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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

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
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I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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Smarajit Reply
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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
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