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# Mapping coefficient of continues transfer function to discre

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#### ali_th

##### Member level 3
Hi.
I want to map coefficient of continues transfer function to discrete with AVR. It means that I have the transfer function in S region and I need to convert it to Z region .I need the optimum algorithm that makes possible this work please help me.
Thanks

You need Bilinear transform in Matlab to convert some S parameters into Z parameters. It's straightforward.

Re: Mapping coefficient of continues transfer function to di

ali_th,
The 2 most commonly used mappings are the bilinear z transform (mentioned by FrankCh), and the impulse-invariant transform.
. Does not suffer from aliasing
. Is scalable
. For filter design, it provides an attenuation
. characteristic superior to the anlaog prototype
. It "warps" the frequency scale
. It does not preserve the phase characteristics
. or time response of the analog prototype
. No "warping": The attenuation charateristic
. closely matches that of the analog prototype
. as long as the frequency is considerably
. lower than the Nyquist frequency
. Phase characteristics closely match those of
. the analog prototype
. Not scaleable
. Suffers from aliasing
Regards,
Kral

thanks Kral
I want to map the filter that the cut of frequency is 2kHz in minimum case. My sampling frequency is 100kHz with Analog Devise A/D. in this situation the Bilinear does not application. Do you have any suggestion for Impulse Invariant in optimum case?

Re: Mapping coefficient of continues transfer function to di

ali_th,
The procedure is as follows:
Determine the s-domain transfer function of the analog filter prototype that you want to implement as a digital filter.
.
Replace every occurence of [s-pn] with [1-exp(pnT)Z^(-1)]
Where:
. pn represents a pole or zero of the s-domain transfer function
.
. T is the sampling interval
For example, a single pole low pass filter with a transfer function of K/(s-2.7) would map to K/(1-exp(2.7*1/100000)z^-1), assuming a sampling frequency of 100KHz.
Regards,
Kral

Re: Mapping coefficient of continues transfer function to di

Hi,

I have a similar problem where I want to "map" an analog filter to a digital filter where both the phase and amplitude responses of the analog filter should be preserved (in the digital filter). In essence I want a digital "representation" of an analog filter.

For certain analog filter types for example elliptic, using impinvar may result in erroneous results. While bilinear may produce a digital filter that follows closely the analog filter (in-band) response however it will produce slightly different phase (group delay) mag response. Bilinear may also produce large errors in the digital filter stop-band response (compared to the analog filter response in the same freq).

Re: Mapping coefficient of continues transfer function to di

ali_th,
Tigana123 raises some good points regarding the impulse invariant method to implement the elliptic filter. These comments also apply to the Chebyshev type 2 (Inverse Chebyshev) filter, or, for that matter, any filter that has real zeros in the stopband. Antoniou describes a modified impulse invariant method that overcomes the probelems of direct application of the impulse invariant transformation to these filters. For more information, see the following:
Ntoniou, Abdreas, "Digital Filters Analysis, Design and Applications.
Regards,
Kral

Sorry; I had "finger blight" when typing the author's name. It is Antoniou, Andreas.

Another point to keep in mind; because of the aliasing problem, the impulse invariant method can only be used for low-pass and band-pass filters. It does not work for high-pass or band-reject filters.

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