A query on the non-uniform quantization

Question

I have read that non-uniform quantization boosts the smaller amplitude signals by a large amount. However the larger amplitude signals receive a small gain. As shown in the below diagram (Compressor Input and Output).

The Input - Output Characteristic diagram of compressor is given as below.

My query is as seen from the Input - Characteristic diagram, the step size is very less for smaller amplitude signal and hence the gain has to be less. On the other hand the step size is more for the large amplitude signal. Hence the gain has to be more. But the compressor input and output are in reverse way. Can you please help where I was going wrong?

Answer 1

To make it more clear, I suppose your question is

Why it is said that the compressor gain at low input amplitudes is higher, while the step size of a nonuniform quantizer is small in that region. Similarly, Why it is said that the gain of the compressor is higher for high input amplitudes, while the step size is larger for those inputs.

First, notice that nonuniform quantization is used to improve the average SNR by adapting the quantizer to the source probability density function. Of course, when the source pdf is not uniform (e.g. when it is Gaussian), then uniform quantization is not a good choice. To better understand it, assume that you want to quantize a Gaussian source into $K$ levels. Let's first assume we have a uniform quantizer. One way to reduce the distortion caused by qunantization is to increase the number of steps ($K$). Increasing $K$ means the step size is reduced which eventually leads to a higher SNR (finer quantization). Increasing the number of levels means that the total number of bits to represent each sample is increased. For instance if we have 256 levels we need 8 bits to represent each level. But for 1024 levels we will require 10 bits.

Now in the above example assume that we want to work with a fixed $K$ (e.g. 256) and improve the SNR. To do this, notice that a Gaussian source is more likely to produce samples around the mean. So for example, a zero-mean Gaussian source is very likely to generate values close to zero. So we allocate smaller step size in the area around zero, and instead make the step size larger at the tail of the pdf (where it is less likely to happen). So technically, we quantize with a smaller step size on average (but without increasing the number of levels).

Any scalar nonuniform quantizer can be implemented by using a chain of

1- A compressor: a memoryless monotonic nonlinearity $G$

2- A uniform quantizer

3- An expander: inverse of nonlinearity $G$

In your example, the nonlinearity has a large gain at the origin (the characteristic function is steep) while it causes some sort of saturation effect at large amplitudes. But this gain is applied on the input signal not on the quantizer. Still the result should be quantized by a uniform quantizer afterwards. Although it is a uniform quantizer, but the preliminary change caused by the nonlinearity $G$ and then $G^{-1}$ makes it seem as if different step sizes are used: The low amplitude bins are magnified first, then there is a uniform grid, then they are compressed. They seem as they are quantized by a small step size. Something similar also applies to the high amplitude bins, except they are compressed first, then they are expanded. So the overall result is equivalent to the effect of a large step size.

Answer 2

[EDIT: improved the graphics and code] If you call $S$ the sigmoid function (left), and $Q_{\text{u}}$ the uniform quantization operator, the right plot is obtained by:

$$Q_{\text{nu}} =S^{-1}(Q_{\text{u}}(S(\cdot)))$$

as illustrated in

from this basic Matlab code:

time = linspace(-1,1,1000);
Q=4; % Number of bits (almost)
Qu = round(time*2^Q)/2^Q; % Uniform quantization
%%% Choice of companding/expanding
%% Square-root
S =  @(x) sign(x).*sqrt(abs(x));
Sinv =  @(x) sign(x).*(x.^2);
%% Mu-law
mu = 2^5-1;
S =  @(x) sign(x).*log(1+mu*abs(x))/log(1+mu);
Sinv =  @(x) sign(x).*((1+mu).^abs(x)-1)/mu;

Qnu = sign(time).*Sinv(round(S(abs(time))*2^Q)/2^Q);  % Non-uniform quantization
subplot(2,2,1)
plot(time,S(time)); 
xlabel('Sigmoid function')
subplot(2,2,2)
plot(time,Qu);
xlabel('Uniform quantization')
subplot(2,2,3)
plot(time,Sinv(time));
xlabel('Inverse sigmoid function')
subplot(2,2,4)
plot(time,Qnu);
xlabel('Non-uniform quantization')

The main idea is to keep a quantization step "almost proportional" to the input. So that the relative quantization error $(x-x_Q)/x$ does not vary too much across the signal. You can plot the relative error diagram too. So your options are:

• use a non-uniform quantizer (right plot): more precise, less easy to implement
• use a shaping function (left plot), that acts like a variance-stabilizing transform (short discussion) like Anscombe or Box-Cox: a sigmoid, a square toot, a logarithm (like the A-law and $\mu$-law), and then apply a uniform quantizer, and when needed, apply the inverse of the shaping function

The latter is sometimes called companding (or compansion), a merger of compressing and expanding It is also related to dynamic range compression. While such designs have been largely heuristic, I would like to mention the recent paper Scalar Quantization for Relative Error, Data compression conference, 2011:

Quantizers for probabilistic sources are usually optimized for mean-squared error. In many applications, maintaining low relative error is a more suitable objective. This measure has previously been heuristically connected with the use of logarithmic companding in perceptual coding. We derive optimal companding quantizers for fixed rate and variable rate under high-resolution assumptions. The analysis shows logarithmic companding is optimal for variable-rate quantization but generally not for fixed-rate quantization. Naturally, the improvement in relative error from using a correctly optimized quantizer can be arbitrarily large. We extend this framework for a large class of nondifference distortions.