Вопрос 12 Error Minimization

The system identification problem consists of the estimation of the AR parameters

^a_i from s½n], with the estimates being the LPCs. To perform the estimation, a criter- ion must be established. In the present case, the mean-squared prediction error

8

J ¼ E^.e²½n]^. ¼ E^<

:

M

s½n]þ ^X a_is½n — i]

i¼1

_!29

=

;

ð4:3Þ

is minimized by selecting the appropriate LPCs. Note that the cost function J is precisely a second-order function of the LPCs. Consequently, we may visualize the dependence of the cost function J on the estimates a₁, a₂, .. ., a_M as a bowl- shaped (M þ 1)-dimensional surface with M degrees of freedom. This surface is characterized by a unique minimum. The optimal LPCs can be found by setting the partial derivatives of J with respect to a_i to zero; that is,

qJ qa_k ^¼

(

2E s½n]þ

M

^X a_is½n —

i¼1

! )

i] s½n — k] ¼ 0

ð4:4Þ

for k ¼ 1; 2; ... ; M. At this point, it is maintained without proof that when (4.4) is satisfied, then a_i ¼ ^a_i; that is, the LPCs are equal to the AR parameters. Justification

of this claim appears at the end of the section. Thus, when the LPCs are found, the system used to generate the AR signal (AR process synthesizer) is uniquely identified.

Normal Equation

Equation (4.4) can be rearranged to give

M

Efs½n]s½n — k]g þ ^X a_iEfs½n — i]s½n — k]g ¼ 0ð4:5Þ

i¼1

or

for k ¼ 1; 2; ... ; M, where

M

^X a_iR_s½i — k] ¼ —R_s½k] ð4:6Þ

i¼1

R_s½i — k]¼ Efs½n — i]s½n — k]g; ð4:7Þ

R_s½k]¼ Efs½n]s½n — k]g: ð4:8Þ

Equation (4.6) defines the optimal LPCs in terms of the autocorrelation R_s½l] of the signal s½n]. In matrix form, it can be written as

R_sa ¼ —r_s; ð4:9Þ

where

⁰ R_s½0] R_s½1] ··· R_s½M — 1] ^{1 B} R_s½1] R_s½0] ··· R_s½M — 2] ^C

R_s ¼ ^B

_.. _.. . _.

_.. ^C; ð4:10Þ

B C

B C

.

.

.

.

@ A

R_s½M — 1] R_s½M — 2] ··· R_s½0]

a ¼ ½a₁ a₂ ··· a_M]^T ; ð4:11Þ

^rs ^{¼ ½} R_s½1] R_s½2] ··· R_s½M]]^{T : ð4:12Þ}

Equation (4.9) is known as the normal equation. Assuming that the inverse of the correlation matrix R_s exists, the optimal LPC vector is obtained with

a ¼ —R_s^—1r_s: ð4:13Þ

Equation (4.13) allows the finding of the LPCs if the autocorrelation values of s½n]

are known from l ¼ 0 to M.

Prediction Gain

The prediction gain of a predictor is given by

PG ¼ 10 log₁₀

._s2.

s

s

2

e

¼ 10 log₁₀

^.E^.s²½n]^..

Efe²½n]g

ð4:14Þ

and is the ratio between the variance of the input signal and the variance of the prediction error in decibels (dB). Prediction gain is a measure of the predictor’s per- formance. A better predictor is capable of generating lower prediction error, leading to a higher gain.

ВОЗМОЖНОЕ ДОПОЛНЕНИЕ К 12 ВОПРОСУ!!Minimum Mean-Squared Prediction Error

From Figure 4.1 we can see that when a_i ¼ ^a_i, e½n] ¼ x½n]; that is, the prediction error is the same as the white noise used to generate the AR signal s½n]. Indeed, this is the optimal situation where the mean-squared error is minimized, with

x

J_min ¼ E^.e²½n]^. ¼ E^.x²½n]^. ¼ s²; ð4:15Þ

or equivalently, the prediction gain is maximized.

The optimal condition can be reached when the order of the predictor is equal to or higher than the order of the AR process synthesizer. In practice, M is usually unknown. A simple method to estimate M from a signal source is by plotting the prediction gain as a function of the prediction order. In this way it is possible to determine the prediction order for which the gain saturates; that is, further increas- ing the prediction order from a certain critical point will not provide additional gain. The value of the predictor order at the mentioned critical point represents a good estimate of the order of the AR signal under consideration.

As was explained before, the cost function J in (4.3) is characterized by a unique minimum. If the prediction order M is known, J is minimized when a_i ¼ ^a_i, leading

to e½n]¼ x½n]; that is, prediction error is equal to the excitation signal of the AR process synthesizer. This is a reasonable result since the best that the prediction- error filter can do is to ‘‘whiten’’ the AR signal s½n]. Thus, the maximum prediction

gain is given by the ratio between the variance of s½n] and the variance of x½n] in decibels.

Taking into account the AR parameters used to generate the signal s½n], we have

M

x

J_min ¼ s² ¼ R_s½0]þ ^X a_iR_s½i]; ð4:16Þ

i¼1

which was already derived in Chapter 3. The above equation can be combined with (4.9) to give

s

^{. R}_s^{½0] rT ..} 1 ^.

^{. J}min ^.

4:17Þ

^r_s ^R_s a ^¼ 0 ^ð

and is known as the augmented normal equation, with 0 the M × 1 zero vector. Equation (4.17) can also be written as

_M .

_{i s}½ — ] ¼

^X _{a R i k} J_min; k ¼ 0

0; k ¼ 1; 2; ... ; M

ð4:18Þ

where a₀ ¼ 1.

i¼0