# Implementation of Linear phase IIR filter

Started by September 26, 2003
 Sir, Iam trying to implement Linear phase IIR filter. My problem is that i want to reset the state variable of the Elliptic lowpass filter i want your suggestions in doing that task. If anyone came across ftpsite of Linear phase iir filter source code please send me. Thanks in advance, shamganth.k
Hello Friends,

The topic of designing linear phase IIR filters is very interesting. I am
attaching some relevant material from "Multirate Systems and Filter Banks"
by Vaidyanathan (1993), page 37. It says that causal stable IIR linear phase
filters are impossible to design ! (Of course, we are asked to prove it !)

Regards,
Nalin

-----Original Message-----
From: Chatonda Mtika [mailto:]
Sent: Sunday, September 28, 2003 12:12 AM
To: ; sham ganth;
Subject: RE: [matlab] Implementation of Linear phase IIR filter Feng,

You are confusing phase response with impulse response. You *can* have an
infinite impulse response but still have a linear phase response. Try
plotting the phase and magnitude response of the following IIR function:
h(z) = (1 + 2*z^-1 + 2*z^-2 + z^-3)/(1 + 4*z^-2).

Or better yet, look at the response of a Butterworth IIR filter.

Thanks,

Chatonda

--- Feng Ouyang <> wrote:
> impossible. A linear phase filter must be symmetric about some point in
> time domain. If it is infinitely long (as in an IIR), it must have
infinite
> delay as well (i.e., the symmetry center is infinitely away from time 0).
> Do I miss something?
>
> Feng > -----Original Message-----
> From: sham ganth [mailto:]
> Sent: Friday, September 26, 2003 3:35 AM
> To:
> Subject: [matlab] Implementation of Linear phase IIR filter
>
> Sir,
> Iam trying to implement Linear phase IIR filter. My problem is that i
> want to reset the state variable of the Elliptic lowpass filter i
> want your suggestions in doing that task. If anyone came across
> ftpsite of Linear phase iir filter source code please send me.
> shamganth.k >
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 Attachment (not stored) Linear Phase IIR.doc Type: application/msword

 Dear ALL,   This is the first time am writing something to this group........   Coming straight to the point of linear phase IIR Filter design.   Yes, it is possible to design linear phase IIR Filter.   To prove above point......lets start with the very basic thing which is pretty clear to all.....The Linear Phase FIR Filter.   Here let say A0, A1, A2, A3, A4,.....Ap (0 <= p < N) are the coefficients. Then.........               N-1               _ Yn =       \       (Ap * Xn-p)               /_              p=0   Taking Z Transform on both sides we have.........                    N-1                  _                              -p Y(z)  =       \       [     Ap * X(z) * z      ]                  /_                 p=0                               N-1                              _                     -p H(z)  =  Y(z)  =       \       [     Ap * z      ]              X(z)          /_                             p=0                              -jkw/N H(w)  =     M(w) * e      .....................Here M(w) is Magnitude response                                                       and other is Linear Phase response                                                       also k = 0......N-1   Now lets invert the above equation.....                                                       -jkw/N Hi(w)  =  Y(w)   =   1  /   [ M(w)   *   e         ]               X(z)                                                                               N-1                                              _              -p Hi(z)  =  Y(z)  =   1  /  {           \   [  Ap * z    ]           }               X(z)                         /_                                            p=0   N-1  _                              -p  \       [     Ap * Y(z) * z      ]  =  X(z)  /_ p=0     N-1  _  \       (Ap * Yn-p)        =   Xn  /_  p=0                        N-1                       _ A0*Yn    +     \       (Ap * Yn-p)        =       Xn                     /_                     p=1                                   N-1                                  _ Yn   =     B0*Xn    -     \       (Bp * Yn-p)    ..........Here Bp = Ap/A0                                  /_                                    where   0 <= p < N                                 p=1   Above equation is a linear Phase IIR.   Hence, a LPF in FIR can be represented as a HPF in IIR a BPF in FIR can be represented as a BSF in IIR   and vice versa........(As frequency domain response is reversed)   Hope this explanation suffice your requirement.   Well thats it!.......this is tirthankar signin off.......till then take care.....BYE    Want to chat instantly with your online friends?Get the FREE Yahoo! Messenger
 Ooops, the example I gave the other day was incorrect. This is what it should have been: Let N(z) = (1 + z^-1)^p = b(z^2) + c(z^2)/z (for p >= 2) Then H(z) = N(z)/b(z^2) is linear phase IIR. example: take p=3 N(z) = 1 + 3*z^-1 + 3*z^-2 + z^-3 = (1 + 3*z^-2) + (3 + z^-2)/z b(z^2) = 1 + 3*z^-2 Butterworth wavelets (Martin Viterli, et al) fall into this group. -chatonda --- Tirthankar De <> wrote: > Dear ALL, > > This is the first time am writing something to this group........ > > Coming straight to the point of linear phase IIR Filter design. > > Yes, it is possible to design linear phase IIR Filter. > > To prove above point......lets start with the very basic thing which is > pretty clear to all.....The Linear Phase FIR Filter. > > Here let say A0, A1, A2, A3, A4,.....Ap (0 <= p < N) are the coefficients. > Then......... > > N-1 > _ > Yn = \ (Ap * Xn-p) > /_ > p=0 > > Taking Z Transform on both sides we have......... > N-1 > _ -p > Y(z) = \ [ Ap * X(z) * z ] > /_ > p=0 > N-1 > _ -p > H(z) = Y(z) = \ [ Ap * z ] > X(z) /_ > p=0 > > -jkw/N > H(w) = M(w) * e .....................Here M(w) is Magnitude response > and other is Linear Phase > response > also k = 0......N-1 > > Now lets invert the above equation..... > > -jkw/N > Hi(w) = Y(w) = 1 / [ M(w) * e ] > X(z) > > N-1 > _ -p > Hi(z) = Y(z) = 1 / { \ [ Ap * z ] } > X(z) /_ > p=0 > > N-1 > _ -p > \ [ Ap * Y(z) * z ] = X(z) > /_ > p=0 > N-1 > _ > \ (Ap * Yn-p) = Xn > /_ > p=0 > > N-1 > _ > A0*Yn + \ (Ap * Yn-p) = Xn > /_ > p=1 > N-1 > _ > Yn = B0*Xn - \ (Bp * Yn-p) ..........Here Bp = Ap/A0 > /_ where 0 <= p < > N > p=1 > > Above equation is a linear Phase IIR. > > Hence, > a LPF in FIR can be represented as a HPF in IIR > a BPF in FIR can be represented as a BSF in IIR > > and vice versa........(As frequency domain response is reversed) > > Hope this explanation suffice your requirement. > > Well thats it!.......this is tirthankar signin off.......till then take care.....BYE > > --------------------------------- > Want to chat instantly with your online friends?Get the FREE Yahoo!Messenger ===== __________________________________