# 0.2 Fast fourier aliasing-based sparse transform

 Page 1 / 1
A brief description of the FFAST algorithm.

## Fast fourier aliasing-based sparse transform

Fast Fourier Aliasing-based Sparse Transform (FFAST) is a sparse FFT algorithm developed by Sameer Pawar and Kannan Ramchandran in May 2013 [link] . We present a formulation of the algorithm that is specialized for our digital multitone scheme.

FFAST consists of three modules: the downsampling module, the FFT module, and the peeling module. See [link] for a diagram of this architecture.

## Front end

The downsampling module consists of three stages. In each stage $i$ , the signal and a delayed version are both downsampled by a sampling coefficient, ${n}_{i}\in {\mathbb{Z}}_{+}$ . This introduces aliasing in the frequency domain, which will be a key component in the peeling module. It is necessary that the sampling coefficients ${n}_{i}$ are coprime factors of the singal length $N$ ; that is, ${n}_{0}\phantom{\rule{0.166667em}{0ex}}{n}_{1}\phantom{\rule{0.166667em}{0ex}}{n}_{2}=N$ where ${n}_{0}$ , ${n}_{1}$ , ${n}_{2}$ , are all relatively prime.

These smaller subsignals are passed to the FFT module, which computes the DFT of each subsignal. Any FFT algorithm may be used in this stage with the condition that it works for a general signal length $N$ . The DFTs of the subsignals at stage $i$ are then paired together. We denote such a pair as ${\stackrel{\to }{y}}_{l}^{\phantom{\rule{0.166667em}{0ex}}i}=\left({x}_{i}\left[l\right],{\stackrel{˜}{x}}_{i}\left[l\right]\right)$ , where ${x}_{i}\left[l\right]$ and ${\stackrel{˜}{x}}_{i}\left[l\right]$ are the ${l}_{th}$ values of the DFT of the normal and delayed subsignals of stage $i$ , respectively.

## Back end

The peeling module takes these smaller DFT pairs and backsolves a bipartite graph to obtain the DFT coefficients of the original signal. To understand the structure of this graph, recall that the aliasing caused by downsampling “mixes" frequency domain components. More precisely, the coefficients of the smaller DFTs are a linear combination of the original DFT coefficients. Consider a graph with two types of vertices: the smaller DFT pair coefficients ${\stackrel{\to }{y}}_{l}^{\phantom{\rule{0.166667em}{0ex}}i}$ and original DFT coefficients $X\left[p\right]$ . If an original DFT coefficient contributes to the value of a smaller DFT coefficient, an edge is placed between the two vertices. It is easy to see that this is a bipartite graph because the vertex set can be partitioned into smaller DFT coefficients and original DFT coefficients.

We denote a smaller DFT coefficient vertex as a zero-ton if no nodes are connected to it, a singleton if exactly one node is connected to it, and a multi-ton if it is neither a zero-ton nor a singleton.

If a vertex ${\stackrel{\to }{y}}_{l}^{\phantom{\rule{0.166667em}{0ex}}i}=\left({x}_{i}\left[l\right],{\stackrel{˜}{x}}_{i}\left[l\right]\right)$ is a pair of zeros, then it is a zero-ton. Otherwise, to determine whether a vertex is a zero-ton, a singleton, or a multi-ton, the algorithm uses a “Ratio Test"  [link] . Recall that a circular shift in the time domain is a multiplication by a complex exponential in the frequency domain so that we may use the values in ${\stackrel{\to }{y}}_{l}^{\phantom{\rule{0.166667em}{0ex}}i}$ to determine whether the vertex is a singleton. To perform this ratio test we may check if the quantity

$q=\frac{N}{2\pi }\left(\angle ,\frac{{\stackrel{˜}{x}}_{i}\left[l\right]}{{x}_{i}\left[l\right]}\right)$

is an integer. If $q$ is an integer, then the vertex in question is a singleton and thus, $X\left[q\right]={x}_{i}\left[l\right]$ ; otherwise, the vertex in question is a multi-ton.

We now describe the process of backsolving this bipartite graph to get the DFT coefficients of the original signal. If a vertex is a zero-ton, we may remove it from the graph because it provides no relevant information. If a vertex is a singleton, we have obtained a DFT coefficient $X\left[q\right]$ . By the “mixing” process of aliasing, we know which smaller DFT pairs ${\stackrel{\to }{y}}_{l}^{\phantom{\rule{0.166667em}{0ex}}i}$ that $X\left[q\right]$ contributes to. With this information, we may subtract $X\left[q\right]$ from these smaller DFT pairs, thus removing edges from the graph. We repeat these steps until all edges are removed from the graph and $X\left[q\right]$ is known completely. This process is known as peeling and is reminiscent of decoding Low Density Parity Check codes.

## Convergence conditions

In general, FFAST is a robust algorithm that can handle noise, many signal lengths $N$ , and sparsity factors $k$ , where $k$ is the number of nonzero DFT coefficients. We presented a specific noiseless version of the algorithm that requires the sparsity constraint $k<{N}^{1/3}$ . As previously mentioned, FFAST also requires that the subsampling coefficients are coprime factors of $N$ . With these conditions, FFAST is guaranteed to converge to a solution almost surely.

#### Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Using ffast to decrease computation time in digital multitone communication. OpenStax CNX. Dec 17, 2014 Download for free at http://legacy.cnx.org/content/col11731/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Using ffast to decrease computation time in digital multitone communication' conversation and receive update notifications?

 By Sandy Yamane By Joli Julianna By Katherina jennife... By Rylee Minllic By Marion Cabalfin By Madison Christian By Donyea Sweets By Stephen Voron By Keyaira Braxton By Jazzycazz Jackson