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

Started by Unknown 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