Also, the magnitude response cannot be constant over an interval.

We will see shortly why.

And therefore, we will have to specify some tolerances over the passbands and

stopbands, within which we allow the frequency response to move.

In general, the lesson is the following.

If we want very small transmission bands,

we will need to use a filter of high order.

And similarly, if we want small error tolerances, we need a high order filter.

Now, a high order means a high polynomial degree either at the numerator or

at the denominator of the transfer function.

Which means that we will need to use more computational power to implement

that filter, and

that the delay introduced by the filter in a causal realization will be larger.

We can plot the realistic lowpass specifications graphically as follows.

Instead of having one cutoff frequency, we have a transition band between a frequency

omega p, which specifies the end of the pass band to a frequency omega s,

which specifies the beginning of the stopband.

And instead of having one desired value for stopband and passband, we have

tolerance regions within which the frequency response can wiggle.

The fact that we cannot have an arbitrarily sharp transition from

passband to stopband should be sufficiently clear.

The transfer function is a rational transfer function, and as such,

it cannot have a discontinuity point.

The reason why we cannot have a flat response, ie,

a response that is identical to a constant over a certain interval.

Maybe it requires a little bit more of an explanation.

So, again, we start from a rational transfer function, and now suppose

that the frequency response H of e to the j omega is constant over an interval.

No matter how small that interval.

Well, in that case, the z transform will be constant over an interval as well.

And therefore, we can write that the denominator minus

the constant times the numerator is identically zero over an interval but

this guy here, B of Z minus C-A of Z is a polynomial in z.

So, if it is constant over an interval,

it has an infinite number of roots over that interval.

But we know, from the fundamental theorem of algebra,

that if a polynomial has an infinite number of roots,

it is identically 0 over the entire complex plane.

Which means that if the fourier transform is constant over an interval, no matter

how small that interval, it will be constant over the entire frequency range.

As a consequence, you can think of the frequency response of a filter as a shark.

It must always move and it can never be at rest.

An important case is what we call the equiripple error, where the error, for

instance, in this case in the passband, oscillates between a maximum and

a minimum and the local extrema of the frequency response coincides with

the upper and lower limit of the tolerance region.

Once the specs are in place, the three big questions are,

well first are we going to design an IIR or FIR?

And once we have answered that question, how will we determine the coefficients of

the transfer function and how do we evaluate the performance of the filter?

In order to answer the first question, we have to consider the pros and

the cons of one design versus another.

IIRs are computationally efficient.

They can achieve a very strong attenuation in the stop band rather easily.

And they're good for

audio because they can achieve a monotonic characteristic in the pass band.

On the other hand, they might have stability issues specially in numerical

implementations that are prone to overflow or underflow.

They're difficult to design for arbitrary responses,

namely a low-pass or high-pass characteristic is easy to obtain.

But it's not so easy to obtain an arbitrary response with an IIR.

And they have no linear phase.