Documentation

# wt

Continuous wavelet transform with a filter bank

## Syntax

``cfs = wt(fb,x)``
``[cfs,f] = wt(fb,x)``
``[cfs,f,coi] = wt(fb,x)``
``[cfs,f,coi,scalcfs] = wt(fb,x)``
``[cfs,p] = wt(fb,x)``
``[cfs,p,coi] = wt(fb,x)``
``[cfs,p,coi,scalcfs] = wt(fb,x)``

## Description

example

````cfs = wt(fb,x)` returns the continuous wavelet transform (CWT) coefficients of the signal `x`, using the CWT filter bank `fb`. `x` is a double-precision real- or complex-valued vector. `x` must have at least 4 samples. If `x` is real-valued, `cfs` is a 2-D matrix where each row corresponds to one scale. The column size of `cfs` is equal to the length of `x`. If `x` is complex-valued, `cfs` is a 3-D matrix, where the first page is the CWT for the positive scales (analytic part or counterclockwise component) and the second page is the cwt for the negative scales (anti-analytic part or clockwise component).```
````[cfs,f] = wt(fb,x)` returns the frequencies, `f`, corresponding to the scales (rows) of `cfs` if the `SamplingPeriod` property is not specified in the CWT filter bank `fb`. If you do not specify a sampling frequency, `f` is in cycles/sample.```
````[cfs,f,coi] = wt(fb,x)` returns the cone of influence, `coi` for the CWT. `coi` is in the same units as `f`. If the input `x` is complex, the `coi` applies to both pages of `cfs`.```

example

````[cfs,f,coi,scalcfs] = wt(fb,x)` returns the scaling coefficients, `scalcfs` for the wavelet transform if the analyzing wavelet is `'Morse'` or `'amor'`. Scaling coefficients are not supported for the bump wavelet.```
````[cfs,p] = wt(fb,x)` returns the periods, `p`, corresponding to the scales (rows) of `cfs` if you specify a sampling period in the CWT filter bank, `fb`. `p` has the same units and format as the duration scalar sampling period.```
````[cfs,p,coi] = wt(fb,x)` returns the cone of influence `coi` in periods for the CWT. `coi` is an array of durations with the same format property as the sampling period. If the input `x` is complex, the `coi` applies to both pages of `cfs`.```
````[cfs,p,coi,scalcfs] = wt(fb,x)` returns the scaling coefficients, `scalcfs`, for the wavelet transform if the analyzing wavelet is `'Morse'` or `'amor'`. Scaling coefficients are not supported for the bump wavelet.```

## Examples

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Load a real-valued signal. Create a CWT filter bank that can be applied to the signal.

```load noisdopp.mat sig = noisdopp; fb = cwtfilterbank('SignalLength',numel(sig));```

Use the filter bank to obtain the continuous wavelet transform of the signal.

`cfs = wt(fb,sig);`

Create and plot a signal sampled at 1000 Hz. Create a CWT filter bank that can be used on the signal. Since the signal is periodic, set the boundary extension property of the filter bank to `'periodic'`.

```Fs = 1000; t = 0:1/Fs:1-1/Fs; sig = 3*sin(2*pi*20*t)+cos(2*pi*2*t); fb = cwtfilterbank('SignalLength',length(sig),'SamplingFrequency',Fs,'Boundary','periodic'); plot(t,sig) xlabel('Time (sec)') title('Signal')``` Take the CWT of the signal. Return the wavelet and scaling coefficients.

`[cfs,~,~,scalcfs] = wt(fb,sig);`

Reconstruct the signal two ways. First use the mean of the signal, then use the scaling coefficients. Plot the difference between the original signal and both reconstructions.

```xrec0 = icwt(cfs,'SignalMean',mean(sig)); xrec1 = icwt(cfs,'ScalingCoefficients',scalcfs); plot(t,sig-xrec0) hold on plot(t,sig-xrec1) grid on legend('Using mean(sig)','Using scalcfs') title('Difference Between Reconstructions')``` Observe that using the scaling coefficients results in a significantly more accurate reconstruction. To investigate the source of the dramatic improvement, create a second signal consisting of the 2 Hz component of the original signal. Compare the scaling coefficients with the 2 Hz signal.

```figure sig2hz = cos(2*pi*2*t); plot(t,sig2hz) hold on plot(t,scalcfs) grid on title('Comparing Scaling Coefficients with 2 Hz Component') xlabel('Time (sec)') legend('2 Hz Component', 'Scaling Coefficients')``` Note that the scaling coefficients and 2 Hz signal are virtually identical. Using the scaling coefficients helps with the reconstruction because the 2 Hz component is not representable by a wavelet with this sampling frequency and length.

## Input Arguments

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Continuous wavelet transform (CWT) filter bank, specified as a `cwtfilterbank` object.

Input signal, specified as a double-precision real- or complex-valued vector. `x` must have at least four samples.

## Output Arguments

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Continuous wavelet transform, returned as a matrix of complex values. If `x` is real-valued, `cfs` is a 2-D matrix where each row corresponds to one scale. The column size of `cfs` is equal to the length of `x`. If `x` is complex-valued, `cfs` is a 3-D matrix, where the first page is the CWT for the positive scales (analytic part or counterclockwise component), and the second page is the CWT for the negative scales (anti-analytic part or clockwise component).

Frequencies, returned as a vector, corresponding to the scales (rows) of `cfs` if the `'SamplingPeriod'` is not specified in `fb`. If you specify a sampling frequency, `f` is in hertz. If you do not specify a frequency, `f` is in cycles/sample.

Data Types: `double`

Periods, returned as an array of durations, corresponding to the scales (rows) of `cfs` if you specify a sampling period in `fb`. `p` has the same units and format as the duration scalar sampling period.

Data Types: `duration`

Cone of influence for the CWT, returned as either an array of doubles or array of durations. The cone of influence indicates where edge effects occur in the CWT. If you specify a sampling frequency, `coi` is an array of doubles in the same units as `f`. If you specify a sampling period, `coi` is an array of durations with the same format property as the sampling period. Due to the edge effects, give less credence to areas that are outside or overlap the cone of influence.

For additional information, see Boundary Effects and the Cone of Influence.

Data Types: `double` | `duration`

Scaling coefficients for the wavelet transform, returned as a vector with length equal to the length of `x`. If `x` is real-valued, `scalcfs` is real-valued. If `x` is complex-valued, `scalcfs` is complex-valued.

Data Types: `double`

## See Also

### Topics

#### Introduced in R2018a

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