1. In the first chapter need to keep introduction of spread spectrum objective of the project, motivation of the project

2. Literature survey on the project

3. Second chapter should contain the the fhss and DSSs techniques with block diagrams

4. Third chapter contains the list of algorithms use for finding the range with more detailed manner

5. Fourth chapter should contains the results u have got

6. Fifth chapter should contains the conclusion and futer scope

7. Appendix should contain the code

CROSS CORRELATION

Insignal processing,cross-correlationis a measure of similarity of twowaveformsas a function of a time-lag applied to one of them. This is also known as aslidingdot productorsliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications inpattern recognition,single particle analysis,electron tomography,averaging,cryptanalysis, andneurophysiology.

For continuous functionsfandg, the cross-correlation is defined as:

wheref* denotes thecomplex conjugateoffandis the time lag.

Similarly, for discrete functions, the cross-correlation is defined as:

Visual comparison ofconvolution, cross-correlation andautocorrelation.

The cross-correlation is similar in nature to theconvolutionof two functions.

In anautocorrelation, which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero unless the signal is a trivial zero signal.

Inprobabilityandstatistics, the termcross-correlationsis used for referring to thecorrelationsbetween the entries of tworandom vectorsXandY, while theautocorrelationsof a random vectorXare considered to be thecorrelationsbetween the entries ofXitself, those forming thecorrelation matrix(matrix of correlations) ofX. This is analogous to the distinction betweenautocovarianceof a random vector andcross-covarianceof two random vectors. One more distinction to point out is that inprobabilityandstatisticsthe definition ofcorrelationalways includes a standardising factor in such a way that correlations have values between 1 and +1.

Ifandare twoindependentrandom variableswithprobability density functionsfandg, respectively, then the probability density of the differenceis formally given by the cross-correlation (in the signal-processing sense); however this terminology is not used in probability and statistics. In contrast, theconvolution(equivalent to the cross-correlation off(t) andg(t) ) gives the probability density function of the sum.

Explanation[edit]

As an example, consider two real valued functionsanddiffering only by an unknown shift along the x-axis. One can use the cross-correlation to find how muchmust be shifted along the x-axis to make it identical to. The formula essentially slides thefunction along the x-axis, calculating the integral of their product at each position. When the functions match, the value ofis maximized. This is because when peaks (positive areas) are aligned, they make a large contribution to the integral. Similarly, when troughs (negative areas) align, they also make a positive contribution to the integral because the product of two negative numbers is positive.

Withcomplex-valued functionsand, taking theconjugateofensures that aligned peaks (or aligned troughs) with imaginary components will contribute positively to the integral.

Ineconometrics, lagged cross-correlation is sometimes referred to as cross-autocorrelation.[1]Properties[edit]

The cross-correlation of functionsf(t) andg(t) is equivalent to theconvolutionoff*(t) andg(t). I.e.:

IffisHermitian, then Analogous to theconvolution theorem, the cross-correlation satisfies:

wheredenotes theFourier transform, and an asterisk again indicates the complex conjugate. Coupled withfast Fourier transformalgorithms, this property is often exploited for the efficient numerical computation of cross-correlations. (seecircular cross-correlation)

The cross-correlation is related to thespectral density. (seeWienerKhinchin theorem)

The cross correlation of a convolution offandhwith a functiongis the convolution of the cross-correlation offandgwith the kernelh:

Time series analysis[edit]

Intime series analysis, as applied instatisticsandsignal processing, the cross-correlation between two time series describes the normalized cross-covariance function.

Letrepresent a pair ofstochastic processesthat are jointlywide sense stationary. Then the cross-covariance and the cross-correlation are given by

cross-covariance

cross-correlation

whereandare the mean and variance of the process, which are constant over time due to stationarity; and similarly for, respectively.The cross-correlation of a pair of jointlywide sense stationarystochastic processcan be estimated by averaging the product of samples measured from one process and samples measured from the other (and its time shifts). The samples included in the average can be an arbitrary subset of all the samples in the signal (e.g., samples within a finite time window or asub-samplingof one of the signals). For a large number of samples, the average converges to the true cross-correlation.

Time delay analysis[edit]

Cross-correlationsare useful for determining the time delay between two signals, e.g. for determining time delays for the propagation of acoustic signals across a microphone array.[2]

HYPERLINK "http://en.wikipedia.org/wiki/Cross-correlation" \l "cite_note-3" [3]After calculating thecross-correlationbetween the two signals, the maximum (or minimum if the signals are negatively correlated) of the cross-correlation function indicates the point in time where the signals are best aligned, i.e. the time delay between the two signals is determined by the argument of the maximum, orarg maxof thecross-correlation, as in

Normalized cross-correlation[edit]

For image-processing applications in which the brightness of the image and template can vary due to lighting and exposure conditions, the images can be first normalized. This is typically done at every step by subtracting the mean and dividing by thestandard deviation. That is, the cross-correlation of a template,with a subimageis

.

whereis the number of pixels inand,is the average offandisstandard deviationoff. Infunctional analysisterms, this can be thought of as the dot product of twonormalized vectors. That is, if

and

then the above sum is equal to

whereis theinner productandis theL norm. Thus, iffandtare real matrices, their normalized cross-correlation equals the cosine of the angle between the unit vectorsFandT, being thus1if and only ifFequalsTmultiplied by a positive scalar.

Normalized correlation is one of the methods used fortemplate matching, a process used for finding incidences of a pattern or object within an image. It is also the 2-dimensional version ofPearson product-moment correlation coefficient.

Nonlinear systems[edit]

Caution must be applied when using cross correlation for nonlinear systems. In certain circumstances, which depend on the properties of the input, cross correlation between the input and output of a system with nonlinear dynamics can be completely blind to certain nonlinear effects.[4]This problem arises because some moments can go to zero and this can incorrectly suggest that there is little correlation between two signals when in fact the two signals are strongly related by nonlinear dynamics.

An Introduction to Spread-Spectrum Communications

Introduction

As spread-spectrum techniques become increasingly popular, electrical engineers outside the field are eager for understandable explanations of the technology. There are books and websites on the subject, but many are hard to understand or describe some aspects while ignoring others (e.g., the DSSS technique with extensive focus on PRN-code generation).

The following discussion covers the full spectrum (pun intended).

A Short History

Spread-spectrum communications technology was first described on paper by an actress and a musician! In 1941 Hollywood actress Hedy Lamarr and pianist George Antheil described a secure radio link to control torpedos. They received U.S. Patent #2.292.387. The technology was not taken seriously at that time by theU.S.Army and was forgotten until the 1980s, when it became active. Since then the technology has become increasingly popular for applications that involve radio links in hostile environments.

Typical applications for the resulting short-range data transceivers include satellite-positioning systems (GPS), 3G mobile telecommunications, W-LAN (IEEE 802.11a, IEEE 802.11b, IEEE 802.11g), and Bluetooth. Spread-spectrum techniques also aid in the endless race between communication needs and radio-frequency availabilitysituations where the radio spectrum is limited and is, therefore, an expensive resource.

Theoretical Justification for Spread Spectrum

Spread-spectrum is apparent in the Shannon and Hartley channel-capacity theorem:

C = B log2(1 + S/N)(Eq. 1)

In this equation, C is the channel capacity in bits per second (bps), which is the maximum data rate for a theoretical bit-error rate (BER). B is the required channel bandwidth in Hz, and S/N is the signal-to-noise power ratio. To be more explicit, one assumes that C, which represents the amount of information allowed by the communication channel, also represents the desired performance. Bandwidth (B) is the price to be paid, because frequency is a limited resource. The S/N ratio expresses the environmental conditions or the physical characteristics (i.e., obstacles, presence of jammers, interferences, etc.).

There is an elegant interpretation of this equation, applicable for difficult environments, for example, when a low S/N ratio is caused by noise and interference. This approach says that one can maintain or even increase communication performance (high C) by allowing or injecting more bandwidth (high B), even when signal power is below the noise floor. (The equation does not forbid that condition!)

Modify Equation 1 by changing the log base from 2 to e (the Napierian number) and by noting that ln = loge. Therefore:

C/B = (1/ln2) ln(1 + S/N) = 1.443 ln(1 + S/N)(Eq. 2)

Applying the MacLaurin series development for

ln(1 + x) = x - x/2 + x/3 - x4/4 + ... + (-1)k+1xk/k + ...:

C/B = 1.443 (S/N - 1/2 (S/N) + 1/3 (S/N) - ...)(Eq. 3)

S/N is usually low for spread-spectrum applications. (As just mentioned, the signal power density can even be below the noise level.) Assuming a noise level such that S/N