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Theory:cccccsConvolution .

convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function ofthe amount that one of the original functions is translated.

compute a sliding, weighted-average of function f(T), where the weighting function is g(-T).

Correlation .

Correlation is the statistical relationship between two random variables or twosets of data. It indicates a predictive relationship that can be exploited in practice.

correlation coefficients, measures the degree of correlation or linear relationship between two variables

"Pearson's correlation." is obtained by dividing the covariance of the two variables by the product of their standard deviations.

1. The Pearson correlation is +1 in the case of a perfect positive (increasing)linear relationship (correlation),2. -1 in the case of a perfect decreasing (negative) linear relationship (anticorrelation),3. and some value between -1 and 1 in all other cases, indicating the degree oflinear dependence between the variables.4. zero there is less of a relationship (closer to uncorrelated).5. The closer the coefficient is to either -1 or 1, the stronger the correlationbetween the variables.

Cross-correlation

cross-correlation is a measure of similarity of two waveforms as a function of atime-lag applied to one of them. It is commonly used for searching a long-signal for a shorter, known feature. It also has applications in pattern recognition.

1. The cross-correlation is similar in nature to the convolution of two functions.2. Correlation is always used to include a standardising factor in such a way that correlations have values between -1 and +1, and the term cross-correlation isused for referring to the correlation corr(X, Y) between two random variables X

and Y, while the "correlation" of a random vector X is considered to be the correlation matrix (matrix of correlations) between the scalar elements of X.3. consider two real valued functions f and g differing only by an unknown shiftalong the x-axis. One can use the cross-correlation to find how much g must beshifted along the x-axis to make it identical to f. The formula essentially slides the g function along the x-axis, calculating the integral of their product ateach position. When the functions match, the value of conjugate of f and g is 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

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two negative numbers is positive.4. With complex-valued functions f and g, taking the conjugate of f ensures thataligned peaks (or aligned troughs) with imaginary components will contribute positively to the integral.

(f \star g)(t)\ \stackrel{\mathrm{def}}{=} \int_{-\infty}^{\infty} f^*(\tau)\ g(t+\tau)\,d\tau,

(f \star g)[n]\ \stackrel{\mathrm{def}}{=} \sum_{m=-\infty}^{\infty} f^*[m]\g[n+m].

Autocorrelation

Autocorrelation is the cross-correlation of a signal with itself. It is the similarity between observations as a function of the time separation between them. It is a mathematical tool for finding repeating patterns, such as the presence ofa periodic signal which has been buried under noise, or identifying the missingfundamental frequency in a signal implied by its harmonic frequencies.

Given a signal f(t), the continuous autocorrelation R_{ff}(\tau) is most often defined as the continuous cross-correlation integral of f(t) with itself, at lag\tau.

When the autocorrelation function is normalized by mean and variance,, that is,without subtracting the mean and dividing by the variance, it is sometimes referred to as the autocorrelation coefficient.

Fourier transform

The Fourier transform, is a mathematical transform, transforming a mathematicalfunction of time domain, f(t), into a new function, in frequency domain with units of cycles/sec (hertz) or radians per second. The new function is then known a

s the Fourier transform and/or the frequency spectrum of the function f. The Fourier transform is also a reversible operation.

In the case of a periodic function (for example, a continuous but not necessarily sinusoidal musical sound), the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier series coefficients. Also, when a time-domain function is sampled to facilitate storage or computer-processing, it is still possible to recreate a version of the original Fourier transform according to the Poisson summation formula, also known as discrete-time Fourier transform.

Periodic functions are written as the sum of simple waves mathematically represented by sines and cosines. The Fourier transform is an extension of the Fourier

series that results when the period of the represented function is lengthened and allowed to approach infinity.

Due to the properties of sine and cosine, it is possible to recover the amplitude of each wave in a Fourier series using an integral. In many cases it is desirable to use Euler's formula, which states that e2pi?= cos(2p?) + isin(2p?), to write Fourier series in terms of the basic waves e2pi?. This has the advantage ofsimplifying many of the formulas involved, and provides a formulation for Fourier series.

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Discrete Fourier transform

It convert the sampled function from its original domain (often time or positionalong a line) to the frequency domain.

The input samples are complex numbers (in practice, usually real numbers), and the output coefficients are complex too. The frequencies of the output sinusoidsare integer multiples of a fundamental frequency, whose corresponding period isthe length of the sampling interval. The combination of sinusoids obtained through the DFT is therefore periodic with that same period. The DFT differs from thediscrete-time Fourier transform (DTFT) in that its input and output sequences are both finite; it is therefore said to be the Fourier analysis of finite-domain(or periodic) discrete-time functions.

X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-i 2 \pi k n / N}.

Discrete-time Fourier transform

the discrete-time Fourier transform (DTFT) is one of the specific forms of Fouri

er analysis that transforms one function into another, which is called the frequency domain representation, of the original function (which is often a functionin the time-domain). The DTFT requires an input function that is discrete ie sampling a continuous function, like a person's voice.

The DTFT frequency-domain representation is always a periodic function. Since one period of the function contains all of the unique information, it is sometimesconvenient to say that the DTFT is a transform to a "finite" frequency-domain (the length of one period), rather than to the entire real line.

X(\omega) = \sum_{n=-\infty}^{\infty} x[n] \,e^{-i \omega n}.

Often the x[n] sequence represents the values (aka samples) of a continuous-time

function, x(t), at discrete moments in time: t = nT, where T is the sampling interval (in seconds), and fs = 1/T is the sampling rate (samples per second). Then the DTFT provides an approximation of the continuous-time Fourier transform:

X(f) = \int_{-\infty}^\infty x(t)\cdot e^{- i 2\pi f t} dt.

Fast Fourier transform

A fast Fourier transform (FFT) is an algorithm to compute the discrete Fourier transform (DFT) and its inverse in a fast (computationally efficient) way.

By making use of periodicities in the sines that are multiplied to do the transforms, the FFT greatly reduces the amount of calculation required.1. Functionally, the FFT decomposes the set of data to be transformed into a series of smaller data sets to be transformed.2. Then, it decomposes those smaller sets into even smaller sets.3. At each stage of processing, the results of the previous stage are combined in special way.4. Finally, it calculates the DFT of each small data set.

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It turns out that it is possible to take the DFT of the first N/2 points and combine them in a special way with the DFT of the second N/2 points to produce a single N-point DFT. Each of these N/2-point DFTs can be calculated using smaller DFTs in the same way. These are combined to form N/4 4-point DFTs. The next stageproduces N/8 8-point DFTs, and so on, until a single N-point DFT is produced.

efficient is the FFT?

The DFT takes N^2 operations for N points. Since at any stage the computation required to combine smaller DFTs into larger DFTs is proportional to N, and thereare log2(N) stages (for radix 2), the total computation is proportional to N * log2(N). Therefore, the ratio between a DFT computation and an FFT computation for the same N is proportional to N / log2(n). In cases where N is small this ratio is not very significant, but when N becomes large, this ratio gets very large.(Every time you double N, the numerator doubles, but the denominator only increases by 1.)

Are FFTs limited to sizes that are powers of 2?

No. The most common and familiar FFTs are "radix 2". However, other radices aresometimes used, which are usually small numbers less than 10. For example, radix

-4 is especially attractive because the "twiddle factors" are all 1, -1, j, or -j, which can be applied without any multiplications at all.

What is an FFT "radix"?

The "radix" is the size of an FFT decomposition. In the example above, the radix was 2. For single-radix FFTs, the transform size must be a power of the radix. In the example above, the size was 32, which is 2 to the 5th power.

What are "twiddle factors"?

"Twiddle factors" are the coefficients used to combine results from a previo

us stage to form inputs to the next stage.

What is an "in place" FFT?

An "in place" FFT is simply an FFT that is calculated entirely inside its original sample memory. In other words, calculating an "in place" FFT does not require additional buffer memory (as some FFTs do.)

What is "bit reversal"?

"Bit reversal" is just what it sounds like: reversing the bits in a binary word from left to right. Therefore the MSBs become LSBs and the LSBs become MSBs.But what does that have to do with FFTs? Well, the data ordering required by ra

dix-2 FFTs turns out to be in "bit reversed" order, so bit-reversed indexes areused to combine FFT stages. It is possible (but slow) to calculate these bit-reversed indices in software; however, bit reversals are trivial when implemented in hardware. Therefore, almost all DSP processors include a hardware bit-reversalindexing capability (which is one of the things that distinguishes them from other microprocessors.)

What is "decimation-in-time" versus "decimation-in-frequency"?

FFTs can be decomposed using DFTs of even and odd points, which is called a

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Decimation-In-Time (DIT) FFT, or they can be decomposed using a first-half/second-half approach, which is called a "Decimation-In-Frequency" (DIF) FFT. Generally, the user does not need to worry which type is being used.

How do I implement an FFT?

Many good FFT implementations are available in C, Fortran and other languages, and microprocessor manufacturers generally provide free optimized FFT implementations in their processors' assembly code, Therefore, it is not so important to understand how the FFT really works, as it is to understand how to use it.

Laplace transform

The Laplace transform is a generalized Fourier transform. It allows a transformof any system or signal because it is a transform into the complex plane insteadof just the j? line like the Fourier transform. The major difference is that the Laplace transform has a region of convergence for which the transform is valid. This implies that a signal in frequency may have more than one signal in time;the correct time signal for the transform is determined by the region of convergence. If the region of convergence includes the j? axis, j? can be substituted

into the Laplace transform for s and it's the same as the Fourier transform. TheLaplace transform is:

X(s) = \int^\infty_{0^-} x(t)e^{-s t}\, dt

and the inverse Laplace transform, if all the singularities of X(s) are in the left half of the complex plane, is:

x(t)=\frac{1}{2\pi}\int^\infty_{-\infty} X(s )e^{s t}\, d s

Bode plots

Bode plots are plots of magnitude vs. frequency and phase vs. frequency for a system. The magnitude axis is in Decibel (dB). The phase axis is in either degreesor radians. The frequency axes are in a logarithmic scale. These are useful because for sinusoidal inputs, the output is the input multiplied by the value of the magnitude plot at the frequency and shifted by the value of the phase plot atthe frequency.

Radius of convergence

The radius of convergence of a power series is the radius of the largest disk inwhich the series converges. It is either a non-negative real number or 8. Whenit is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence.

Z Transform

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Z-transform converts a discrete time-domain signal, which is a sequence of realor complex numbers, into a complex frequency-domain representation.

It can be considered as a discrete-time equivalent of the Laplace transform

The Z Transform has a strong relationship to the DTFT, and is incredibly usefulin transforming, analyzing, and manipulating discrete calculus equations.

Stochastic process

a stochastic process (pronunciation: /sto?'kst?k/), or sometimes random process (widely used) is a collection of random variables; this is often used to represent the evolution of some random value, or system, over time.In the simple case ofdiscrete time, a stochastic process amounts to a sequence of random variables known as a time series. Another basic type of a stochastic process is a random field, whose domain is a region of space, in other words, a random function whosearguments are drawn from a range of continuously changing values. One approach to stochastic processes treats them as functions of one or several deterministicarguments (inputs, in most cases regarded as time) whose values (outputs) are random variables: non-deterministic (single) quantities which have certain probability distributions.