# Documentation

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# zpkrateup

Zero-pole-gain complex bandpass frequency transformation

## Syntax

```[Z2,P2,K2,AllpassNum,AllpassDen] = zpkrateup(Z,P,K,N) ```

## Description

```[Z2,P2,K2,AllpassNum,AllpassDen] = zpkrateup(Z,P,K,N)``` returns zeros, `Z`2, poles, `P`2, and gain factor, `K`2, of the target filter being transformed from any prototype by applying an `N`th-order rateup frequency transformation, where `N` is the upsample ratio. Transformation creates `N` equal replicas of the prototype filter frequency response.

It also returns the numerator, `AllpassNum`, and the denominator, `AllpassDen`, of the allpass mapping filter. The original lowpass filter is given with zeros, `Z`, poles, `P`, and gain factor, `K`.

Relative positions of other features of an original filter do not change in the target filter. This means that it is possible to select two features of an original filter, F1 and F2, with F1 preceding F2. Feature F1 will still precede F2 after the transformation. However, the distance between F1 and F2 will not be the same before and after the transformation.

## Examples

Design a prototype real IIR halfband filter using a standard elliptic approach:

```[b, a] = ellip(3,0.1,30,0.409); z = roots(b); p = roots(a); k = b(1); % Upsample the prototype filter 4 times [z2,p2,k2] = zpkrateup(z, p, k, 4); % Compare prototype filter with target filter fvtool(b, a, k2*poly(z2), poly(p2));```

## Arguments

VariableDescription
`Z`

Zeros of the prototype lowpass filter

`P`

Poles of the prototype lowpass filter

`K`

Gain factor of the prototype lowpass filter

`N`

Integer upsampling ratio

`Z2`

Zeros of the target filter

`P2`

Poles of the target filter

`K2`

Gain factor of the target filter

`AllpassNum`

Numerator of the mapping filter

`AllpassDen`

Denominator of the mapping filter

Frequencies must be normalized to be between -1 and 1, with 1 corresponding to half the sample rate.