Recursive filter with truncated polynomial impulse response

Question

How can we recursively implement a causal discrete-time truncated infinite impulse response (TIIR) filter with an arbitrary truncated polynomial impulse response:

$$h[n] = \begin{cases}\displaystyle\sum_{k=0}^K c_k n^k&\text{if }0\le n<N,\\ 0&\text{otherwise,}\end{cases}$$

where real $c_k$ are the arbitrary coefficients of the polynomial, nonnegative integer $K$ is the degree of the polynomial, $0^0 = 1$, and positive integer $N$ is the length of the part of the impulse response that may be non-zero?

Perhaps a recursive form allows time complexity to be not dependent on $N$, giving an efficient implementation when $K$ is small, even when $N$ is large. Moving average can be implemented recursively as an integrator with tail cancellation. That's a $K=0$ degree truncated polynomial impulse response. Perhaps higher degree polynomials are also possible.

A cascade of identical moving average filters gives a B-spline impulse response, which is a piece-wise polynomial of any desired degree but lacking arbitrary control of the polynomial coefficients. Perhaps some form of tail cancellation allows to isolate a single piece of a higher degree B-spline impulse response and to linearly combine the piece with the same of lower degree B-splines to get arbitrary polynomials of any degree. Or maybe there is another approach.

Some promising literature I found: Oscar G. Ibarra-Manzano, Yuriy S. Shmaliy, "Implementation of Digital Unbiased FIR Filters with Polynomial Impulse Responses", Circuits Systems and Signal Processing, April 2012.

Answer 1

I don't think you can, I don't see any special property of the truncated polynomial representation that would help here.

The impulse response coefficients are related to the polynomial coefficients through a set of linear equations, so we can simply write this as a matrix multiplication.

$$h = M \cdot c$$ where $h$ is the impulse response vector, $c$ the polynomial coefficient vector and $M$ a matrix that looks something like $$M = \begin{bmatrix} 1 & 0 & 0 & ... \\ 1 & 1 &1 & ... \\ 1 & 2 & 4 & ... \\ 1 & ... & ... & ... \end{bmatrix}$$

That matrix appears to be invertible, so we simply express the polynomial coefficients as

$$c = M^{-1} \cdot h$$

for any given impulse response. That means that every FIR impulse response can be represented in truncated polynomial form and there is really nothing special about it. If there were a way to implement this recursively, it would mean that every FIR filter can be implemented recursively which, to the best of my knowledge, is not the case.

Answer 2

Consider transfer function of FIR filter with N-taps:

$$ \begin{align} H(z) = \sum\limits_{n = 0}^{N - 1} h_n z^{-n} \end{align} $$

We can change it to IIR filter by adding pair of pole-zero with unity gain at $-a_1$:

$$ \begin{align} H(z) &= \sum\limits_{n = 0}^{N - 1} h_n z^{-n} \\ &= \frac{(1 + a_1 z^{-1}) \sum\limits_{n = 0}^{N - 1} h_n z^{-n}}{1 + a_1 z^{-1}} \\ &= \frac{\sum\limits_{n = 0}^{N - 1} h_n z^{-n} + \sum\limits_{n = 0}^{N - 1} a_1 h_n z^{-(n + 1)}} {1 + a_1 z^{-1}} \\ &= \frac{\sum\limits_{n = 0}^{N - 1} h_n z^{-n} + \sum\limits_{n = 1}^{N} a_1 h_{n - 1} z^{-n}} {1 + a_1 z^{-1}} \\ &= \frac{h_0 + a_1 h_{N-1} z^{-N} + \sum\limits_{n = 1}^{N - 1} (h_n + a_1 h_{n-1}) z^{-n}} {1 + a_1 z^{-1}} \end{align} $$

So, the required condition is for $ 1 \le n < N $:

$$ \begin{align} h_n + a_1 h_{n-1} &= 0 \\ h_n &= -a_1 h_{n-1} \\ h_n &= (-a_1)^n h_0 \end{align} $$

When $ a_1 = - 1 $, it is moving average filter. But it seems that the required condition of $ h_n $ is unrelated with truncated polynomial impulse response definition.

We can also expand it with higher order polynomial denumerator $ 1 + \sum\limits_{m=1}^{M} a_m z^{-m}$. In that case, the required condition is:

$$ \begin{align} h_n + \sum\limits_{m=1}^{M}a_m h_{n-m} &= 0 \\ h_n &= - \sum\limits_{m=1}^{M}a_m h_{n-m} \end{align} $$