## The World's Most Interesting FIR Filter Equation: Why FIR Filters Can Be Linear Phase

●8 commentsThis article discusses a little-known filter characteristic that enables real- and complex-coefficient tapped-delay line FIR filters to exhibit linear phase behavior. That is, this article answers the question: What is the constraint on real- and complex-valued FIR filters that guarantee linear phase behavior in the frequency domain?

## Comments:

wenkaixu

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11

4 years ago

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DirkDoesDSP

Said:

Hi Rick,

Interesting paper. One thing missing. Equation (2) is not the only equation for a real-coefficient FIR filter with linear phase. As you point out, equation (4) with beta=0 gives equation (2) for real coefficients. (2) gives even-symmetric coefficients. However equation (4) with beta = pi/2 can give an equation for linear phase odd-symmetric real-coefficient filters.

Interesting paper. One thing missing. Equation (2) is not the only equation for a real-coefficient FIR filter with linear phase. As you point out, equation (4) with beta=0 gives equation (2) for real coefficients. (2) gives even-symmetric coefficients. However equation (4) with beta = pi/2 can give an equation for linear phase odd-symmetric real-coefficient filters.

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DirkDoesDSP

Said:

Another attempt at getting a post to show up.

Hi Rick,

Interesting paper.

Equation (2) is a symmetric real coefficient linear filter. Another real coefficient linear filter is an antisymmetric real coefficient filter, which doesn't satify equation (2). This falls out from equation (4) if you use beta=pi/2.

Dirk

Hi Rick,

Interesting paper.

Equation (2) is a symmetric real coefficient linear filter. Another real coefficient linear filter is an antisymmetric real coefficient filter, which doesn't satify equation (2). This falls out from equation (4) if you use beta=pi/2.

Dirk

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