Audio Engineering

Recording Consoles 803

However, the 1/4, 1/2, 1/4 fi lter is really a very rough approximation indeed. So
let’s now imagine a better estimate to the true sinc function and design a relatively
simple fi lter using a 7-tap FIR circuit. I have derived the values for this fi lter in
Figure F27.2. This suggests a circuit with the following tap values:

0.3, 0.6, 0.9, 1, 0.9, 0.6, 0.3

The only problem these values present is that they total to a value greater than

1. If the input was the unity step-function input, the output would take on a fi nal
   amplitude of 4.6. This might overload the digital system, so we normalize the
   values so that the fi lter’s response at DC (zero frequency) is unity. This leads to the
   following, scaled values:

0.07, 0.12, 0.2, 0.22, 0.2, 0.12, 0.07

The time-domain response of such an FIR fi lter to a step and impulse response
is illustrated in Figure F27.3. The improvement over the three-tap fi lter is already
obvious.

A

Fs/8 Fs/2

7-tap filter response

Ideal response

0

0.2

0.4

0.4

Step response Impulse response

0.2

0.6

0.8

1.0

Figure F27.3 : Finite impulse response fi lter.