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Sinusoidal Response of Finite Impulse Response Filter

Prerequisites

Discrete signal | \(x[n] = x(nT_{s}) = a \cos(\omega n T_{s} + \phi) = a \cos(\hat \omega n + \phi)\)
General form of a Finite Impulse Response Filter | \(h[n] = \sum_{k = 0}^{M} b_{k} x[n - k]\)

Description

This is the sinusoidal form of a finite impulse response filter. It describes a fir filter in the frequency domain. Considering that the impulse response sequence is identical to the sequence of filter coefficients, the frequency response can either be denoted in terms of the filter coeffients, \( \htmlId{tooltip-2const}{b} \)_{\( \htmlId{tooltip-integer}{k} \)}, or in terms of the frequency response, \( \htmlId{tooltip-firFilter}{h} \)[\( \htmlId{tooltip-integer}{k} \)].

Equation

\[\htmlId{tooltip-sinusoidalResponseFirFilter}{H}(\htmlId{tooltip-euler}{e}^{\htmlId{tooltip-imag}{j} \htmlId{tooltip-normalizedRadianFrequency}{\hat \omega}}) = \htmlId{tooltip-summation}{\sum}_{\htmlId{tooltip-integer}{k} = 0}^{\htmlId{tooltip-orderDifferenceEquation}{M}} \htmlId{tooltip-2const}{b}_{\htmlId{tooltip-integer}{k}} \htmlId{tooltip-euler}{e}^{-\htmlId{tooltip-imag}{j} \htmlId{tooltip-normalizedRadianFrequency}{\hat \omega} \htmlId{tooltip-integer}{k}} = \htmlId{tooltip-summation}{\sum}_{\htmlId{tooltip-integer}{k} = 0}^{\htmlId{tooltip-orderDifferenceEquation}{M}} = \htmlId{tooltip-firFilter}{h}[\htmlId{tooltip-wholeNumber}{n}] \htmlId{tooltip-euler}{e}^{\htmlId{tooltip-imag}{j}e \htmlId{tooltip-normalizedRadianFrequency}{\hat \omega}}\]

Symbols Used

\(k\)

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

This is the symbol for normalized radian frequency. It is measured in radians (\( \htmlId{tooltip-u-radians}{rad} \))
\(b\)

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

This symbol represents Euler's constant. It is approximately \(2.718\).
\(M\)

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.
\(H\)

This symbol denotes the sinusoidal response of a finite impulse response 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, \( n \in \htmlId{tooltip-setOfWholeNumbers}{\mathbb{W}}\).
\(j\)

This symbol represents the imaginary unit, which is defined as the square root of \(-1\). \( j = \sqrt{-1}\). It is the most fundamental unit in the field of complex numbers, allowing for the expression of numbers that cannot be represented on the real number line.

Derivation

  1. Consider the definition of the general form of a finite impulse response Filter:
    \[\htmlId{tooltip-firFilter}{h}[\htmlId{tooltip-wholeNumber}{n}] = \htmlId{tooltip-summation}{\sum}_{\htmlId{tooltip-integer}{k} = 0}^{\htmlId{tooltip-orderDifferenceEquation}{M}} \htmlId{tooltip-2const}{b}_{\htmlId{tooltip-integer}{k}} \htmlId{tooltip-discreteFunction}{x}[\htmlId{tooltip-wholeNumber}{n} - \htmlId{tooltip-integer}{k}]\]
  2. And the definition of a discrete-time exponential signal:
    \[\htmlId{tooltip-signal}{x}[\htmlId{tooltip-wholeNumber}{n}] = \htmlId{tooltip-signal}{x}(\htmlId{tooltip-wholeNumber}{n}\htmlId{tooltip-samplingPeriod}{T_{s}}) = \htmlId{tooltip-const}{a} \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-radianFrequency}{\omega} \htmlId{tooltip-wholeNumber}{n} \htmlId{tooltip-samplingPeriod}{T_{s}} + \htmlId{tooltip-2angle}{\phi}) = \htmlId{tooltip-const}{a} \htmlId{tooltip-cosine}{\cos}(\htmlId{tooltip-normalizedRadianFrequency}{\hat \omega} \htmlId{tooltip-wholeNumber}{n} + \htmlId{tooltip-2angle}{\phi})\]

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