Digital Signal Processing Reference
In-Depth Information
Illustration 49:
Pulse response (Si-function) with different lowpass bandwidths
As already suggested, the lowpass filter has (at best) a rectangular progression. Up to now we have dealt
mainly with rectangular progressions in the time domain. Now look closely at the Si-function in the time
domain and compare it with the progression of the frequency spectrum of a rectangular pulse (see in this
connection Illustration 48 bottom).
You will probably have noticed that with the Si-functions the time T' = 1/
f is entered which appears to
describe something like the period length visually. But there cannot be a period length because the
function is not repeated exactly after the time T'. However, each of the Si-functions represented have a
different ripple content: it depends on the band width
Δ
f of the lowpass. This ripple content is equivalent to
the ripple content of the highest frequency which passes through the lowpass. The pulse response can
never change faster than the highest occurring frequency in the signal. The progression of the Si-function
is determined precisely by this highest frequency.
Δ
The pulse response of an ideal "rectangular" lowpass filter (which - as already pointed out
- is physically impossible) has a special importance and is called an Si-function. It is like
a sine which is "compressed or bundled in time". For this reason it cannot consist of only
one frequency because of
UP
.
