# 0.1 Theory

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The Theory module within the Sparse Signal Reconstruction in the Presence of Noise collection.

## Motivation

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.

Introduction about quantum dots in nanotechnology
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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
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it is a goid question and i want to know the answer as well
Maciej
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for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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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
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is Bucky paper clear?
CYNTHIA
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Do you know which machine is used to that process?
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for screen printed electrodes ?
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What is lattice structure?
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or in general
Ebrahim
in general
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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
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China
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types of nano material
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many many of nanotubes
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what is the function of carbon nanotubes?
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I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
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