Hi, friends! Is there any good method for "Polynomial Decomposition" being used in Filter Design? For example, if I use Park-McClellan algorithm designing a 10-order FIR filter,then want to decompose the polynomial into five 2-order filters, and keep the overall magnitude response. Does anyone has good suggestion or known some papers about this issue, please tell me.
Polynomial Decomposition in Filter Design
Started by ●May 30, 2010
Reply by ●May 30, 20102010-05-30
On 30 Mai, 10:47, "Shen Zhi" <markk...@hotmail.com> wrote:> Hi, friends! > > � Is there any good method for "Polynomial Decomposition" being used in > Filter Design? > For example, if I use Park-McClellan algorithm designing a 10-order FIR > filter,then want to decompose the polynomial into five 2-order filters, and > keep the overall magnitude response. > � Does anyone has good suggestion or known some papers about this issue, > please tell me.Any reason, other than numerical accuracy issues [*], why you can't try polynomial rooting? Rune [*] Numerical accuracy issues might well be severe enough to destroy your results, if the polynomial order is too high.
Reply by ●May 30, 20102010-05-30
On 5/30/2010 4:47 AM, Shen Zhi wrote:> Hi, friends! > > Is there any good method for "Polynomial Decomposition" being used in > Filter Design? > For example, if I use Park-McClellan algorithm designing a 10-order FIR > filter,then want to decompose the polynomial into five 2-order filters, and > keep the overall magnitude response. > Does anyone has good suggestion or known some papers about this issue, > please tell me.Question: why do you want the roots? Aren't the coefficients that P-M gives you all you really need? (Some FIR filters can reasonably have 100 taps. Would you still want to find the roots?) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 30, 20102010-05-30
Haha, Rune! I forget the roots of polynomial. By the way, do you think any high order FIR filter could be decomposed into many low order filters by the roots? If yes, is that means we doesn't need to implement a high order FIR filter but use many low order filters? "Rune Allnor" <allnor@tele.ntnu.no> ??????:d512d56c-f38d-40b0-948b-b95fd7cce808@32g2000prq.googlegroups.com... On 30 Mai, 10:47, "Shen Zhi" <markk...@hotmail.com> wrote:> Hi, friends! > > Is there any good method for "Polynomial Decomposition" being used in > Filter Design? > For example, if I use Park-McClellan algorithm designing a 10-order FIR > filter,then want to decompose the polynomial into five 2-order filters, > and > keep the overall magnitude response. > Does anyone has good suggestion or known some papers about this issue, > please tell me.Any reason, other than numerical accuracy issues [*], why you can't try polynomial rooting? Rune [*] Numerical accuracy issues might well be severe enough to destroy your results, if the polynomial order is too high.
Reply by ●May 30, 20102010-05-30
Hi, Jerry. I'm thinking of differents effections or different magnitude response errors, which comes from the limited wordlength quantiziton, between the single high-order FIR filter and several its decomposed lower FIR filters. "Jerry Avins" <jya@ieee.org> ??????:EYpMn.82856$gv4.41042@newsfe09.iad...> On 5/30/2010 4:47 AM, Shen Zhi wrote: >> Hi, friends! >> >> Is there any good method for "Polynomial Decomposition" being used in >> Filter Design? >> For example, if I use Park-McClellan algorithm designing a 10-order FIR >> filter,then want to decompose the polynomial into five 2-order filters, >> and >> keep the overall magnitude response. >> Does anyone has good suggestion or known some papers about this issue, >> please tell me. > > Question: why do you want the roots? Aren't the coefficients that P-M > gives you all you really need? (Some FIR filters can reasonably have 100 > taps. Would you still want to find the roots?) > > Jerry > -- > Engineering is the art of making what you want from things you can get. > �����������������������������������������������������������������������
Reply by ●May 30, 20102010-05-30
On 5/30/2010 5:21 AM, Shen Zhi wrote:> Haha, Rune! > I forget the roots of polynomial. > By the way, do you think any high order FIR filter could be decomposed into > many low order filters by the roots? > If yes, is that means we doesn't need to implement a high order FIR filter > but use many low order filters?You could do that, but it would take longer to compute and probably be less accurate than a single tapped delay line. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 30, 20102010-05-30
I think only few real roots could be decomposed from the original polynomial to implement a low-order filters, and most of the rest are complex roots. "Shen Zhi" <markkknd@hotmail.com> д����Ϣ����:httanl$k43$1@www.shinco.com...> Haha, Rune! > I forget the roots of polynomial. > By the way, do you think any high order FIR filter could be decomposed > into many low order filters by the roots? > If yes, is that means we doesn't need to implement a high order FIR filter > but use many low order filters? > > "Rune Allnor" <allnor@tele.ntnu.no> > ??????:d512d56c-f38d-40b0-948b-b95fd7cce808@32g2000prq.googlegroups.com... > On 30 Mai, 10:47, "Shen Zhi" <markk...@hotmail.com> wrote: >> Hi, friends! >> >> Is there any good method for "Polynomial Decomposition" being used in >> Filter Design? >> For example, if I use Park-McClellan algorithm designing a 10-order FIR >> filter,then want to decompose the polynomial into five 2-order filters, >> and >> keep the overall magnitude response. >> Does anyone has good suggestion or known some papers about this issue, >> please tell me. > > Any reason, other than numerical accuracy issues [*], why you > can't try polynomial rooting? > > Rune > > [*] Numerical accuracy issues might well be severe enough to > destroy your results, if the polynomial order is too high. >
Reply by ●May 30, 20102010-05-30
On 5/30/2010 5:29 AM, Shen Zhi wrote:> Hi, Jerry. > I'm thinking of differents effections or different magnitude response > errors, which comes from the limited wordlength quantiziton, between the > single high-order FIR filter and several its decomposed lower FIR filters.I don't see the improvement in accuracy or stability that factoring gives in IIRs. In a single tapped delay line, each element is multiplied by a coefficient and added to an accumulator. The outputs of cascaded smaller filters need to be multiplied together. Why do you believe that the product of many terms, each of which needs to ve calculated separately, might be more accurate than a sum of terms closer to the raw data? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 30, 20102010-05-30
On 5/30/2010 5:43 AM, Shen Zhi wrote:> I think only few real roots could be decomposed from the original polynomial > to implement a low-order filters, and most of the rest are complex roots.Of course. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●May 30, 20102010-05-30
A factor that the high-order FIR filter coefficients dynamic range is quite large. For example, its center coefficient may be 0.5~1, but the most outside (or the smallest) coefficient may be 0.00...001. That means in fix-point system, the wordlength to be used must be long, or the small coefficient will be quantized to zero. So I'm thinking whether the discussion in 'coefficient dynamic range' also has contribution to improve the FIR filter design. "Jerry Avins" <jya@ieee.org> ??????:BtqMn.82969$gv4.62139@newsfe09.iad...> On 5/30/2010 5:29 AM, Shen Zhi wrote: >> Hi, Jerry. >> I'm thinking of differents effections or different magnitude response >> errors, which comes from the limited wordlength quantiziton, between the >> single high-order FIR filter and several its decomposed lower FIR >> filters. > > I don't see the improvement in accuracy or stability that factoring gives > in IIRs. In a single tapped delay line, each element is multiplied by a > coefficient and added to an accumulator. The outputs of cascaded smaller > filters need to be multiplied together. Why do you believe that the > product of many terms, each of which needs to ve calculated separately, > might be more accurate than a sum of terms closer to the raw data? > > Jerry > -- > Engineering is the art of making what you want from things you can get. > �����������������������������������������������������������������������
