# scipy.signal.tf2zpk

scipy.signal.tf2zpk(b, a)

  Return zero, pole, gain (z, p, k) representation from a numerator, denominator representation of a linear filter.

  Parameters:
    b array_like

      Numerator polynomial coefficients.

    a array_like

      Denominator polynomial coefficients.

  Returns:
    z ndarray

      Zeros of the transfer function.

    p ndarray

      Poles of the transfer function.

    k float

      System gain.

  Notes

  If some values of b are too close to 0, they are removed. In that case, a BadCoefficients warning is emitted.

  The b and a arrays are interpreted as coefficients for positive, descending powers of the transfer function variable. So the inputs \(b = [b_0, b_1, ..., b_M]\) and \(a =[a_0, a_1, ..., a_N]\) can represent an analog filter of the form:

  \[H(s) = \frac {b_0 s^M + b_1 s^{(M-1)} + \cdots + b_M} {a_0 s^N + a_1 s^{(N-1)} + \cdots + a_N}\]

  or a discrete-time filter of the form:

  \[H(z) = \frac {b_0 z^M + b_1 z^{(M-1)} + \cdots + b_M} {a_0 z^N + a_1 z^{(N-1)} + \cdots + a_N}\]

  This “positive powers” form is found more commonly in controls engineering. If M and N are equal (which is true for all filters generated by the bilinear transform), then this happens to be equivalent to the “negative powers” discrete-time form preferred in DSP:

  \[H(z) = \frac {b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}} {a_0 + a_1 z^{-1} + \cdots + a_N z^{-N}}\]

  Although this is true for common filters, remember that this is not true in the general case. If M and N are not equal, the discrete-time transfer function coefficients must first be converted to the “positive powers” form before finding the poles and zeros.