combining direct spectral estimators:...
TRANSCRIPT
Combining Direct Spectral Estimators: I
• tapering introduced as means of creating SDF estimator S(d)(·)with potentially better bias properties than periodogram S(p)(·)• both S(d)(·) & S(p)(·) inherently noisy – smoothing across fre-
quencies leads to lag window estimator S(lw)(·)• large-sample approximation to variance of S(lw)(f ) points out
price of tapering, namely, inflation of variance by Ch > 1:
var {S(lw)(f )} ≈ ChS2(f )
BWN ∆t
− Ch = 1 if and only if rectangular taper used (i.e., no real ta-pering done, leading to periodogram & use of prewhitening)
− Ch ≈ 2 for Hanning data taper
SAPA2e–371 VIII–1
Combining Direct Spectral Estimators: II
• Q: can we compensate for increase in variance due to tapering?
• A: yes, by combining together different S(d)(·)’s formed usingsame time series
• two schemes
− multitapering: use multiple orthogonal tapers on {Xt}−Welch’s overlapped segment averaging (WOSA): use a single
taper multiple times on {Xt}• like lag window estimators, multitaper and WOSA estimators
reduce variance over that given by S(d)(·), but do so withoutextracting price similar to Ch
• will discuss multitaper estimators first and then WOSA
SAPA2e–371 VIII–2
Multitaper Spectral Estimation: I
• tapering useful for S(·) with large dynamic range, but increases
variance of S(lw)(f ) by Ch > 1
• alternatives to S(lw)(·), i.e., smoothing S(d)(·), include prewhiten-ing (see Chapter 6), WOSA and multitapering (Thomson, 1982)
• why multitapering?
− works automatically on high dynamic range SDFs
− natural definition of resolution
− can tradeoff bias/variance/resolution easily
− for some processes, can argue
∗ superior to prewhitening (Thomson, 1990a)
∗ superior to WOSA (Bronez, 1992)
SAPA2e–375, 376, 377 VIII–3
Multitaper Spectral Estimation: II
− produces ‘S(d)(·)’ with EDOF ν = 2K, where K is thenumber of tapers used (2 to 10 typically; recall that widthof 95% CIs shrinks considerably as ν increases from 2 to 10)
− can get internal estimate of variance (‘jackknifing’)
− can handle mixed spectra (i.e., line components)
− extends naturally to irregularly sampled processes
SAPA2e–375, 376, 377 VIII–4
Widths of 95% CIs on dB Scale versus EDOF ν
100 101 102 103
ν
−10
0
10
20
dB
3 dB
2 dB
1 dB
SAPA2e–294 VIII–5
Basics of Multitapering: I
• basic multitaper estimator is average of K direct spectral esti-mators:
S(mt)(f ) ≡ 1
K
K−1∑k=0
S(mt)k (f ),
where
S(mt)k (f ) ≡ ∆t
∣∣∣∣∣∣N−1∑t=0
hk,tXte−i2πft∆t
∣∣∣∣∣∣2
is called kth eigenspectrum and uses kth taper {hk,t} normal-
ized by∑t h
2k,t = 1
SAPA2e–372 VIII–6
Basics of Multitapering: II
• generalization of interest later on: weighted multitaper estima-tor
S(wmt)(f ) ≡K−1∑k=0
dkS(mt)k (f ),
where weights dk are nonnegative and sum to unity
• basic multitaper estimator is a special case of the above: justset dk = 1/K
SAPA2e–372 VIII–7
Basics of Multitapering: III
• spectral window for kth eigenspectrum:
Hk(f ) ≡ ∆t
∣∣∣∣∣∣N−1∑t=0
hk,te−i2πft∆t
∣∣∣∣∣∣2
• an eigenspectrum is just a direct spectral estimator, so
E{S(mt)k (f )} =
∫ fN
−fNHk(f − f ′)S(f ′) df ′
• for the basic multitaper estimator, we thus have
E{S(mt)(f )} =
∫ fN
−fNH(f−f ′)S(f ′) df ′ with H(f ) ≡ 1
K
K−1∑k=0
Hk(f )
SAPA2e–372, 373 VIII–8
Basics of Multitapering: IV
• corresponding result for weighted multitaper estimator is
E{S(wmt)(f )} =
∫ fN
−fNH(f−f ′)S(f ′) df ′ with H(f ) ≡
K−1∑k=0
dkHk(f )
• leakage for S(mt)(·) or S(wmt)(·) OK if Hk(·)’s all have smallsidelobes
• if all K eigenspectra are approximately unbiased, then multi-taper estimators will be also approximately so
SAPA2e–372, 373 VIII–9
Basics of Multitapering: V
• recall standard measure of effective bandwidth for direct spec-tral estimator:
BHk ≡ widtha {Hk(·)} =
(∫ fN−fNHk(f ) df
)2
∫ fN−fNH2k(f ) df
=∆t∑N−1
τ=−(N−1)(h ? hk,τ )2
• for basic and weighted multitaper estimators, measures are
BH =∆t∑N−1
τ=−(N−1)
(1K
∑K−1k=0 h ? hk,τ
)2
and
BH =∆t∑N−1
τ=−(N−1)
(∑K−1k=0 dkh ? hk,τ
)2
SAPA2e–372, 373 VIII–10
Basics of Multitapering: VI
• rationale for averaging eigenspectra is to produce estimator ofS(f ) with variance smaller than any single eigenspectrum
• making use of Exercise [2.1e] (Problem 4), have
var {S(mt)(f )} =1
K2
K−1∑k=0
var {S(mt)k (f )} +
2
K2
∑j<k
cov {S(mt)j (f ), S
(mt)k (f )}
• assume S(·) locally constant about f
• for j 6= k, can argue cov {S(mt)j (f ), S
(mt)k (f )} ≈ 0 if
N−1∑t=0
hj,thk,t = 0, i.e., if tapers are orthogonal
• var {S(mt)k (f )} ≈ S2(f ) =⇒ var {S(mt)(f )} ≈ S2(f )/K
SAPA2e–373, 374 VIII–11
Basics of Multitapering: VII
• corresponding result for weighted multitaper estimator is
var {S(wmt)(f )} ≈ S2(f )
K−1∑k=0
d2k,
which reduces to
var {S(mt)(f )} ≈ S2(f )
K
by setting dk = 1/K
SAPA2e–374 VIII–12
Basics of Multitapering: VIII
• to determine approximate distribution of S(mt)(f ), note that,since each eigenspectrum is a direct spectral estimator,
S(mt)k (f )
d= S(f )χ2
2/2 for 0 < f < fNasymptotically as N →∞• factoid: if χ2
ν1& χ2
ν2are independent chi-square RVs with ν1
& ν2 DOFs, then χ2ν1
+χ2ν2
is chi-square RV with ν1 + ν2 DOF
• assuming mutual independence, implies
S(mt)(f )d= S(f )χ2
2K/2K for 0 < f < fNasymptotically as N →∞
SAPA2e–374, 375 VIII–13
Basics of Multitapering: IX
• corresponding result for weighted multitaper estimator:
S(mt)(f )d= S(f )χ2
ν/ν for 0 < f < fNasymptotically as N →∞, where
ν =2∑K−1
k=0 d2k
(note that ν = 2K when dk = 1/K)
• two sets of orthogonal tapers in common use
− Slepian (DPSS) tapers (Thomson, 1982)
− sinusoidal tapers (Riedel and Sidorenko, 1995)
• both are termed constructive multitaper estimators (as op-posed to deconstructed weighted multitaper estimators to bediscussed later on)
SAPA2e–375 VIII–14
Slepian Multitapers: I
• tapers minimize spectral window sidelobes
• for fixed design bandwidth 2W , measure sidelobes via
β2k(W ) ≡
∫W−W Hk(f ) df∫ fN−fNHk(f ) df
• for given W and N , {h0,t} maximizes β20(W )
• {hk,t} maximizes β2k(W ) amongst sequences orthogonal to
{h0,t}, {h1,t}, . . ., {hk−1,t}
SAPA2e–378, 95 VIII–15
Slepian Multitapers: II
• computation of {hk,t}’s requires solution of Ahk = β2k(W )hk,
where hTk =[hk,0, · · · , hk,N−1
]and
At′,t =sin(2πW (t′ − t))
π(t′ − t)is (t′, t)th element of N ×N matrix A
• can solve using inverse iteration (stable, but slow), numericalintegration (Thomson, 1982) or tridiagonal formulation (fast!)
SAPA2e–95 VIII–16
Slepian Multitapers: III
• number of {hk,t}’s with good leakage protection is 2NW ∆t−1or less
• strategy & considerations for picking K
− set design half-bandwidthW as multiple J of spacing 1/N ∆tbetween Fourier frequencies, where J = 2, 3, 4, . . .
−W = J/N ∆t expressed as NW∆t = J (or just NW = J)
− set K < 2NW ∆t = 2J , noting that, as K increases, vari-ance decreases, but leakage gets worse
− increasing W implies a decrease in resolution, but gain moreleakage-free tapers to work with (i.e., can increase K)
• following plots use NW = 4 (i.e, 2NW = 8), for which Kshould be no greater than 7, but eighth taper {h7,t} also shown
SAPA2e–378, 379 VIII–17
AR(4) Series, NW = 4 Slepian {h0,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.08
0.00
0.08
Sle
pian
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–380 VIII–18
AR(4) Series, NW = 4 Slepian {h1,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.08
0.00
0.08
Sle
pian
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–380 VIII–19
AR(4) Series, NW = 4 Slepian {h6,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.08
0.00
0.08
Sle
pian
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–380 VIII–20
NW = 4 Slepian H0(·), S(mt)0 (·), H(·) and S(mt)(·), K = 1
−60
−40
−20
0
20
dB
k = 0
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 1
SAPA2e–381, 384 VIII–21
NW = 4 Slepian H1(·), S(mt)1 (·), H(·) and S(mt)(·), K = 2
−60
−40
−20
0
20
dB
k = 1
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 2
SAPA2e–381, 384 VIII–22
NW = 4 Slepian H6(·), S(mt)6 (·), H(·) and S(mt)(·), K = 7
−60
−40
−20
0
20
dB
k = 6
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 7
SAPA2e–383, 385 VIII–23
Sinusoidal Multitapers: I
• recall notion of smoothing window bias:
bW ≡∫ fN
−fNWm(f − f ′)S(f ′) df ′ − S(f )
≈ S′′(f )
2
∫ fN
−fNφ2Wm(φ) dφ =
S′′(f )
24β2W
(part of criterion used to derive Papoulis lag window)
• recall notion of spectral window bias (Exer. [211]):
b(f ) ≡ E{S(d)(f )} − S(f )
≈ S′′(f )
2
∫ fN
−fNφ2H(φ) dφ ≡ S′′(f )
24β2H
SAPA2e–413 VIII–24
Sinusoidal Multitapers: II
• for given N , {h0,t} minimizes β2H0
• {hk,t} minimizes β2Hk amongst sequences orthogonal to
{h0,t}, {h1,t}, . . . , {hk−1,t}
• scheme due to Riedel & Sidorenko (1995), who actually workedwith continuous parameter processes (similiar to Papoulis, 1973)
• can approximate solutions well using
hk,t =
{2
N + 1
}1/2
sin
{(k + 1)π(t + 1)
N + 1
}, t = 0, 1 . . . , N−1,
which is very easy to compute!
SAPA2e–414, 415 VIII–25
Sinusoidal Multitapers: III
• all {hk,t}’s give moderate leakage protection (like Hanning)
• strategy & considerations for picking K
− bandwidth BH ≈ (K+1)/(N+1) increases with K (dashed& solid red lines show BH/2 & (K + 1)/(2N + 2) on plots)
− leakage relatively unchanged as K increases
− can trade off variance and resolution by increasing K, whichdecreases variance and resolution (i.e., increases bandwidth)
• sinusoidal tapers vs. Slepian tapers
− 1 parameter (K) vs. 2 parameters (2W & K)
− moderate vs. adjustable leakage protection
− juggle resolution/variance vs. leakage/resolution/variance
− simple expression vs. need software to compute
SAPA2e–423 VIII–26
AR(4) Series, Sinusoidal {h0,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
sine
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–417 VIII–27
AR(4) Series, Sinusoidal {h1,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
sine
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–417 VIII–28
AR(4) Series, Sinusoidal {h6,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
sine
tape
r
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–417 VIII–29
Sinusoidal H0(·), S(mt)0 (·), H(·) and S(mt)(·), K = 1
−60
−40
−20
0
20
dB
k = 0
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 1
VIII–30
Sinusoidal H1(·), S(mt)1 (·), H(·) and S(mt)(·), K = 2
−60
−40
−20
0
20
dB
k = 1
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 2
VIII–31
Sinusoidal H6(·), S(mt)6 (·), H(·) and S(mt)(·), K = 7
−60
−40
−20
0
20
dB
k = 6
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 7
VIII–32
Recovery of ‘Lost Information”: I
• Slepian tapers are solutions to Ahk = β2k(W )hk
• N orthonormal solutions h0, . . . , hN−1
• can order via eigenvalues (concentration measure):
1 > β20(W ) > β2
1(W ) > · · · > β2N−1(W ) > 0
• only first K < 2NW ∆t have β2k(W ) ≈ 1
• form V =[h0, h1, . . . , hN−1
]• V TV = IN restates orthonormality, where IN is N ×N iden-
tity matrix:N−1∑t=0
hj,thk,t =
{1, j = k;
0, j 6= k
SAPA2e–388 VIII–33
Recovery of ‘Lost Information’: II
• since V T = V −1, also have V V T = IN , yielding
N−1∑k=0
hk,thk,t′ =
{1, t = t′;
0, t 6= t′.
• thus have
N−1∑k=0
N−1∑t=0
(hk,tXt
)2=
N−1∑t=0
X2t
N−1∑k=0
h2k,t︸ ︷︷ ︸
(∗)
=
N−1∑t=0
X2t
because (∗) – the energy across tapers – is unity
• following figures shows relative influence of Xt’s by plotting∑K−1k=0 h2
k,t versus t for K = 1, . . . , 8 (here NW ∆t = 4)
SAPA2e–388 VIII–34
Decomposition of Slepian Taper Energy for K = 1
0 256 512 768 1024
t
0
1
SAPA2e–387 VIII–35
Decomposition of Slepian Taper Energy for K = 2
0 256 512 768 1024
t
0
1
SAPA2e–387 VIII–36
Decomposition of Slepian Taper Energy for K = 7
0 256 512 768 1024
t
0
1
SAPA2e–387 VIII–37
Multitapering of White Noise: I
• assume X0, . . . , XN−1 is zero mean Gaussian white noise withunknown variance s0 and SDF S(f ) = s0 (here we set ∆t tounity for convenience)
• by several criteria, best estimate of s0 is∑N−1t=0 X2
t /N = s(p)0
• implies best estimate of S(f ) is s(p)0
• can obtain best estimator from S(p)(·) via∫ 1/2
−1/2S(p)(f ) df = s
(p)0 ;
i.e., ‘smoothing’ with Wm(f ) = 1
• Equation (392) says var {s(p)0 } = 2s2
0/N
SAPA2e–392 VIII–38
Multitapering of White Noise: II
• let S(d)(·) be direct spectral estimator based upon {h0,t}
• smoothing S(d)(·) with Wm(f ) = 1 yields∫ 1/2
−1/2S(d)(f ) df =
N−1∑t=0
h20,tX
2t = s
(d)0 .
• since var {X2t } = 2s2
0 (Isserlis – Equation (32a)), have
var {s(d)0 } =
N−1∑t=0
var {h20,tX
2t } = 2s2
0
N−1∑t=0
h40,t = 2s2
0Ch/N,
where, as before, Ch ≡ N∑N−1t=0 h4
0,t
SAPA2e–392 VIII–39
Multitapering of White Noise: III
• use Cauchy inequality∣∣∣∣N−1∑t=0
atbt
∣∣∣∣2 ≤ N−1∑t=0
|at|2N−1∑t=0
|bt|2 ,
with at = h20,t and bt = 1 to argue
∑N−1t=0 h4
0,t ≥ 1/N ; i.e.,
Ch ≥ 1, with equality if and only if h0,t = 1/√N
• can conclude
var {s(d)0 } = 2s2
0Ch/N > 2s20/N = var {s(p)
0 }for any nonrectangular taper
SAPA2e–392 VIII–40
Multitapering of White Noise: IV
• claim: multitapering reclaims best estimator
• let {h0,t}, {h1,t}, . . . , {hN−1,t} be orthonormal
• let V be the N ×N matrix given by
V ≡
h0,0 h1,0 . . . hN−1,0
h0,1 h1,1 . . . hN−1,1... ... . . . ...
h0,N−1 h1,N−1 . . . hN−1,N−1
• orthonormality says V T V = IN & hence V V T = IN
• kth eigenspectrum:
S(mt)k (f ) ≡
∣∣∣∣∣∣N−1∑t=0
hk,tXte−i2πft
∣∣∣∣∣∣2
SAPA2e–393 VIII–41
Multitapering of White Noise: V
• form S(mt)(·) by averaging all S(mt)k (·)’s:
S(mt)(f ) ≡ 1
N
N−1∑k=0
S(mt)k (f )
=1
N
N−1∑k=0
N−1∑t=0
hk,tXte−i2πft
N−1∑u=0
hk,uXuei2πfu
=
1
N
N−1∑t=0
N−1∑u=0
XtXu
N−1∑k=0
hk,thk,u
︸ ︷︷ ︸
1 if t = u; 0 if t 6= u
e−i2πf (t−u)
=1
N
N−1∑t=0
X2t = s
(p)0
SAPA2e–393 VIII–42
Multitapering of White Noise: VI
• note: holds for any set of orthonormal tapers!
• as K increases, can study rate of decay
var {S(mt)(f )} = var
1
K
K−1∑k=0
S(mt)k (f )
=
1
K2
K−1∑j=0
K−1∑k=0
cov {S(mt)j (f ), S
(mt)k (f )}
• Exercise [8.8b] indicates how to compute this for white noise
SAPA2e–394, 395 VIII–43
Multitapering of White Noise: VII
• following plot shows example for f = 1/4 using Slepian tapers
− N = 64; NW = 4; s0 = 1; S(f ) = 1
− thick curve: var {S(mt)(1/4)} vs. K
∗K = 1: var {S(mt)(1/4)} = S2(f ) = 1
∗K = N : var {S(mt)(1/4)} = 2/N.= 0.03
∗ curve agrees with these values
− thin curve: computed assuming
cov {S(mt)j (f ), S
(mt)k (f )} = 0 when j 6= k
• dashed vertical line marks Shannon number 2NW = 8
• two curves agree closely for K ≤ 2NW
• variance decreases slowly for K > 2NW (bias then can be badfor nonwhite processes)
SAPA2e–394, 395 VIII–44
var {S(mt)(1/4)} versus Number K of Eigenspectra
0 16 32 48 64
K
10−2
10−1
100
varia
nce
of m
ultit
aper
est
imat
e
SAPA2e–394 VIII–45
Multitapering of White Noise: VIII
• following plot shows 2nd example, which differs from 1st onlyin that NW = 16 rather than 4
• two examples similar in some aspects, but differ in others
− NW = 4 achieves minimum only when K = 64, whereasNW = 16 achieves minimum at both K = 32 and K = 64
− NW = 4 after Shannon number has smaller values, whereasNW = 16 after Shannon has larger values (except K = 64)
• 2nd example shows that, if U0, U1, . . . are correlated RVs withcommon mean µ & common variance, variance of sample mean
µN =1
N
N−1∑n=0
Un
can actually increase as N increases!
SAPA2e–394, 395 VIII–46
var {S(mt)(1/4)} versus Number K of Eigenspectra
0 16 32 48 64
K
10−2
10−1
100
varia
nce
of m
ultit
aper
est
imat
e
SAPA2e–394 VIII–47
Multitapering of White Noise: IX
• following plot shows 3rd example, which differs from 1st & 2ndonly in that sinusoidal tapers are used rather than Slepian
• after accounting for numerical precision, R declares that, atall K, var {S(mt)(1/4)} is identical for Slepian tapers withNW = 16 and sinusoidal tapers!
• Q: why?
• A: hmmm . . .
SAPA2e–426 VIII–48
var {S(mt)(1/4)} versus Number K of Eigenspectra
0 16 32 48 64
K
10−2
10−1
100
varia
nce
of m
ultit
aper
est
imat
e
SAPA2e–426 VIII–49
Quadratic Spectral Estimators: I
• provides important motivation for multitapering
• let X0, . . . , XN−1 be portion of real-valued stationary processwith zero mean 0, SDF S(·) and ACVS {sτ}• for fixed f , define Zt ≡ Xte
i2πft∆t
• Exercise [5.13a]: {Zt} stationary with
SZ(f ′) = S(f − f ′) and sZ,τ = sτei2πfτ ∆t
(recall that S(·) is periodic with a period of 2fN )
• note: SZ(0) = S(f ), so can estimate S(f ) by estimating SZ(·)at f = 0
SAPA2e–395, 396, 177 VIII–50
Quadratic Spectral Estimators: II
• let Z be vector with tth element Zt
• let ZH be its Hermitian transpose:
ZH ≡[Z∗0 , . . . , Z
∗N−1
]note: if A real-valued matrix, then AH = AT
• since XtXt′∆t has same units as S(f ), consider
S(q)(f ) ≡ S(q)Z (0) ≡ ∆t
N−1∑s=0
N−1∑t=0
Z∗sQs,tZt = ∆tZHQZ,
where Qs,t is (s, t)th element of weight matrix Q
• S(q)(f ) called quadratic spectral estimator
SAPA2e–396 VIII–51
Quadratic Spectral Estimators: III
• assumptions about N ×N matrix Q:
− Qs,t is real-valued
− Q is symmetric; i.e., Qs,t = Qt,s− Qs,t does not depend on {Zt}
• if Q positive semidefinite (PSD), then S(q)(f ) ≥ 0, which isobviously a desirable property in view of S(f ) ≥ 0
• three examples of quadratic estimators
SAPA2e–396 VIII–52
Quadratic Spectral Estimators: IV
1. lag window estimator (need not be PSD):
S(lw)(f ) ≡ ∆t
N−1∑τ=−(N−1)
wm,τ s(d)τ e−i2πfτ ∆t
= ∆t
N−1∑s=0
N−1∑t=0
wm,t−shsXshtXte−i2πf (s−t) ∆t
= ∆tN−1∑s=0
N−1∑t=0
Z∗s hswm,s−tht︸ ︷︷ ︸= Qs,t
Zt
SAPA2e–396 VIII–53
Quadratic Spectral Estimators: V
2. direct spectral estimator (always PSD – special case of S(lw)(f )with wm,τ = 1):
S(d)(f ) ≡ ∆t
∣∣∣∣∣∣N−1∑t=0
htXte−i2πft∆t
∣∣∣∣∣∣2
= ∆t
N−1∑s=0
N−1∑t=0
Z∗s hsht︸︷︷︸= Qs,t
Zt
3. basic multitaper estimator (always PSD):
S(mt)(f ) ≡ ∆t
K
K−1∑k=0
∣∣∣∣∣∣N−1∑t=0
hk,tXte−i2πft∆t
∣∣∣∣∣∣2
= ∆t
N−1∑s=0
N−1∑t=0
Z∗s
K−1∑k=0
hk,shk,tK︸ ︷︷ ︸
= Qs,t
Zt
SAPA2e–397, 372 VIII–54
Quadratic Spectral Estimators: VI
• goal: set Q so S(q)(·) unbiased and has small variance
• to get S(q)(f ) ≥ 0, assume Q is PSD: let K = rank of Q, andassume 1 ≤ K ≤ N (rules out K = 0, which is not of interest)
• matrix theory: there exists an N ×N orthonormal matrix HNsuch that HT
NQHN = DN , where DN is a diagonal matrixwith diagonal elements
d0 ≥ d1 ≥ · · · ≥ dK−1 > 0 and dK = · · · = dN−1 = 0
with each dk being an eigenvalue of Q
• Exercise [397a]: as a result of the above, can write
Q = HDKHT ,
where H is N×K matrix (the 1st K columns of HN ), and DKis a diagonal matrix with diagonal entries d0, d1, . . . , dK−1
SAPA2e–397 VIII–55
Quadratic Spectral Estimators: VII
• Exercise [397b]: substituting HDKHT for Q in
S(q)(f ) = ∆tZHQZ
yields
S(q)(f ) = ∆tZHHDKHTZ = ∆t
K−1∑k=0
dk
∣∣∣∣∣∣N−1∑t=0
hk,tXte−i2πft∆t
∣∣∣∣∣∣2
,
where hk,t is the (k, t)th element of H
• reusing definition S(mt)k (f ) = ∆t
∣∣∣∑N−1t=0 hk,tXte
−i2πft∆t∣∣∣2,
can write
S(q)(f ) =
K−1∑k=0
dkS(mt)k (f ) = S(wmt)(f )
SAPA2e–397, 398 VIII–56
Quadratic Spectral Estimators: VIII
• conclusion: can write any PSD quadratic estimator
S(q)(f ) = ∆tZHQZ
with weight matrix Q of rank K as a weighted multitaper es-timator with
− weights dk given by positive eigenvalues of Q
− tapers {hk,t} given by associated eigenvectors of Q (tapersare mutually orthonormal)
• S(q)(f ) formulated as S(wmt)(f ) termed a deconstructed mul-titaper estimator
• opposing concept is constructive multitaper estimator for whichtapers are formulated explicitly (e.g., Slepian or sinusoidal)
SAPA2e–398 VIII–57
Deconstructed Multitaper Spectral Estimators: I
• as an example of a deconstructed multitaper spectral estimator,consider an m = 179 Parzen lag window estimator used with adirect spectral estimator employing a Hanning data taper
• following plot shows estimate for AR(4) time series of Fig-ure 36(e)
SAPA2e–435 VIII–58
Parzen S(lw)(·) with Hanning S(d)(·) & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
m = 179
SAPA2e–327 VIII–59
Deconstructed Multitaper Spectral Estimators: II
• weight matrix Q for m = 179 Parzen lag window estimator isof full rank so K = N , but, as shown in next plot, eigenval-ues decay rapidly to zero, implying that effective rank is muchsmaller
SAPA2e–435 VIII–60
Eigenvalues of Q for Lag Window Estimator
●
●
●
●
●
●●
●● ● ● ● ● ● ● ●
k
eige
nval
ue
0 5 10 15
0.00
0.05
0.10
0.15
0.20
0.25
0.30
SAPA2e–435 VIII–61
Deconstructed Multitaper Spectral Estimators: III
• following plots show first seven eigenvectors of Q – these serveas data tapers in weighted multitaper representation for lagwindow estimator
SAPA2e–435, 437, 438 VIII–62
AR(4) Series, Deconstructed {h0,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–63
AR(4) Series, Deconstructed {h1,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–64
AR(4) Series, Deconstructed {h6,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–65
Deconstructed Multitaper Spectral Estimators: IV
• following plots show eigenspectra based on seven data tapers,along with weighted multitaper estimates of increasing order K
• weights dk determined by eigenvalues dk, but renormalized foreach K to sum to unity:
dk =dk∑K−1
k′=0dk′, k = 0, . . . , K − 1
(renormalization required because∑N−1k=0 dk = 1)
• plots also show spectral windows for eigenspectra and weightedmultitaper estimates, with vertical red dashed lines indicatingstandard bandwidth measures (either BHk or BH)
SAPA2e–435, 437, 438 VIII–66
Deconstructed H0(·), S(mt)0 (·), H(·) and S(wmt)(·), K = 1
−60
−40
−20
0
20
dB
k = 0
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 1
VIII–67
Deconstructed H1(·), S(mt)1 (·), H(·) and S(wmt)(·), K = 2
−60
−40
−20
0
20
dB
k = 1
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 2
VIII–68
Deconstructed H6(·), S(mt)6 (·), H(·) and S(wmt)(·), K = 7
−60
−40
−20
0
20
dB
k = 6
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 7
SAPA2e–437 VIII–69
Deconstructed Multitaper Spectral Estimators: V
• because eigenvalues decrease to zero rapidly, K = 7 multitaperestimate is a reasonable approximation to lag window estimate,as next plot shows
SAPA2e–437, 438 VIII–70
Parzen S(lw)(·) with K = 7 S(wmt)(·) Approximation
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
SAPA2e–437, 327 VIII–71
Deconstructed Multitaper Spectral Estimators: VI
• lag window estimators are sometimes well approximated by low-order weighted multitaper estimators, suggesting that
‘lag-windowing and multiple-data-windowing are roughlyequivalent for smooth spectrum estimation’
(title of article by McCloud et al.,1999)
SAPA2e–438 VIII–72
Quadratic Spectral Estimators: XI
• Q: what conditions on Q ensure S(q)(f ) has good bias andvariance properties?
• let’s consider line of thought leading to Slepian tapers (Bronez,1985)
SAPA2e–395 VIII–73
First Moment of S(q)(·): I
• since S(q)(·) has a weighted multitaper representation, an ap-peal to results stated for the latter says that
E{S(q)(f )} =
∫ fN
−fNH(f − f ′)S(f ′) df ′,
where
H(f ) ≡K−1∑k=0
dkHk(f ) with Hk(f ) ≡ ∆t
∣∣∣∣∣∣N−1∑t=0
hk,te−i2πft∆t
∣∣∣∣∣∣2
SAPA2e–398 VIII–74
First Moment of S(q)(·): II
• Exercise [8.12] gives equivalent ‘time domain’ expression:
E{S(q)(f )} = ∆t tr {QΣZ} = ∆t tr {HDKHTΣZ} = ∆t tr {DKHTΣZH},where tr = trace and ΣZ = covariance matrix for Zt’s,
− last equation above uses following result: if A and B havedimensions M ×N and N ×M , then tr {AB} = tr {BA}
• equating frequency and time domain expressions yields∫ fN
−fNH(f − f ′)S(f ′) df ′ = ∆t tr {DKHTΣZH}
SAPA2e–398 VIII–75
First Moment of S(q)(·): III
• for general {Xt}, can get handle on first moment by incorpo-rating notion of resolution (key idea!)
• given resolution bandwidth 2W > 0, seek Q’s so
E{S(q)(f )} ≈ 1
2W
∫ f+W
f−WS(f ′) df ′ ≡ S(f ),
i.e., no longer seek E{S(q)(f )} ≈ S(f )
• rationale
− ‘regularizes’ SDF estimation problem: S(·) smooth to somedegree, whereas S(·) need not be
− incorporates filter bandwidth in filtering interpretation ofS(·) (Section 5.6)
SAPA2e–398, 399 VIII–76
First Moment of S(q)(·): IV
• strategy
− set resolution bandwidth 2W appropriately
− optimize bias/variance within limitations imposed by choiceof 2W
• basically we are giving up finest possible resolution of 1/N ∆tto get handle on bias/variance
SAPA2e–399 VIII–77
First Moment of S(q)(·): V
• definition for bias that incorporates notion of resolution is
b{S(q)(f )} = E{S(q)(f )} − S(f ) = E{S(q)(f )} − 1
2W
∫ f+W
f−WS(f ′) df ′
• since
E{S(q)(f )} =
∫ fN
−fNH(f − f ′)S(f ′) df ′,
insisting upon b{S(q)(f )} = 0 would require
H(f ′) =
{1/(2W ), |f ′| ≤ W ;
0, W < |f ′| ≤ fN
• H(·) is sum of functions Hk(·) that are based on Fourier trans-
forms of index-limited sequences – hence H(·) cannot be exactlyband-limited
SAPA2e–399 VIII–78
First Moment of S(q)(·): VI
• game plan
− insist that S(q)(f ) be unbiased for white noise
− break up bias into two components (yet to be defined, butwe’ll end up calling them ‘broad-band bias’ and ‘local bias’)
− develop bounds for both components for colored noise
SAPA2e–399 VIII–79
Unbiasedness for White Noise: I
• for white noise S(f ) = s0 ∆t for all f , so
S(f ) =1
2W
∫ f+W
f−WS(f ′) df ′ =
1
2W
∫ f+W
f−Ws0 ∆t df ′ = s0 ∆t
• since
E{S(q)(f )} =
∫ fN
−fNH(f − f ′)S(f ′) df ′ = s0 ∆t
∫ fN
−fNH(f − f ′) df ′,
have unbiasedness if ∫ fN
−fNH(f ′) df ′ = 1
SAPA2e–399 VIII–80
Unbiasedness for White Noise: II
• noting that∫ fN
−fNH(f ′) df ′ =
∫ fN
−fN
K−1∑k=0
dkHk(f ′) df ′ =
K−1∑k=0
dk
because ∫ fN
−fNHk(f ′) df ′ =
N−1∑t=0
h2k,t = 1,
requirement for unbiasedness is
K−1∑k=0
dk = 1
SAPA2e–399, 400 VIII–81
Unbiasedness for White Noise: III
• alternative approach: since ΣZ = s0IN for white noise, can use
E{S(q)(f )} = ∆t tr {DKHTΣZH}= s0 ∆t tr {DKHTH}= s0 ∆t tr {DK}
because columns {hk,t} of H are orthonormal
• use of the fact that
tr {DK} =
K−1∑k=0
dk
leads to the same requirement for white noise unbiasedness:K−1∑k=0
dk = 1
SAPA2e–400 VIII–82
Broad-Band & Local Bias: I
• using new notion of bias, have
b{S(q)(f )} ≡ E{S(q)(f )} − S(f )
=
∫ fN
−fNH(f − f ′)S(f ′) df ′ − 1
2W
∫ f+W
f−WS(f ′) df ′
=
∫ f+W
f−W
[H(f − f ′)− 1
2W
]S(f ′) df ′
+
∫f ′ 6∈[f−W,f+W ]
H(f − f ′)S(f ′) df ′
≡ b(l){S(q)(f )}︸ ︷︷ ︸local bias
+ b(b){S(q)(f )}︸ ︷︷ ︸broad-band bias
• to bound bias terms, assume S(·) bounded by Smax; i.e., S(f ) ≤Smax <∞ for all f
SAPA2e–400 VIII–83
Broad-Band and Local Bias: II
• bound on magnitude of local bias:∣∣∣b(l){S(q)(f )}∣∣∣ =
∣∣∣∣∣∫ f+W
f−W
[H(f − f ′)− 1
2W
]S(f ′) df ′
∣∣∣∣∣≤∫ f+W
f−W
∣∣∣∣H(f − f ′)− 1
2W
∣∣∣∣S(f ′) df ′
≤ Smax
∫ W
−W
∣∣∣∣H(f ′′)− 1
2W
∣∣∣∣ df ′′;integral gives useful measure of local bias
• local bias small if H(f ) ≈ 1/2W over [−W,W ]
SAPA2e–400 VIII–84
Broad-Band and Local Bias: III
• bound on broad-band bias (must be positive!):
b(b){S(q)(f )} =
∫f ′ 6∈[f−W,f+W ]
H(f − f ′)S(f ′) df ′
≤ Smax
∫f ′ 6∈[f−W,f+W ]
H(f − f ′) df ′
= Smax
∫f 6∈[−W,W ]
H(f ′′) df ′′
= Smax
(∫ fN
−fNH(f ′′) df ′′ −
∫ W
−WH(f ′′) df ′′
)= Smax
(tr {DK} − tr {DKHTΣ(bl)H}
),
where Σ(bl) arises from the following argument
SAPA2e–401 VIII–85
Broad-Band and Local Bias: IV
• let {Xt} be band-limited white noise; i.e., has SDF and ACVS
S(bl)(f ) ≡
{∆t, |f | ≤ W ;
0, W < |f | ≤ fN ,and s
(bl)τ ≡
{2W ∆t, τ = 0;sin (2πWτ ∆t)
πτ τ 6= 0.
• for this SDF (and letting f = 0 so Zt = Xt), have
E{S(q)(0)} =
∫ fN
−fNH(0− f ′)S(bl)(f ′) df ′
= ∆t
∫ W
−WH(f ′′) df ′′ = ∆t tr {DKHTΣ(bl)H},
where (j, k)th element of Σ(bl) is s(bl)j−k
SAPA2e–401 VIII–86
Minimizing Broad-Band Bias Measure: I
• measure of broad-band bias (similar concept to leakage) is thus
tr {DK} − tr {DKHTΣ(bl)H}
• insisting on tr {DK} = 1 ensures unbiasedness for white noise
• to minimize broad-band bias under this restriction,
maximize tr {DKHTΣ(bl)H} subject to tr {DK} = 1
SAPA2e–401, 402 VIII–87
Minimizing Broad-Band Bias Measure: II
• Exercsie [8.13] gives solution:
− set K = 1
−H = h0 is normalized eigenvector associated with largesteigenvalue λ0(N,W ) of Σ(bl)
− eigenvector is zeroth order Slepian sequence (technically: fi-nite subsequence of DPSS)
− broad-band bias measure = 1−λ0(N,W ), where λ0(N,W )is the concentration ratio (Exercise [402])
• solution conflicts with variance in white noise case: as K in-creases, variance decreases
• reasonable balance: use K orthonormal Slepian tapers
SAPA2e–402, 403 VIII–88
Managing Bias and Variance
• broad-band bias: Exercise [8.14] says this can be measured by
1− 1
K
K−1∑k=0
λk(N,W );
recall that λk(N,W ) is close to unity as long as K < 2NW ∆t
• variance: previous argument says
S(mt)(f )d=S(f )
2Kχ2
2K
approximately if S(·) not rapidly varying over [f −W, f +W ];
thus have var {S(mt)(f )} ≈ S2(f )/K
• local bias: small if H(f ) ≈ 12W over [−W,W ]; as following
figures show, decreases as K increases; here NW = 4 withN = 1024 for which 1
2W = N8 = 128
.= 21 dB
SAPA2e–403 VIII–89
NW = 4 Slepian H0(·), S(mt)0 (·), H(·) and S(mt)(·), K = 1
−60
−40
−20
0
20
dB
k = 0
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 1
SAPA2e–381, 384 VIII–90
NW = 4 Slepian H1(·), S(mt)1 (·), H(·) and S(mt)(·), K = 2
−60
−40
−20
0
20
dB
k = 1
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 2
SAPA2e–381, 384 VIII–91
NW = 4 Slepian H6(·), S(mt)6 (·), H(·) and S(mt)(·), K = 7
−60
−40
−20
0
20
dB
k = 6
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 7
SAPA2e–383, 385 VIII–92
Adaptive Multitaper Estimation: I
• Section 8.5 discusses data-adaptive refinement to basic multi-tapering (developed for Slepian tapers)
• idea: weight eigenspectra adaptively according to need for leak-age suppression at each f
− if S(f ) relatively large, leakage not a concern, so can makeK large
− if S(f ) relatively small, leakage might be a concern, so shouldmake K small
SAPA2e–412 VIII–93
Adaptive Multitaper Estimation: II
• adaptive multitaper estimator given by
S(amt)(f ) ≡∑K−1k=0 b2k(f )λkS
(mt)k (f )∑K−1
k=0 b2k(f )λk
where λk ≈ 1− 1/10j (with j ↓ as k ↑) &
bk(f ) =1
λk + (1− λk)s0 ∆tS(f )
≈ 1
1 + s0 ∆t10jS(f )
− λk’s downweight higher eigenspectra (slightly)
− s0 ∆t = average value of S(·)− bk(f ) small if 10jS(f )� s0 ∆t & large if 10jS(f )� s0 ∆t
• determine bk(f ) using preliminary estimate of S(·); can iterateto refine bk(f )’s if desired
SAPA2e–412 VIII–94
Adaptive Multitaper Estimation: III
• assume
− S(mt)k (f )
d= S(f )χ2
2/2 for each eigenspectrum
− S(mt)k (f )’s are pairwise uncorrelated
• as before, assume S(amt)(f )d= aχ2
ν
• EDOF argument similar to S(lw)(·) and S (WOSA)(·) yields
ν =2(E{S(amt)(f )}
)2
var {S(amt)(f )}≈
2(∑K−1
k=0 b2k(f )λk
)2
∑K−1k=0 b4k(f )λ2
k
SAPA2e–412 VIII–95
Motivation for WOSA: I
• letX0, . . . , XN−1 be sample of zero mean Gaussian white noise
• partition into NB blocks of size NS = N/NB:
X0, . . . , XNS−1; XNS, . . . , X2NS−1; . . . ; X(NB−1)NS, . . . , XN−1
b = 0 b = 1 b = 2 b = 3
0 256 512 768 1024
t
−4
−2
0
2
4
whi
te n
oise
SAPA2e–412 VIII–96
Motivation for WOSA: II
• form periodograms for blocks b = 0, . . . , NB − 1:
S(p)b (fk) ≡ ∆t
NS
∣∣∣∣∣∣NS−1∑t=0
XbNS+te−i2πfkt∆t
∣∣∣∣∣∣2
0.0 0.5f
0
2
4
6
8
perio
dogr
am
b = 0
0.0 0.5f
b = 1
0.0 0.5f
b = 2
0.0 0.5f
b = 3
SAPA2e–412 VIII–97
Motivation for WOSA: III
• form average of NB periodograms:
S(p)(fk) ≡ 1
NB
NB−1∑b=0
S(p)b (fk)
0.0 0.5f
0
2
4
6
8av
erag
ed p
erio
dogr
am
SAPA2e–412 VIII–98
Motivation for WOSA: IV
• S(p)b (fk)
d= S(fk)χ2
2/2 for 0 < fk < fN
• S(p)b (fk) independent of S
(p)b′ (fk) for b′ 6= b
• χ2ν = Y 2
0 + Y 21 + · · · + Y 2
ν−1 for IID N(0, 1) RVs Yj implies
χ2ν1
+ χ2ν2
= χ2ν1+ν2
if χ2ν1
and χ2ν2
are independent
• with ν ≡ 2NB, have
NB−1∑b=0
S(p)b (fk)
d=S(fk)
2χ2
2NB=⇒ S(p)(fk) =
1
NB
NB−1∑b=0
S(p)b (fk)
d=S(fk)
νχ2ν
• note similarity to S(lw)(f )d= S(f )χ2
ν/ν
SAPA2e–412 VIII–99
Definition of WOSA: I
•Welch’s overlapped segment averaging (WOSA)
− generalization applicable to other {Xt}’s− break time series into NB blocks
∗ each block has NS points
∗ blocks now allowed to overlap
− apply data taper {h0, . . . , hNS−1} to each block
− form direct spectral estimator for each block
− average NB estimators together
SAPA2e–427, 428 VIII–100
Definition of WOSA: II
• for 0 ≤ l ≤ N −NS, let
S(d)l (f ) ≡ ∆t
∣∣∣∣∣∣NS−1∑t=0
htXt+le−i2πft∆t
∣∣∣∣∣∣2
•WOSA spectral estimator given by
S (WOSA)(f ) ≡ 1
NB
NB−1∑j=0
S(d)jn (f ),
where n is integer such that n(NB − 1) = N −NS• overlapping recovers ‘information’ lost in tapering (not obvi-
ously useful if ht = 1/√NS, i.e., S
(d)l (f ) = S
(p)l (f ))
• plots show example N = 100, NS = 32, NB = 5 and n = 17
SAPA2e–427, 428 VIII–101
Block j = 4 Before and After Tapering
0 50 100t
SAPA2e–428 VIII–102
First Moment Properties of WOSA: I
• first moment:
E{S (WOSA)(f )} =1
NB
NB−1∑j=0
E{S(d)jn (f )};
however,
E{S(d)jn (f )} =
∫ fN
−fNH(f − f ′)S(f ′) df ′
is the same for all j, where H(f ) ≡ |H(f )|2 is the spectralwindow associated with
{h0, . . . , hNS−1} ←→ H(·)
SAPA2e–428, 429 VIII–103
First Moment Properties of WOSA: II
• thus have
E{S (WOSA)(f )} =
∫ fN
−fNH(f − f ′)S(f ′) df ′
• note that this expected value
− depends on just NS and data taper
− does not depend on N , NB or n
− can be biased if we make NS too small
SAPA2e–428, 429 VIII–104
Variance of WOSA Estimator: I
• assume E{S (WOSA)(f )} ≈ S(f )
• variance is given by
var {S (WOSA)(f )} = cov
1
NB
NB−1∑j=0
S(d)jn (f ),
1
NB
NB−1∑k=0
S(d)kn (f ),
=
1
N2B
NB−1∑j=0
var {S(d)jn (f )}
+2
N2B
∑j<k
cov {S(d)jn (f ), S
(d)kn (f )}
• for 0 < f < fN can use var {S(d)jn (f )} ≈ S2(f )
SAPA2e–429 VIII–105
Variance of WOSA Estimator: II
• assume:
− S(·) is locally constant
− f is not too close to 0 or fN− ht = 0 for t ≥ NS
• can argue (Exercise [8.25]):
cov {S(d)jn (f ), S
(d)kn (f )} ≈ S2(f )
∣∣∣∣∣∣NS−1∑t=0
htht+|k−j|n
∣∣∣∣∣∣2
;
i.e., depends on autocorrelation of {ht}
SAPA2e–429 VIII–106
Variance of WOSA Estimator: III
• leads to the useful expression
var {S (WOSA)(f )} ≈ S2(f )
NB
1 +2
NB
∑j<k
∣∣∣∣∣∣NS−1∑t=0
htht+|k−j|n
∣∣∣∣∣∣2
=S2(f )
NB
1 + 2
NB−1∑m=1
(1− m
NB
) ∣∣∣∣∣∣NS−1∑t=0
htht+mn
∣∣∣∣∣∣2
via a ‘diagonalization’ argument
SAPA2e–429 VIII–107
Distribution of WOSA Estimator
• assuming S (WOSA)(f )d= aχ2
ν, usual EDOF argument yields
ν =2(E{S (WOSA)(f )}
)2
var {S (WOSA)(f )}≈ 2NB
1 + 2∑NB−1m=1
(1− m
NB
) ∣∣∣∑NS−1t=0 htht+mn
∣∣∣2• specialize to Hanning data taper
− following plot shows EDOF versus percentage overlap
− 50% overlap gets close to maximum EDOF
− for Hanning + 50% overlap (i.e., n = NS/2):
ν ≈ 2NB
1 + 2 (1− 1/NB)∣∣∣∑NS/2−1
t=0 htht+NS/2
∣∣∣2 ≈36N2
B
19NB − 1≈ 3.79N
NS
• need 65% overlap for Slepian with NW = 4
SAPA2e–429, 430 VIII–108
DOFs ν versus % of Block Overlap for N = 1024
0 50 100percentage of overlap
0
10
20
30
40
50
60
70
ν
NS = 256
NS = 128
NS = 64
SAPA2e–430 VIII–109
S (WOSA)(·) with Hanning & 50% Overlap & AR(2) SDF
0.0 0.5f
−20
−10
0
10
20
AR
(2)
spec
tra
(dB
)
NS = 4
SAPA2e–431 VIII–110
S (WOSA)(·) with Hanning & 50% Overlap & AR(2) SDF
0.0 0.5f
−20
−10
0
10
20
AR
(2)
spec
tra
(dB
)
NS = 16
SAPA2e–431 VIII–111
S (WOSA)(·) with Hanning & 50% Overlap & AR(2) SDF
0.0 0.5f
−20
−10
0
10
20
AR
(2)
spec
tra
(dB
)
NS = 32
SAPA2e–431 VIII–112
S (WOSA)(·) with Hanning & 50% Overlap & AR(2) SDF
0.0 0.5f
−20
−10
0
10
20
AR
(2)
spec
tra
(dB
)
NS = 64
SAPA2e–431 VIII–113
S (WOSA)(·) with Hanning & 50% Overlap & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
NS = 64
SAPA2e–432 VIII–114
S (WOSA)(·) with Hanning & 50% Overlap & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
NS = 128
SAPA2e–432 VIII–115
S (WOSA)(·) with Hanning & 50% Overlap & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
NS = 256
SAPA2e–432 VIII–116
S (WOSA)(·) with Hanning & 50% Overlap & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
NS = 512
SAPA2e–432 VIII–117
Advantages/Disadvantages of WOSA
• widely used in spectrum analyzers
• advantages
− computationally efficient
− can handle large N
− can handle ‘locally stationary’ processes
− can be ‘robustified’ (Chave et al., 1987)
• disadvantages
− leakage if NS too small
− loss of resolution if NS too small
SAPA2e–434 VIII–118
Deconstructed Multitaper Spectral Estimators: VII
• as 2nd example of deconstructed multitaper estimator, considerWOSA estimator using a Hanning data taper for N = 1024with block size NS = 256 and 50% overlap, yielding NB = 7blocks
• following plot redisplays estimate for AR(4) time series of Fig-ure 36(e)
SAPA2e–435, 438 VIII–119
S (WOSA)(·) with Hanning & 50% Overlap & AR(4) SDF
0.0 0.5f
−60
−40
−20
0
20
AR
(4)
spec
tra
(dB
)
NS = 256
SAPA2e–432 VIII–120
Deconstructed Multitaper Spectral Estimators: VIII
• weight matrix Q for WOSA estimator has rank K = NB = 7,as following plot demonstrates because there are only 7 nonzeroeigenvalues
SAPA2e–435, 438 VIII–121
Eigenvalues of Q for WOSA Estimator
++
++
++
+
+ + + + + + + + +
k
eige
nval
ue
0 5 10 15
0.00
0.05
0.10
0.15
SAPA2e–435 VIII–122
Deconstructed Multitaper Spectral Estimators: IX
• following plots show eigenvectors associated with 7 nonzeroeigenvalues – these serve as data tapers in weighted multita-per representation for WOSA estimator
SAPA2e–435, 438 VIII–123
AR(4) Series, Deconstructed {h0,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–124
AR(4) Series, Deconstructed {h1,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–125
AR(4) Series, Deconstructed {h6,t} & Tapered Series
−5
0
5
AR
(4)
serie
s
−0.05
0.00
0.05
deco
n ta
per
0 512 1024t
−0.2
−0.1
0.0
0.1
0.2
tape
red
serie
s
SAPA2e–436 VIII–126
Deconstructed Multitaper Spectral Estimators: X
• following plots show eigenspectra based on seven data tapers,along with weighted multitaper estimates of increasing order K
• because Q is of rank 7, weighted multitaper estimate of orderK = 7 is exactly the same as WOSA estimate
• weights dk determined by eigenvalues dk, but renormalized forK < 7 to sum to unity:
dk =dk∑K−1
k′=0dk′, k = 0, . . . , K − 1
(renormalization required because∑6k=0 dk = 1)
• plots also show spectral windows for eigenspectra and weightedmultitaper estimates, with vertical red dashed lines indicatingstandard bandwidth measures (either BHk or BH)
SAPA2e–435, 437, 438 VIII–127
Deconstructed H0(·), S(mt)0 (·), H(·) and S(wmt)(·), K = 1
−60
−40
−20
0
20
dB
k = 0
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 1
VIII–128
Deconstructed H1(·), S(mt)1 (·), H(·) and S(wmt)(·), K = 2
−60
−40
−20
0
20
dB
k = 1
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 2
VIII–129
Deconstructed H6(·), S(mt)6 (·), H(·) and S(wmt)(·), K = 7
−60
−40
−20
0
20
dB
k = 6
0.00 0.01f
−60
−40
−20
0
20
dB
0.0 0.5f
K = 7
VIII–130
Ocean Wave Data: I
• sea level time series for which N = 1024, ∆t = 1/4 second andfN = 2 cycles per second
0 50 100 150 200 250
−10
000
500
1000
t (seconds)
rela
tive
heig
ht
SAPA2e–248 VIII–131
Ocean Wave Data: II
• arguably best SDF estimate seen so far is Gaussian S(lw)(·)based on S(d)(·) using NW = 2 Slepian data taper
• effective bandwidth of S(lw)(·) given by BU.= 0.135 Hz
• following plot shows basic multitaper estimate S(mt)(·)− set NW = 4 (resolution not main concern)
− maximum of 7 possible reasonable tapers, but S(mt)6 (·) poor
at high frequencies
− set K = 6, yielding ν = 12 EDOF
SAPA2e–439, 441, 443 VIII–132
S(mt)(·), NW = 4, K = 6 Slepian Tapers
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–442 VIII–133
Ocean Wave Data: III
• next plots compare another S(mt)(·) estimate with Parzen lag
window estimate S(lw)(·)− set NW = 6 and K = 10 so ν = 20
− bandwidth of S(lw)(·) is 0.049 Hz ≈ 2W.= 0.047 Hz
− good agreement between S(mt)(·) and S(lw)(·)
SAPA2e–443 VIII–134
S(mt)(·), NW = 6, K = 10 Slepian Tapers
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–442 VIII–135
Parzen S(lw)(·), m = 150, using NW = 2 Slepian S(d)(·)
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–341 VIII–136
S(mt)(·), NW = 6, K = 10 Slepian Tapers
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–442 VIII–137
Ocean Wave Data: IV
• next two plots show
1. adaptive multitaper estimate S(amt)(·) with NW = 4 andK = 7, along with 95% confidence intervals (CIs)
2. EDOFs ν versus f
• wider CIs for f > 1 Hz due to decreasing degrees of freedom
• final plot compares S(amt)(·) with Parzen S(lw)(·) – good over-all agreement between estimates
SAPA2e–443 VIII–138
S(amt)(·), NW = 4, K = 7 Slepian Tapers
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–442 VIII–139
Equivalent Degrees of Freedom ν for S(amt)(·)
0.0 0.5 1.0 1.5 2.0f
0
4
8
12
16
degr
ees
of fr
eedo
m
SAPA2e–442 VIII–140
S(amt)(·) and Parzen S(lw)(·)
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
VIII–141
Ocean Wave Data: V
• recall that effective bandwidth of Gaussian-based S(lw)(·) isBU
.= 0.135 Hz
• for WOSA estimator, effective bandwidth is BH, which de-pends upon selected data taper {ht} and block size NS
• following plot shows BH versus NS for Hanning data taper,along with dashed line showing BU for Gaussian lag windowestimate
SAPA2e–440 VIII–142
BH versus NS for Hanning Data Taper (∆t = 0.25 sec)
40 60 80 100 120block size
0.10
0.15
0.20
0.25
band
wid
th
SAPA2e–440 VIII–143
Ocean Wave Data: VI
• NS = 61 yields BH.= 0.134 Hz (closest to BU
.= 0.135 Hz)
• setting n = 30 gives blocks that overlap by ≈ 50%
• let X ′t = Xt −X , i.e., centered time series
− block 1: X ′0, X′1, . . . , X
′60
− block 2: X ′30, X′31, . . . , X
′90
− block 3: X ′60, X′91, . . . , X
′120
− ...
− block 33: X ′960, X′961, . . . , X
′1020
• redefining last block to be X ′963, X′964, . . . , X
′1023 allows use of
entire time series
SAPA2e–440 VIII–144
Hanning-based S (WOSA)(·) and Gaussian S(lw)(·)
0.0 0.5 1.0 1.5 2.0f
−40
−20
0
20
40
60
80
ocea
n w
ave
spec
tra
(dB
)
SAPA2e–441 VIII–145
Ocean Wave Data: VII
• S (WOSA)(f ) has EDOF ν.= 62.6, whereas S(lw)(f ) has ν
.= 34.4
• recall of that variance is inversely proportional to ν
• 95% CIs based on S (WOSA)(f ) are tighter
• greater variability on explanation for S(lw)(·) being somewhatmore wobbly in appearance than S (WOSA)(f )
SAPA2e–440, 441 VIII–146