synsq_cwt_iw {SynchWave}

Invese Synchrosqueezing Transform

Description

This function performs inverse synchrosqueezing transform.

This code is translated from MATLAB Synchrosqueezing Toolbox, version 1.1 developed by Eugene Brevdo (http://www.math.princeton.edu/~ebrevdo/).

Usage

synsq_cwt_iw(Tx, fs, opt=NULL)

Arguments

Tx

synchrosqueezed output of x (columns associated with time t)

fs

frequencies associated with rows of Tx

opt

list of options. See synsq_cwt_fw. opt.type: type of wavelet used in synsq_cwt_fw, other wavelet options (opt.mu, opt.s) should also match those used in synsq_cwt_fw

Details

This function performs inverse synchrosqueezing transform of Tx with associated frequencies in fs. This implements Eq. 5 of [1].

Value

x

reconstructed signal

References

[1] Thakur, G., Brevdo, E., Fuckar, N. S. and Wu, H-T. (2013) The Synchrosqueezing algorithm for time-varying spectral analysis: Robustness properties and new paleoclimate applications. Signal Processing, 93, 1079–1094.

See Also

synsq_cwt_fw, synsq_filter_pass.

Examples

set.seed(7)
n <- 2048
tu <- seq(0,10,, n)
dt <- tu[2]-tu[1]

feq1 <- function(t) (1+0.2*cos(t))*cos(2*pi*(2*t+0.3*cos(t)))
feq2 <- function(t) (1+0.3*cos(2*t))*exp(-t/15)*cos(2*pi*(2.4*t+0.5*t^(1.2)+0.3*sin(t)))
feq3 <- function(t) cos(2*pi*(5.3*t-0.2*t^(1.3)))
feq <- function(t) feq1(t) + feq2(t) + feq3(t)
s2 <- 2.4
noise <- sqrt(s2)*rnorm(length(tu))

fu0 <- feq(tu);
fu <- fu0 + noise;
fus <- cbind(feq1(tu), feq2(tu), feq3(tu))

# Continuous wavelet transform
nv <- 32
opt <- list(type = "bump")

cwtfit <- cwt_fw(fu, opt$type, nv, dt, opt)
thresh <- est_riskshrink_thresh(cwtfit$Wx, nv)

# Hard thresholding and Reconstruction
cwtfit$Wx[which(abs(cwtfit$Wx) < thresh)] <- 0.0
fur <- cwt_iw(cwtfit$Wx, opt$type, opt)

# Synchrosqueezed wavelet transform using denoised signal
sstfit <- synsq_cwt_fw(tu, fur, nv, opt)

#par(mfrow=c(1,2))
#image.plot(list(x=tu, y=sstfit$fs, z=t(abs(sstfit$Tx))), log="y",
#    xlab="Time", ylab="Frequency", main="Time-Frequency Representation by SST", 
#    col=designer.colors(64, c("azure", "cyan", "blue", "darkblue")), ylim=c(1, 8))

# Extracting the second component by filtering of synchrosqueezed wavelet transform
fm <- fM <- (2.4+0.5*1.2*tu^0.2+0.3*cos(tu))

#lines(tu, 0.88*fm, col="red", lty=3, lwd=2)
#lines(tu, 1.22*fM, col="red", lty=3, lwd=2)

tmp <- synsq_filter_pass(sstfit$Tx, sstfit$fs, 0.88*fm, 1.12*fM);
fursst <- synsq_cwt_iw(tmp$Txf, w, opt);

#plot(tu, fursst, type="l", main="SST", xlab="time", ylab="f", col="red", 
#    xlim=c(1.5,8.5), ylim=c(-1,1))
#lines(tu, feq2(tu), col="blue")

# Example:
# tmp <- synsq_cwt_fw(t, x, 32)  # Synchrosqueezing
# Txf <- synsq_filter_pass(tmp$Tx, tmp$fs, -Inf, 1) # Pass band filter
# xf <- synsq_cwt_iw(Txf, fs)  # Filtered signal reconstruction