This document discusses several topics related to Fourier transforms including: 1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT). 2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively. 3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.

1. From Fast Multiplication to Compressed Image - Fourier Rules
CSA Showcase Illumination
Polynomials: To determine a polynomial f(x) of
Choosing a Value Representation Speeding Up DFT: Image Compression :
degree n, suffices to know its value for n+1 distinct
While we can choose any n+1 values to
values. Thus two points uniquely represents a line, FFT is a technique that reduces the complexity of An image is a two dimensional signal. The signal
represent a polynomial in value representation,
three points represents a quadratic equation. DFT computation from O(n2) to O(n.logn). The idea varies in space , instead of time. Converting from
it is useful to compute f(x) at the n roots of unity.
is to formulate DFT recursively spatial domain to frequency domain is advantages
ie f(1), f(ω), f(ω2)..... If f(x) is a0 + a1x + a2
Thus a polynomial ( 6t- 5t2 + t3 ) can be because we can analyze frequencies better. DCT
x2........
represented uniquely in coefficient representation converts an array of pixel values into an equally
as [ 0 6 -5 1 ] or in value representation eg. values or sized array of frequency components that also
f(1) = a0 + a1 + a2 ........
at t=0,1,2,3 are [ 0 2 0 0 ] represents the image
f(ω) = a0 + a1 ω + a2 ω2........
f(ω2) = a0 + a1 ω2 + a2 ω4........
Why n+1 points uniquely determine the polynomial? These can be represented in form of matrix.
suppose f(x) and g(x) (of degree n) both agrees A Note on Fourier Transform
on n+1 points, then f(x) – g(x) is a polynomial with
degree n or less and has n+1 zeros. This happens
only if f(x)- g(x)=0 or f=g. A signal f(t) can be expressed as the sum of sum of
cosines and sinusoids. In the Euclidean space. Any
This can be used to find the polynomial given its point can be given in terms of weighted sum of
values at n+1 points say f(0)=1, f(1)=4 , f(2)=13, then [ f(1), f(ω), f(ω2).. ] has same information as basis. Likewise any function f(t) can be considered
f(3)=34, f(4)=73 [ a0 , a1 , a2 ]. This representation is called the as a point in the vector space of functions whose
Discrete Fourier Transform basis are the cosines and sinusoids.
The human eye, cannot detect high frequency
using finite differencing we can find components. Using the process of quantization,
the polynomial. Babbage used it in we retain all low frequency components. (The high
his Difference Engine Polynomial Multiplication using DFT frequency components are retained only if they
Suppose we have to multiply f(x) = x2 + x + 2 and have large coefficients). On the resulting
g(x)= x+3 . then DFT of g(x) ( whose coefficient coefficients run length encoding and huffman
Polynomial Multiplication representation is [3 1 0 0]) is given by compression is applied
The advantages of converting it from time domain
If f(x) and g(x) has to be multiplied using coefficient representation to frequency domain representation
representation, the kth coefficient would be is that many manipulations (like filtering out high
calculated as Σ ai*bk-i where i varies from 0 to k. frequencies) can be easily done in frequency
Thus we get (a0b0, a0b1+a1b0, a0b2+a1b1+a2b0....) . domain. FT handles continuous signals. DFT can
This is also know as vector convolution of ai and bi
Applications
be viewed as forming a continuous function of ω
DFT of f(x) is [4, i+3, 2 , -i+3] (calculation not shown) from discrete values
A better method is to use value representation. DFT of g(x) is [4, i+1, 2 , 1-i ] (calculated above) SEQUENCE RETRIEVAL: We have a database of
Suppose f(x) is of degree 10 and g(x) is of degree p(x).q(x)= [16, 4i+2,4, 2-4i] (multiplying pointwise) sequences and a query sequence. Each sequence
20, then we can compute f(x) and g(x) at 31 points is represented by its first few coefficients from
and multiply them. since f(a)*g(a) = (f*g) (a) , we The Real Matrix: DFT. The query sequence is also transformed
would get the value representation of f*g from which using DFT and its first few coefficients are
since ? is a complex root of unity compared.
we can get the coefficient representation.
OTHER APPLICATIONS include filtering,
convolution, audio processing, medical imaging,
If the function is even, the the sin component is pattern recognition, partial differential equations,
zero and then the entries of the DFT matrix are multiplication of large numbers, analysis of time
computing set in 2, c3 similarly we get, equivalent, italic, 18 to 24 points,
Captions to be c1,c Times or Times New Roman or real. Such a transformation is called Discrete
to the length of the column in case a figure takes more than 2/3 of column width. series, solving system of linear equations
Cosine Transformation.
Presented by D.Arvind