<< Chapter < Page Chapter >> Page >
The Theory module within the Sparse Signal Reconstruction in the Presence of Noise collection.



In theory, we should never have to 'recover' a signal – it should merely pass from one location to another, undisturbed. However, all real-world signals pass through the infamous “channel” – a path between the transmitter and the receiver that includes a variety of hazards, including attenuation , phase shift , and, perhaps most insidiously, noise . Nonetheless, we depend upon precise signal transmission daily – in our watches, computer networks, and advanced defense systems. Therefore, the field of signal processing concerns itself not only with the deployment of a signal, but also with its recovery in the most efficient and most accurate manner.

Types of noise

Noise takes many forms. The various 'colors' of noise are used to refer to the different power spectral density curves that types of noise exhibit. For example, the power density of pink noise falls off at 10dB per decade. The power density spectrum of pink noise is flat in logarithmic space. The most common type of noise, however, is white noise . White noise exhibits a flat power density spectrum in linear space. In many physical process (and in this report), we deal primarily with Additive White Gaussian Noise – abbreviated AWGN . As a reminder, the Gaussian distribution has the following PDF (Probability Density Function):

μ is the mean; σ 2 ≥ 0 is the variance.

Sparse signals

An additional constraint we imposed upon our input signals was that they were required to be sparse . A signal that is sparse in a given basis can be reconstructed using a small number of the basis vectors in that basis. In the standard basis for R n , for example, the signal (1,0,0,0,...,0) would be as sparse as possible – it requires only the basis vector e 1 for reconstruction (in fact, e 1 is the signal!). By assuming that the original signals are sparse, we are able to employ novel recovery methods and minimize computation time.

Typical reconstruction approaches

We have a number of choices for the recovery of sparse signals. As a first idea, we could “ optimally select ” the samples we use for our calculations from the signal. However, this is a complicated and not always fruitful process.

Another approach is Orthogonal Matching Pursuit (OMP) . OMP essentially involves projecting a length-n signal into the space determined by the span of a k-component “nearly orthonormal” basis (a random array of 1/sqrt(n) and (-1)/sqrt(n) values). Such a projection is termed a Random Fourier Projection . Entries in the projection that do not reach a certain threshold are assigned a value of zero. This computation is iterated and the result obtained is an approximation of the original sparse signal. Unfortunately, OMP itself can be fairly complicated, as the optimal basis is often a wavelet basis. Wavelets are frequency “packets” - that is, localized in both time and frequency; in contrast, the Fourier transform is only localized in frequency.

Signal reconstruction: our method

The fundamental principle for our method of signal analysis is determining where the signal is not, rather than finding where it is. This information is stored in a mask that, when multiplied with the running average of the signal, will provide the current approximation of the signal. This mask is built up by determining whether a given value in the signal is above a threshold, which is determined by the standard deviation of the noise; if so, the value is most likely a signal element. This process is repeated until the signal expected is approximately equal to a signal stored in a library on the device. While this operation is naturally more noticeable at each iteration with sparse signals, even for non-sparse signals the only limiting factor is the minimum value of the signal. For reasons of application, the primary limiting factor is the number of samples required to recover the signal. This is because the raw mathematical operations take fractions of a second to a few seconds to execute (which is more than enough for conventional applications). The signal itself may be transmitted for a very short period; the requisite number of samples must be garnered before transmission halts. Further, given an arbitrary amount of computation time, our algorithm can reconstruct a sparse signal contaminated with any level of AWGN – there is no mathematical limit on the recovery process. This is an impressive and surprising feat.

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Sparse signal recovery in the presence of noise. OpenStax CNX. Dec 14, 2009 Download for free at http://cnx.org/content/col11144/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Sparse signal recovery in the presence of noise' conversation and receive update notifications?