Minimax Threshold for Denoising Complex Signals
with Waveshrink
Sylvain Sardy
26个英文字母表
Abstract—For the problem of signal extraction from noisy data,Waveshrink has proven to be a powerful tool,both from an empir-ical and an asymptotic point of view.Waveshrink is especially effi-cient at estimating spatially inhomogeneous signals.A key step of the procedure is the lection of the threshol
d parameter.Donoho and Johnstone propo a lection of the threshold bad on a min-imax principle.Their derivation is specifically for real signals and real wavelet transforms.In this paper,we propo to extend the u of Waveshrink to denoising complex signals with complex wavelet transforms.We illustrate the problem of denoising complex signals with an electronic surveillance application.
Index Terms—Complex signals,complex wavelet transform,electronic surveillance,minimax,nonparametric denoising,waveshrink.
I.B ACKGROUND
S
UPPOSE we obrve a univariate real
signal
where
the are identically and independently distributed stan-dard Gaussian random variables.Therefore,we assume that the
variance of the noi is known and unity
(i.e.,
In prac-tice,the variance can be estimated by taking the median abso-lute deviation of the high level wavelet coefficients,as propod in [1].Our goal is to find a good estimate of the underlying
signal
(1)
where
known basis
functions
,
where
where
or the soft
shrinkage
function
(5)
where
is
初2英语作文
is
t to zero,hence,enforcing sparsity.Other shrinkage methods have been ,non-negative Garrote [3],Firm [4],or Bayesian [5].However,more important than the choice of a shrinkage function,the lection
of
for the soft shrinkage function
(5).
B.Threshold Selection The threshold
parameter
diag
,
where meta pa-rameters:one for each least squares coefficient.It would be difficult in practice to estimate
the
,and considered
lecting
(6)
In practice,we would like to cloly approach this ideal risk.
•Universal threshold :They considered the Waveshrink es-timate (3)with the soft shrinkage function,and lect the single meta
parameter
for all possible true
coefficients,
namely
overboard
百度机器翻译
,then the signal will be estimated as zero with high probability
since
to
achieve the minimum
bound
,
namely,
llss,
and
The minimax threshold does not usually give an estimate with a nice visual appearance.However,it has the advantage of giving good predictive performance.For details on the properties of the two thresholds,e [1]and [2].
II.W A VESHRINK FOR C OMPLEX S IGNALS
Thus far,it has been assumed that the signal is real valued.In some applications,however,the signal is complex valued (e,for instance,our application in Section III),and the wavelets have real and imaginary parts.Examples of complex wavelets include the complex Daubechies wavelets [6],chirplets [7],and brushlets [8].
To denoi the signal,both the real and imaginary parts of the least squares coefficients have to be shrunk toward zero.A simple shrinkage procedure would consist of independently shrinking the real and imaginary parts of the least squares wavelet coefficients (2).One drawback of this procedure is that the underlying signal estimate is not guaranteed to have a spar wavelet reprentation (a shrunk coefficient may have its real part t to zero but not its imaginary part,and vice versa).Moreover,the phas of the least squares coefficients are changed by this kind of shrinkage.If instead the moduli of the least squares coefficients are shrunk,then both the real and imaginary parts are guaranteed to be t to zero together,and the phas remain the same.Shrinking the moduli is the natural generalization of Waveshrink to complex signals since it forces sparsity on the wavelet reprentation.
To be more specific,suppo we obrve a univariate complex
signal
where
the are identically and inde-pendently distributed complex random variables
with
Normal
,where now,
the
is the (complex)least squares
estimate.We generalize the soft shrinkage function to complex values
as
(8)
where
SARDY:MINIMAX THRESHOLD FOR DENOISING COMPLEX SIGNALS WITH WA VESHRINK 1025
the least squares coefficients are unchanged.Any complex least squares coefficient which modulus is less
than
that achieves the optimal
factor
of the risk (4).Let us
call
Normalgrosso
,
where
For any estimate
of
rl是什么意思啊
,
and the corresponding oracle risk
is
This is analogous to (6)for complex signals,and it constitutes our reference predictive performance.
2)Universal Threshold:In Appendix A,we rewrite the risk defined in (9)in the convenient form of (14).Note that the three terms of the cond and third lines of (14)are negative.Therefore,
for
Putting both inequalities together,we
have
small enough such
that
becomes large,and only the predom-inant features of a signal remain after denoising.Indeed,
for
,we
have
aslovely
attaining
above.
Then,the overall risk for
the
,consider the analogous quantity,where the
supremum
over
,
namely
be the
largest
and
1026IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.48,NO.4,APRIL2000 TABLE I
C OEFFICIENT AN
D R ELATED Q UANTITIES
,and
is strictly decreasing in
粉饰
Additionally,this last equation defines uniquely.
Table I lists the universal thresholds
and the corresponding bounds for
between6and16.The table is analogous to[2,Tab.II].
III.A PPLICATION
In the previous ction,we have derived the universal
and minimax thresholds for denoising complex signals with
Waveshrink.We now u the procedure for an electronic
surveillance application.In this electronic surveillance applica-
tion,we are interested in the problem of passive detection and
fingerprinting of incoming radar signatures from an electronic
surveillance platform.The obrved complex signal plotted in
Fig.1is
is larger than the number of obrvations
has more columns than rows and is there-
fore not orthonormal.Chen et al.[9]propod an extension
of soft-Waveshrink,called basis pursuit,to estimate the coef-
ficients in the over-complete situation.The coefficient estimate
is or-
thonormal.The nontrivial basis pursuit optimization problem
(when
SARDY:MINIMAX THRESHOLD FOR DENOISING COMPLEX SIGNALS WITH WA VESHRINK
1027
Fig. 2.Spectrogram for a chirped RF source at 3dB with jamming interference.The top spectrogram is of the original signal,and the bottom spectrogram is of the denoid signal using the universal meta parameter ( =2:22):
IV .C ONCLUSIONS
We have generalized Waveshrink and basis pursuit to com-plex signals.Both procedures need a lection of the meta pa-rameter that controls the amount of denoising.For using the two procedures on complex signals,we have derived the universal and minimax thresholds lection
of
(13)
is the Gaussian density function with
mean
We deduce the following properties of the risk:
P1)
;
let
P2)
at
verna
