804 Chapter 27
Frequency Response
But how does this response relate to the frequency domain response? For it is
usually with a desired frequency response requirement that fi lter design begins.
This important question is really asking, how can we express something in the
time domain in terms of the frequency domain? It should be no surprise by
now that such a manipulation involves the Fourier transform. Normal textbook
design methods involve defi ning a desired frequency response and computing (via
the Fourier transform) the required impulse response, thereby defi ning the tap
coeffi cients of the FIR fi lter.
However, this is a little labor intensive and is not at all intuitive, so here’s a little
rule of thumb that helps when you’re thinking about digital fi lters. If you count the
number of sample periods in the main lobe of the sinc curve and give this the value,
n , then, very roughly, the cutoff frequency of the low-pass fi lter will be the sampling
frequency divided byn ,
FFncs /
So, for our 7-term, FIR fi lter above, n 8 and Fc is roughly Fs /8. In audio terms, if
the sample rate is 48 kHz, the fi lter will show a shallow roll-off with the turnover at
about 6 kHz. The frequency response of this fi lter (and an ideal response) is shown
in Figure F27.3. In order to approach the ideal response, a fi lter of more than 30
taps would be required.Derivation of Band-Pass and High-Pass Filters
All digital fi lters start life as low-pass fi lters and are then transformed into band-
pass or high-pass types. A high-pass is derived by multiplying each term in the fi lter
by alternating values of 1 and 1. So, our low-pass fi lter,
0.07, 0.12, 0.2, 0.22, 0.2, 0.12, 0.07
is transformed into a high-pass like this,
0.07, 0.12, 0.2, 0.22, 0.2, 0.12, 0.07
The impulse response and the frequency of this fi lter are illustrated in Figure F27.4.
If you add up these high-pass fi lter terms, you’ll notice that they come nearly to
zero. This demonstrates that the high-pass fi lter has practically no overall gain at
DC, as you’d expect. Note too how the impulse response looks “ right, ” in other
words, as you’d anticipate from an analogue type.