# scipy.signal.invresz#

scipy.signal.invresz(r, p, k, tol=0.001, rtype='avg')[source]#

Compute b(z) and a(z) from partial fraction expansion.

If M is the degree of numerator b and N the degree of denominator a:

```        b(z)     b + b z**(-1) + ... + b[M] z**(-M)
H(z) = ------ = ------------------------------------------
a(z)     a + a z**(-1) + ... + a[N] z**(-N)
```

then the partial-fraction expansion H(z) is defined as:

```        r                   r[-1]
= --------------- + ... + ---------------- + k + kz**(-1) ...
(1-pz**(-1))         (1-p[-1]z**(-1))
```

If there are any repeated roots (closer than tol), then the partial fraction expansion has terms like:

```     r[i]              r[i+1]                    r[i+n-1]
-------------- + ------------------ + ... + ------------------
(1-p[i]z**(-1))  (1-p[i]z**(-1))**2         (1-p[i]z**(-1))**n
```

This function is used for polynomials in negative powers of z, such as digital filters in DSP. For positive powers, use `invres`.

Parameters:
rarray_like

Residues corresponding to the poles. For repeated poles, the residues must be ordered to correspond to ascending by power fractions.

parray_like

Poles. Equal poles must be adjacent.

karray_like

Coefficients of the direct polynomial term.

tolfloat, optional

The tolerance for two roots to be considered equal in terms of the distance between them. Default is 1e-3. See `unique_roots` for further details.

rtype{‘avg’, ‘min’, ‘max’}, optional

Method for computing a root to represent a group of identical roots. Default is ‘avg’. See `unique_roots` for further details.

Returns:
bndarray

Numerator polynomial coefficients.

andarray

Denominator polynomial coefficients.