Analysis by Synthesis LPC Coding - Digital Speech: Coding for Low Bit Rate Communication Systems - page 227

s(n)

W(z)

s w (n)

+

−

Pulse
Generator

1/A w (z) +

^

s(n)

e(n)

Error
Minimization

Figure 7.17 Block diagram of a simple MPLPC coder

diagram of a simple MPLPC is shown in Figure 7.17. Assuming that h(n) is
the impulse response of the combined LPC and perceptual weighting filters,
the squared error for a single pulse excitation at location m i with amplitude
g i canbewrittenas,

L − 1

m i ) ] 2

= ˜ − −

E [ s(n) g i h(n (7.47)

n = 0

where s(n) is the perceptually-weighted target signal with the combined

LPC and perceptual-weighting filter memory effect subtracted. The optimum
pulse location is obtained by differentiating equation (7.47) with respect to g i
and setting the derivative to zero,

˜

L − 1

∂E
∂g i =−

2 [ ˜ s(n) − g i h(n − m i ) ] × h(n − m i ) = 0 (7.48)

n = 0

This yields,

L − 1

0 ˜

s(n)h(n − m i )

n =

g i = (7.49)

−

L 1

h 2 (n − m i )

n = 0