Scalar Quantization - Introduction to Data Compression - page 277

c(x)

3

2

1

− 4 − 3 − 2 − 1 1 2 3 4

x

−1

− 2

−3

−4

F I GU R E 9 . 23 Compressor mapping.

c −1 (x)

3

2

1

− 4 −3 −2 −1 23 4 1

x

−1

−2

−3

−4

F I GU R E 9 . 24 Expander mapping.

The mapping is shown graphically in Figure 9.23 . The inverse mapping is given by

⎧
⎨

x
2

if − 2 x 2

c − 1

3 x
2

( x ) = − 2 x > 2 (36)

⎩

3 x
2

+ 2 x < − 2

The inverse mapping is shown graphically in Figure 9.24 .
Let's see how using these mappings affects the quantization error both near and far from the
origin. Suppose we had an input of 0.9. If we quantize directly with the uniform quantizer, we
get an output of 0.5, resulting in a quantization error of 0.4. If we use the companded quantizer,
we first use the compressor mapping, mapping the input value of 0.9 to 1.8. Quantizing this