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Definition of a Finite Impulse Response

Prerequisites

Kronecker Delta Function | \( \delta \)
Finite Impulse Response | \( h \)

Description

This is the the definition of a Finite Impulse Response. The impulse response of a filter with the Kronecker Delta function essentially represents the coefficients of the the filter. This means that if the input of a given FIR filter is \(\htmlClass{sdt-0000000054}{\delta} [n]\), the output \(\htmlClass{sdt-0000000131}{X}discreteOutput [n]\) will be the coefficients, \(\htmlClass{sdt-0000000033}{b}_{\htmlClass{sdt-0000000015}{k}}\) of the filter itself.

\[\htmlClass{sdt-0000000058}{h}[\htmlClass{sdt-0000000117}{n}] = \htmlClass{sdt-0000000080}{\sum}_{\htmlClass{sdt-0000000015}{k} = 0}^{\htmlClass{sdt-0000000053}{M}} \htmlClass{sdt-0000000033}{b}_{\htmlClass{sdt-0000000015}{k}} \htmlClass{sdt-0000000054}{\delta}[\htmlClass{sdt-0000000117}{n} - \htmlClass{sdt-0000000015}{k}]\]

Symbols Used:

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.
\( k \)

This symbol represents any given integer, \( k \in \htmlClass{sdt-0000000122}{\mathbb{Z}}\).
\( b \)

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

This is the symbol for the Kronecker Delta Function. It is the discrete version of the Dirac Delta Function.
\( h \)

This is the symbol for a Finite Impulse Response (FIR), the unit Impulse Response (\( \htmlClass{sdt-0000000113}{h} \)) 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.
\( \sum \)

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

This symbol represents any given whole number, \( n \in \htmlClass{sdt-0000000014}{\mathbb{W}}\).

Derivation

  1. Consider the Finite Impulse Response:

    The symbol \(h\) symbolizes a Finite Impulse Response (FIR), the unit Impulse Response (\( \htmlClass{sdt-0000000113}{h} \)) 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:
    \[[\htmlClass{sdt-0000000121}{a}, \htmlClass{sdt-0000000033}{b}, ..., \htmlClass{sdt-0000000004}{c}]\]
  3. Let's consider a list of these constants, B:
    \[B = [\htmlClass{sdt-0000000033}{b}_0, \htmlClass{sdt-0000000033}{b}_1, \htmlClass{sdt-0000000033}{b}_{\htmlClass{sdt-0000000015}{k}}]\]
  4. Let us now consider the Kronecker Delta Function:

    The symbol for the Kronecker Delta Function is \(\delta\). It is the discrete version of the Dirac Delta Function. It is defined to be \(1\) when \(\htmlClass{sdt-0000000117}{n} = 0\) and \(0\) for all other values of \(\htmlClass{sdt-0000000117}{n}\).

    As a piecewise function, it is defined as:

    \[ \delta[\htmlClass{sdt-0000000117}{n}] = \begin{cases} 1 &\text{if n is 0} \\ 0 & \text{for all other n} \end{cases} \]

  5. We can therefore say that the function is:
    \[\htmlClass{sdt-0000000058}{h}[\htmlClass{sdt-0000000117}{n}] = \htmlClass{sdt-0000000080}{\sum}_{\htmlClass{sdt-0000000015}{k} = 0}^{\htmlClass{sdt-0000000121}{a}} \htmlClass{sdt-0000000033}{b}_{\htmlClass{sdt-0000000015}{k}} \htmlClass{sdt-0000000054}{\delta}[\htmlClass{sdt-0000000117}{n} - \htmlClass{sdt-0000000015}{k}]\]

as required.

Example

Coming soon...