code: def group_delay(system, w=512, whole=False, fs=2 * cupy.pi):
      r"""Compute the group delay of a digital filter.
  
      The group delay measures by how many samples amplitude envelopes of
      various spectral components of a signal are delayed by a filter.
      It is formally defined as the derivative of continuous (unwrapped) phase::
  
                 d        jw
       D(w) = - -- arg H(e)
                dw
  
      Parameters
      ----------
      system : tuple of array_like (b, a)
          Numerator and denominator coefficients of a filter transfer function.
      w : {None, int, array_like}, optional
          If a single integer, then compute at that many frequencies (default is
          N=512).
  
          If an array_like, compute the delay at the frequencies given. These
          are in the same units as `fs`.
      whole : bool, optional
          Normally, frequencies are computed from 0 to the Nyquist frequency,
          fs/2 (upper-half of unit-circle). If `whole` is True, compute
          frequencies from 0 to fs. Ignored if w is array_like.
      fs : float, optional
          The sampling frequency of the digital system. Defaults to 2*pi
          radians/sample (so w is from 0 to pi).
  
      Returns
      -------
      w : ndarray
          The frequencies at which group delay was computed, in the same units
          as `fs`.  By default, `w` is normalized to the range [0, pi)
          (radians/sample).
      gd : ndarray
          The group delay.
  
      See Also
      --------
      freqz : Frequency response of a digital filter
  
      Notes
      -----
      The similar function in MATLAB is called `grpdelay`.
  
      If the transfer function :math:`H(z)` has zeros or poles on the unit
      circle, the group delay at corresponding frequencies is undefined.
      When such a case arises the warning is raised and the group delay
      is set to 0 at those frequencies.
  
      For the details of numerical computation of the group delay refer to [1]_.
  
      References
      ----------
      .. [1] Richard G. Lyons, "Understanding Digital Signal Processing,
             3rd edition", p. 830.
  
      """
      if w is None:
          # For backwards compatibility
          w = 512
  
      if _is_int_type(w):
          if whole:
              w = cupy.linspace(0, 2 * cupy.pi, w, endpoint=False)
          else:
              w = cupy.linspace(0, cupy.pi, w, endpoint=False)
      else:
          w = cupy.atleast_1d(w)
          w = 2 * cupy.pi * w / fs
  
      b, a = map(cupy.atleast_1d, system)
      c = cupy.convolve(b, a[::-1])
      cr = c * cupy.arange(c.size)
      z = cupy.exp(-1j * w)
      num = cupy.polyval(cr[::-1], z)
      den = cupy.polyval(c[::-1], z)
      gd = cupy.real(num / den) - a.size + 1
      singular = ~cupy.isfinite(gd)
      gd[singular] = 0
  
      w = w * fs / (2 * cupy.pi)
      return w, gd
  
imports: import operator
  from math import pi
  import warnings
  import cupy
  from cupy.polynomial.polynomial import polyval, polyvalfromroots
  import cupyx.scipy.fft
  from cupyx import jit
  from cupyx.scipy._lib._util import float_factorial
  from cupyx.scipy.signal._polyutils import roots
label_desc: Use image node_id 6 for calling a global function with example usage: group_delay(system, w, whole, fs) and returns: w,  gd