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I have some doubts about the meaning of order for digital filters. All I'll put here is taken from these slides.

1) Let's start from the definition of FIR filters:

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From this definition we see that order means the number of delayed input values in the input - output equation of the filter.

So my question is: is there a link with the concept of "order" in control theory? For instance, in control theory a second order system (or filter) has a transfer function with a second - degree polynomial at the denominator, and this happens also in analog filters (where it is sometimes equal to the number of reactive components of the circuit). Is it applicable also here?

2) Let's consider the so called windows design method:

enter image description here

Basically it tries to realize (in an approximate way) a filter by selecting a window of the desired pulse response of the filter, and you may see from the expression of w(n) that the length of the window is equal to the order of the filter (M-1).

So my question is: which is the link between the order of the filter (as described in 1) and the number of samples of the window?

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  • \$\begingroup\$ FIR : No direct connection ... IMO teh term "order" for FIR is misleading in this respect. For IIR filters : yes there is; and you can approximately translate a continuous time filter into an TTR equivalent (e.g. biliniar transform). \$\endgroup\$ – Brian Drummond Apr 24 at 13:25
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In general, the order refers to the underlying degree of the polynomial. Just like a polynomial, there are the powers of \$x\$ and the coefficients. If the order is 2, then the polynomial has 3 powers of \$x\$ and 3 coefficients, but since \$x^0=1\$, we say it has 2 powers and 3 coefficients.


For filters without feedback (FIR), the powers of \$x^n\$ are the delays, \$z^{-n}\$, and the transfer function has only zeroes (the denominator is 1, all the poles are at 0), so the numerator's coefficients form the polynomial:

$$h_nz^{-n}+h_{n-1}z^{-(n-1)}+...+h_1z^{-1}+h_0$$

The samples do not require to have a memory (e.g. zeroth order sample&hold). But if you need to work with samples from memory, then you'd need one sample for the current value, and \$n\$ samples for the ones in memory. This means that the number of samples is given by the length of the filter, \$n+1\$.

As a side comment, you'll see many notations for length and order, such as the length, \$L=N+1\$, being the order plus one, and \$M=\frac N2\$ being the midpoint, or half the order.


For filters with feedback (IIR), the transfer function should be proper, so then the degree represents the order of the denominator:

$$\prod_{k=0}^N\frac{z-z_k}{z-p_k}$$

which expands into (see this for the formula):

$$\sum_{k=0}^N(-1)^{N-k}\left(\sum_{0\leq i_1<i_2<\dots<i_{N-k}\leq N-1}\prod_{j=1}^{N-k}p_{i_j}\right)z^{-k}$$

So, yes, since the control theory makes use of filters, it obeys their definitions, in general.

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  • \$\begingroup\$ So, for instance, if I have a 4th order transfer function which represents a Butterworth filter, will its corresponding FIR filter have only a pulse response of 4 samples? I'm a bit confused \$\endgroup\$ – Kinka-Byo Apr 24 at 12:57
  • \$\begingroup\$ @Kinka-Byo There is no direct correspondence between a Butterworth filter and a FIR: one has feedback, the other has not. What you're probably talking about is an approximation for a Butterworth response, achieved with frequency sampling, or other such methods, but then you're comparing apples and oranges. Also, in the answer I said that the number of samples is given by the length of the filter, so whether you want a 4th order IIR, or FIR, the order of each will be given by the number of delays. For analog filters, the order is given by the number of reactive elements. \$\endgroup\$ – a concerned citizen Apr 24 at 14:13
  • \$\begingroup\$ (cont'd) For IIRs, the number of delays are those at the denominator, because there can be at the numerator, too, though this is dependent on the realization topology. For FIRs it's straightforward. \$\endgroup\$ – a concerned citizen Apr 24 at 14:18
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1) FIR filter order N only means it has length of M=N+1 coefficients. It's frequence response (transfer function) can be anything really that can be realized with the coefficients value and amount of coefficients. It's not related to say analog filter of order X where it means the slope after cutoff.

2) Order N means length M=N+1 coefficients. You basically have a infinitely long ideal filter, and if you only use the rectangular window of length M to truncate the infinite filter to have only M coefficients then you end up with order N=M-1 filter.

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