Linkwitz-Riley filter implementation (issue with 3dB peaks at cut-off frequency)

Hi,

My name is Daniel. I'm a Master Thesis student from Spain. At the time I'm trying to implement a 3 way linkwitz-riley crossover in Matlab. I'm doing it because the use of normal butterworth filters chained produces a 3dB peak at the cutoff frequencies. The thing is that I'm actually getting those peaks anyway.

To do the Linkwitz-Riley implemetation I'm following an explanation that I've found on dsprelated ( http://www.dsprelated.com/showmessage/112896/1.php )

There it is said...

You need for the three outputs to sum to the cascade of the two allpass filters,

.  Lo + Mid + Hi = apH*apL

_ _ _Well,

.  apH = lpH + hpH
.  apL = lpL + hpL

So,

.  apH*apL = (lpH + hpH)*(lpL + hpL)
.           = lpH*lpL + lpH*hpL + hpH*(lpL + hpL)
.           = lpH*lpL + lpH*hpL + hpH*apL

And the crossover must be configured like this:

.       +---+  +---+
.    +--|apL|--|hpH|-- Hi
.    |  +---+  +---+
.         |
.    |  +---+  +---+
.  --+--|hpL|--|lpH|-- Mid
.    |  +---+  +---+
.    |
.    |  +---+  +---+
.    +--|lpL|--|lpH|-- Lo
.       +---+  +---+_ _ _

So... My code, reproducing this fact is the following...

[b0 a0]=butter(4,(2*pi*500)/22050);
[b1 a1]=butter(4,(2*pi*2000)/22050);
[b2 a2]=butter(4,(2*pi*500)/22050,'high');
[b3 a3]=butter(4,(2*pi*2000)/22050,'high');

H1=dfilt.df2t(b0,a0);
H2=dfilt.df2t(b1,a1);
H3=dfilt.df2t(b2,a2);
H4=dfilt.df2t(b3,a3);

paral=dfilt.parallel(H1,H3)

hi=dfilt.cascade(paral,H4);
mid=dfilt.cascade(H3,H2);
low=dfilt.cascade(H1,H2);

sum=dfilt.parallel(mid,hi,low)
fvtool(sum)

Can anyone find my mistake?

Thank you and best regards,

Dani