**Definition:**An FIR (finite impulse response) filter is a type of digital filter that is commonly used to process signals in a wide range of applications, including audio processing, image processing, and biomedical signal processing. An FIR filter is characterized by a finite impulse response, which means that the output of the filter is a weighted sum of a finite number of past and present input samples. The impulse response of an FIR filter is typically a finite sequence of coefficients used to compute the output of the filter. The input signal is convolved with the impulse response to produce the output signal.In fNIRS, FIR is used to process and analyze the measured optical density data. FIR filters are often used in fNIRS analysis because they have several desirable properties. One of the main advantages of FIR filters is that they have a linear phase response, which means they do not introduce any phase distortion in the filtered signal. This is important in fNIRS analysis because changes in hemoglobin concentration can be affected by changes in the optical path length, which can introduce phase shifts in the measured signals. The mathematical representation of an FIR filter can be expressed as: y[n] = b[0]x[n] + b[1]x[n-1] + b[2]x[n-2] + … + b[M]x[n-M] where: y[n] is the output signal at time n x[n] is the input signal at time n b[0], b[1], …, b[M] are the filter coefficients M is the filter order The coefficients b[0], b[1], …, b[M] are typically determined by windowing, frequency sampling, and least squares techniques.

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**References:**https://doi.org/10.3389/fnins.2019.00737 https://doi.org/10.3389/fnhum.2018.00505

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