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CODE.PPT 6.9
CODE.PPT 6.10
Quantising a Gaussian Signal
Adaptive Quantization
a 1 a 2
x 1 x 2 … … x n–1
Input values from a i–1 to a i are converted to x i .
For minimum error we must have a i = ½(x i + x i+1 )
x n
For this range of x, the quantisation error is q(x) = x i – x
The mean square quantisation error is
+∞
∫
n
a
2 2
∑ ∫ ( i )
ai−1
i = 1
i
E = p( x) q ( x) dx = p( x)
x − x dx
−∞
Differentiating w.r.t. x i : ∂E
ai
= px( x x)
dx
∂x
∫ −2
( ) −
a
i
i−1
i
ai
ai
= 2x p( x) dx−2
xp( x)
dx
Find minimum error by setting the derivative to zero
x
i
=
∫
∫
ai
ai−1
ai
ai−1
xp( x)
dx
pxdx ( )
To find optimal {x i }, take an initial guess and iteratively use
the above equation to improve their estimates.
For 15 bins, SNR = 19.7dB
i
∫
∫
ai−1 ai−1
⇔ x i is at the centroid of its bin.
– We want to use smaller steps when the signal level is
small
– We can adjust the step sizes automatically according
to the signal level:
• If almost all samples correspond to the central few
quantisation levels, we must reduce the step size
• If many samples correspond to high quantisation levels,
we must increase the step sizes
Page 6. Speech Coding E.4.14 – Speech Processing