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Two autoregressive identities
Signal Processing 19 (1990) 337-342 Elsevier
337
TWO AUTOREGRESSIVE IDENTITIES Jack-Kang C H A N Norden Systems, Inc., Anti-Submarine Warfare Engineering, 75 Maxess Road, Melville, New York 11747, U.S.A. Received 10 July 1989 Revised 11 October 1989
Abstract. Two useful autoregressive (AR) identities relating the reflection coefficients (the Ks), the AR coefficients (the as) and two particular AR spectral values (one at d.c., and the other at half of the sampling frequency) are obtained from a geometric interpretation of a recursive algorithm to compute both the magnitude and the phase spectra of an AR model directly from the complex reflection coefficients. These identities can be used to check for correct a-to-K and K-to-a conversions in the Levinson recursions. Furthermore, it is shown here that if the K s are not available, some stability information can be provided directly from the as.
Zusammenfassung. Zwei niitzliche autoregressive (AR) Identit/iten verkniipfen die Reflexions-Koeffizienten, die Pr~idiktorKoeffizienten und zwei spezielle Werte des AR-Spektrums (den bei der Frequenz Null sowie den bei der halben Abtastfrequenz). Man erh~ilt sie aus einer geometrischen Interpretation eines rekursiven AIgorithmus" zur Berechnung sowohl des Betrags- als auch des Phasen-Spektrums eines AR-Modells unmittelbar aus den komplexen Reflexionskoeffizienten. Diese Identit~iten k/Snnen dazu benutzt werden, in der Levinson-Rekursion die Korrektheit der Umrechnungen von Reflexions- in Pr~idiktorKoeffizienten und umgekehrt zu iiberpriifen. Weiterhin wird hier gezeigt, daJ] auch ohne die Reflexionskoeffizienten eine gewisse Stabilit~itsinformation unmittelbar aus den Pr~idiktorkoeffizienten gewonnen werden kann. R6sum6. Cet article pr6sente deux identit6s utiles dans le cas d'une mod61isation autoregressive, et reliant les coefficients de r6flexion (K~) et les coefficients AR (ai) ~ deux valeurs spectrales particuli~res (une ~t la fr6quence nulle, et l'autre ~ la fr6quence moiti6 de la fr6quence d'6chantillonnage). Ces deux identit6s sont obtenues ~ i'aide d'une interpr6tation g6om6trique d'un algorithme r6cursif afin de calculer ~ la fois les spectres d'amplitude et de phase d'un module AR directement ~ partir des coefficients complexes de r6flexion. Ces identit6s peuvent ~tre utilis6es pour corriger de conversions K e n a ou a en K dans un calcul r6cursif de type Levinson. De plus, il est montr6 que si les Kt ne sont pas connus, il est quand m~me possible d'avoir une information sur la stabilit6 directement ~t partir des av
Keywords. Autoregressive (AR) identities, AR coefficients, reflection coefficients, AR spectrum.
1. Introduction Autoregressive (AR) model is an important modern spectrum analysis technique, and it can either be characterized by its AR coefficients (the as, in general complex) for a direct form filter realization, or equivalently, by its reflection coefficients (the Ks, in general complex) for a lattice filter realization. Note that the as and the K s are related by the famous Levinson recursions. We have assumed that the lattice filter is stable, that is, all the poles are inside the unit circle, or, 0165-1684/90/$3.50 © 1990, Elsevier Science Publishers B.V.
all the K s are in magnitude less than unity. An order-recursive algorithm can be derived from the Levinson recursions to compute both the magnitude and the phase AR spectra of any order at any frequency directly from the K s [2, 4]. The recursive formulas have an interesting geometric interpretation, that is, successive order AR spectra can be constructed geometrically by two vector rotations and a symmetric (diagonal, for real AR model) matrix multiplication and a complex number inversion in the two-dimensional complex plane [3, 4]. From this novel geometric interpretation we will
338
J . - K . C h a n / T w o a u t o r e g r e s s i v e identities
derive two important identities relating the as, the Ks, and the AR spectral values at d.c. and at half of the sampling frequency, respectively. These identities can be used to check for correct a - t o - K and K - t o - a conversions. Furthermore, if the Ks are not available, some information can be provided directly from the as.
The above recursive formulas lead to the following geometric interpretation [3,4] of the AR spectral computations: Let
'
I m C~o') J '
(5)
A~o,,)- [Re A(")(&°)] Im A(")(eJ°) J'
(6)
2. R e s u l t s
Let an AR model of order M, denoted by AR(M), be represented by the following AR polynomial:
D _[I+K,,, 0
0 1-Krn
cos(O/Z) Ro/2-= I_sin(0/2)
] (7)
'
-sin(O/Z) ], cos(0/2).]
(8)
a(M M) . -M ,
A(M)(z)~I+a~M)z-I+...+,..
(1)
a~oM) =-- 1,
where the (real) AR coefficients (the as) are related to the reflection coefficients (the Ks) by the well.known Levinson recursions:
r cos(toO~Z) R-molE =- [ . - s i n ( m O / 2 )
sin(mO/2) ] cos(mO/2)J"
(9)
Then, for m = 1 , . . . , M, (10)
c(m) -- n D ~(m-1) 0 - - JL"m~" 0 / 2 t ' 0
a ~'' ) = K,. ,
a~m)
(10')
= D,,Ro/2" • " DlRo/2Co,
m=l,...,M,
(2)
_ ( m - - l ) + /,.,- / ~ ( m - - 1 ) ' ~ * = t4 i lX-ml, t~m-- i J
i=l,...,m-1. All the Ks are assumed to have magnitudes less than unity. Suppose the sampling period is T and f is any frequency between 0 and 1/T. Let z = e j° where 0 = 2"rrfT. The following recursive algorithm computes the AR spectra of successive orders directly from the Ks [2, 4]: Define c~o°)--- 1.
A
TM)= D. . . . 0/2~0 ~(m) •
(11)
Note that Ro/2 and R-,,o/2 are orthogonal matrices which represent planar rotation of angles 0/2 and - mO / 2, respectively. Set 0 = 0 and ~ in (1). Then we have A(M)(e j°) = 1 + a~M)+ - . . + a~ff ) = x/{1/S~o~)}, A(M)(e j~) = 1 - a ~ M ) + . • " + ( - - 1 ) M a ~
(12) )
For m = 1, 2 , . . . , M, we have = x/{1 / S ~ ) } , c ( m ) - - ~ ( m - - l ) ~ j O / 2 £. lp,- / ~ ( m - 1 )
- co
....
,,~0
,~jO/2"~*
,.
/
and A(")(eJO ) = C(o,,,) e-J,,o/:.
(4)
Thus, all the AR magnitude and phase (or, real and imaginary) spectra can be computed recursively in terms of the Ks. Signal Processing
(13)
(3)
where S(oM) and S(~ ) are the power spectral values at d.c. and half of the sampling frequency respectively. The positivity of (12) and (13) will be jus!ified shortly. Two identities relating the as, the Ks, and these two particular spectral values can be derived from the geometric formulas (10) and (11).
J.-K. Chan / Two autoregressive identities
Theorem (First identity) M
Let
(1 - K,)
,/{ s(_M~} -- I
M
i
1+ E al r~)= 1-[ (I+K,) i~l
339
i=1
and
= x/{1/S(o M)} > 0.
(14) p(M) = (1/2"rr)
(Second identity)
I1/ACM)(eJ°)l 2 dO. --or
M
M
= total power of the A R ( M ) model.
1+ 2 ( - 1 ) ' a l ~a)= [I (I+(-I)'K,) i=l
i=I
= ~ ( ~ M ) } > 0.
(15)
Proof. See Appendix.
Then, using the A s t r o m - G r a y - M a r k e l recursions [1, 5], we have
p(M)m.J{S(oM).s(_M)} =
These two identities can be used to check for correct a - t o - K or K - t o - a conversions using the Levinson recursions. It is shown here that if the K s are not available, some stability information can be provided directly from the as. Since the K s have magnitudes less than unity, the products of 1 + K and 1 - K must be positive. Hence (14) and (15) imply that the AR polynomial is positive for 0 = 0 and ~r (or, z = + l and - 1 ) :
A(M)(+I) = A(M)(ej°) > 0,
(16)
A(M)(-1) = A(M)(e ~ ) > 0.
(17)
The First identity states that the sum of all AR coefficients is equal to the product of one plus the reflection coefficients, and it is also a necessary condition for all poles to lie inside the unit circle. Therefore, we conclude from these identities the following corollaries. Corollary 1. A n y stable A R polynomial A ( z ) with real coefficients is positive at z = ±1 ( Eqs. (16) and (17)). Corollary 2. I f the sum o f all A R coefficients o f a real A R model is not positive (Eq. (14)), (or if the left hand sum in (15) is not positive), then there exists at least one pole on or outside the unit circle.
1
i=~1 I (I --IK,12).
Thus, the total power is the geometric mean of S(oM) and S(Y), or, the area under the AR power spectral curve is equal to that of a rectangle with sides x~{S~o~a)} and ~ .
3. Example As an illustration of the two AR identities, let M = 8, and K, = - 0 . 6 6 6 7 6 ,
K2 = 0.89417,
-0.37076,
/ ( 4 = 0.61572,
K 3=
K5 = -0.26256,
K 6=
0.82470,
K 7 =-0.19676,
K8 = 0.49803.
The corresponding AR coefficients (with superscripts suppressed for simplicity) are a, = -2.46122,
a2 = 4.45897,
a 3 = -5.43748,
a 4 = 5.83586,
a5 = -4.64787,
a 6 = 3.16649,
a 7 = -1.37372,
a8 = 0.49803.
We have Note that the products of 1 + K, 1 - K, and 1 IKI2, etc. are important parameters for a real AR model. In fact, we can prove the following:
1 + a, + a2+ a3+ a 4 + a s + a6-F a 7 + a s = 1.03906 > 0, Vol. 19, No. 4, April 1990
J.-K. Chan / Two autoregressiveidentities
340
(1 + KI)(1 + g2)(1 + K3)(1 + K4)(1 + K5) × (1 + K6)(1 + K7)(1 + Ks) = 1.03906, 1 - a~ + a2 - a3 + an - a5 + a6 - a7 +
as
= 28.8796 > 0, (1 - K , ) ( 1 + K 2 ) ( 1 - K 3 ) ( 1 + K , ) ( 1 -
× (1 + K6)(1
-
-
Ks)
K7)(1 + Ks) = 28.8796.
4. C o n c l u s i o n s
We have derived two AR identities from the geometric interpretation of the recursive AR spectral formulas. They can be used as quick checks for a - K conversions as illustrated in our numerical example, and they also give stability information on the AR model.
Hence the two identities (14) and (15) are verified.
Appendix.
Proof of the two AR identities
First identity From (8) and (9), set 0 = 0, we have
Hence (10') implies
e~o"~)= DmDm-i • " • D2DIeo =[(I+K~)'''(I+K0],0 which together with (11) implies (14).
Second identity From (8) and (9), set 0 = ~r, we have
and I cosM~r/2
R-Mo/2= 1--sin M.tr/2
sinM~r/2] cos M~r/2 J"
When M is even, (10') implies
e~M)=([ I+K~O
[0 1 Signal Processing
--KM
1 -K~O ][01 - 1 0 ] ) . . . ( [ 1 +KI0
1ol][
lo M,] [1°2 loK2,][ 0
-- K 1
J.-K. Chan/ Two autoregressive identities
:[o
341
[_~1+~>~,_~,> o ][1] 0 - ( 1 - K2)(1 + K1) 0 o (_1)M/2[(1 + KM)(1 -- K ~ _ , ) " "" (1 + K2)(1 - K1) 1-
_~,o~>]
KM
L
]
(1 - K M ) ( 1 + K M _ I ) ( 1 - K~)(1 + K 0
0
=(_I)M~[(I+K~,)(1-k:M_~)'''0(I+K~)(1-K,)]. Now
R-~oJ2:[~-l~Mj' ~_l~J,], and (11) implies r(1 + KM)(1 -- K M - 1 ) " " " (1 + K2)(1 - K I ) ] . 0 J
L
A~o~) =
Hence (15) is true when M is even. When M is odd, (10') implies e'~)=([l+0K~
x
x
1_OM][01
--10] )
([1+~ o][o lo] [1+~ , °][°1 0
1 - KM-I
1
"""
0
o
[ ( 1 + K M _ I ) ( 1 - K M - 2 ) " ° " (1 + K 2 ) ( 1 - K 1 )
k
0
lo]tIlo] ]
(1 - K M _ ~ ) ( 1 + K M _ : ) ( 1 -- K : ) ( 1 + K1)
[
0 Kl)]" = (--1)(M-~)/EL(1 -- K ~ ) ( 1 + K~_~) • "" (1 + K:)(1 Now _ (_I)(M-I)/2 and (11) implies
A~oM)=[(1--KM)(I+ KM_,)'' O
(1 + K2)(1 - K1)].
Hence (15) is also true when M is odd.
[] Vol. 19, No. 4, April 1990
342
J.-K. Chan / Two autoregressive identities
References [1] K.J. Astrom, Introduction to Stochastic Control Theory, Academic Press, New York, 1970. [2] J.P. Burg, Maximum entropy spectral analysis, PhD dissertation, Department of Geophysics, Stanford University, May 1975. [3] J.-K. Chan, Geometric interpretation of recursive AR spectral computations directly from complex reflection coefficients, Norden Systems ASW Technical Notes, NS-ASW-TN-8902, Melville, New York, 1989.
Signal Processing
[4] J.-K. Chan, Recursive autoregressive spectral computations directly from reflection coefficients, IEEE Proc. lnternat. Conf. Acoust. Speech Signal Process., New York, 11-14 April 1988, pp. 2364-2367. [5] A.H. Gray Jr. and J.D. Markel, Digital lattice and ladder filter synthesis, IEEE Trans. Audio Electroacoust., Vol. AU21, No. 6, December 1973, pp. 491-500.
337
TWO AUTOREGRESSIVE IDENTITIES Jack-Kang C H A N Norden Systems, Inc., Anti-Submarine Warfare Engineering, 75 Maxess Road, Melville, New York 11747, U.S.A. Received 10 July 1989 Revised 11 October 1989
Abstract. Two useful autoregressive (AR) identities relating the reflection coefficients (the Ks), the AR coefficients (the as) and two particular AR spectral values (one at d.c., and the other at half of the sampling frequency) are obtained from a geometric interpretation of a recursive algorithm to compute both the magnitude and the phase spectra of an AR model directly from the complex reflection coefficients. These identities can be used to check for correct a-to-K and K-to-a conversions in the Levinson recursions. Furthermore, it is shown here that if the K s are not available, some stability information can be provided directly from the as.
Zusammenfassung. Zwei niitzliche autoregressive (AR) Identit/iten verkniipfen die Reflexions-Koeffizienten, die Pr~idiktorKoeffizienten und zwei spezielle Werte des AR-Spektrums (den bei der Frequenz Null sowie den bei der halben Abtastfrequenz). Man erh~ilt sie aus einer geometrischen Interpretation eines rekursiven AIgorithmus" zur Berechnung sowohl des Betrags- als auch des Phasen-Spektrums eines AR-Modells unmittelbar aus den komplexen Reflexionskoeffizienten. Diese Identit~iten k/Snnen dazu benutzt werden, in der Levinson-Rekursion die Korrektheit der Umrechnungen von Reflexions- in Pr~idiktorKoeffizienten und umgekehrt zu iiberpriifen. Weiterhin wird hier gezeigt, daJ] auch ohne die Reflexionskoeffizienten eine gewisse Stabilit~itsinformation unmittelbar aus den Pr~idiktorkoeffizienten gewonnen werden kann. R6sum6. Cet article pr6sente deux identit6s utiles dans le cas d'une mod61isation autoregressive, et reliant les coefficients de r6flexion (K~) et les coefficients AR (ai) ~ deux valeurs spectrales particuli~res (une ~t la fr6quence nulle, et l'autre ~ la fr6quence moiti6 de la fr6quence d'6chantillonnage). Ces deux identit6s sont obtenues ~ i'aide d'une interpr6tation g6om6trique d'un algorithme r6cursif afin de calculer ~ la fois les spectres d'amplitude et de phase d'un module AR directement ~ partir des coefficients complexes de r6flexion. Ces identit6s peuvent ~tre utilis6es pour corriger de conversions K e n a ou a en K dans un calcul r6cursif de type Levinson. De plus, il est montr6 que si les Kt ne sont pas connus, il est quand m~me possible d'avoir une information sur la stabilit6 directement ~t partir des av
Keywords. Autoregressive (AR) identities, AR coefficients, reflection coefficients, AR spectrum.
1. Introduction Autoregressive (AR) model is an important modern spectrum analysis technique, and it can either be characterized by its AR coefficients (the as, in general complex) for a direct form filter realization, or equivalently, by its reflection coefficients (the Ks, in general complex) for a lattice filter realization. Note that the as and the K s are related by the famous Levinson recursions. We have assumed that the lattice filter is stable, that is, all the poles are inside the unit circle, or, 0165-1684/90/$3.50 © 1990, Elsevier Science Publishers B.V.
all the K s are in magnitude less than unity. An order-recursive algorithm can be derived from the Levinson recursions to compute both the magnitude and the phase AR spectra of any order at any frequency directly from the K s [2, 4]. The recursive formulas have an interesting geometric interpretation, that is, successive order AR spectra can be constructed geometrically by two vector rotations and a symmetric (diagonal, for real AR model) matrix multiplication and a complex number inversion in the two-dimensional complex plane [3, 4]. From this novel geometric interpretation we will
338
J . - K . C h a n / T w o a u t o r e g r e s s i v e identities
derive two important identities relating the as, the Ks, and the AR spectral values at d.c. and at half of the sampling frequency, respectively. These identities can be used to check for correct a - t o - K and K - t o - a conversions. Furthermore, if the Ks are not available, some information can be provided directly from the as.
The above recursive formulas lead to the following geometric interpretation [3,4] of the AR spectral computations: Let
'
I m C~o') J '
(5)
A~o,,)- [Re A(")(&°)] Im A(")(eJ°) J'
(6)
2. R e s u l t s
Let an AR model of order M, denoted by AR(M), be represented by the following AR polynomial:
D _[I+K,,, 0
0 1-Krn
cos(O/Z) Ro/2-= I_sin(0/2)
] (7)
'
-sin(O/Z) ], cos(0/2).]
(8)
a(M M) . -M ,
A(M)(z)~I+a~M)z-I+...+,..
(1)
a~oM) =-- 1,
where the (real) AR coefficients (the as) are related to the reflection coefficients (the Ks) by the well.known Levinson recursions:
r cos(toO~Z) R-molE =- [ . - s i n ( m O / 2 )
sin(mO/2) ] cos(mO/2)J"
(9)
Then, for m = 1 , . . . , M, (10)
c(m) -- n D ~(m-1) 0 - - JL"m~" 0 / 2 t ' 0
a ~'' ) = K,. ,
a~m)
(10')
= D,,Ro/2" • " DlRo/2Co,
m=l,...,M,
(2)
_ ( m - - l ) + /,.,- / ~ ( m - - 1 ) ' ~ * = t4 i lX-ml, t~m-- i J
i=l,...,m-1. All the Ks are assumed to have magnitudes less than unity. Suppose the sampling period is T and f is any frequency between 0 and 1/T. Let z = e j° where 0 = 2"rrfT. The following recursive algorithm computes the AR spectra of successive orders directly from the Ks [2, 4]: Define c~o°)--- 1.
A
TM)= D. . . . 0/2~0 ~(m) •
(11)
Note that Ro/2 and R-,,o/2 are orthogonal matrices which represent planar rotation of angles 0/2 and - mO / 2, respectively. Set 0 = 0 and ~ in (1). Then we have A(M)(e j°) = 1 + a~M)+ - . . + a~ff ) = x/{1/S~o~)}, A(M)(e j~) = 1 - a ~ M ) + . • " + ( - - 1 ) M a ~
(12) )
For m = 1, 2 , . . . , M, we have = x/{1 / S ~ ) } , c ( m ) - - ~ ( m - - l ) ~ j O / 2 £. lp,- / ~ ( m - 1 )
- co
....
,,~0
,~jO/2"~*
,.
/
and A(")(eJO ) = C(o,,,) e-J,,o/:.
(4)
Thus, all the AR magnitude and phase (or, real and imaginary) spectra can be computed recursively in terms of the Ks. Signal Processing
(13)
(3)
where S(oM) and S(~ ) are the power spectral values at d.c. and half of the sampling frequency respectively. The positivity of (12) and (13) will be jus!ified shortly. Two identities relating the as, the Ks, and these two particular spectral values can be derived from the geometric formulas (10) and (11).
J.-K. Chan / Two autoregressive identities
Theorem (First identity) M
Let
(1 - K,)
,/{ s(_M~} -- I
M
i
1+ E al r~)= 1-[ (I+K,) i~l
339
i=1
and
= x/{1/S(o M)} > 0.
(14) p(M) = (1/2"rr)
(Second identity)
I1/ACM)(eJ°)l 2 dO. --or
M
M
= total power of the A R ( M ) model.
1+ 2 ( - 1 ) ' a l ~a)= [I (I+(-I)'K,) i=l
i=I
= ~ ( ~ M ) } > 0.
(15)
Proof. See Appendix.
Then, using the A s t r o m - G r a y - M a r k e l recursions [1, 5], we have
p(M)m.J{S(oM).s(_M)} =
These two identities can be used to check for correct a - t o - K or K - t o - a conversions using the Levinson recursions. It is shown here that if the K s are not available, some stability information can be provided directly from the as. Since the K s have magnitudes less than unity, the products of 1 + K and 1 - K must be positive. Hence (14) and (15) imply that the AR polynomial is positive for 0 = 0 and ~r (or, z = + l and - 1 ) :
A(M)(+I) = A(M)(ej°) > 0,
(16)
A(M)(-1) = A(M)(e ~ ) > 0.
(17)
The First identity states that the sum of all AR coefficients is equal to the product of one plus the reflection coefficients, and it is also a necessary condition for all poles to lie inside the unit circle. Therefore, we conclude from these identities the following corollaries. Corollary 1. A n y stable A R polynomial A ( z ) with real coefficients is positive at z = ±1 ( Eqs. (16) and (17)). Corollary 2. I f the sum o f all A R coefficients o f a real A R model is not positive (Eq. (14)), (or if the left hand sum in (15) is not positive), then there exists at least one pole on or outside the unit circle.
1
i=~1 I (I --IK,12).
Thus, the total power is the geometric mean of S(oM) and S(Y), or, the area under the AR power spectral curve is equal to that of a rectangle with sides x~{S~o~a)} and ~ .
3. Example As an illustration of the two AR identities, let M = 8, and K, = - 0 . 6 6 6 7 6 ,
K2 = 0.89417,
-0.37076,
/ ( 4 = 0.61572,
K 3=
K5 = -0.26256,
K 6=
0.82470,
K 7 =-0.19676,
K8 = 0.49803.
The corresponding AR coefficients (with superscripts suppressed for simplicity) are a, = -2.46122,
a2 = 4.45897,
a 3 = -5.43748,
a 4 = 5.83586,
a5 = -4.64787,
a 6 = 3.16649,
a 7 = -1.37372,
a8 = 0.49803.
We have Note that the products of 1 + K, 1 - K, and 1 IKI2, etc. are important parameters for a real AR model. In fact, we can prove the following:
1 + a, + a2+ a3+ a 4 + a s + a6-F a 7 + a s = 1.03906 > 0, Vol. 19, No. 4, April 1990
J.-K. Chan / Two autoregressiveidentities
340
(1 + KI)(1 + g2)(1 + K3)(1 + K4)(1 + K5) × (1 + K6)(1 + K7)(1 + Ks) = 1.03906, 1 - a~ + a2 - a3 + an - a5 + a6 - a7 +
as
= 28.8796 > 0, (1 - K , ) ( 1 + K 2 ) ( 1 - K 3 ) ( 1 + K , ) ( 1 -
× (1 + K6)(1
-
-
Ks)
K7)(1 + Ks) = 28.8796.
4. C o n c l u s i o n s
We have derived two AR identities from the geometric interpretation of the recursive AR spectral formulas. They can be used as quick checks for a - K conversions as illustrated in our numerical example, and they also give stability information on the AR model.
Hence the two identities (14) and (15) are verified.
Appendix.
Proof of the two AR identities
First identity From (8) and (9), set 0 = 0, we have
Hence (10') implies
e~o"~)= DmDm-i • " • D2DIeo =[(I+K~)'''(I+K0],0 which together with (11) implies (14).
Second identity From (8) and (9), set 0 = ~r, we have
and I cosM~r/2
R-Mo/2= 1--sin M.tr/2
sinM~r/2] cos M~r/2 J"
When M is even, (10') implies
e~M)=([ I+K~O
[0 1 Signal Processing
--KM
1 -K~O ][01 - 1 0 ] ) . . . ( [ 1 +KI0
1ol][
lo M,] [1°2 loK2,][ 0
-- K 1
J.-K. Chan/ Two autoregressive identities
:[o
341
[_~1+~>~,_~,> o ][1] 0 - ( 1 - K2)(1 + K1) 0 o (_1)M/2[(1 + KM)(1 -- K ~ _ , ) " "" (1 + K2)(1 - K1) 1-
_~,o~>]
KM
L
]
(1 - K M ) ( 1 + K M _ I ) ( 1 - K~)(1 + K 0
0
=(_I)M~[(I+K~,)(1-k:M_~)'''0(I+K~)(1-K,)]. Now
R-~oJ2:[~-l~Mj' ~_l~J,], and (11) implies r(1 + KM)(1 -- K M - 1 ) " " " (1 + K2)(1 - K I ) ] . 0 J
L
A~o~) =
Hence (15) is true when M is even. When M is odd, (10') implies e'~)=([l+0K~
x
x
1_OM][01
--10] )
([1+~ o][o lo] [1+~ , °][°1 0
1 - KM-I
1
"""
0
o
[ ( 1 + K M _ I ) ( 1 - K M - 2 ) " ° " (1 + K 2 ) ( 1 - K 1 )
k
0
lo]tIlo] ]
(1 - K M _ ~ ) ( 1 + K M _ : ) ( 1 -- K : ) ( 1 + K1)
[
0 Kl)]" = (--1)(M-~)/EL(1 -- K ~ ) ( 1 + K~_~) • "" (1 + K:)(1 Now _ (_I)(M-I)/2 and (11) implies
A~oM)=[(1--KM)(I+ KM_,)'' O
(1 + K2)(1 - K1)].
Hence (15) is also true when M is odd.
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J.-K. Chan / Two autoregressive identities
References [1] K.J. Astrom, Introduction to Stochastic Control Theory, Academic Press, New York, 1970. [2] J.P. Burg, Maximum entropy spectral analysis, PhD dissertation, Department of Geophysics, Stanford University, May 1975. [3] J.-K. Chan, Geometric interpretation of recursive AR spectral computations directly from complex reflection coefficients, Norden Systems ASW Technical Notes, NS-ASW-TN-8902, Melville, New York, 1989.
Signal Processing
[4] J.-K. Chan, Recursive autoregressive spectral computations directly from reflection coefficients, IEEE Proc. lnternat. Conf. Acoust. Speech Signal Process., New York, 11-14 April 1988, pp. 2364-2367. [5] A.H. Gray Jr. and J.D. Markel, Digital lattice and ladder filter synthesis, IEEE Trans. Audio Electroacoust., Vol. AU21, No. 6, December 1973, pp. 491-500.