Why are numerator coefficients named $b$ and denominator coefficients named $a$?

This is not a technical question, it is rather a question about the implicit notation used in MATLAB and some digital signal processing books to refer to the numerator and denominator coefficients of impulse responses or system functions.

When I think of a fraction the first thing that comes into my mind and the first thing I write is the numerator and then the denominator. When I spell the alphabet the first letter is a and the second is b.

When I started reading DSP books and working with Matlab I found:

$$Y(z)=\frac{b(1)+b(2)z^{-1}+\cdots+b(n_b+1)z^{-n_b}}{1+a(2)z^{-1}+\cdots+a(n_a+1)z^{-n_a}}X(z)$$

and y = filter(b,a,x) respectively

In these cases the first things to write were labeled b (numerator coefficients) and the second things to write were label a (denominator coefficients). This caused me a little bit of confusion at first as it seemed a little bit counter-intuitive. Is there any reason why this was selected in this way?

I'm thinking that maybe poles are considered more important than zeros (due to stability, etc.) so they most be though of first and a comes first in the alphabet... but this is just an educated guess.

• It's arbitrary. Here is an old report where the opposite is the case: pdfs.semanticscholar.org/3bdc/… . See equations I-1 and I-2 in the report. It probably really boils down to what references texts and papers the implementers of the MatLab software were using at the time. Also a majority of EE's I know all refer to Oppenheim and Schafer's book as their reference DSP text, which uses the convention you point out, so those EE's tend to use the conventions from that text. – Andy Walls Mar 23 '18 at 16:29
• Thanks Andy! Yes Oppenheim and Schafer's book uses this notation and it's definitely a reference. – VMMF Mar 23 '18 at 16:31

My suspicion is that this ordering comes from the difference equation, which in most texts precedes the $\mathcal Z$-transform:

$$a_0 y[n] + a_1 y[n-1] + \cdots = b_0 x[n] + b_1 x[n-1] + \cdots$$

• Then again, input coefficients x come first than output coefficients y. Why are they labeled b? – VMMF Mar 23 '18 at 15:37
• @VMMF: Because output is traditionally on the left-hand side: output = ... – Matt L. Mar 23 '18 at 17:15
• texts that put $a_m$ in the numerator of the transfer function have the $a_m$ multiplying $x[n-m]$ term. so this doesn't answer the question. – robert bristow-johnson Mar 24 '18 at 9:34
• @robertbristow-johnson What you point is so interesting! I haven't seen many books with this notation. Could you name a few? – VMMF Mar 26 '18 at 14:57

i think it's because poles are more important than zeros. the location of the poles determine the stability of the system. when partial fraction expansion, it's the poles that survive in the partial fractions. it's the dominant poles that determine the decay rate of the impulse response (or any response after the input goes to zero).

and it's in the denominator of the transfer function where you find the poles. so the denominator is more "primary" than the numerator. "$a_m$" is more "primary" than "$b_m$".

• +1This is exactly what I though! However @Laurent Duval has a very nice theory as well. – VMMF Mar 26 '18 at 14:59

Possibly a pseudo-Darwinian effect, related to the autoregressive or all-pole models, and the initial letter 'a', the first of the alphabet. Details follow.

This question made me dig into early works related to the (re)-discovery and usage of the $z$-transform for the representation of systems. Apparently, the concept of the $z$-transform was known to Laplace, re-introduced by W. Hurewicz in 1947, modified and popularized by E. I. Jury. Considering transfer functions as ratios of polynomials in $z$ was seemingly published first by R. H. Barker, in a report I could not access to, yet. References follow:

I could not find any prevalence of $a/b$ or $b/a$ notations. Sometimes $P/Q$ and $P/R$ were used, following standard notations for rational functions.

Hence, my guess relates to the initial autoregressive model as given in P. Whittle, 1952, Tests of Fit in Time Series, Biometrika:

Let $x_t$ be a stationary, purely non-deterministic process, the reciprocal of whose spectrum may be expanded in a Fourier series. The process may then be expressed $x_t+a_1 x_{t-1}+a_2x_{t-2}+\ldots=e_t$, i.e. as an autoregression, generally infinite.

The above is a model in the form of a "stochastic difference equation", with a stochastic term $e_t$. However, it can be interpreted at the output of an all-pole infinite impulse response filter whose input is white noise.

In other words, using the first letter of the alphabet $a$ made sense as a series of coefficients, and perhaps remained, in a sort of crypto-darwinian meme, as the "A" letter was also the initial of Autoregressive and All-pole.

• what's "$e_t$"? input or output? – robert bristow-johnson Mar 24 '18 at 11:24
• I don't really consider this equation as a system with input and output, but more generically as a "stochastic difference equation" – Laurent Duval Mar 24 '18 at 15:28
• Sure, not fully conclusive though... – Laurent Duval Mar 26 '18 at 21:25