old_contents: import numpy as np
  from scipy.fftpack import fft, ifft
  
  def periodogram(x, nfft=None, fs=1):
      """Compute the periodogram of the given signal, with the given fft size.
  
      Parameters
      ----------
      x: array-like
          input signal
      nfft: int
          size of the fft to compute the periodogram. If None (default), the
          length of the signal is used. if nfft > n, the signal is 0 padded.
      fs: float
          Sampling rate. By default, is 1 (normalized frequency. e.g. 0.5 is the
          Nyquist limit).
  
      Returns
      -------
      pxx: array-like
          The psd estimate.
      fgrid: array-like
          Frequency grid over which the periodogram was estimated.
  
      Notes
      -----
      Only real signals supported for now.
  
      Returns the one-sided version of the periodogram.
  
      Discrepency with matlab: matlab compute the psd in unit of power / radian /
      sample, and we compute the psd in unit of power / sample: to get the same
      result as matlab, just multiply the result from talkbox by 2pi"""
      # TODO: this is basic to the point of being useless:
      #   - support windowing
      #   - normalization/frequency unit + definition
      #   - one-sided vs two-sided
      #   - plot
      #   - support complex input
      #   - trend/mean handling
      x = np.atleast_1d(x)
      n = x.size
  
      if x.ndim > 1:
          raise ValueError("Only rank 1 input supported for now.")
      if not np.isrealobj(x):
          raise ValueError("Only real input supported for now.")
      if not nfft:
          nfft = n
      if nfft < n:
          raise ValueError("nfft < signal size not supported yet")
  
      pxx = np.abs(fft(x, nfft)) ** 2
      if nfft % 2 == 0:
          pn = nfft / 2 + 1
      else:
          pn = (nfft + 1 )/ 2
  
      fgrid = np.linspace(0, fs * 0.5, pn)
      return pxx[:pn] / (n * fs), fgrid
  
new_contents: import numpy as np
  from scipy.fftpack import fft, ifft
  
  def periodogram(x, nfft=None, fs=1):
      """Compute the periodogram of the given signal, with the given fft size.
  
      Parameters
      ----------
      x : array-like
          input signal
      nfft : int
          size of the fft to compute the periodogram. If None (default), the
          length of the signal is used. if nfft > n, the signal is 0 padded.
      fs : float
          Sampling rate. By default, is 1 (normalized frequency. e.g. 0.5 is the
          Nyquist limit).
  
      Returns
      -------
      pxx : array-like
          The psd estimate.
      fgrid : array-like
          Frequency grid over which the periodogram was estimated.
  
      Notes
      -----
      Only real signals supported for now.
  
      Returns the one-sided version of the periodogram.
  
      Discrepency with matlab: matlab compute the psd in unit of power / radian /
      sample, and we compute the psd in unit of power / sample: to get the same
      result as matlab, just multiply the result from talkbox by 2pi"""
      # TODO: this is basic to the point of being useless:
      #   - support Daniel smoothing
      #   - support windowing
      #   - trend/mean handling
      #   - one-sided vs two-sided
      #   - plot
      #   - support complex input
      x = np.atleast_1d(x)
      n = x.size
  
      if x.ndim > 1:
          raise ValueError("Only rank 1 input supported for now.")
      if not np.isrealobj(x):
          raise ValueError("Only real input supported for now.")
      if not nfft:
          nfft = n
      if nfft < n:
          raise ValueError("nfft < signal size not supported yet")
  
      pxx = np.abs(fft(x, nfft)) ** 2
      if nfft % 2 == 0:
          pn = nfft / 2 + 1
      else:
          pn = (nfft + 1 )/ 2
  
      fgrid = np.linspace(0, fs * 0.5, pn)
      return pxx[:pn] / (n * fs), fgrid
  
subject: Update TODO for periodogram.