Audio Engineering

Recording Consoles 805

A band-pass fi lter is derived by multiplying the low-pass coeffi cients with samples
of a sine wave at the center frequency of the band pass. Let’s take our band pass to
be centered on the frequency ofFs /4. Samples of a sine wave at this frequency will
be at the 0 degree point, the 90 degree point, the 180 degree point, the 270 degree
point, and so on. In other words,

0, 1, 0,  1, 0, 1, 0,  1

If we multiply the low-pass coeffi cients by this sequence we get the following,

0, 0.12, 0,  0.22, 0,  0.12, 0

The impulse response of this circuit is illustrated in Figure F27.5. This looks
intuitively right too, because the output can be seen to “ ring ” at Fs /4, which is what
you’d expect from a resonant fi lter. The derived frequency response is also shown in
the diagram.

Digital Frequency Domain Analysis—The Z -Transform

The z -transform of a digital signal is identical to the Fourier transform except for
a change in the lower summation limit. In fact, you can think of ‘ z ’ as a frequency

Fs/8

A

Fs/2

0.4 Impulse response

0.2

0

0.2

0.4

Figure F27.4 : Digital high-pass fi lter.

Fs/8

A

Impulse response

0.4

0.2

0

0.2

0.4 F

s/4 Fs/2

Figure F27.5 : Digital band-pass fi lter.