# The history of FIR filters

Does anyone know the name of the first DSP (sampled data) practitioner who discovered that real-valued symmetrical-coefficient FIR filters exhibit linear phase in the frequency domain?

• Do you know who first talked about smoothing filters, and who first coined "finite impulse response" or even "digital filter"? Because the history may go back to moving average in time series. I recently found an analog of the Dirac delta function in Joseph Fourier "Théorie de la chaleur" – Laurent Duval Aug 4 '15 at 6:49
• SE.DSP wishes you a happy new year 2017, with a kind reminding signal that your question or its answers may require some action from you (edit, update, votes, acceptance, etc.) – Laurent Duval Jan 15 '17 at 16:24

Let me propose a majorizing date for this discovery, at least in written form: Rader, C. M. and Gold, B. (1967) Digital filter design techniques in the frequency domain, Proceedings of the IEEE. Such a discussion appears on page 164, under the name of "frequency sampling filters", involving the combination of "elemental" and "comb filter". They say "The phase versus frequency is exactly linear except for discontinuities of $\pi$ radians. These discontinuities occur where the magnitude response is zero".

A lower bound could be Lerner, R. M. (1964) Band-pass filters with linear phase, Proceedings of the IEEE, which seems to keep analog.

• thanks for your thoughts. I studied the Rader & Gold paper years ago when I was learning about 'frequency sampling FIR filters'. In their Part C they mention "Lerner" filters and say that those Lerner filters "have a high degree of phase linearity." But they say nothing about why linear phase is a good thing. – Richard Lyons Aug 5 '15 at 9:59
• What I seek is the name of the first guy who determined that for a nonrecursive FIR to have linear phase the filter's h(n) coefficients must satisfy: ∑h(n)sin[(n-D)w] = 0. (D is the time delay through the filter measured in samples and w is frequency measured in radians/sample.) The earliest mention of this idea that I can find is the 1975 book "Theory and Application of DSP" by Rabiner and Gold. – Richard Lyons Aug 5 '15 at 10:18
• I understand the specific question, yet I have no answer yet. Do I chat with the author of "Understanding DSP" and the columnist in IEEE Signal Processing Magazine? – Laurent Duval Aug 5 '15 at 17:32
• Yes you do. I am that person. – Richard Lyons Aug 6 '15 at 9:14

I believe that this question doesn't have an answer in terms of a name and a date, simply because we're not talking about a discovery here. From the theory of Fourier series it is a well-known fact that the Fourier coefficients of a real-valued periodic function exhibit Hermitian symmetry. Analogously, purely imaginary periodic functions have anti-symmetric Fourier coefficients.

Since the filter coefficients of a transversal filter are essentially the Fourier series coefficients of the corresponding spectrum (which is also nothing that needed to be discovered), it is obvious that (conjugate) symmetric filter coefficients correspond to a real-valued spectrum (i.e., zero phase spectrum). A linear phase is of course obtained by shifting the coefficients such that the filter becomes causal. A similar thing is true for anti-symmetric coefficients, which correspond to generalized linear phase filters.

So in sum, there's nothing that needed to be discovered, and, consequently, no single researcher can be associated with "discovering" this fact.

EDIT:

In answer to Richard Lyons' comment I'll explain what I meant by writing "it is obvious that symmetric filter coefficients correspond to a real-valued spectrum". By "symmetric filter coefficients" I meant filter coefficients that satisfy

$$h[n]=h^*[-n]\tag{1}$$

(i.e., hermitian symmetry), which for real-valued coefficients means simple symmetry. The frequency response of a length $2N+1$ symmetric FIR filter is

$$H(e^{j\omega})=\sum_{n=-N}^{N}h[n]e^{-jn\omega}\tag{2}$$

With $(1)$, this can be rewritten as

$$H(e^{j\omega})=h[0]+\sum_{n=1}^{N}\left(h[n]e^{-jn\omega}+h^*[n]e^{jn\omega}\right)=h[0]+2\Re\left\{\sum_{n=1}^{N}h[n]e^{-jn\omega}\right\}\tag{3}$$

where from the last equality it can be seen that $H(e^{j\omega})$ is real-valued. Note that for $(1)$ to be satisfied, $h[0]$ needs to be real-valued. If you shift the center of symmetry away from $n=0$ you get a linear phase instead of a zero phase.

• I've found papers from the late 1960s that mention linear phase with regards to FIR filters. Surely there must have been someone who first described that property in a paper. Matt L., you wrote "it is obvious that symmetric filter coefficients correspond to a real-valued spectrum." Did you really mean to write that? Can you give us an example of symmetrical FIR filter coefficients whose discrete Fourier transform is real-valued? – Richard Lyons Aug 6 '15 at 9:50
• @RichardLyons: Yes, I meant it. Please see my edited answer. – Matt L. Aug 6 '15 at 10:09
• Ah. I see the confusion. My original question referred to "real-valued symmetrical-coefficients." So when you used the phrase "symmetrical coefficients" I assumed you meant real-valued coefficients, but for some reason you were thinking about complex-valued coefficients. Your 'EDIT:' cleared things up. I've been studying complex-coefficient FIR filters recently. I'll mention here that, so far, I haven't encountered any real-world (practical) complex-coeff FIR filters whose center coefficient is real-valued. – Richard Lyons Aug 7 '15 at 13:24
• @RichardLyons: Of course everything I wrote remains true for real-valued symmetrical coefficients; it's just a special case of the more general hermitian symmetry. – Matt L. Aug 7 '15 at 16:45
• There are DSP beginners reading our words here, so we have to be very clear and specific in what we write. The bottom line is: real-valued symmetrical FIR filter coefficients produce a complex-valued freq response, and hermitian (conjugate-symmetric complex) coefficients produce a real-valued freq response. – Richard Lyons Aug 8 '15 at 0:32