# filtic

Initial conditions for transposed direct-form II filter implementation

## Description

z = filtic(b,a,y,x) finds the initial conditions, z, for the delays in the transposed direct-form II filter implementation given past outputs y and inputs x. The vectors b and represent the numerator and denominator coefficients, respectively, of the filter's transfer function.

example

z = filtic(b,a,y) assumes that the input x is 0 in the past..

## Examples

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Determine the zero input response of the following system: $\mathit{y}\left(\mathit{n}\right)+1.12\mathit{y}\left(\mathit{n}-1\right)=0.1\mathit{x}\left(\mathit{n}\right)+0.2\mathit{x}\left(\mathit{n}-1\right)$ with initial condition $\mathit{y}\left(-1\right)=1$. Set the numerator and denominator coefficients and the initial conditions for the output.

b = [0.1 0.2];
a = [1 1.12];
Y = 1;

Calculate the zero input initial conditions for the system.

xic = filtic(b,a,Y);

Compute the zero input response.

yzi = filter(b,a,zeros(1,20),xic);
stem(yzi)

## Input Arguments

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Transfer function coefficients, specified as vectors.

Example: b = [1 3 3 1]/6 and a = [3 0 1 0]/3 specify a third-order Butterworth filter with a normalized 3-dB frequency of 0.5π rad/sample.

Past output, specified as a vector. The vector y contains the most recent output first, and oldest output last as in

$y=\left[y\left(-1\right),y\left(-2\right),y\left(-3\right),\dots ,y\left(-m\right)\right]$

where m is length(a)-1 (the denominator order); if length(y) is less than m, filtic pads it with zeros to length m.

Past input, specified as a vector. The vector x contains the most recent input first, and oldest input last as in

$x=\left[x\left(-1\right),x\left(-2\right),x\left(-3\right),\dots ,x\left(-n\right)\right]$

where n is length(b)-1 (the numerator order). If length(x) is less than n, filtic pads it with zeros to length n

## Output Arguments

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Initial conditions, returned as a vector. The output z is a column vector of length equal to the larger of n and m. z describes the state of the delays given past inputs x and past outputs y.

## Tips

If any of the input arguments y, x, b, or a is not a vector (that is, if any argument is a scalar or array), filtic gives the following error message:

Requires vector inputs.

## Algorithms

filtic performs a reverse difference equation to obtain the delay states z. Elements of x beyond x(n-1) and elements of y beyond y(m-1) are unnecessary so filtic ignores them.

The transposed direct-form II structure is shown in the following illustration.

n – 1 is the filter order.

## References

[1] Oppenheim, A.V., and R.W. Schafer, Discrete-Time Signal Processing, Prentice-Hall, 1989, pp. 296, 301-302.

Introduced before R2006a