B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

442
FUNDAMENTALS OF PROBABILITY THEORY
It can be rewritten into Yule-Walker equations
(8.88)
where
Differentiating E 2 with respect to a1, a2, ... , an and equating to zero, we obtain n simultaneous
equations of the form shown in Eq. (8.88). The desired constants a1, a2, ... , an can be found
from these equations by matrix inversion
[
a1 Ru
a2
]
_ R21
.
l
. ..
Rn1
an
... Rin ]-l [ : 0 1 ]
. . .
R2n
o2
. . . . . . .
· · · Rnn
R On
(8.89)
Equation (8.87) shows that E (the error) is orthogonal to data (x1 , x 2 , ... , Xn) for optimum
estimation. This gives the more general form for the principle of orthogonality in mean square
estimation. Consequently, the mean square error (under optimum conditions) is
Because EX; = 0 (i = 1, 2, ... , n),
= xo[xo - (a1x1 + a2x 2 + · · · + anX n )l
= Rao - (a1Ro1 + a2Ro2 + · · · + anRon) (8.90)
Example 8 .24 In differential pulse code modulation (DPCM), instead of transmitting sample values directly,
we estimate (predict) the value of each sample from the knowledge of previous n samples.
The estimation error Ek , the difference between the actual value and the estimated value of the
kth sample, is quantized and transmitted (Fig. 8.19). Because the estimation error Ek is smaller
than the sample value mk, for the same number of quantization levels (the same number of
PCM code bits), the SNR is increased. It was shown in Sec. 6.5 that the SNR improvement
is equal to m 2 / E 2 , where m 2 and E 2 are the mean square values of the speech signal and the
estimation error E, respectively. In this example, we shall find the optimum linear second-order
predictor and the corresponding SNR improvement.
The equation of a second-order estimator (predictor), shown in Fig. 8.19, is
where mk is the best linear estimate of mk, The estimation error Ek is given by