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General form of a Finite Impulse Response Filter

Prerequisites

Discrete function | \(x\)
Finite Impulse Response | \(h\)

Description

This is the general form of a finite impulse response filter. It can describe a single Finite Impulse Response (FIR) or any linear combination of a finite number of FIRs.

Equation

\[\htmlId{tooltip-firFilter}{h}[\htmlId{tooltip-wholeNumber}{n}] = \htmlId{tooltip-summation}{\sum}_{\htmlId{tooltip-integer}{k} = 0}^{\htmlId{tooltip-orderDifferenceEquation}{M}} \htmlId{tooltip-2const}{b}_{\htmlId{tooltip-integer}{k}} \htmlId{tooltip-discreteFunction}{x}[\htmlId{tooltip-wholeNumber}{n} - \htmlId{tooltip-integer}{k}]\]

Symbols Used

\(k\)

This symbol represents any given integer, \( k \in \htmlId{tooltip-setOfIntegers}{\mathbb{Z}}\).
\(b\)

This is a symbol for any secondary generic constant. It can hold any numerical value
\(M\)

This is the symbol for the order of a difference equation. It refers to the maximum number of points back in a filter (or lags) that are used. In the case of a digital filter, it usually refers to the number of elements in the filter.
\(h\)

This is the symbol for a Finite Impulse Response (FIR), the unit Impulse Response (\( \htmlId{tooltip-impulseResponse}{h} \)) of a FIR filter. Because the result of this happens to be equal to the coefficients of the FIR filter, it is commonly also used to represent the FIR filter.
\(x\)

This symbol describes a discrete function. Discrete meaning that it only has a valid output for inputs from the set of integers \( \htmlId{tooltip-setOfIntegers}{\mathbb{Z}} \).
\(\sum\)

This is the summation symbol in mathematics, it represents the sum of a sequence of numbers.
\(n\)

This symbol represents any given whole number, \( n \in \htmlId{tooltip-setOfWholeNumbers}{\mathbb{W}}\).

Derivation

  1. Consider the Finite Impulse Response:

    The symbol \(h\) symbolizes a Finite Impulse Response (FIR), the unit Impulse Response (\( \htmlId{tooltip-impulseResponse}{h} \)) of a FIR filter. Because the result of this happens to be equal to the coefficients of the FIR filter, it is commonly also used to represent the FIR filter.. FIR Filters have no feedback, meaning that its value at any point only depends on current and previous input values, not output values. This makes them inherently stable, meaning that if the input is bounded (will not grow indefinitely), then so is the output. It acts on the input signal by convolution as can be seen on the associated definition equation page:

  2. Based on this definition, we would expect it to be a discrete function in that looks something like:
    \[[\htmlId{tooltip-const}{a}, \htmlId{tooltip-2const}{b}, ..., \htmlId{tooltip-3const}{c}]\]
  3. Let's consider a list of these constants, B:
    \[B = [\htmlId{tooltip-2const}{b}_0, \htmlId{tooltip-2const}{b}_1, ...,\htmlId{tooltip-2const}{b}_{\htmlId{tooltip-integer}{k}}]\]
  4. Let us now consider the Kronecker Delta Function:

    The symbol \(x\) describes a discrete function. Discrete meaning that it only has a valid output for inputs from the set of integers \( \htmlId{tooltip-setOfIntegers}{\mathbb{Z}} \). It is often used in the signal processing of discrete signals.


    In this context it describes a finite signal of length N.
  5. We can therefore say that the function is:
    \[\htmlId{tooltip-firFilter}{h}[\htmlId{tooltip-wholeNumber}{n}] = \htmlId{tooltip-summation}{\sum}_{\htmlId{tooltip-integer}{k} = 0}^{\htmlId{tooltip-const}{a}} \htmlId{tooltip-2const}{b}_{\htmlId{tooltip-integer}{k}} \htmlId{tooltip-discreteFunction}{x}[\htmlId{tooltip-wholeNumber}{n} - \htmlId{tooltip-integer}{k}]\]

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