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# scipy.signal.cascade

`scipy.signal.``cascade`(hk, J=7)

  (x,phi,psi) at dyadic points K/2**J from filter coefficients.

  Inputs:
    hk – coefficients of low-pass filter J – values will be computed at grid points $K/2^J$
  Outputs:
    x – the dyadic points $K/2^J$ for $K=0...N*(2^J)-1$
      where len(hk)=len(gk)=N+1
    phi – the scaling function phi(x) at x
      $phi(x) = sum_{k=0}^{N} h_k phi(2x-k)$
    psi – the wavelet function psi(x) at x

      > $psi(x) = sum_{k=0}^N g_k phi(2x-k)$

      Only returned if gk is not None

  Algorithm:

  > Uses the vector cascade algorithm described by Strang and Nguyen in “Wavelets and Filter Banks”

  > Builds a dictionary of values and slices for quick reuse. Then inserts vectors into final vector at then end

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