So, let us start with ah a a new topic here
in multi rate signal processing itself, which
is digital filter banks. A digital filter
bank is a collection of filters with a common
input and a common output . So, the idea is
very simple here, we have a discrete time
signal and we are filtering it through a bank
of filters and that is why it is called a
filter bank because it is a bank of filters
. So, this is H naught of Z which is H 1 of
Z dot dot dot till H m minus 1 of Z.
And let us designate the signal is x naught
of n this is x 1 of n so on till x m minus
1 of n. To these bank or filters they are
called the analysis bank, where they called
analysis bank of filters because you are taking
the signal x of n and you analysing this signal
x of n through these filters right this could
be perhaps h H naught H naught could be basically
your low pass filter H m minus 1 could be
your high pass filter and all of these other
filters could be possibly be band pass versions
right.
So, this is basically analysis analysis bank;
and these signal x naught of n x 1 of n. So,
ion till x m minus 1 of n they are called
sub band signals . So, very important they
are called sub band signals and the name is
pretty intuitive and obvious there called
band because you are filtering them in different
frequency bands and this is why they are called
sub band filters.
So, like the way we have this analysis bank
one can think of synthesis bank .
So, which is again pretty straightforward.
So, we have signals y y naught of n, y 1 of
n so on 
and the past through some sort of filters
these are basically synthesis filters, and
our hope is we can reconstruct x back right.
I mean you can think about is this as basically
you are sab band signals 
and these sab band filters or filtered through
a bank of synthesis filters, which is termed
as synthesis bank and then you basically sum
the output of all the signals coming out of
the synthesis filters and hopefully you get
your reconstructed signal ok. This is a very
crude, crude crude form there are a lot of
details which we will explore ah during our
journey through this model.
So, a question that naturally comes to us
is, what could be the nature of frequency
responses for H i of Z and this is a question
that you would probably asked to yourself.
So, H naught could basically be the low pass
version, H 1 could be some band pass version
H 2 can be probably ah next version of this
filter basically with some frequency translation.
As you can see this is basically overlapping,
there is a spectral overlap. An alternative
way to think about these filters could be
to treat them as 
non-overlapping .
We may wonder what is the connection to having
overlapping frequency responses versus non-overlapping
frequency responses right. So, if if you are
to think about having filters that have non-overlapping
responses it is beneficial for us, because
there is you know aliasing that you are introducing.
So, therefore, when you are trying to reconstruct
these things by passing them through these
filters there overlapping the spectral domain.
So, therefore, you will get a clean signal.
But you think about the kv hat for of this
of this structure, you will have to really
design filters that have very steep responses,
because you really have to notch and null
them off at certain points in the frequency
spectrum right and this is a a a critical
parameter because if you have to have such
sharp frequency ah responses of the transitions
then your filter order just goes up. And if
you just look at the formula from bellanger
I mean this is the empirical formula is inversely
filter order is inversely related to the normalised
transition bandwidth right in the if if the
response is to be very steep, then you know
you are you are order really jacks up.
So, this is 1 thing. So, it is easy to visualize
conceptually having filters that are non-overlapping
in the frequency responses, it helps us overcome
aliasing and all these things, but we have
a problem because orders can be quite high.
But if I have aliasing I mean if I if I bring
in this overlapping nature of frequency responses,
then ok my filter order is less etcetera I
mean I am able to tolerate because I can I
can have a more steeper curt in this in this
transition band, in the in the transition
bandwidth I can basically have a more gradual
slope right ah it is basically less stringent
in terms of my steepness right .
So; however, that would cause aliasing and
I have to overcome this alias better, through
some means in the process of doing this ah
this this filtering through a bank of analysis
filters, followed by synthesis filters and
when I call this is a filter bank and have
to do something to this filter bank to overcome
aliasing errors; introduce because of overlapping
frequency responses. So, I think the the message
is very clear.
So, non-overlapping means, steeper ah frequency
ah transitions, frequency transition ah bandwidth
slope is a transition ah that is a transition
bandwidth and if you look at the slope of
the transition bandwidth it is it is steep,
and here is more relaxed slope in the transition
bandwidth region, which means ah it is easy
to design filters , but aliasing is an issue
and in this case filter orders could be prohibited
and no aliasing ok. Here filter orders could
be prohibited, but no aliasing here because
aliasing, but easy to design the filters.
So, there is some compromise that we will
have to do.
But fortunately we have solutions when we
have designed digital filter banks, we can
overcome these aliasing errors and have overlapping
frequency response, but the design our synthesis
filters somehow to cancel aliasing using our
multi rate operations. So, I think that is
the whole idea ah in in in in the design of
this filter banks ok. And you may wonder why
even should be take a signal and decompose
this into analysis and synthesis filter banks.
That means, we have a signal, we have an analysis
bank let us say we have the sub band signals,
and we take the sub band signals passed them
through a synthesis bank and we get back 
our reconstructed signal . So, sum x hat of
n this is sum x of n right at this step, which
is very important there is a transmission
that is happening over a communication channel.
When we say we are transmitting over a communication
channel, this means that we are claimed with
a bandwidth of the channel in some sense right
and if we have stringent constraints on the
bandwidth of the channel, then we may have
to possibly compress the signal right.
Now, we have different frequency components
that get filtered as we go through this analysis
bank, I mean this could be essentially a low
pass version and all of these could be band
pass BP 1 dot dot dot some band pass say n
and this could be our high pass right. And
depending upon the information content in
each of the bands I may have 2 choose my bit
rate in such a way that I can throw away certain
samples in each of the banks; that means,
I can adjust my sampling rate in the sub band
signals and ah basically quantized and do
anything that I want to do to compress and
again I restore this sampling rate in the
process of web sampling then I just get a
set of signals here at the input of the synthesis
bank and then I try to reconstruct the signal.
So, the journey here is under what conditions
can I do perfect reconstruction. So, if I
do not do quantisation and I do down sampling
here at the output of the analysis bank followed
by up sampling, can I reconstruct perfectly,
what are the conditions for filtering ah or
choosing ah you know, how do I choose my analysis
filters and synthesis filters to overcome
errors due to aliasing due to errors in you
know phase distortion possibly magnitude distortion,
in the frequency response etcetera etcetera
right
So, these are all various questions 1 would
get, but I think you have to appreciate here
that this is this practical because ah one
1 when one thinks about ah sending the sub
band signals over a communication channel
then we will have to basically work with the
limitations of the channel, and that is the
real channel challenge here how to design
this this filter banks ok.
So, with this we will stop here, we will look
into the discrete fourier transform or the
DFT as a filter bank, we will consider ah
d DFT as a case study and and then we will
generalise this ah towards other other filter
banks ok we stop here.
