6.5 Dft-based filterbanks

 Page 1 / 1

One common application of multirate processing arises in multirate, multi-channel filter banks ( [link] ).

One application is separating frequency-division-multiplexed channels. If the filters are narrowband, the output channelscan be decimated without significant aliasing.

Such structures are especially attractive when they can be implemented efficiently. For example, if the filters are simplyfrequency modulated (by $e^{-(i\frac{2\pi k}{L}n)}$ ) versions of each other, they can be efficiently implemented using FFTs!

Furthermore, there are classes of filters called perfect reconstruction filters which are of finite length but from which the signal can be reconstructed exactly (using all $M$ channels), even though the output of each channel experiences aliasing in the decimation step. These types of filterbankshave received a lot of research attention, culminating in wavelet theory and techniques.

Uniform dft filter banks

Suppose we wish to split a digital input signal into $N$ frequency bands, uniformly spaced at center frequencies ${\omega }_{k}=\frac{2\pi k}{N}$ , for $0\le k\le N-1$ . Consider also a lowpass filter $h(n)$ , $H(\omega )\approx \begin{cases}1 & \text{if \left|\omega \right|< \frac{\pi }{N}}\\ 0 & \text{otherwise}\end{cases}$ . Bandpass filters can be constructed which have the frequency response ${H}_{k}(\omega )=H(\omega +\frac{2\pi k}{N})$ from ${h}_{k}(n)=h(n)e^{-(i\frac{2\pi kn}{N})}$ The output of the $k$ th bandpass filter is simply (assume $h(n)$ are FIR)

$(x(n), {h}_{k}(n))=\sum_{m=0}^{M-1} x(n-m)h(m)e^{-(i\frac{2\pi km}{N})}={y}_{k}(n)$
This looks suspiciously like a DFT, except that $M\neq N$ , in general. However, if we fix $M=N$ , then we can compute all ${y}_{k}(n)$ outputs simultaneously using an FFT of $x(n-m)h(m)$ : The $\text{kth FFT frequency output}={y}_{k}(n)$ ! So the cost of computing all of these filter banks outputs is $O(N\lg N)$ , rather than $N^{2}$ , per a given $n$ . This is very useful for efficient implementation of transmultiplexors (FDM to TDM).

How would we implement this efficiently if we wanted to decimate the individual channels ${y}_{k}(n)$ by a factor of $N$ , to their approximate Nyquist bandwidth?

Simply step by $N$ time samples between FFTs.

Do you expect significant aliasing? If so, how do you propose to combat it? Efficiently?

Aliasing should be expected. There are two ways to reduce it:

1. Decimate by less ("oversample" the individual channels) such as decimating by a factor of $\frac{N}{2}$ . This is efficiently done by time-stepping by the appropriate factor.
2. Design better (and thus longer) filters, say of length $LN$ . These can be efficiently computed by producing only $N$ (every $L$ th) FFT outputs using simplified FFTs.

How might one convert from $N$ input channels into an FDM signal efficiently? ( [link] )

Such systems are used throughout the telephone system, satellite communication links, etc.

Use an FFT and an inverse FFT for the modulation (TDM to FDM) and demodulation (FDM to TDM), respectively.

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
what's the program
Jordan
?
Jordan
what chemical
Jordan
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!