科学文献

WSS x n] ?Байду номын сангаас rx k] ?! Px (ej! )
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N ?jkj?1 X n=0
x n] ?! N ? j k j
Note that where
^ x n]x n + k] ?! Px (ej! )
1 rx k] = N xN n] xN ?n] ^
R stands for the rectangular window. Take the Fourier transform of rx k], we obtain ^ 1 ^ ^ Px (ej! ) = PPER (ej! ) = N X (ej! )
n=0
k=?N
x(n + k)x (n)
r0x k] = N ?1j k j ^
where
N ?jkj?1 X n=0
x n+ j k j] x n]
Properties of this are { Unbiased { Asymptotically unbiased { Consistent { Might give invalid autocorrelated sequence 5. Note that
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2.6 Periodogram spectrum estimation
Problem is to estimate Px ( ) given x n] for n = 0; 1;
; N ? 1.
The applications are Wiener ltering, system identi cation, detection of narrow band signals in noise. Problem formulation
2. Consistent estimator
rx k] is asymptotically unbiased ^
2 SPECTRUM ESTIMATION
V ar frx k]g = E (rx k] ? rx k])2 ?! 0, as N ! 1 ^ ^ varfr k g ^ 3. Relative e ciency of two unbiased estimators, varfr1 k]]g ^2 4. Notes
X
k
X
rx (k)e?jk! rx (k)e?jk! ^
k
N X 1 x(n + k)x(n) rx (k) = Nlim 2N + 1 !1 k=?N
2 SPECTRUM ESTIMATION
Classical Modern y n] y n] rx k ] ^ constrained models with a priori information ^x (!) ^ P Px (!) periodogram signal modeling periodogram averaging maximum entropy method modi ed periodogram minimum cross entropy periodogram smoothing Pisarenko: subspace decomposition data/correlation extension Table 2: Summary of the spectrum estimation methods
N ?1 X n=0
x n + k]x n]
n=0
;N ? 1
With rx ?k] = rx k], we have ^
X 1 N ?jkj?1 x n+ j k j] x n] rx k] = N ^
n=0
Note that x n] is a random variable or random process. Therefore, rx k] is also a random variable or random process.
{ Stationary observation and desired signal { Known statistics
Kalman Filter: Linear optimum (MMSE) recursive lter
{ Nonstationary environment { Known statistics
2.3 Approaches to Spectrum Estimation
Some representative methods are summarized in Table 2 Spectrum estimation: towards a solution
Px (!) =
so we use If x(n) is ergodic, then ^ Px (!) =
k=0
wn k]x n ? k]
2 Spectrum Estimation
2.1 Power Spectrum
Take a look at the Table 1. The problem is that we can only measure the observed data y n] instead of the original data x n], but we want to nd Px (!). Usually, y n] = x n] + w n], which is noise contaminated and has limited number of observations.
2
xN n] = x n]wR n] = x n] 0 n N 0 otherwise
(
(1)
This is periodogram spectrum estimation. ^ Note that Px (ej! ) is estimated using a DFT. The accuracy is based on:
Is this a good estimator?
2.5 Properties of a good estimator
1. Bias
B = rx k] ? E frx k]g = 0 gives the unbiased estimate ^ limN !1 B = 0 gives asymptotically unbiased estimate.
2
2 Therefore, periodogram is asymptotically unbiased due, in part, to the windowing of rx k] while rx k] = 0; j k j> N . The DTFT of windowed autocorrelation gives the smoothed power spectrum expectation. The drawbacks are Loss of resolution in narrow band applications
{ Bias
^ E PPER (ej! )
n
o
= =
X
jkX j<N jkj<N
E frx k]g e?j!k ^ wB k]rx k]e?j!k
where and its Fourier transform is
= 21 Px (ej! ) WB (ej! )
? wB k] = N N k
N! 32 1 6 sin 2 7 WB (ej! ) = N 6 ! 7 4 sin 5
3
2.4 Approach 1
Given x n] for n 2 0; N ? 1], form an estimate of rx k]: 1 rx k ] = N ^ exclude x n] falling outside 0; N ? 1] to get
X 1 N ?k?1 x n + k]x n]; k = 0; 1; N
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- EE381K-9 Advanced Signal Processing -
Lecture #13: Spectrum Estimation
Lecture by Prof. F. Ayhan Sakarya (UT Austin)
1 Review
Wiener Filter: Linear optimum (MMSE) lter
P
FIR Adaptive Filters
{ FIR Wiener Filters
y n] =
X
T ^ y n] = d n] = wn x n] ^ e n] = d n] ? d n] n o = E j e n] j2 Rx n]w n] = rdx n] optimum at time n 2 min n] = d ? rT n]wopt n] MMSE for optimum w at time n dx
rx k] = N ? j k j r0x k] ^ N ^
E frx k]g = N ? j k j rx k] = wB k]rx k] ^ N , where wB k] is the triangular window. If rx k] is consistent, then rx k] = rx k] in the sense ^ ^ of mean square. Px (!) is the Fourier transform of rx k]. The question is whether Px (!) ^
Signal restoration
j H (!) j = Px(!)
2
^ segment of x n] ?! estimate of Px (!) ?! j H (!) j2 Detection of narrow band signals in noise: estimate amplitude/frequencies of sinusoids. Detect hidden periodicities in noise.
y n] = x n] + w n] noisy observations x n] = y n] h n] Wiener Filter ^ ( H (!) = P (!Px+!) (!) Need Powerspectrum x ) Pw System identi cation: In this case, the input to the system is w n], white Gaussian noise, and the output from the system is x n]. The system is unknown. The system function is
is reliable. The answer is NO. Because: rx k] ?! rx k] for a xed lag k ^ rx k] ?! 0 for j k j> N for any N ^
var rx k]] is large for k N ^
2 SPECTRUM ESTIMATION
Ergodic
n o
4
{ convergent in the mean { convergent in the mean square
rx(k) = Nlim 2N1+ 1 !1
X
N
Problem here is that we do not have unlimited observations. Finite observation: X 1 N ?jkj?1 x n+ j k j] x n] rx k ] = N ^ Properties of this are { Biased { Asymptotically unbiased { Consistent Finite observation:
Adaptive lters: Linear recursive lters
{ Nonstationary environment { Unknown statistics
LMS ! Steepest descent RLS (deterministic) ! Kalman lter (stochastic) LMS ( ): slow convergence ! RLS (R?1 ): fast convergence x
2 SPECTRUM ESTIMATION
Stochastic Deterministic x n] x n] r k] = E fx n + k]x n]g r k] = x k] x ?k] Px (!) = R(!) Px(!) = R(!) = X (!) Table 1:
2
2.2 Applications