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Discrete Fourier transformFast Fourier transform& their application in

Signal Processing

A presentation bySujoy ketan Saha

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DFT Definition: The sequence ofN complex numbers x0, ..., xN1 is

transformed into the sequence ofN complex numbers X0, ..., X

N

1by the DFT according to the formula

[1]:

The inverse discrete Fourier transform (IDFT) is given by

[2]:

NB:1.the normalization factor multiplying the DFT and IDFT (here 1 and 1/N)and the signs of the exponents are merely conventions.

2.A normalization of for both the DFT and IDFT makes thetransforms unitary, which has some theoretical advantages.

3.The convention of a negative sign in the exponent is often convenientbecause it means that X

k

is the amplitude of a "positive frequency" 2k/N. Equivalently, the DFT is often thought of as a matched filter: whenlooking for a frequency of +1, one correlates the incoming signal with afrequency of1.

http://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Matched_filterhttp://en.wikipedia.org/wiki/Matched_filterhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Complex_number

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DFT: Properties:1. Completeness :

The discrete Fourier transform is an invertible, linear transformation

With denoting the set ofcomplex numbers. In other words, for anyN > 0, an n-dimensional complex vector has a DFT and an IDFTwhich are in turn n-dimensional complex vectors.

2. Orthogonality :The vectors form an orthogonal basis over the set of N-

dimensional complex vectors:

where is the Kronecker delta. This orthogonality condition canbe used to derive the formula for the IDFT from the definition of

the DFT.

http://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Kronecker_deltahttp://en.wikipedia.org/wiki/Kronecker_deltahttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Linear_transformation

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DFT: Properties:3. Periodicity :

If the expression that defines the DFT is evaluated for all integers kinstead of just for then the resulting infinite

sequence is a periodic extension of the DFT, periodic with period N.

The periodicity can be shown directly from the definition:

where we have used the fact that e 2i = 1. In the same way it can

be shown that the IDFT formula leads to a periodic extension.

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DFT: Properties:4. The shift theorem :

Multiplying xn by a linear phase for some integerm

corresponds to a circular shiftof the output Xk: X

kis replaced by

Xkm, where the subscript is interpreted modulo N(i.e. periodically).

Similarly, a circular shift of the input xn

corresponds to multiplying

the output Xk

by a linear phase. Mathematically, if {xn} represents the

vector x then

If

then

And

Where, the transform [DFT] is denoted by the symbol , as in

http://en.wikipedia.org/wiki/Modulohttp://en.wikipedia.org/wiki/Modulo

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DFT: Properties:5. Circular convolution theorem and cross-correlation

theorem :The cyclic or circular convolution x*y of the two

vectorsx = xk and y = yn is the vector x*y with components

where we continue y cyclically so that

The DFT turns cyclic convolutions into component-wisemultiplication.

That is, if , then

where capital letters (X, Y, Z) represent the DFTs of sequencesrepresented by small letters (x, y, z).

http://en.wikipedia.org/wiki/Circular_convolutionhttp://en.wikipedia.org/wiki/Circular_convolution

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DFT: Properties:

Circular convolution theorem and cross-correlationtheorem (contd.) :

NB:if a different normalization convention is adopted for the DFT

(e.g., the unitary normalization), then there will in general be aconstant factor multiplying the above relation

The direct evaluation of the convolution summation, above,would require O(N2) operations, but the DFT (via an FFT)provides an O(NlogN) method to compute the same thing.

It can be shown that ifzn is the cross-correlation ofxn and yn:

where the sum is again cyclic in m, then the discrete Fouriertransform ofzn is:

where capital letters are again used to signify the discrete Fouriertransform.

http://en.wikipedia.org/wiki/Cross-correlationhttp://en.wikipedia.org/wiki/Cross-correlation

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DFT: Properties:6.The unitary DFT :

Another way of looking at the DFT is to note that in the above

discussion, the DFT can be expressed as a Vandermonde matrix:

Where: is a primitive Nth root of unity.

The inverse transform is then given by the inverse of the above

matrix:

With unitary normalization constants the DFT becomes a

unitary transformation, defined by a unitary matrix: ;

; .

http://en.wikipedia.org/wiki/Vandermonde_matrixhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Unitary_operatorhttp://en.wikipedia.org/wiki/Unitary_transformationhttp://en.wikipedia.org/wiki/Unitary_transformationhttp://en.wikipedia.org/wiki/Unitary_operatorhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Vandermonde_matrix

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DFT: Properties:7. Expressing the inverse DFT in terms of the DFT :

Can be easily done via several well-known "tricks.1st: we can compute the inverse DFT by reversing the inputs:

2nd:one can also conjugate the inputs and outputs:

3rd:a variant of this conjugation trick, which is sometimes preferable becauseit requires no modification of the data values, involves swapping real andimaginary parts (which can be done on a computer simply by modifyingpointers). Define swap(xn) as xn with its real and imaginary parts

swappedthat is, ifxn = a + b i then swap(xn) is b + ai. Equivalently,swap(xn) equals.

then :

i.e. the inverse transform is the same as the forward transform with the realand imaginary parts swapped for both input and output, up to anormalization

http://en.wikipedia.org/wiki/Pointerhttp://en.wikipedia.org/wiki/Pointer

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DFT: Properties:8. Eigenvalues and eigenvectors:

The eigenvalues of the DFT matrix are simple and well-known,whereas the eigenvectors are complicated, not unique, and are the

subject of ongoing research.

Consider the unitary form defined above for the DFT of length N,

where

This matrix satisfies the equation:

operating twice gives the original data in reverse order, so

operating four times gives back the original data and is thusthe identity matrix. This means that the eigenvalues satisfy acharacteristic equation: 4 = 1.

SO,the eigenvalues of are the fourth roots of unity:

is +1, 1, +i, or i.

http://en.wikipedia.org/wiki/Eigenvaluehttp://en.wikipedia.org/wiki/Eigenvectorhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Characteristic_equationhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Characteristic_equationhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Eigenvectorhttp://en.wikipedia.org/wiki/Eigenvalue

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DFT:Properties:

Eigenvalues and eigenvectors (Contd):

Since there are only four distinct eigenvalues for this

matrix, they have some multiplicity. The multiplicity gives thenumber oflinearly independent eigenvectors corresponding toeach eigenvalues.

The multiplicity depends on the value ofN modulo 4

http://en.wikipedia.org/wiki/Algebraic_multiplicityhttp://en.wikipedia.org/wiki/Linearly_independenthttp://en.wikipedia.org/wiki/Modulohttp://en.wikipedia.org/wiki/Modulohttp://en.wikipedia.org/wiki/Linearly_independenthttp://en.wikipedia.org/wiki/Algebraic_multiplicity

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DFT: Properties: The real-input DFT:

If are real numbers, as they often are in practicalapplications, then the DFT obeys the symmetry:

where the star denotes complex conjugation and the subscripts are

interpreted modulo N.

Therefore, the DFT output for real inputs is half redundant, and oneobtains the complete information by only looking at roughly halfof the outputs . In this case, the "DC" element X0 is purely real,

and for evenN

the "Nyquist" elementX

N/ 2 is also real, so thereare exactly N non-redundant real numbers in the first half +Nyquist element of the complex output X.

Using Euler's formula, the interpolating trigonometric polynomial

can then be interpreted as a sum of sine and cosine functions.

http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Euler%27s_formula_in_complex_analysishttp://en.wikipedia.org/wiki/Euler%27s_formula_in_complex_analysishttp://en.wikipedia.org/wiki/Real_number

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Generalized/shifted DFT: It is possible to shift the transform sampling in time and/or

frequency domain by some real shifts a and b, respectively.This is known as generalized DFT (or GDFT), also called the

shifted DFT or offset DFT , and has analogous properties to theordinary DFT:

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Generalized/shifted DFT: Most often, shifts of 1 / 2 (half a sample) are used. While the

ordinary DFT corresponds to a periodic signal in both time andfrequency domains, a = 1 / 2 produces a signal that is anti-periodic in frequency domain (Xk + N = Xk) and vice-versa for b= 1 / 2. Thus, the specific case ofa = b = 1 / 2 is known as anodd-time odd-frequency discrete Fourier transform (or O2 DFT).

Such shifted transforms are most often used forsymmetric data, to represent different boundarysymmetries, and for real-symmetric data they correspondto different forms of the discrete cosine and sinetransforms.

Another interesting choice is a = b = (N 1) / 2, which iscalled the centered DFT (or CDFT). The centered DFT has theuseful property that, when N is a multiple of four, all four of itseigenvalues have equal multiplicities

http://en.wikipedia.org/wiki/Discrete_cosine_transformhttp://en.wikipedia.org/wiki/Discrete_sine_transformhttp://en.wikipedia.org/wiki/Discrete_sine_transformhttp://en.wikipedia.org/wiki/Discrete_cosine_transform

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Applications of DFT:

The DFT has seen wide usage across a large number of fields :

Spectral analysis,

Data compression,

Partial differential equations,

Multiplication of large integers,

Outline of DFT polynomial multiplication algorithm.

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FFT: A Fast Fourier Transform (FFT) is an efficient algorithm to

compute the discrete Fourier transform (DFT) and its inverse.

Let x0, ...., xN-1 be complex numbers. The DFT is defined by theformula :

Evaluating these sums directly would take O(N2) arithmeticaloperations.

An FFT is an algorithm to compute the same result in only O(N logN) operations. In general, such algorithms depend upon thefactorization ofN, but (contrary to popular misconception) thereare O(N log N) FFTs for all N, even prime N.

http://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Discrete_Fourier_transformhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Factorizationhttp://en.wikipedia.org/wiki/Prime_numberhttp://en.wikipedia.org/wiki/Prime_numberhttp://en.wikipedia.org/wiki/Factorizationhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Discrete_Fourier_transformhttp://en.wikipedia.org/wiki/Algorithm

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FFT: Many FFT algorithms only depend on the fact that is a

primitive root of unity, and thus can be applied to analogoustransforms over any finite field, such as number-theoretictransforms.

Since the inverse DFT is the same as the DFT, but with theopposite sign in the exponent and a 1/N factor, any FFTalgorithm can easily be adapted for it as well.

http://en.wikipedia.org/wiki/Primitive_root_of_unityhttp://en.wikipedia.org/wiki/Finite_fieldhttp://en.wikipedia.org/wiki/Number-theoretic_transformhttp://en.wikipedia.org/wiki/Number-theoretic_transformhttp://en.wikipedia.org/wiki/Number-theoretic_transformhttp://en.wikipedia.org/wiki/Number-theoretic_transformhttp://en.wikipedia.org/wiki/Number-theoretic_transformhttp://en.wikipedia.org/wiki/Number-theoretic_transformhttp://en.wikipedia.org/wiki/Number-theoretic_transformhttp://en.wikipedia.org/wiki/Finite_fieldhttp://en.wikipedia.org/wiki/Primitive_root_of_unity

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The Cooley-Tukey algorithm:

By far the most common FFT is the Cooley-Tukey algorithm. This is adivide and conquer algorithm that recursively breaks down a DFT of any

composite size N = N1N2 into many smaller DFTs of sizes N1 and N2,along with O(N) multiplications by complex roots of unity traditionallycalled twiddle factors (after Gentleman and Sande, 1966).

This method (and the general idea of an FFT) was popularized by apublication ofJ. W. Cooley and J. W. Tukey in 1965, but it was later

discovered that those two authors had independently re-invented analgorithm known to Carl Friedrich Gauss around 1805 (and subsequentlyrediscovered several times in limited forms).

The most well-known use of the Cooley-Tukey algorithm is to divide thetransform into two pieces of size N/ 2 at each step, and is therefore

limited to power-of-two sizes, but any factorization can be used ingeneral (as was known to both Gauss and Cooley/Tukey). These arecalled the radix-2 and mixed-radix cases, respectively (and othervariants such as the split-radix FFT have their own names as well).Although the basic idea is recursive, most traditional implementationsrearrange the algorithm to avoid explicit recursion.

http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithmhttp://en.wikipedia.org/wiki/Divide_and_conquer_algorithmhttp://en.wikipedia.org/wiki/Recursionhttp://en.wikipedia.org/wiki/Composite_numberhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Twiddle_factorhttp://en.wikipedia.org/wiki/J._W._Cooleyhttp://en.wikipedia.org/wiki/J._W._Tukeyhttp://en.wikipedia.org/wiki/1965http://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/1805http://en.wikipedia.org/wiki/Split-radix_FFThttp://en.wikipedia.org/wiki/Split-radix_FFThttp://en.wikipedia.org/wiki/1805http://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/1965http://en.wikipedia.org/wiki/J._W._Tukeyhttp://en.wikipedia.org/wiki/J._W._Cooleyhttp://en.wikipedia.org/wiki/Twiddle_factorhttp://en.wikipedia.org/wiki/Roots_of_unityhttp://en.wikipedia.org/wiki/Composite_numberhttp://en.wikipedia.org/wiki/Recursionhttp://en.wikipedia.org/wiki/Divide_and_conquer_algorithmhttp://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm

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Other FFT algorithms:

Pr im e- fac to r FFT a lgor i t hm ,

Br u u n ' s FFT a lg o r i t h m , Rader 's FFT algo r i t h m ,

Bluest e in ' s FFT a lgo r i t hm .

http://en.wikipedia.org/wiki/Prime-factor_FFT_algorithmhttp://en.wikipedia.org/wiki/Bruun%27s_FFT_algorithmhttp://en.wikipedia.org/wiki/Rader%27s_FFT_algorithmhttp://en.wikipedia.org/wiki/Bluestein%27s_FFT_algorithmhttp://en.wikipedia.org/wiki/Bluestein%27s_FFT_algorithmhttp://en.wikipedia.org/wiki/Rader%27s_FFT_algorithmhttp://en.wikipedia.org/wiki/Bruun%27s_FFT_algorithmhttp://en.wikipedia.org/wiki/Prime-factor_FFT_algorithm

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Digital signal processing:

Digital signal processing (DSP) is the study ofsignals in a

digital representation and the processing methods of thesesignals. DSP inculdes subfields like: audio signal processing,control engineering, digital image processing and speechprocessing. RADAR Signal processing and communications signalprocessing are two other important subfields of DSP.

Since the goal of DSP is usually to measure or filter continuousreal-world analog signals, the first step is usually to convert thesignal from an analog to a digital form, by using an analog todigital converter. Often, the required output signal is another

analog output signal, which requires a digital to analogconverter.

http://en.wikipedia.org/wiki/Signal_%28information_theory%29http://en.wikipedia.org/wiki/Digitalhttp://en.wikipedia.org/wiki/Audio_signal_processinghttp://en.wikipedia.org/wiki/Control_engineeringhttp://en.wikipedia.org/wiki/Digital_image_processinghttp://en.wikipedia.org/wiki/Speech_processinghttp://en.wikipedia.org/wiki/Speech_processinghttp://en.wikipedia.org/wiki/Speech_processinghttp://en.wikipedia.org/wiki/Speech_processinghttp://en.wikipedia.org/wiki/Speech_processinghttp://en.wikipedia.org/wiki/Speech_processinghttp://en.wikipedia.org/wiki/Analog_to_digital_converterhttp://en.wikipedia.org/wiki/Analog_to_digital_converterhttp://en.wikipedia.org/wiki/Digital_to_analog_converterhttp://en.wikipedia.org/wiki/Digital_to_analog_converterhttp://en.wikipedia.org/wiki/Digital_to_analog_converterhttp://en.wikipedia.org/wiki/Digital_to_analog_converterhttp://en.wikipedia.org/wiki/Analog_to_digital_converterhttp://en.wikipedia.org/wiki/Analog_to_digital_converterhttp://en.wikipedia.org/wiki/Speech_processinghttp://en.wikipedia.org/wiki/Speech_processinghttp://en.wikipedia.org/wiki/Speech_processinghttp://en.wikipedia.org/wiki/Speech_processinghttp://en.wikipedia.org/wiki/Digital_image_processinghttp://en.wikipedia.org/wiki/Control_engineeringhttp://en.wikipedia.org/wiki/Audio_signal_processinghttp://en.wikipedia.org/wiki/Digitalhttp://en.wikipedia.org/wiki/Signal_%28information_theory%29

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Digital signal processing:

The algorithms required for DSP are sometimes performed using

specialized computers, which make use of specializedmicroprocessors called digital signal processors (also abbreviatedDSP). These process signals in real time and are generallypurpose-designed application-specific integrated circuits (ASICs).When flexibility and rapid development are more important than

unit costs at high volume, DSP algorithms may also beimplemented using field-programmable gate arrays (FPGAs).

http://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Computer_hardwarehttp://en.wikipedia.org/wiki/Digital_signal_processorhttp://en.wikipedia.org/wiki/Real_timehttp://en.wikipedia.org/wiki/Application-specific_integrated_circuithttp://en.wikipedia.org/wiki/Field-programmable_gate_arrayhttp://en.wikipedia.org/wiki/Field-programmable_gate_arrayhttp://en.wikipedia.org/wiki/Application-specific_integrated_circuithttp://en.wikipedia.org/wiki/Real_timehttp://en.wikipedia.org/wiki/Digital_signal_processorhttp://en.wikipedia.org/wiki/Computer_hardwarehttp://en.wikipedia.org/wiki/Algorithm

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DSP domains

In DSP, engineers usually study digital signals in one of the

following domains:

time domain (one-dimensional signals),

spatial domain (multidimensional signals),

frequency domain, autocorrelation domain, and

wavelet domains.

http://en.wikipedia.org/wiki/Time_domainhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Autocorrelationhttp://en.wikipedia.org/wiki/Wavelethttp://en.wikipedia.org/wiki/Wavelethttp://en.wikipedia.org/wiki/Autocorrelationhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Time_domain

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DSP domains

A sequence of samples from a measuring device produces a time

or spatial domain representation, whereas a discrete Fourier transform produces the frequency

domain information, that is the frequency spectrum.

Autocorrelation is defined as the cross-correlation of the signal

with itself over varying intervals of time or space.

http://en.wikipedia.org/wiki/Discrete_Fourier_transformhttp://en.wikipedia.org/wiki/Frequency_spectrumhttp://en.wikipedia.org/wiki/Cross-correlationhttp://en.wikipedia.org/wiki/Cross-correlationhttp://en.wikipedia.org/wiki/Frequency_spectrumhttp://en.wikipedia.org/wiki/Discrete_Fourier_transform