So this mean that the ideal lowpass has a perfectly flat passband which,
is identical to 1.
Has infinite attenuation in the stopband and again has zero-phase and no delay.
Let's try and compute the impulse response from the frequency response by
taking the inverse Fourier transform of big H of e to the j omega.
So We can write the inverse DTFT formula, which we know by heart,
and since big H is a 0 from omega C to pi and
from minus pi to minus omega C, we can change the integration limits and
just integrate from minus omega C to plus omega C,
incidentally over that interval, big H is equal to 1.
So this simplifies to just the integral of e to the j omega n.
Between minus omega c and omega c.
And if you solve that integral, which is rather easy to do, we get one over pi n.
That multi Multiplies e to the j omega cn minus e to the minus j omega cn,
divided by 2j.
We know from Mueller's formula that this is equal to the sine of omega cn.
And we are left also with pi n at the denominator.
So we plot this impulse response and
we find out that it has a nice oscillatory shape.
We also notice that it is an infinite support impulse response and
as opposed to the other infinite support impulse responses we've seen so far.
Now the support is infinite both to the right and to the left.
Now this is a problem because it means that no matter how we compute
the convolution we will always have an infinite number of operation to compute.
So this lowpass filter cannot be exactly implemented in practice, and
we will spend quite some time in the rest of the module to find ways
to approximate this ideal behavior with computable filters.
Another problem with the impulse response is that it decays very slowly in time
as 1 over n in as a matter of fact.
And so we need a lot of samples to get a good approximation
of the impulse response if we were to implement the filter directly.
In spite of the shortcomings, the ideal low pass filter is so
important that we use two special functions to describe
its impulse response and its frequency response.
So we start with a rect function that has name suggests, looks like a rectangle.
And it is indeed the indicator function
on the real line from one-half to minus one-half.
The sinc function, in this particular case it's a normalized sinc function.
Is simply the ratio between that sine of pi of x,
divided by pi of x for all real numbers other than zero.
And it is prolonged by continuity in zero to the value of one.
Note that the sinc function is equal to zero
whenever its argument is a null zero integer.
Because in that case the sign of pi x becomes zero.
With this notation in place we can express the ideal lowpass filter in
canonical form.
The frequency response, H of e to the j omega
will be simple the rect of omega divided by 2 omega c,
where omega c is the cutoff frequency of the filter.
So it will look like that.
The impulse response will be the sinc function, normalized by omega c over pi,
both in terms of its argument and in terms of managing it.
So here's an example with omega c equal to pi over 3,
we have a low pass characteristic with cutoff frequency pi over 3.
And in the time domain, we have sinc(n/3).
Multiply by one-third.
Which means every third sample, as we said, is going to be 0, because
every time n is a multiple of 3, the argument to the sinc will be an integer.
So here's a little known fact about the sinc function.
The sinc function is not absolutely summable.
The proof is not too complicated, it only involves basic calculus, and
you can find it in the references.
As a consequence, the ideal lowpass is not BIBO stable.
We can use a technique that we developed in the proof of the stability theorem.
And so for instance if we take omega c equal to pi over 3 The impulse
response of the ideal filter will be one-third sinc of n over 3.
We build an artificial bounded input signal,
which is simply the sine of sinc of minus n over 3.
And so if we compute the convolution in n equal to 0,
so we have the convolution product here.
And it is one-third, sum from minus infinity to plus infinity,
of the absolute value of sinc of k over 3, which
because of the non absolute summability of the sinc diverges to infinity.
Paolo PrandoniLecturer
Martin VetterliProfessor
