Two-Channel Critically Sampled Filter Banks
Figure 11.15 shows a simple two-channel band-splitting filter bank,
followed by the corresponding synthesis filter bank which
reconstructs the original signal (we hope) from the two channels. The
analysis filter
is a half-band lowpass filter, and
is a complementary half-band highpass filter. The synthesis filters
and
are to be derived. Intuitively, we expect
to be a lowpass that rejects the upper half-band due to the
upsampler by 2, and
should do the same but then also
reposition its output band as the upper half-band, which can be
accomplished by selecting the upper of the two spectral images in the
upsampler output.
The outputs of the two analysis filters in Fig.11.15 are
![]() |
(12.16) |
Using the results of §11.1, the signals become, after downsampling,
![]() |
(12.17) |
After upsampling, the signals become
![]() |
![]() |
![]() |
|
![]() |
![]() |
After substitutions and rearranging, we find that the output
![$ \hat{x}$](http://www.dsprelated.com/josimages_new/sasp2/img1992.png)
For perfect reconstruction, we require the aliasing term to be zero. For ideal half-band filters cutting off at
![$ \omega=\pi/2$](http://www.dsprelated.com/josimages_new/sasp2/img1233.png)
![$ F_0=H_0$](http://www.dsprelated.com/josimages_new/sasp2/img1997.png)
![$ F_1=H_1$](http://www.dsprelated.com/josimages_new/sasp2/img1998.png)
![$ H_0(-z)F_0(z)=H_1(z)H_0(z)=0$](http://www.dsprelated.com/josimages_new/sasp2/img1999.png)
![$ H_1(-z)F_1(z)=H_0(z)H_1(z)=0$](http://www.dsprelated.com/josimages_new/sasp2/img2000.png)
In this case, synthesis filter
![$ F_0(z)$](http://www.dsprelated.com/josimages_new/sasp2/img1984.png)
![$ \pi$](http://www.dsprelated.com/josimages_new/sasp2/img192.png)
![$ z$](http://www.dsprelated.com/josimages_new/sasp2/img3.png)
![$ F_1(z)$](http://www.dsprelated.com/josimages_new/sasp2/img1985.png)
![$ \pi$](http://www.dsprelated.com/josimages_new/sasp2/img192.png)
![$ H_0(z)$](http://www.dsprelated.com/josimages_new/sasp2/img1982.png)
![$ F_0$](http://www.dsprelated.com/josimages_new/sasp2/img58.png)
![$ F_1$](http://www.dsprelated.com/josimages_new/sasp2/img998.png)
![$ H_0$](http://www.dsprelated.com/josimages_new/sasp2/img2005.png)
![$ H_1$](http://www.dsprelated.com/josimages_new/sasp2/img2006.png)
Referring again to (11.18), we see that we also need the
non-aliased term to be of the form
where
![$ A(z)$](http://www.dsprelated.com/josimages_new/sasp2/img1756.png)
![]() |
(12.21) |
That is, for perfect reconstruction, we need, in addition to aliasing cancellation, that the non-aliasing term reduce to a constant gain
![$ g$](http://www.dsprelated.com/josimages_new/sasp2/img2010.png)
![$ d$](http://www.dsprelated.com/josimages_new/sasp2/img2011.png)
Let
denote
. Then both constraints can be expressed in
matrix form as follows:
![]() |
(12.22) |
Substituting the aliasing-canceling choices for
and
from
(11.19) into the filtering-cancellation constraint (11.20), we
obtain
The filtering-cancellation constraint is almost satisfied by ideal zero-phase half-band filters cutting off at
![$ \pi/2$](http://www.dsprelated.com/josimages_new/sasp2/img2016.png)
![$ H_1(-z)=H_0(z)$](http://www.dsprelated.com/josimages_new/sasp2/img2017.png)
![$ H_0(-z)=H_1(z)$](http://www.dsprelated.com/josimages_new/sasp2/img2018.png)
![$ \omega=\pi/2$](http://www.dsprelated.com/josimages_new/sasp2/img1233.png)
Next Section:
Amplitude-Complementary 2-Channel Filter Bank
Previous Section:
Multirate Noble Identities