Amplitude response of a Biquad Filter

in article -e2dnb04YcARFW7enZ2dnUVZ_tGdnZ2d@giganews.com, rkthebad at
raviyenduri@gmail.com wrote on 02/15/2006 15:01:

> I implemented the EQ with the DF-1 structure with first order noise
shaping.
> There appears to be some problem though. A desired output is
obtained only for
> some settings of the EQ. If the settings (coefficients)
are changed, the
> output goes wild. Any suggestions?

this is the simulation of the 16 bit filter with a 32 or 40 bit accumulator?
or is this with full precision floating-point?

have you tried it both ways: same input signal and same coefficients but
different word widths/formats?  if it behaves with the floating-point and
not with the fixed-point, then, i guess, at least we know it's a numerical
problem.
-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

>in article -e2dnb04YcARFW7enZ2dnUVZ_tGdnZ2d@giganews.com, rkthebad at
>raviyenduri@gmail.com wrote on 02/15/2006 15:01:
>
>> I implemented the EQ with the DF-1 structure with first order noise
>shaping.
>> There appears to be some problem though. A desired output is
>obtained only for
>> some settings of the EQ. If the settings (coefficients)
>are changed, the
>> output goes wild. Any suggestions?
>
>this is the simulation of the 16 bit filter with a 32 or 40 bit
accumulator?
>or is this with full precision floating-point?

This is a 16-bit filter with 32 bit accumulator.

>have you tried it both ways: same input signal and same coefficients but
>different word widths/formats?  if it behaves with the floating-point
and
>not with the fixed-point, then, i guess, at least we know it's a
numerical
>problem.

I have not come across any problems with the floating point
implementation. The first order noise shaping does help for some settings
of the EQ. But, everything remaining same, just changing the gains of the
bands throws my system into instability. Specifically, the BASS band.

>-- 
>
>r b-j                  rbj@audioimagination.com
>
>"Imagination is more important than knowledge."
>
>
>

I was even wondering whether having a very low freq. band at a high
sampling rate is creating any ringing? My bands are with center
frequencies 100 , 300, 1000, 3000 and 10000 Hz.

And, if these sound wrong, can you suggest any specifications for a 5-band
eq.??

~ R K

in article GaqdnTAzoZb3k2nenZ2dnUVZ_v6dnZ2d@giganews.com, rkthebad at
raviyenduri@gmail.com wrote on 02/16/2006 00:32:

> I have not come across any problems with the floating point
implementation.
> The first order noise shaping does help for some settings
of the EQ. But,
> everything remaining same, just changing the gains of the
bands throws my
> system into instability. Specifically, the BASS band.

i guess i would suggest comparing the fixed-point coefficients to their
floating-point counterparts for filters given identical parameters.  see if
coefficient rounding or truncation is giving you trouble.  the poles will be
at:

    p1 = -a1/2 + sqrt((a1/2)^2 - a2)
    p2 = -a1/2 - sqrt((a1/2)^2 - a2)

if a1/2)^2 < a2, then you have to factor a -1 out of the inside of the
square root and you get complex conjugate poles:

    p1 = -a1/2 + j*sqrt(a2 - (a1/2)^2)
    p2 = -a1/2 - j*sqrt(a2 - (a1/2)^2)

then you need to make sure that the poles have magnitude less than 1 for
stability.  in the later (complex conj) case:

    |p1| = |p2| = sqrt(a2)

in the former case (real poles)

    max{|p1|, |p2|} = |a1/2| + sqrt((a1/2)^2 - a2)

so to put the two together, to insure stability

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

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

using your *quantized* coefficients, you must check to see that whichever
criteria 1. or 2. (whichever condition applies) is satisfied.  it the
criterion that is applicable is not satisfied, you cannot expect stability.

now, i am convinced that the cookbook equations result in stable filters at
least until there is coefficient quantization, and after quantization i am
not so sure.  and 16 bit quantization is pretty severe.


... 
> I was even wondering whether having a very low freq. band at a high
sampling
> rate is creating any ringing? My bands are with center
frequencies 100 , 300,
> 1000, 3000 and 10000 Hz.

because cos(w0) gets so close to 1 for 0 < w0 << pi (Nyquist), the low bands
might be the worst problem from a numerical POV.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

>in article GaqdnTAzoZb3k2nenZ2dnUVZ_v6dnZ2d@giganews.com, rkthebad at
>raviyenduri@gmail.com wrote on 02/16/2006 00:32:
>
>> I have not come across any problems with the floating point
>implementation.
>> The first order noise shaping does help for some settings
>of the EQ. But,
>> everything remaining same, just changing the gains of the
>bands throws my
>> system into instability. Specifically, the BASS band.
>
>i guess i would suggest comparing the fixed-point coefficients to their
>floating-point counterparts for filters given identical parameters.  see
if
>coefficient rounding or truncation is giving you trouble.  the poles will
be
>at:
>
>    p1 = -a1/2 + sqrt((a1/2)^2 - a2)
>    p2 = -a1/2 - sqrt((a1/2)^2 - a2)
>
>if a1/2)^2 < a2, then you have to factor a -1 out of the inside of the
>square root and you get complex conjugate poles:
>
>    p1 = -a1/2 + j*sqrt(a2 - (a1/2)^2)
>    p2 = -a1/2 - j*sqrt(a2 - (a1/2)^2)
>
>then you need to make sure that the poles have magnitude less than 1 for
>stability.  in the later (complex conj) case:
>
>    |p1| = |p2| = sqrt(a2)
>
>in the former case (real poles)
>
>    max{|p1|, |p2|} = |a1/2| + sqrt((a1/2)^2 - a2)
>
>so to put the two together, to insure stability
>
>   1.  if  (a1/2)^2 >= a2    then    |a1/2| + sqrt((a1/2)^2 - a2) < 1
>
>   2.  if  (a1/2)^2 < a2     then    a2 < 1
>
>using your *quantized* coefficients, you must check to see that
whichever
>criteria 1. or 2. (whichever condition applies) is satisfied.  it the
>criterion that is applicable is not satisfied, you cannot expect
stability.
>
>now, i am convinced that the cookbook equations result in stable filters
at
>least until there is coefficient quantization, and after quantization i
am
>not so sure.  and 16 bit quantization is pretty severe.
>
>
>... 
>> I was even wondering whether having a very low freq. band at a high
>sampling
>> rate is creating any ringing? My bands are with center
>frequencies 100 , 300,
>> 1000, 3000 and 10000 Hz.
>
>because cos(w0) gets so close to 1 for 0 < w0 << pi (Nyquist), the low
bands
>might be the worst problem from a numerical POV.
>
>-- 
>
>r b-j                  rbj@audioimagination.com
>
>"Imagination is more important than knowledge."
>
>
>

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.

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

~ R K

>>in article GaqdnTAzoZb3k2nenZ2dnUVZ_v6dnZ2d@giganews.com, rkthebad at
>>raviyenduri@gmail.com wrote on 02/16/2006 00:32:
>>
>>> I have not come across any problems with the floating point
>>implementation.
>>> The first order noise shaping does help for some settings
>>of the EQ. But,
>>> everything remaining same, just changing the gains of the
>>bands throws my
>>> system into instability. Specifically, the BASS band.
>>
>>i guess i would suggest comparing the fixed-point coefficients to their
>>floating-point counterparts for filters given identical parameters. 
see
>if
>>coefficient rounding or truncation is giving you trouble.  the poles
will
>be
>>at:
>>
>>    p1 = -a1/2 + sqrt((a1/2)^2 - a2)
>>    p2 = -a1/2 - sqrt((a1/2)^2 - a2)
>>
>>if a1/2)^2 < a2, then you have to factor a -1 out of the inside of the
>>square root and you get complex conjugate poles:
>>
>>    p1 = -a1/2 + j*sqrt(a2 - (a1/2)^2)
>>    p2 = -a1/2 - j*sqrt(a2 - (a1/2)^2)
>>
>>then you need to make sure that the poles have magnitude less than 1
for
>>stability.  in the later (complex conj) case:
>>
>>    |p1| = |p2| = sqrt(a2)
>>
>>in the former case (real poles)
>>
>>    max{|p1|, |p2|} = |a1/2| + sqrt((a1/2)^2 - a2)
>>
>>so to put the two together, to insure stability
>>
>>   1.  if  (a1/2)^2 >= a2    then    |a1/2| + sqrt((a1/2)^2 - a2) < 1
>>
>>   2.  if  (a1/2)^2 < a2     then    a2 < 1
>>
>>using your *quantized* coefficients, you must check to see that
>whichever
>>criteria 1. or 2. (whichever condition applies) is satisfied.  it the
>>criterion that is applicable is not satisfied, you cannot expect
>stability.
>>
>>now, i am convinced that the cookbook equations result in stable
filters
>at
>>least until there is coefficient quantization, and after quantization i
>am
>>not so sure.  and 16 bit quantization is pretty severe.
>>
>>
>>... 
>>> I was even wondering whether having a very low freq. band at a high
>>sampling
>>> rate is creating any ringing? My bands are with center
>>frequencies 100 , 300,
>>> 1000, 3000 and 10000 Hz.
>>
>>because cos(w0) gets so close to 1 for 0 < w0 << pi (Nyquist), the low
>bands
>>might be the worst problem from a numerical POV.
>>
>>-- 
>>
>>r b-j                  rbj@audioimagination.com
>>
>>"Imagination is more important than knowledge."
>>
>>
>>
>
>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.
>
>But, wouldn't it be sufficient to see whether the poles of the quantized
>filter are inside the unit circle? 
>
>~ R K
>

Typo in my previous post.

Condition is : |a2| < 1 and |a1| < 1 + a2.

Also, when I compare the spectrum of the filter before and after
quantization, I see a considerable change in the spectrum of the first two
bands as the sampling frequency increases. This was *practically"
eliminated if I had the flexibility to use 32 bit coefficients.

Looking at the spectrum, I also notice that when I apply a boost of 12 dB
on my first band, I do not see the required boost. I can see a boost of
only 9 dB. The same is the case with the second band. Upon decreasing the
Q, I could see the 12 dB increase. Why is this so?

~ R K

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

>
>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

"rkthebad" <raviyenduri@gmail.com> wrote in
news:J8KdncyPSKDCkGveRVn-ug@giganews.com: 

>>
>>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

>"rkthebad" <raviyenduri@gmail.com> wrote in
>news:J8KdncyPSKDCkGveRVn-ug@giganews.com: 
>
>>>
>>>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?

Thanks,
~ R K

>>"rkthebad" <raviyenduri@gmail.com> wrote in
>>news:J8KdncyPSKDCkGveRVn-ug@giganews.com: 
>>
>>>>
>>>>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?
>
>Thanks,
>~ R K
>
>
Everyone,

   I am able to get a *decent* output through my 5-band equalizer. But, I
am sacrificing the bass content till 500 Hz. Now, I know for sure that
there is serious problem with a high sampling frequency and my first band
(if below 500 Hz). I would have to find a way around this. Any suggestions
will be definitely helpful.

Thanks to everyone.
~ R K