IIR Filter – openfnirs

Definition: An IIR (Infinite Impulse Response) filter is a type of digital filter that uses feedback to achieve its filtering effect. It is called “infinite impulse response” because the filter’s impulse response does not decay to zero in a finite amount of time, but rather continues indefinitely.The mathematical representation of an IIR filter can be expressed in terms of its transfer function, which relates the input signal to the output signal. The transfer function of an IIR filter can be represented in the z-domain as: $H(z) = \frac{Y(z)}{X(z)} = \frac{(b0 + b1 z^{-1} + b2 z^{-2} + \ldots + bm z^{-m}}{1 + a1 z^{-1} + a2 z^{-2} + \ldots + an z^{-n}}$ where: $H(z)$ is the transfer function of the IIR filter $X(z)$ is the z-transform of the input signal $Y(z)$ is the z-transform of the output signal $b0, b1, \ldots, bm$ are the filter coefficients of the numerator (feedforward) part of the transfer function $a1, a2, \ldots, an$ are the filter coefficients of the denominator (feedback) part of the transfer function $z^{-1}, z^{-2}, \ldots, z^{-m}, z^{-n}$ are the z-domain delay operators The coefficients $b0, b1, \ldots, bm$ and $a1, a2, \ldots an$ , can determine the frequency response and other characteristics of the filter. In general, the order of the filter is equal to the maximum of m and n. The transfer function of an IIR filter can be converted to the time domain using the inverse z-transform, which gives the filter’s impulse response. The impulse response of an IIR filter typically decays slowly and does not reach zero, due to the feedback loop in the filter’s design. In terms of fNIRS analysis, the Butterworth filter is a commonly used type of IIR filter. It is a type of low-pass filter that provides a maximally flat frequency response in the passband, which means that the gain is constant up to a certain cutoff frequency. The Butterworth filter is often used in fNIRS analysis to remove high-frequency noise from the measured signals, which can be caused by factors such as movement artifacts or electrical interference. Furthermore, it may be helpful to learn about the concept of filter design and how to select appropriate filter parameters, such as the cutoff frequency and filter order, based on the characteristics of the input signal and the desired filtering effect. Additionally, it may be useful to learn about the limitations and potential drawbacks of IIR filters, such as the potential for instability and phase distortion. Additionally, it may be useful to learn about the limitations and potential drawbacks of IIR filters, such as their susceptibility to instability and phase distortion. For instance, a combination of IIR and zero-phase filtering is only suitable for offline analysis, as correcting the nonlinear phase distortion requires access to the entire time series. This allows the data to be processed both forward and backward, effectively eliminating phase distortion.

Category: Math

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Abbreviation or symbol: IIR

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References:

https://doi.org/10.1109/ICEC.1996.542373

Alternative definition: An IIR filter requires past output data values in addition to input data. An IIR filter has a frequency response that includes zeros and poles due to having a feedback mechanism that might allow the filter becomes unstable compared to an FIR filter. The property of infinite impulse response (IIR) applies to many linear time-invariant systems that are distinguished by having an impulse response that does not become exactly zero beyond a given point but instead continues indefinitely.

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