Scicos Diagram
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Second order Infinite Impulse Response filter

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Description

The Fig. [*] presents the block diagram of an ideal second order digital filter.

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Figure : Block diagram of a linear second order digital recursive filter

The differential discret equation that described the system is given by:

$\displaystyle y\left(z\right)=y\left(z\right)\left(az^{-1}+bz^{-2}\right)+x\left(z\right)
$

This linear system is traditionaly studed by the Z function transfert :

$\displaystyle H\left(z\right)=\dfrac{y\left(z\right)}{x\left(z\right)}=\dfrac{1}{1+az^{-1}+bz^{-2}}
$

With $ H\left(Z\right)$, we determinate the stability domain of the filter in accord to the two gain paramter a et b. The Fig.[*] represents the stability domain of the second order recursive filter.
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Figure : Stability domain of the second order recursive filter

Now, we replace the ideal operators (adder and multiplactor) by real operaters that are presents in Digital Signal Processors (DSP) when realising recursive filters. The associated non-linear function is the modulo function, wich represents the effect of the overflow that occurs in digital implementation. The non-linear system is then described by a non-linear discret state equations system :

Context


Te = 1
Tfin = 1e4*Te
Nbit = 31
a=0.5
b=-1
ci=0.675
Intmax = 2^(Nbit-1);
ci1=-int(ci*Intmax)
ci2=int(ci*Intmax)
 

Scope Results

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Figure : (a) Temporal wave forms of discrete state variables
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Figure : (b) Phase plan

Mod_num blocks

Authors

IRCOM Group Alan Layec