sinusoidal input)

    For a sinusoidal x , can show x max / σ x = 2 , so that SNR = 6 . 02 R + 1 . 76 . (Interesting since sine waves are often used as test signals).

    Snr for uniform quantization of gaussian input

    Though not truly bounded, Gaussian x might be considered as approximately bounded if we choose x max = 4 σ x and ignore residual clipping.In this case SNR = 6 . 02 R - 7 . 27 .
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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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