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anti to conjugate #327

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Nov 15, 2023
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anti to conjugate #327

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18 changes: 9 additions & 9 deletions clouddrift/signal.py
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Original file line number Diff line number Diff line change
Expand Up @@ -13,16 +13,16 @@ def analytic_signal(
time_axis: Optional[int] = -1,
) -> Union[np.ndarray, Tuple[np.ndarray, np.ndarray]]:
"""Return the analytic signal from a real-valued signal or the analytic and
anti-analytic signals from a complex-valued signal.
conjugate analytic signals from a complex-valued signal.

If the input is a real-valued signal, the analytic signal is calculated as
the inverse Fourier transform of the positive-frequency part of the Fourier
transform. If the input is a complex-valued signal, the anti-analytic signal
is additionally calculated as the inverse Fourier transform of the conjugate of
the negative-frequency part of the Fourier transform.
transform. If the input is a complex-valued signal, the conjugate analytic signal
is additionally calculated as the inverse Fourier transform of the positive-frequency
part of the Fourier transform of the complex conjugate of the input signal.

For a complex-valued signal, the mean is evenly divided between the analytic and
anti-analytic signals.
conjugate analytic signal.

The calculation is performed along the last axis of the input array by default.
Alternatively, the user can specify the time axis of the input. The user can also
Expand All @@ -44,7 +44,7 @@ def analytic_signal(
xa : np.ndarray
Analytic signal. It is a tuple if the input is a complex-valed signal
with the first element being the analytic signal and the second element
being the anti-analytic signal.
being the conjugate analytic signal.

Examples
--------
Expand All @@ -54,7 +54,7 @@ def analytic_signal(
>>> x = np.random.rand(99)
>>> xa = analytic_signal(x)

To obtain the analytic and anti-analytic signals of a complex-valued signal:
To obtain the analytic and conjugate analytic signals of a complex-valued signal:

>>> w = np.random.rand(99)+1j*np.random.rand(99)
>>> wp, wn = analytic_signal(w)
Expand Down Expand Up @@ -121,7 +121,7 @@ def analytic_signal(

# analytic signal
xap = np.fft.fft(xa)
# anti-analytic signal
# conjugate analytic signal
xan = np.fft.fft(np.conj(xa))

# time dimension of extended time series
Expand All @@ -142,7 +142,7 @@ def analytic_signal(
xap = np.fft.ifft(xap)
xan = np.fft.ifft(xan)

# return central part plus hlaf the mean
# return central part plus half the mean
xap = xap[..., int(N + 1) - 1 : int(2 * N + 1) - 1] + 0.5 * mx_
xan = xan[..., int(N + 1) - 1 : int(2 * N + 1) - 1] + 0.5 * np.conj(mx_)

Expand Down
8 changes: 4 additions & 4 deletions clouddrift/wavelet.py
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Expand Up @@ -64,7 +64,7 @@ def morse_wavelet_transform(
``radian_frequency``. However, if the optional argument ``complex=True``
is specified, the wavelets will be divided by 2 so that the total
variance of the input complex signal is equal to the sum of the
variances of the returned analytic (positive) and anti-analiyic
variances of the returned analytic (positive) and conjugate analytic
(negative) parts. See examples below. The other option is ``"energy"``
which uses the unit energy normalization. In this last case, the
time-domain wavelet energies ``np.sum(np.abs(wave)**2)`` are always
Expand Down Expand Up @@ -102,7 +102,7 @@ def morse_wavelet_transform(

Apply a wavelet transform with a Morse wavelet with gamma parameter 3, beta parameter 4,
for a complex input signal at radian frequency 0.2 cycles per unit time. This case returns the
analytic and anti-analytic components:
analytic and conjugate analytic components:

>>> z = np.random.random(1024) + 1j*np.random.random(1024)
>>> wtz_p, wtz_n = morse_wavelet_transform(z, 3, 4, np.array([2*np.pi*0.2]), complex=True)
Expand All @@ -113,7 +113,7 @@ def morse_wavelet_transform(
>>> wtz_imag = morse_wavelet_transform(np.imag(z)), 3, 4, np.array([2*np.pi*0.2]))
>>> wtz_p, wtz_n = (wtz_real + 1j*wtz_imag) / 2, (wtz_real - 1j*wtz_imag) / 2

For the "energy" normalization, the analytic and anti-analytic components are obtained as follows
For the "energy" normalization, the analytic and conjugate analytic components are obtained as follows
with this alternative method:
>>> wtz_real = morse_wavelet_transform(np.real(z)), 3, 4, np.array([2*np.pi*0.2]))
>>> wtz_imag = morse_wavelet_transform(np.imag(z)), 3, 4, np.array([2*np.pi*0.2]))
Expand Down Expand Up @@ -170,7 +170,7 @@ def morse_wavelet_transform(

# apply the wavelet transform, distinguish complex and real cases
if complex:
# imaginary case, divide by 2 the wavelet and return analytic and anti-analytic
# imaginary case, divide by 2 the wavelet and return analytic and conjugate analytic
if normalization == "bandpass":
wtx_p = wavelet_transform(
0.5 * x, wavelet, boundary="mirror", time_axis=time_axis
Expand Down