# scipy.signal.morlet2

scipy.signal.morlet2(M, s, w=5)

  Complex Morlet wavelet, designed to work with `cwt`.

  Returns the complete version of morlet wavelet, normalised according to s:

      exp(1j*w*x/s) * exp(-0.5*(x/s)**2) * pi**(-0.25) * sqrt(1/s)

  Parameters:
    M int

      Length of the wavelet.

    s float

      Width parameter of the wavelet.

    w float, optional

      Omega0. Default is 5

  Returns:
    morlet(M,) ndarray

  See also

  `morlet`

    Implementation of Morlet wavelet, incompatible with `cwt`

  Notes

  New in version 1.4.0.

  This function was designed to work with `cwt`. Because `morlet2` returns an array of complex numbers, the dtype argument of `cwt` should be set to complex128 for best results.

  Note the difference in implementation with `morlet`. The fundamental frequency of this wavelet in Hz is given by:

      f = w*fs / (2*s*np.pi)

  where `fs` is the sampling rate and s is the wavelet width parameter. Similarly we can get the wavelet width parameter at `f`:

      s = w*fs / (2*f*np.pi)

  Examples

      >>> import numpy as np
      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt

      >>> M = 100
      >>> s = 4.0
      >>> w = 2.0
      >>> wavelet = signal.morlet2(M, s, w)
      >>> plt.plot(abs(wavelet))
      >>> plt.show()

  This example shows basic use of `morlet2` with `cwt` in time-frequency analysis:

      >>> t, dt = np.linspace(0, 1, 200, retstep=True)
      >>> fs = 1/dt
      >>> w = 6.
      >>> sig = np.cos(2*np.pi*(50 + 10*t)*t) + np.sin(40*np.pi*t)
      >>> freq = np.linspace(1, fs/2, 100)
      >>> widths = w*fs / (2*freq*np.pi)
      >>> cwtm = signal.cwt(sig, signal.morlet2, widths, w=w)
      >>> plt.pcolormesh(t, freq, np.abs(cwtm), cmap='viridis', shading='gouraud')
      >>> plt.show()