taper: Oracle function for obtaining a particular taper/window
In sapa: Spectral Analysis for Physical Applications
Description
Usage
Arguments
Details
Value
S3 METHODS
References
See Also
Examples
Description
Develop signal processing tapers or windows.
Usage
1
2
3
4
Arguments
bandwidth
bandwidth for DPSS tapers.
See Details for more information. Default: 4.
beta
kaiser window shape factor (must be positive or zero).
See Details for more information. Default: 4*pi*(n.sample-1)/n.sample.
cutoff
parzen or Papoulis window cutoff (must be greater than unity).
See Details for more information. Default: floor(n.sample/2).
flatness
raised cosine taper flatness fraction (must be on [0,1]).
See Details for more information. Default: 0.3.
n.sample
an integer denoting the number of samples.
Default: 1000.
n.taper
an integer defining the multitaper order (number of orthogonal tapers) to use
in a multitaper scheme. The taper order directly impacts the quality of
the SDF estimate. Low taper orders are usually associated with SDF estimates with low bias and high variance,
while high taper orders attenuate the variance of the estimate at the risk of incurring a large bias.
This tradeoff between bias and variance is unavoidable but taper order allows you to tune the SDF
to meet the needs of your application. Studies show that a multitaper order of 5 typically provides a good balance
with reasonably low bias and variance properties (see the references for more details). Default: NULL,
which serves as a flag to set the default taper order depending on the type of taper chosen for the analysis.
If sine or dpss multitapers are chosen, the default taper order is 5, otherwise is set to unity.
normalize
a logical value. If TRUE,
the taper is normalized to have unit energy. Default: TRUE.
roughness
daniell windown roughness factor (must be positive).
See Details for more information. Default: n.sample/2.
sidelobedB
chebyshev sidelobedB bandwidth in decibels
(must be positive).
See Details for more information. Default: 80.
sigma
standard deviation for Guassian taper. Default: 0.3.
type
a character string denoting the type of taper to create.
Supported types are "rectangle", "triangle", "raised cosine",
"hanning", "hamming", "blackman", "nuttall", "gaussian",
"kaiser", "chebyshev", "born jordan", "sine",
"parzen", "papoulis", "daniell", and "dpss".
See Details for more information. Default: "rectangle".
Details
Let w() and h(t) for t = 0, ..., N-1
be a lag window and taper, respectively.
The following lag window or taper types are supported.
- rectangular
A rectangular taper is defined as
h(t) = 1.
- triangle
A triangular taper is defined as
h(t)=h(N-t-1)=2*(t+1)/(N+1)
for t < M where M=floor(N/2)
and h(M) = 1 if N is evenly divisible by 2.
- raised cosine
A raised cosine is a symmetric
taper with a flat mid-plateau. Let p on [0,1]
be the fraction of the length of the taper that is flat,
M=floor(p*N), and
B=2*pi/(M+1).
A raised cosine taper is defined as
h(t) = h(N-t-1) = 0.5 * (1 - cos(B*(t+1))) for 0 <= t < floor(M/2) and
h(t) = 1 for floor(M/2) <= t < N - floor(M/2).
- hanning
Let B=2*pi/(N+1).
A Hanning taper is defined as
h(t) = 0.5 * (1 - cos(B*(t+1))).
- hamming
Let B=2*pi/(N-1).
A Hamming taper is defined as
h(t) = 0.54 - 0.46 * cos(B*t).
- blackman
Let B=2*pi/(N+1).
A Blackman taper is defined as
h(t) = 0.42 - 0.5 * cos( B * ( t + 1 ) ) +
0.08 * cos( 2 * B * ( t + 1 ) ).
- nuttall
Let B=2*pi/(N-1).
A Nuttall taper is defined as
h(t) = 0.3635819 - 0.4891775 * cos( B*t ) + 0.1365995 * cos( 2*B*t ) -
0.0106411 * cos( 3*B*t )
- gaussian
Let S be the standard deviation
of a Gaussian distribution. Let B(t)=2*S*(0.5 - t/(N-1)).
A Gaussian taper is defined as
h(t) = h(N-t-1) = exp(-B*B/2) for 0 <= t < floor(N/2)
.
h(N/2)=1 if N is evenly divisible by 2
.
- kaiser
Let V(t) = (2t-1-N)/N and I0() be the zeroth-order modified
Bessel function of the first kind. Given the shape factor beta > 0,
a Kaiser taper is defined as
h(t) = I0(beta*sqrt(1-V(t)\eqn{\mbox{\textasciicircum}}{^}2))/I0(beta).
- chebyshev
The Dolph-Chebyshev taper is a function of
both the desired length N and the desired sidelobe level
(our routine accepts a sidelobe attenuation factor expressed
in decibels). See the Mitra reference for more details.
- born jordan
Let M=(N-1)/2.
A Born-Jordan taper is defined as
h(t) = h(N-t-1) = 1 / ( M - t + 1 ).
- sine
Sine multitapers are defined as
h(k,t) = sqrt(2/(N+1)) * sin((k+1)*pi*(t+1)/(N+1))
for t = 0, ..., N-1 and k = 0, ..., .. This simple equation
defines a good approximation to the discrete prolate
spheroidal sequences (DPSS) used in multitaper SDF
estimation schemes.
- parzen
A Parzen lag window is defined as
w(t,m)= {1 - 6(t/m)^2 + 6(|t|/m)^3} for |t| <= m/2;
{2(1 - |t|/m)^3} for m/2 < |t| <= m; and 0 otherwise
for -(N-1) <= t <= (N - 1).
The variable m is referred to as the cutoff since
all values beyond that point are zero.
- papoulis
A Papoulis lag window is defined as
w(t,m)= {(1/pi) * |sin(pi*t/m)| + (1 - |t|/m) * cos(pi*t/m)} for |t| < m;
and 0 otherwise
for -(N-1) <= t <= (N - 1).
The variable m is referred to as the cutoff since
all values beyond that point are zero.
- daniell
A Daniell lag window is defined as
w(t,m)= {sin(pi*t/m)/(pi*t/m)} for |t| < N;
and 0 for |t| >= N
for -(N-1) <= t <= (N - 1).
The variable m is referred to as the roughness factor,
since, in the context of spectral density function (SDF) estimation,
it controls the degree of averaging that is performed
on the preliminary direct SDF estimate. The smaller the roughness,
the greater the amount of smoothing.
- dpss
Discrete prolate spheroidal sequences are (typically) used
for multitaper spectral density function estimation. The first order
DPSS can be defined (to a good approximation) as
h(t,k) = C * I0( W' * sqrt(1 - (1 - g(t))^2) / I0(W')
for t=1,...,N,
where C is a scaling constant used
to force the normalization sum[t=0,...,N-1]{h(t,k)^2}=1;
W'=pi*W*(N-1)*dt where
dt is the sampling interval;
g(t)=(2*t-1)/N; and
I0() is the modified Bessel function
of the first kind and zeroth order. The parameter W
is related to the resolution bandwidth since
it roughly defines the desired half-width of the central
lobe of the resulting spectral window. Higher order
DPSS tapers (i.e., h(t,k) for k > 0)
can be calculated using a relatively simple tridiagonalization
formulation (see the references for more information). Finally,
we note that the sampling interval dt can
be set to unity without any loss of generality.
Value
an object of class taper.
S3 METHODS
- as.matrix
converts output to a matrix.
- plot
plots the output. Optional arguments are:
- ylab
Character string denoting the y-axis label for
the plot. Default: upperCase(attr(x,"type")).
- type
Line type (same as the type argument of the par function). Default: "l".
- ...
Additional plot arguments (set internally by the par function).
- print
prints a summary of the output object.
References
A. T. Walden, “Accurate Approximation of a 0th Order Discrete Prolate
Spheroidal Sequence for Filtering and Data Tapering", Signal Processing,
18, 341–8 (1989).
Percival, Donald B. and Constantine, William L. B. (2005)
“Exact Simulation of Gaussian Time Series
from Nonparametric Spectral Estimates
with Application to Bootstrapping",
Journal of Computational and Graphical Statistics,
accepted for publication.
D.B. Percival and A. Walden (1993),
Spectral Analysis for Physical Applications: Multitaper
and Conventional Univariate Techniques,
Cambridge University Press, Cambridge, UK.
S.K.Mitra, J. Kaiser (1993),
Handbook for Digital Signal Processing, John Wiley and Sons, Inc.
See Also
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
## change plot layout
gap <- 0.11
old.plt <- splitplot(4,4,1,gap=gap)
## create a plot of all supported tapers and
## windows
nms <- c("rectangle", "triangle", "raised cosine",
"hanning", "hamming", "blackman",
"nuttall", "gaussian", "kaiser",
"chebyshev", "born jordan", "sine",
"parzen", "papoulis", "daniell", "dpss")
for (i in seq(along=nms)){
if (i > 1) splitplot(4,4,i,gap=gap)
plot(taper(type=nms[i]))
}
## restore plot layout to initial state
par(old.plt)
sapa documentation built on May 2, 2019, 6:50 p.m.
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Description Usage Arguments Details Value S3 METHODS References See Also Examples
Description
Develop signal processing tapers or windows.
Usage
1 2 3 4 |
Arguments
bandwidth |
bandwidth for DPSS tapers.
See |
beta |
kaiser window shape factor (must be positive or zero).
See |
cutoff |
parzen or Papoulis window cutoff (must be greater than unity).
See |
flatness |
raised cosine taper flatness fraction (must be on [0,1]).
See |
n.sample |
an integer denoting the number of samples. Default: 1000. |
n.taper |
an integer defining the multitaper order (number of orthogonal tapers) to use
in a multitaper scheme. The taper order directly impacts the quality of
the SDF estimate. Low taper orders are usually associated with SDF estimates with low bias and high variance,
while high taper orders attenuate the variance of the estimate at the risk of incurring a large bias.
This tradeoff between bias and variance is unavoidable but taper order allows you to tune the SDF
to meet the needs of your application. Studies show that a multitaper order of 5 typically provides a good balance
with reasonably low bias and variance properties (see the references for more details). Default: |
normalize |
a logical value. If |
roughness |
daniell windown roughness factor (must be positive).
See |
sidelobedB |
chebyshev sidelobedB bandwidth in decibels
(must be positive).
See |
sigma |
standard deviation for Guassian taper. Default: 0.3. |
type |
a character string denoting the type of taper to create.
Supported types are |
Details
Let w() and h(t) for t = 0, ..., N-1 be a lag window and taper, respectively. The following lag window or taper types are supported.
- rectangular
A rectangular taper is defined as h(t) = 1.
- triangle
A triangular taper is defined as h(t)=h(N-t-1)=2*(t+1)/(N+1) for t < M where M=floor(N/2) and h(M) = 1 if N is evenly divisible by 2.
- raised cosine
A raised cosine is a symmetric taper with a flat mid-plateau. Let p on [0,1] be the fraction of the length of the taper that is flat, M=floor(p*N), and B=2*pi/(M+1). A raised cosine taper is defined as
h(t) = h(N-t-1) = 0.5 * (1 - cos(B*(t+1))) for 0 <= t < floor(M/2) and
h(t) = 1 for floor(M/2) <= t < N - floor(M/2).
- hanning
Let B=2*pi/(N+1). A Hanning taper is defined as h(t) = 0.5 * (1 - cos(B*(t+1))).
- hamming
Let B=2*pi/(N-1). A Hamming taper is defined as h(t) = 0.54 - 0.46 * cos(B*t).
- blackman
Let B=2*pi/(N+1). A Blackman taper is defined as h(t) = 0.42 - 0.5 * cos( B * ( t + 1 ) ) + 0.08 * cos( 2 * B * ( t + 1 ) ).
- nuttall
Let B=2*pi/(N-1). A Nuttall taper is defined as h(t) = 0.3635819 - 0.4891775 * cos( B*t ) + 0.1365995 * cos( 2*B*t ) - 0.0106411 * cos( 3*B*t )
- gaussian
Let S be the standard deviation of a Gaussian distribution. Let B(t)=2*S*(0.5 - t/(N-1)). A Gaussian taper is defined as
h(t) = h(N-t-1) = exp(-B*B/2) for 0 <= t < floor(N/2)
.
h(N/2)=1 if N is evenly divisible by 2
.
- kaiser
Let V(t) = (2t-1-N)/N and I0() be the zeroth-order modified Bessel function of the first kind. Given the shape factor beta > 0, a Kaiser taper is defined as h(t) = I0(beta*sqrt(1-V(t)\eqn{\mbox{\textasciicircum}}{^}2))/I0(beta).
- chebyshev
The Dolph-Chebyshev taper is a function of both the desired length N and the desired sidelobe level (our routine accepts a sidelobe attenuation factor expressed in decibels). See the Mitra reference for more details.
- born jordan
Let M=(N-1)/2. A Born-Jordan taper is defined as h(t) = h(N-t-1) = 1 / ( M - t + 1 ).
- sine
Sine multitapers are defined as
h(k,t) = sqrt(2/(N+1)) * sin((k+1)*pi*(t+1)/(N+1))
for t = 0, ..., N-1 and k = 0, ..., .. This simple equation defines a good approximation to the discrete prolate spheroidal sequences (DPSS) used in multitaper SDF estimation schemes.
- parzen
A Parzen lag window is defined as
w(t,m)= {1 - 6(t/m)^2 + 6(|t|/m)^3} for |t| <= m/2; {2(1 - |t|/m)^3} for m/2 < |t| <= m; and 0 otherwise
for -(N-1) <= t <= (N - 1). The variable m is referred to as the cutoff since all values beyond that point are zero.
- papoulis
A Papoulis lag window is defined as
w(t,m)= {(1/pi) * |sin(pi*t/m)| + (1 - |t|/m) * cos(pi*t/m)} for |t| < m; and 0 otherwise
for -(N-1) <= t <= (N - 1). The variable m is referred to as the cutoff since all values beyond that point are zero.
- daniell
A Daniell lag window is defined as
w(t,m)= {sin(pi*t/m)/(pi*t/m)} for |t| < N; and 0 for |t| >= N
for -(N-1) <= t <= (N - 1). The variable m is referred to as the roughness factor, since, in the context of spectral density function (SDF) estimation, it controls the degree of averaging that is performed on the preliminary direct SDF estimate. The smaller the roughness, the greater the amount of smoothing.
- dpss
Discrete prolate spheroidal sequences are (typically) used for multitaper spectral density function estimation. The first order DPSS can be defined (to a good approximation) as
h(t,k) = C * I0( W' * sqrt(1 - (1 - g(t))^2) / I0(W')
for t=1,...,N, where C is a scaling constant used to force the normalization sum[t=0,...,N-1]{h(t,k)^2}=1; W'=pi*W*(N-1)*dt where dt is the sampling interval; g(t)=(2*t-1)/N; and I0() is the modified Bessel function of the first kind and zeroth order. The parameter W is related to the resolution bandwidth since it roughly defines the desired half-width of the central lobe of the resulting spectral window. Higher order DPSS tapers (i.e., h(t,k) for k > 0) can be calculated using a relatively simple tridiagonalization formulation (see the references for more information). Finally, we note that the sampling interval dt can be set to unity without any loss of generality.
Value
an object of class taper.
S3 METHODS
- as.matrix
converts output to a matrix.
- plot
plots the output. Optional arguments are:
- ylab
Character string denoting the y-axis label for the plot. Default:
upperCase(attr(x,"type")).- type
Line type (same as the
typeargument of theparfunction). Default: "l".- ...
Additional plot arguments (set internally by the
parfunction).
prints a summary of the output object.
References
A. T. Walden, “Accurate Approximation of a 0th Order Discrete Prolate Spheroidal Sequence for Filtering and Data Tapering", Signal Processing, 18, 341–8 (1989).
Percival, Donald B. and Constantine, William L. B. (2005) “Exact Simulation of Gaussian Time Series from Nonparametric Spectral Estimates with Application to Bootstrapping", Journal of Computational and Graphical Statistics, accepted for publication.
D.B. Percival and A. Walden (1993), Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques, Cambridge University Press, Cambridge, UK.
S.K.Mitra, J. Kaiser (1993), Handbook for Digital Signal Processing, John Wiley and Sons, Inc.
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ## change plot layout
gap <- 0.11
old.plt <- splitplot(4,4,1,gap=gap)
## create a plot of all supported tapers and
## windows
nms <- c("rectangle", "triangle", "raised cosine",
"hanning", "hamming", "blackman",
"nuttall", "gaussian", "kaiser",
"chebyshev", "born jordan", "sine",
"parzen", "papoulis", "daniell", "dpss")
for (i in seq(along=nms)){
if (i > 1) splitplot(4,4,i,gap=gap)
plot(taper(type=nms[i]))
}
## restore plot layout to initial state
par(old.plt)
|
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