Analysis by Synthesis LPC Coding - Digital Speech: Coding for Low Bit Rate Communication Systems - page 229

(a)

16000. 0

(b)

11000. 0

(c)

6000. 0

1000. 0

(d)

-4000. 0

0 100 200 300 400 500 600

Time (Samples)

Figure 7.18 Waveform illustrations of the MPLPC coder: (a) original speech, (b)
multi-pulse excitation, (c) synthesized speech, and (d) error signal

Differentiating the above equation with respect to the amplitudes g i gives a
solution for optimum amplitudes,

m i )

L − 1 M − 1

∂E
∂g i =−

˜ − − × − = = −

2 s(n) g i h(n h(n m k ) 0 k 0 , 1 , ... ,M 1

n = 0 i = 0

(7.54)

or,

M − 1

m i )

L − 1 L − 1

0 ˜

−

m k ) =

− × − = −

s(n)h(n g i h(n h(n m k ) k 0 , 1 , ... ,M 1

n = n = 0 i = 0

(7.55)

Rearranging the summation on the right hand side,

g i L − 1 m k )

L − 1 M − 1

0 ˜

s(n)h(n − m k ) = h(n − m i )h(n − k = 0 , 1 , ... ,M − 1

n = i = 0 n = 0

(7.56)