# What does it mean to say that “sinc” filters are ideal?

I suggested to someone that one can create a sinc filter that is ideal enough to be indistinguishable from the ideal, given certain limitations with the data in the first place.

Is that true?

It was only a suggestion yet it was met with unhelpfulness.

• i understand that a sinc filter has some sort of mathematical existence - but how can that be so unless it can be applied to some data ? – user3293056 May 31 '15 at 7:50
• Maybe you should post your comment as a port of the original answer? – jojek May 31 '15 at 8:53
• @jojek this is how i think of it... suppose we know the value of pi thru math proof etc., but our computers for some reason can only compute it to some finite amount. if i have a picture and want to draw a circle round it, i would only need the computer to approximate pi close enough to cope with the resolution. ?? – user3293056 May 31 '15 at 13:58

An ideal filter (system) is the one which can be described mathematically but cannot be implemented (realized) physically.

Coming to a sinc filter, the ideality of this filter results from its frequency domain definition which is an ideal low-pass filter with zero ripple in the pass and stop bands and zero transition width.

Having these specifications, an ideal lowpass filter, can be described mathematically but cannot be realized using any techniques. However you can still find the resulting impulse response of an ideal lowpass filter by using the formal frequency to time transformation of the frequency reponse of the ideal filter.

This kept in mind, therefore, an ideal low-pass filter is seen to have an impulse response in the form of a sinc(x) signal, which extends from minus to plus infinity in time, which is another manifestation of the ideality of the filter being designed.

When such a filter is to be implemented in practice, it can only be approximated. The approximation being better as the length of the filter increases which however is a very undesired condition.

Instead of forcing approximations to ideal sinc filters this way, it is practically more effective to use other techniques such as weighted windowing to realize those filters, which results in the conversion of filter characteristics from those of the ideal one to the realizable one.

• hi Fat32 - is any ad hoc implementation always going to be identifiable - so if i block filter something below 200 hertz, can i really do so without leaving any trace of what was there for spying eyes ? – user3293056 May 31 '15 at 13:53
• possibly not... I'm not sure but. – Fat32 May 31 '15 at 17:40

the "ideal" property of a $\operatorname{sinc}(\cdot)$ filter is in reference to its Fourier Transform

$$\mathcal{F} \left\{ 2 f_0 \ \operatorname{sinc}(2 f_0 t) \right\} \ = \ \operatorname{rect}\left( \frac{f}{2f_0} \right)$$

where

$$\operatorname{sinc}(x) \triangleq \begin{cases} \frac{\sin(\pi x)}{\pi x}, & \text{if }x \ne 0 \\ 1, & \text{if }x = 0 \end{cases}$$

and

$$\operatorname{rect}(x) \triangleq \begin{cases} 1, & \text{if }|x|<\frac{1}{2} \\ 0, & \text{if }|x|>\frac{1}{2} \end{cases}$$

the $\operatorname{sinc}(\cdot)$ filter perfectly and totally excludes all frequency components above $f_0$ and perfectly and totally leaves all frequency components below $f_0$ unmolested. that's why it's ideal. it's an ideal (and impossible to attain) filtering operation.

• I would have thought that the OP knows everything that you wrote down. However, you didn't actually answer the OP's question: whether it is possible to "create a sinc filter that is ideal enough to be indistinguishable from the ideal". – Matt L. Jun 1 '15 at 7:01
• @MattL. please never over-estimate my intelligience haha – user3293056 Jun 2 '15 at 6:08

A sinc filter is unstable and not causal, so as such it can't be implemented. In discrete-time you can in principle approximate it arbitrarily closely by applying a (very long) window to the ideal filter impulse response, and shifting it such that it becomes causal. The latter will add a delay but it will not affect the magnitude response.

• i think you mean by "causal" implementable in real time ? or, not.. ? – user3293056 May 31 '15 at 13:59
• @user3293056: Exactly, it means that the current output sample only depends on the current and on past input samples, but not on future input samples. – Matt L. May 31 '15 at 14:43
• i might agree that a sinc filter is acausal and unrealizable, but i have no idea what is meant by "A sinc filter is unstable ...". i don't think that stability has anything to do with a sinc filter. i've always thunk that stability is a property related to recursion (or feedback). is there some standard recursive theoretical implementation of a sinc filter? i am unaware of it, if there is. – robert bristow-johnson May 31 '15 at 22:29
• Since filter is unstable as mentioned by Matt L. For details please refer to dsp.stackexchange.com/questions/1032/is-ideal-lpf-bibo-unstable – Oliver Jun 1 '15 at 1:44
• "I don't really believe that this is new to you, is it?" ... no, just a sloppy memory. having $\max \left\{ |y[n]| \right\} < \infty$ is not the same as $\max \left\{ |h[n]| \right\} < \infty$ . i stand (or sit) corrected. – robert bristow-johnson Jun 1 '15 at 20:00

If you mean that the filter impulse response is a sinc function, then you are talking about a function with infinite support, i.e, the length of the filter is infinite. This is impractical to implement, but you can approximate it by truncation. The longer the segment resulting from your truncation, the better the approximation.

Therefore, your statement is rather correct.