Re: Amplitude response of a Biquad Filter

"rkthebad" <raviyenduri@xxxxxxxxx> wrote in
news:vOOdnUD8eYVssmveRVn-sw@xxxxxxxxxxxx:

"rkthebad" <raviyenduri@xxxxxxxxx> wrote in
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rkthebad wrote:


Apart from the conditions that you mentioned above, I found one
more stability condition in a textbook.
Condition is : |a2| < 1 and |a1| < 1 + a2.
I did not venture too much into the math, but I am guessing your
conditions might come down to this after simplification.

the second one looks different, but the first is the same.

whatever, i am convinced that the criteria i stated:

1. if (a1/2)^2 >= a2 then |a1/2| + sqrt((a1/2)^2 - a2) < 1

2. if (a1/2)^2 < a2 then a2 < 1

is both necessary and sufficient.

if it's case 1. then

sqrt((a1/2)^2 - a2) < 1 - |a1/2|

(a1/2)^2 - a2 < (1 - |a1/2|)^2 = 1 - |a1| + (a1/2)^2

which comes to

|a1| < 1 + a2

so you're right, it's the same thing (except, it's a "either or" not
*both*). thanks for pointing that out.

But, wouldn't it be sufficient to see whether the poles of the
quantized
filter are inside the unit circle?

that's precisely what i was doing. take a look at the math, that is
all that it is.

r b-j


Here is one strange thing that I observed. Can anyone explain why?

I am trying to see the actual response of the designed 5-band
equalizer. I put a gain of +12 dB for all the bands. Then I vary the
sampling frequency of the system. It is observed that at a sampling
freq. of 48 kHz, the first band (with fc = 300 Hz) does not show a
gain of +12 dB. Instead, it only shows a gain of +9 dB. Why is this
so?

Thanks,
~ R K



I think you are a victim of 16 bit precision. The low frequency bands
with high Q are the first place where the problem is likely to occur.

This kind of problem illustrates why the Motorola 56K (24 bits) and
later

the SHARC (32 bits) became so popular in high performance audio.

I think you already found improvements by changing filter topology
(DFI vs DF2). If you have enough MIPs, you can rewrite the algorithm
with double precision math with your 16 bit processor or move to a
higher precision processor.





--
Al Clark
Danville Signal Processing, Inc.
--------------------------------------------------------------------
Purveyors of Fine DSP Hardware and other Cool Stuff
Available at http://www.danvillesignal.com

Al,
You are right. Because of the 16-bit quantization, I am losing
precision
compared to the floating point architecture. But, the above mentioned
problem, I am experiencing even in the floating point implementation.
The problem is the same when I tried to compute my coefficients using
MATLAB or Microsoft VC++. Is 48 kHz tooo high a sampling freq.
compared to the 300 Hz band?

I'm not sure I completely understand your setup.

DFI works best with fixed point assuming you have a double length
accumulator. In a 16 bit DSP, the accumulator will be 32 bits + guard bits.

In a 32 bit processor such as a SHARC, you have an 80 bit accumulator when
operating in fixed point. This is very good for DFI. I think one of the
reasons that a lot of DSP books suggest DFII, is that they didn't assume
the double width accumulator was available. I think this is true of just
about every DSP, but many of the books were written before there were DSP
chips.

If you are using IEEE float, you have about 24 bits of mantissa and 8 bits
of exponent (discounting the hidden bit). The results of the multiplier and
accumulator are also IEEE float (24 + 8). There is no extended precision in
this mode. DFII might be better for this situation.

My guess is that 300 Hz is too high for 48k sampling with a high Q filter
with only 16 bits of precision.

Al




Thanks,
~ R K




--
Al Clark
Danville Signal Processing, Inc.
--------------------------------------------------------------------
Purveyors of Fine DSP Hardware and other Cool Stuff
Available at http://www.danvillesignal.com
.

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