# scipy.signal.invres#

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

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

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

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

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

```    r       r             r[-1]
= -------- + -------- + ... + --------- + k(s)
(s-p)   (s-p)         (s-p[-1])
```

If there are any repeated roots (closer together than tol), then H(s) has terms like:

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

This function is used for polynomials in positive powers of s or z, such as analog filters or digital filters in controls engineering. For negative powers of z (typical for digital filters in DSP), use `invresz`.

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.