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Continuous 1-D wavelet transform

**See cwt for
information on the older version of the cwt.
The older version is no longer recommended.**

`wt = cwt(x)`

`wt = cwt(x,wname)`

```
[wt,f] =
cwt(___,fs)
```

```
[wt,period]
= cwt(___,ts)
```

```
[wt,f,coi]
= cwt(___,fs)
```

```
[wt,period,coi]
= cwt(___,ts)
```

`[___] = cwt(___,Name,Value)`

`cwt(___)`

returns the
continuous wavelet transform (CWT) of `wt`

= cwt(`x`

)`x`

. The input,
`x`

, is a double-precision real- or complex-valued vector
and must have at least four samples. The CWT is obtained using the analytic
Morse wavelet with the symmetry parameter (gamma) equal to 3 and the
time-bandwidth product equal to 60. `cwt`

uses 10 voices per
octave. The minimum and maximum scales are determined automatically based on the
wavelet's energy spread in frequency and time. If `x`

is
real-valued, `wt`

is a 2-D matrix where each row corresponds
to one scale. The column size of `wt`

is equal to the length
of `x`

. If `x`

is complex-valued,
`wt`

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).

`[`

specifies the sampling frequency, `wt`

,`f`

] =
cwt(___,`fs`

)`fs`

, in Hz as a positive
scalar. `cwt`

uses `fs`

to determine the
scale-to-frequency conversions and returns the frequencies,
`f`

, in Hz. If you do not specify a sampling frequency,
`cwt`

returns `f`

in cycles per
sample. If the input `x`

is complex, the scale-to-frequency
conversions apply to both pages of `wt`

.

`[`

specifies the sampling interval, `wt`

,`period`

]
= cwt(___,`ts`

)`ts`

, as a positive
`duration`

scalar. The
`duration`

can be in years, days, hours, minutes, or
seconds. `cwt`

uses `ts`

to compute the
scale-to period conversion and returns the time periods in
`period`

. The array of durations in
`period`

have the same format property as
`ts`

. If the input `x`

is complex,
the scale-to-period conversions apply to both pages of
`wt`

.

`[___] = cwt(___,`

returns
the CWT with additional options specified by one or more `Name,Value`

)`Name,Value`

pair
arguments.

`cwt(___)`

with no output arguments plots the CWT scalogram,
which is the absolute value of the CWT as a function of time and frequency. The
cone of influence showing where edge effects become significant is also plotted.
Gray regions outside the dashed white line delineate regions where edge effects
are significant. If the input signal is complex-valued, the positive
(counterclockwise) and negative (clockwise) components are plotted in separate
scalograms.

If you do not specify a sampling frequency, `fs`

,
or time interval, `ts`

, the frequencies are plotted
in cycles per sample. If you specify a sampling frequency, `fs`

,
the frequencies are in Hz. If you specify a sampling duration, the
plot is a function of time and periods.

The *y*-axis of the scalogram uses a log_{2} scale.
If you use a data cursor, the actual *y*-value is
displayed. For example, if the axis value is approximately 0.125,
the data cursor *y*-value is –3.01, which
you can verify using `pow2(3.01)`

.

[1] Lilly, J. M., and S. C. Olhede. “Generalized Morse
Wavelets as a Superfamily of Analytic Wavelets.” *IEEE
Transactions on Signal Processing*. Vol. 60, No. 11, 2012,
pp. 6036–6041.

[2] Lilly, J. M., and S. C. Olhede. “Higher-Order
Properties of Analytic Wavelets.” *IEEE Transactions
on Signal Processing*. Vol. 57, No. 1, 2009, pp. 146–160.

[3] Lilly, J. M. *jLab: A data analysis package
for Matlab*, version 1.6.2. 2016. http://www.jmlilly.net/jmlsoft.html.

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