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# zp2tf() - Signal Processing

```zp2tf forms transfer function polynomials
from the zeros, poles, and gains of a system in factored form. [b,a] = zp2tf(z,p,k) finds
a rational transfer function B(s)A(s)=b1s(n−1)+⋯+b(n−1)s+bna1s(m−1)+⋯+a(m−1)s+amgiven a system in factored transfer function formH(s)=Z(s)P(s)=k(s−z1)(s−z2)⋯(s−zm)(s−p1)(s−p2)⋯(s−pn)Column vector p specifies the pole locations,
and matrix z specifies the zero locations, with
as many columns as there are outputs. The gains for each numerator
transfer function are in vector k. The zeros and
poles must be real or come in complex conjugate pairs. The polynomial
denominator coefficients are returned in row vector a and
the polynomial numerator coefficients are returned in matrix b,
which has as many rows as there are columns of z.Inf values can be used as place holders in z if
some columns have fewer zeros than others.```

### Syntax

`[b,a] = zp2tf(z,p,k)`

### Example

```Transfer Function of Mass-Spring SystemOpen This Example
Compute the transfer function of a damped mass-spring system that obeys the differential equation

The measurable quantity is the acceleration,
, and
is the driving force. In Laplace space, the system is represented by
The system has unit gain, a double zero at
, and two complex-conjugate poles.z = [0 0]';
p = roots([1 0.01 1])
k = 1;

p =

-0.0050 + 1.0000i
-0.0050 - 1.0000i

Use zp2tf to find the transfer function.[b,a] = zp2tf(z,p,k)

b =

1     0     0

a =

1.0000    0.0100    1.0000```