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# icwtft

## Syntax

xrec = icwtft(cwtstruct)
xrec = icwtft(cwtstruct,'plot')
xrec = icwtft(cwtstruct,'signal',SIG,'plot')

## Description

xrec = icwtft(cwtstruct) returns the inverse continuous wavelet transform of the CWT coefficients contained in the cfs field of the structure array cwtstruct. Obtain the structure array cwtstruct as the output of cwtft.

xrec = icwtft(cwtstruct,'plot') plots the reconstructed signal.

xrec = icwtft(cwtstruct,'signal',SIG,'plot') places a radio button in the bottom left corner of the plot. Enabling the radio button superimposes the plot of the input signal SIG on the plot of the reconstructed signal. By default the radio button is not enabled and only the reconstructed signal is plotted.

## Input Arguments

 cwtstruct Structure array containing six fields.dt — The sampling periodcfs — CWT coefficient matrixscales — Vector of scaleswav — Analyzing wavelet used in the CWTomega — Angular frequencies used in the Fourier transformmeanSig — Mean of the analyzed signalcwtstruct is the output of cwtft.

## Output Arguments

 xrec Reconstructed signal

## Examples

Compute the CWT and inverse CWT of two sinusoids with disjoint support.

```N = 1024;
t = linspace(0,1,N);
y = sin(2*pi*8*t).*(t<=0.5)+sin(2*pi*16*t).*(t>0.5);
dt = 0.05;
s0 = 2*dt;
ds = 0.4875;
NbSc = 20;
wname = 'morl';
sig = {y,dt};
sca = {s0,ds,NbSc};
wave = {wname,[]};
cwtsig = cwtft(sig,'scales',sca,'wavelet',wave);

% Compute inverse CWT and plot reconstructed signal with original
sigrec = icwtft(cwtsig,'signal',sig,'plot');```

Select the radio button in the bottom left corner of the plot.

Use the inverse CWT to approximate a trend in a time series. Construct a time series consisting of a polynomial trend, a sinewave (oscillatory component), and additive white Gaussian noise. Obtain the CWT of the input signal and use the inverse CWT based on only the coarsest scales to reconstruct an approximation to the trend. To obtain an accurate approximation based on select scales use the default power of two spacing for the scales in the continuous wavelet transform. See cwtft for details.

```t = linspace(0,1,1e3);
% Polynomial trend
x = t.^3-t.^2;
% Periodic term
x1 = 0.25*cos(2*pi*250*t);
% Reset random number generator for reproducible results
rng default
y = x+x1+0.1*randn(size(t));
% Obtain CWT of input time series
cwty = cwtft({y,0.001},'wavelet','morl');
% Zero out all but the coarsest scale CWT coefficients
cwty.cfs(1:16,:) = 0;
% Reconstruct a signal approximation based on the coarsest scales
xrec = icwtft(cwty);
plot(t,y,'k'); hold on;
xlabel('Seconds'); ylabel('Amplitude');
plot(t,x,'b','linewidth',2);
plot(t,xrec,'r','linewidth',2);
legend('Original Signal','Polynomial Trend','Inverse CWT Approximation');
figure
plot(t,x,'b'); hold on;
xlabel('Seconds'); ylabel('Amplitude');
plot(t,xrec,'r','linewidth',2);
legend('Polynomial Trend','Inverse CWT Approximation');```

You can also use the following syntax to plot the approximation. Select the radio button to view the original polynomial trend superimposed on the wavelet approximation.

```% Input the polynomial trend as the value of 'signal'
xrec = icwtft(cwty,'signal',x,'plot');```

## More About

expand all

### Inverse CWT

icwtft computes the inverse CWT based on a discretized version of the single integral formula due to Morlet. The Wavelet Toolbox™ Getting Started Guide contains a brief description of the theoretical foundation for the single integral formula in Inverse Continuous Wavelet Transform. The discretized version of this integral is presented in [5]

## References

[1] Daubechies, I. Ten Lectures on Wavelets, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1992.

[2] Farge, M. "Wavelet Transforms and Their Application to Turbulence", Ann. Rev. Fluid. Mech., 1992, 24, 395–457.

[3] Mallat, S. A Wavelet Tour of Signal Processing, San Diego, CA: Academic Press, 1998.

[4] Sun,W. "Convergence of Morlet's Reconstruction Formula", preprint, 2010.

[5] Torrence, C. and G.P. Compo "A Practical Guide to Wavelet Analysis", Bull. Am. Meteorol. Soc., 79, 61–78, 1998.

## See Also

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