**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) = Y(z) / X(z) = (b0 + b1z^-1 + b2z^-2 + … + bmz^-m) / (1 + a1z^-1 + a2z^-2 + … + anz^-n) z^{-1}z^{-2}z^{-m}z^{-n}$ are the z-domain delay operators The coefficients b0, b1, …, bm and a1, a2, …, 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.

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

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