# Filter design complexity

Started by August 2, 2014
```Hello,

I am brushing up on the filter design method using Z-transform, pole/zero
method. I believe the complexity of the filter is directly proportional to
the number of poles and zeros I use to design a filter. Similarly, the
better (precise) the frequency response, higher is the complexity of the
filter.

Can the experts weigh in here please.

Also, more from a theoretical and conceptual perspective, I look at a
filter more as a system that removes something we don't want. Given this
understanding, when designing a filter, is it not enough if I just play
around with zeros to make sure everything that is not needed is
appropriately supressed?. Why is there a need to use poles?

Thank you

_____________________________
Posted through www.DSPRelated.com
```
```On Saturday, August 2, 2014 6:48:58 PM UTC+12, SBR123 wrote:
> Hello,
>
>
>
> I am brushing up on the filter design method using Z-transform, pole/zero
>
> method. I believe the complexity of the filter is directly proportional to
>
> the number of poles and zeros I use to design a filter. Similarly, the
>
> better (precise) the frequency response, higher is the complexity of the
>
> filter.
>
>
>
> Can the experts weigh in here please.
>
>
>
> Also, more from a theoretical and conceptual perspective, I look at a
>
> filter more as a system that removes something we don't want. Given this
>
> understanding, when designing a filter, is it not enough if I just play
>
> around with zeros to make sure everything that is not needed is
>
> appropriately supressed?. Why is there a need to use poles?

If you require a lower-order filter then poles are the way. You need more zeros to approximate the effect of poles (for digital filters that is - for analogue we usually have poles and/or zeros) You can also get linear phase with FIR filters and this is far harder with poles.
```
```On Saturday, August 2, 2014 2:48:58 AM UTC-4, SBR123 wrote:
> Hello,
>
>
>
> I am brushing up on the filter design method using Z-transform, pole/zero
>
> method. I believe the complexity of the filter is directly proportional to
>
> the number of poles and zeros I use to design a filter. Similarly, the
>
> better (precise) the frequency response, higher is the complexity of the
>
> filter.
>
>
>
> Can the experts weigh in here please.
>
>
>
> Also, more from a theoretical and conceptual perspective, I look at a
>
> filter more as a system that removes something we don't want. Given this
>
> understanding, when designing a filter, is it not enough if I just play
>
> around with zeros to make sure everything that is not needed is
>
> appropriately supressed?. Why is there a need to use poles?
>
>
>
> Thank you
>
>
>
> _____________________________
>
> Posted through www.DSPRelated.com

SBR123,

"Why is there a need to use poles?"

Look in your text about why someone would use IIR filters rather than FIR filters.

Dirk
```
```On Sat, 02 Aug 2014 01:48:58 -0500, "SBR123" <100967@dsprelated>
wrote:

>Hello,

Hello mysterious Mr. SBR123,

>I am brushing up on the filter design method using Z-transform, pole/zero
>method. I believe the complexity of the filter is directly proportional to
>the number of poles and zeros I use to design a filter.

That's a broad statement.  Take care in making
broad (what you think is always true) statements
about digital filters.  As soon as you make a statement
that you think is always true, someone will give you
an example where your statement is NOT true.(This has
happened to me more than once.)

For example, I can think of a linear-phase tapped-delay
line FIR that has 100 zeros on the z-plane.  That
seems like a high-compexity filter, but it's not
complicated at all.  The block diagram of the (100-zeros)
FIR filter I have in mind contains only one adder
and *NO* multipliers.

> Similarly, the
>better (precise) the frequency response, higher is the complexity of the
>filter.
>
>Can the experts weigh in here please.

Well, you have not defined what you mean by the
word "complexity."  Does that word mean "the order of
the numerator and denominator polynomials in
a filter's z-domain transfer function"?   Or perhaps
your word "complexity" means the number of arithmetic
operations needed to compute a single filter output
sample.

The FIR filter I mentioned above has a high-order
polynomial in its z-domain transfer function, but
it requires only one addition operation to compute
a filter output sample.  (Not to be too mysterious
here, ...that filter is called a "comb filter.")

>Also, more from a theoretical and conceptual perspective, I look at a
>filter more as a system that removes something we don't want. Given this
>understanding, when designing a filter, is it not enough if I just play
>around with zeros to make sure everything that is not needed is
>appropriately supressed?. Why is there a need to use poles?

Well, to have high performance (flat passband response.
narrow transition regions, high attenuation in the
stopband) a filter is needed that has a long
time-domain impulse response (many nonzero-valued samples).
So how do we build digital filters having long impulse
responses?  Two ways.  Either (1) a tapped-delay line FIR
filter having many delay elements, or (2) an IIR filter
with poles that are located close to the inside of the
unit circle on the z-plane.

The IIR filter will require fewer arithmetic operations
per output sample than the FIR filter.  However, the FIR
filter can have linear phase which the IIR filter
can never have.  Pick your poison SBR123.

Good Luck,
[-Rick-]
```