# What does "6db per octave" mean in the digital world?

Started by January 29, 2009
```In article <d_WdnQ2KFd8i-
h_UnZ2dnUVZ_jqdnZ2d@giganews.com>, bantone@web.de
says...
>
>
>>Hi,
>>
>>I had a hard time understanding the notion of "6dB per octave".
If I
>>have a single pole IIR low pass with the Fc = 1000 and Fs =
48000, for
>>example,
>>
>>a = [1.00000000000000  -0.87697646299276]
>>b = 0.12302353700724
>>
>>When doing freqz in Matlab, I was expecting
>>-3db at 1000,
>>-6dB at 2000,
>>-9dB at 4000....
>>But what I have is
>>-3@1000
>>-7@2000
>>-12.x@4000
>>-24.x@24000
>>
>>Can anyone give my a hint what was wrong? Why it is -24dB at
24000Hz?
>
>The digital frequency response is warped.
>It doesn't look like a straight line going down
>with 6db per octave (which is what you see
>if you plot an analog response on a logarithmic scale - after
>the rolloff at the beginning),
>but it's a curve.
>Remember that the distance from the pole is what determines
>the amplitude.  And you go from freq 0 to pi in a halfcircle in
>the z-plane.

To add to the above post:

You can approximate 6 dB/octave to any precision you like, but it
takes more than one z-plane pole. At very least, you will need to
add a zero, and you will get a much better approximation if you add
more poles and zeros.

Here are first, second and third-order approximations to your
requirement. They are minimax magnitude approximations from 0 to
20 kHz:

Zero #            Real                 Imag.
1         -0.1698789              0.000000
Pole #            Real                 Imag.
1          0.8720991              0.000000
MAXIMUM ERROR FROM      0.00 Hz TO  20000.00 Hz IS
0.1599127dB

Zero #            Real                 Imag.
1         -0.4590639              0.000000
2         -0.7892787E-01          0.000000
Pole #            Real                 Imag.
1         -0.3661392              0.000000
2          0.8771473              0.000000
MAXIMUM ERROR FROM      0.00 Hz TO  20000.00 Hz IS
0.0042822dB

Zero #            Real                 Imag.
1         -0.6115832              0.000000
2         -0.2722988              0.000000
3         -0.4618174E-01          0.000000
Pole #            Real                 Imag.
1         -0.5712548              0.000000
2         -0.1834612              0.000000
3          0.8773008              0.000000
MAXIMUM ERROR FROM      0.00 Hz TO  20000.00 Hz IS
0.0001147dB

Error of the fifth-order approximation is basically down in the
numerical noise:

1         -0.7513900              0.000000
2         -0.5236606              0.000000
3         -0.3078089              0.000000
4         -0.1286814              0.000000
5         -0.2164216E-01          0.000000
Pole #            Real                 Imag.
1         -0.7370853              0.000000
2         -0.4922007              0.000000
3         -0.2562247              0.000000
4         -0.7020382E-01          0.000000
5          0.8773058              0.000000
MAXIMUM ERROR FROM      0.00 Hz TO  20000.00 Hz IS
0.0000004dB

```