code: def cascade(hk, J=7):
      """
      .. deprecated:: 1.12.0
  
          scipy.signal.cascade is deprecated in SciPy 1.12 and will be removed
          in SciPy 1.14. We recommend using PyWavelets instead.
  
  
      Return (x, phi, psi) at dyadic points ``K/2**J`` from filter coefficients.
  
      Parameters
      ----------
      hk : array_like
          Coefficients of low-pass filter.
      J : int, optional
          Values will be computed at grid points ``K/2**J``. Default is 7.
  
      Returns
      -------
      x : ndarray
          The dyadic points ``K/2**J`` for ``K=0...N * (2**J)-1`` where
          ``len(hk) = len(gk) = N+1``.
      phi : ndarray
          The scaling function ``phi(x)`` at `x`:
          ``phi(x) = sum(hk * phi(2x-k))``, where k is from 0 to N.
      psi : ndarray, optional
          The wavelet function ``psi(x)`` at `x`:
          ``phi(x) = sum(gk * phi(2x-k))``, where k is from 0 to N.
          `psi` is only returned if `gk` is not None.
  
      Notes
      -----
      The algorithm uses the vector cascade algorithm described by Strang and
      Nguyen in "Wavelets and Filter Banks".  It builds a dictionary of values
      and slices for quick reuse.  Then inserts vectors into final vector at the
      end.
  
      """
      warnings.warn(_msg % "cascade", DeprecationWarning, stacklevel=2)
  
      N = len(hk) - 1
  
      if J > 30 - np.log2(N + 1):
          raise ValueError("Too many levels.")
      if J < 1:
          raise ValueError("Too few levels.")
  
      # construct matrices needed
      nn, kk = np.ogrid[:N, :N]
      s2 = np.sqrt(2)
      # append a zero so that take works
      thk = np.r_[hk, 0]
      gk = qmf(hk)
      tgk = np.r_[gk, 0]
  
      indx1 = np.clip(2 * nn - kk, -1, N + 1)
      indx2 = np.clip(2 * nn - kk + 1, -1, N + 1)
      m = np.empty((2, 2, N, N), "d")
      m[0, 0] = np.take(thk, indx1, 0)
      m[0, 1] = np.take(thk, indx2, 0)
      m[1, 0] = np.take(tgk, indx1, 0)
      m[1, 1] = np.take(tgk, indx2, 0)
      m *= s2
  
      # construct the grid of points
      x = np.arange(0, N * (1 << J), dtype=float) / (1 << J)
      phi = 0 * x
  
      psi = 0 * x
  
      # find phi0, and phi1
      lam, v = eig(m[0, 0])
      ind = np.argmin(np.absolute(lam - 1))
      # a dictionary with a binary representation of the
      #   evaluation points x < 1 -- i.e. position is 0.xxxx
      v = np.real(v[:, ind])
      # need scaling function to integrate to 1 so find
      #  eigenvector normalized to sum(v,axis=0)=1
      sm = np.sum(v)
      if sm < 0:  # need scaling function to integrate to 1
          v = -v
          sm = -sm
      bitdic = {"0": v / sm}
      bitdic["1"] = np.dot(m[0, 1], bitdic["0"])
      step = 1 << J
      phi[::step] = bitdic["0"]
      phi[(1 << (J - 1)) :: step] = bitdic["1"]
      psi[::step] = np.dot(m[1, 0], bitdic["0"])
      psi[(1 << (J - 1)) :: step] = np.dot(m[1, 1], bitdic["0"])
      # descend down the levels inserting more and more values
      #  into bitdic -- store the values in the correct location once we
      #  have computed them -- stored in the dictionary
      #  for quicker use later.
      prevkeys = ["1"]
      for level in range(2, J + 1):
          newkeys = [
              "%d%s" % (xx, yy) for xx in [0, 1] for yy in prevkeys
          ]
          fac = 1 << (J - level)
          for key in newkeys:
              # convert key to number
              num = 0
              for pos in range(level):
                  if key[pos] == "1":
                      num += 1 << (level - 1 - pos)
              pastphi = bitdic[key[1:]]
              ii = int(key[0])
              temp = np.dot(m[0, ii], pastphi)
              bitdic[key] = temp
              phi[num * fac :: step] = temp
              psi[num * fac :: step] = np.dot(m[1, ii], pastphi)
          prevkeys = newkeys
  
      return x, phi, psi
  
file_path: scipy/scipy/signal/_wavelets.py
imports: import warnings
  import numpy
  from scipy.linalg import eig
  from scipy.special import comb
  from scipy.signal import convolve
label_desc: Use image node_id 3 for calling a global function with example usage: cascade(hk, J) and returns: x,  phi,  psi