Gk+i(a,uo)

In document Some aspects of time series frequency estimation (Page 92-96)

(5.2.16) W e form th e inverse sp ec tra l e stim a te (27r / ( j 2)|afc(ez<9) |2 an d have th e following

T h e o r e m 5 .2 .2 . U nder th e conditions o f T heorem 5.2.1, th e in verse spectral e stim a te {2x / cr2)\a k{ct9)\2 has th e pro p erties

(i) For 0 ^ u>o, as k —*■ oo, (27r/cr2) |a fc(e ,0) |2 —► (27r/cr2) (ii) as 9 —* ljq, for ß x e d k,

\ak(e id)\2 =

c k , for Some COh*+ov\ts (c^]

so th a t as k —► oo, at 0 — u?o th e inverse spectral e stim a te is converging to zero, and hence th e sp ectra l e stim a te f k ( ß ) will be u n b o u n d e d at $ = u>o.

T h is th e o re m is th e resu lt w hich ensures th a t o u r frequency e stim a tio n pro ­ cedure is sensible as k —► oo. T h e following lem m a a b o u t th e d erivatives of the inverse s p e c tra l e stim ate , will b e used su b seq u en tly in in v estig atin g th e actu al frequency estim ate.

L e m m a 5 .2 .3 . L e t a k (e tuJ°) = a + ib. U nder th e conditions o f T h eo rem 5.2.1, we h ave fo r loq n o t too close to 0, and fo r large k,

t o (i) % 0 ( 1 ) . § 9 — U jQ (ii) J s \a k (e 'a)\2 = 0 ( k ) 9=u>0 2 (act' + bb') = O i ^ 1) 9=u>o (& ) i f 9=u> o o ( f c 2 ) , f i 1 = 0(k) 9—u /0 ( i v ) 3 3 lla * (e '#)l: 2 (aa" +

bb"

+ (a 1)2 + (V )2) = 0 ( k 2 ). 9 =ujq

(b) T h e sam e resu lts h o ld at 9 — £jk , th e freq u en cy at w hich j § \a k ic l&)\2 = 0,

(excep t for (ii), where 0 ( k ~ l ) is replaced b y zero).

5 .2 .3 A l t e r n a t i v e a p p r o a c h

A n a lte rn a tiv e a p p ro a ch to th e pro b lem of analysing th e p ro p e rtie s of a k{z ie)

w hich avoids th e explicit solu tio n of th e Y ule-W alker eq u atio n s a n d th e rep re­ se n ta tio n s of (5.2.8-11). T hese a u th o rs focus on th e singular Y ule-W alker-type e q u atio n s,

[ R (k - 1) - cr2Ik\ dfc = -rfc . (5.2.17) S to ica et al. show th a t th e difference [a* — a*] is 0 (fc - 2 ), a n d use this in estab lish in g , for a.k(z), z = ( p e ld), resu lts equivalent to L em m a 5.2.3, viz:

R e { a k ( e iu/°) } = O ^ " 1)

I m { a - (e ,u,°)} = 0 ( k ~ 2)

ßk = I m { a k {p.e'>)}] ^ ^ = 0 ( 1 )

Sk = - ^ - [ R e { a k( fi e’0)}] „ml

= --L [7 m {a * (/ie ’#)}]

= 0 ( k ) .

O f l o=w0 C/a n=w0

(N o te th a t in o u r n o ta tio n , th e ir L becom es k, th e ir 9 an d 9* becom e a * , C ( z )

becom es a k ( z ) . ) S toica et al. do n o t consider sam pling problem s, bein g concerned only w ith th e bias of th e frequency e stim a to r, as k —► oo w hich we take up in th e n e x t section. We m ake use of th e direct re p re se n ta tio n (5.2.11) for or*, and a lth o u g h ap p ro x im atio n s of leading term s in th e m a n n er of S to ica et al. are also useful it seem s th a t our p ro c e d u re of a ctu a lly solving th e n o n -sin g u lar Yule-W alker eq u atio n s gives a clearer ex p o sitio n of th e a n aly tic situ a tio n th a n does th e indirect ap p ro a ch .

H uzii (1980) is concerned w ith th e case w here o rd er k is fixed, k > 4- 1, w ith am p litu d es pj know n, a n d c e rta in n o t very re stric tiv e con d itio n s on th e noise process r ( n ) , in (5.1.1). He defines a ty p e of noise-to-signal ra tio , 77, w hich in our case £ = 1, x ( n ) = e(n ), becom es 77 = <72/p .

He proves th a t if th e au to co v arian ces r ( j , k) are e stim a te d using th e unw in­ dow ed (u n b iased ) e stim ate s

N- 1

r(j,K) =

(

n - k ) ~ 1 ^ 2

y(m

-

k)y(m

-

j )

in

R( k

1), a n d fjt = [ r ( 0 ,1 ) , . . . , r(0 , k)]T, th e n th e e stim ate s a* o b ta in e d by- solving th e u su al Y ule-W alker eq u atio n s (5.2.6) w ith th ese su b stitu tio n s converge in p ro b a b ility as N —* oo an d r] —♦ 0, to a p a rtic u la r so lu tio n of (5.2.17). He fu r th e r shows th a t u n d e r these assu m p tio n s, /jt( 0 ) converges in p ro b a b ility to a fu n c tio n / 7 (0 ) w hich satisfies lim ^ ^o f v(0) = oo a t 0 = u 0.

H ow ever, w hile it is clear in principle how to im plem ent “increase N ” or “in­ crease k” to ap p ro ach m ore closely th e lim itin g b eh av io u r, it is n o t clear how “d ecrease 77” can be im p lem en ted if one is g a th erin g d a ta on som e process, p a r­ tic u la rly n o t as th e lim it 77 —► 0 is ta k en a fter th e lim it N —► 0 0 . W hile th is line of an aly sis is in terestin g , (a n d can easily b e ded u ced from o u r re su lts above), it a p p e a rs to be a side-issue.

T h e resu lts of Stoica et al. (1986) on th e difference (a * — a * ) suggest th a t dfc m ay b e of th e form of th e first te rm in th e re p re se n ta tio n (5.2.11), since we have also

a k ( j ) + - cos jcjQ

o(k- 2),

a n d in L em m a 5.2.3 we have re su lts equivalent to th eirs b ased on th is re p re se n ta ­ tio n .

5 .3 P r o p e r t i e s o f t h e E s t im a t e d T r a n sfe r F u n c t io n

äk(eld)

5 .3 .1 P r o p e r t ie s o f t h e a u t o c o v a r ia n c e e s t im a t e s r*

y ( N — 1)} on {i/(n)} defined by (5.1.1), form the ‘windowed’ d ata m atrix Y ( k),

2/ (0) 0

y(i)

y (

0)

Y( k )

y(k),

- 1),

y ( N - l ) , y ( N — 2),

. . .

y ( N - l )

0

y(i)

y(0)

y ( N - k - l )

y ( N -

1)

y ( N - 2)

y ( N — 1)

and define the (Toeplitz) autocovariance estimation m atrix and vector

In document Some aspects of time series frequency estimation (Page 92-96)