# Signal Processing Utilities

## Functions

- `signals.corrvis()` ... Visualization for cross- and auto-correlation
- `signals.pSpect()` ... Simple power spectrum from FFT
- `signals.savgol()` ... Savitzky-Golay filter
- `signals.show_se()` ... Show mean and standard error, of a dataset in column form.

## Details

Utilities for signal processing

`signals.``corrvis`(x, y)

  Visualize correlation, by calculating the cross-correlation of two signals. The aligned signals and the resulting cross correlation value are shown, and advanced when the user hits a key or clicks with the mouse.

  Parameters: | X : array (N,) Comparison signal Y : array (M,) Reference signal

  Notes

  Based on an idea from dpwe@ee.columbia.edu

  Examples

      >>> x = np.r_[0:2*pi:10j]
      >>> y = np.sin(x)
      >>> corrvis(y,y)

`signals.``pSpect`(data, rate)

  Power spectrum and frequency

  Parameters: | data : array, shape (N,) measurement data rate : float sampling rate [Hz]
  Returns: | powerspectrum : array, shape (N,) frequency : array, shape (N,)

`signals.``savgol`(x, window_size=3, order=2, deriv=0, rate=1)

  Smooth (and optionally differentiate) data with a Savitzky-Golay filter. The Savitzky-Golay filter removes high frequency noise from data. It has the advantage of preserving the original shape and features of the signal better than other types of filtering approaches, such as moving averages techhniques.

  Parameters: | y : array_like, shape (N,) or (N,m) the values of the time history of the signal. window_size : int the length of the window. Must be an odd integer number. order : int the order of the polynomial used in the filtering. Must be less then window_size - 1. deriv : int the order of the derivative to compute (default = 0 means only smoothing) rate : sampling rate (in Hz; only used for derivatives)
  Returns: | ys : ndarray, shape same as y the smoothed signal (or it’s n-th derivative).

  Notes

  The Savitzky-Golay is a type of low-pass filter, particularly suited for smoothing noisy data. The main idea behind this approach is to make for each point a least-square fit with a polynomial of high order over a odd-sized window centered at the point.

  The data at the beginning / end of the sample are deterimined from the best polynomial fit to the first / last datapoints. This makes the code a bit more complicated, but avoids wild artefacts at the beginning and the end.

  Cutoff-frequencies For smoothing (deriv=0), the frequency where the amplitude is reduced by 10% is approximately given by:

  f_cutoff = sampling_rate / (1.5 * look)

  For the first derivative (deriv=1), the frequency where the amplitude is reduced by 10% is approximately given by:

  f_cutoff = sampling_rate / (4 * look)

  References

  [R1] | A. Savitzky, M. J. E. Golay, Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Analytical Chemistry, 1964, 36 (8), pp 1627-1639.

  [R2] | Numerical Recipes 3rd Edition: The Art of Scientific Computing W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery Cambridge University Press ISBN-13: 9780521880688

  [R3] | Siegmund Brandt, Datenanalyse, pp 435

  Examples

      >>> t = np.linspace(-4, 4, 500)
      >>> y = np.exp( -t**2 ) + np.random.normal(0, 0.05, t.shape)
      >>> ysg = savgol(y, window_size=31, order=4)
      >>> import matplotlib.pyplot as plt
      >>> plt.plot(t, y, label='Noisy signal')
      >>> plt.plot(t, np.exp(-t**2), 'k', lw=1.5, label='Original signal')
      >>> plt.plot(t, ysg, 'r', label='Filtered signal')
      >>> plt.legend()
      >>> plt.show()

`signals.``show_se`(raw)

  Show mean and standard error, of a dataset in column form.

  Parameters: | raw : array (N,M) input data, M sets of N data points
  Returns: | avg : array (N,) average value se : array (N,) standard error

  Notes

  Examples

      >>> t = np.arange(0,20,0.1)
      >>> x = np.sin(t)
      >>> data = []
      >>> for ii in range(10):
      >>>     data.append(x + np.random.randn(len(t)))
      >>> show_se(np.array(data).T)