Transcript Introduction to Transformations

Introduction to Transforms
Rutgers University
Discrete Mathematics for ECE
14:332:202
The Basic Idea

We have a signal


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could be a function or a vector (sampled function)
Given a set of basis functions/vectors, we
wish to find the coefficients that will weight
the functions, so that when summed, they will
add to our original signal.
The signal now has now been transformed
into this set of coefficients
Fourier Series
In the Fourier Series, we transform a continuous (has
a value for every possible t) periodic signal into a
weighted sum of sines and cosines. The more
terms that are added to the sum, the better our
approximation to the original signal.
In a sense, we are “discretizing” the continuous
function by representing as a finite set of coefficients
(really need infinite set to have exact representation)
Square Wave Example
Consider the periodic square wave, that oscillates between +1
and -1 with period T.
Fourier Series
We can make an approximation to the square wave by a sine wave of
period T. fa(t) = b1sin(2π/T)
Fourier Series
Adding another term gives us a better approximation, that term being a
sine with period T/3. fa(t) = b1sin(2π/T) + b3sin(3*2π/T)
Fourier Series
Continuing to add terms our representation becomes better. Here we
have all terms up to the 21st Harmonic.
fa(t) = b1sin(2π/T) + b3sin(3*2π/T) + … + b21sin(21*2π/T)
Fourier Series
With enough terms calculated to adequately represent our signal, we
then have a “discrete representation” of our periodic function.
The original signal can now be represented by the set of coefficients
(bn’s) that multiply by our basis functions to give us the
approximation.
Finding these coefficients is known as the Forward Transform, and
getting back our signal from the coefficients and our known set of
basis functions, is known as the Inverse Transform
Sampled Signals (Vectors)

We will learn the concept of transforms by working with
sampled signals, the N samples being placed in a row vector:
x = [x0 x1 x2 … xN-1]
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We also must have a set of basis vectors, b0, ... , bN-1, each of
length N, that comprise the rows of a matrix B
[
[

B  [

[
[
b0
b1

b N2
b N 1
]
]
]

]
]
Vector Transforms
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Consider the trivial example where we have 4 basis vectors:
e0 = [1 0 0 0]
e1 = [0 1 0 0]
e2 = [0 0 1 0]
e3 = [0 0 0 1]

Obviously, we can form any 4 element vector by a weighted
sum of the unit vectors, which is easy to do:
x = [x0 x1 x2 x3] = x0e0 + x1e1 + x2e2 + x3e3
Vector Transforms
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The set of unit vectors is a trivial basis, so we may
be interested in some other set of basis vectors.
We can represent a signal by a weighted sum of a
(well-chosen) set of basis vectors (of the same size).
x = [x0 x1 x2 … xN-1] = u0b0 + u1b1 + u2b2 + … + uN-1bN-1

The vector, x, has now been transformed into the vector of
coefficients, u. This is also known as “changing the basis”
Vector Transforms
The equation:
x = [x0 x1 x2 … xN-1] = u0b0 + u1b1 + u2b2 + … + uN-1bN-1
can be compactly written in matrix notation as
x  uB
where x and u are row vectors, and B is an NxN matrix whose
rows are the basis vectors, b0 … bN-1
The Forward Transform
We wish to find the vector of coefficients u = xB-1
Computing the inverse of the matrix, B, is not necessary if all of
the bi’s forming the rows of B are orthogonal, that for all
k≠j, bjbkT = dot(bi,bj) = 0
Also, |bk| = bkbkT, and in the case of the Hadamard and FFT
matrices, |bk| = N
Because of the orthogonality property, we can compute the
elements of u by the equation:
uk = (x·bkT) / |bk|
Orthogonality

That two vectors are orthogonal can be interpreted
geometrically as being perpendicular in Ndimensional space, such that their dot product is 0.
The Hadamard Basis Vectors
The rows of the Hadamard Matrix create N orthogonal
vectors of +1’s and -1’s
Hadamard Basis Vectors of Order 8
We can represent any 8 element
vector by a weighted sum of these
8 orthogonal vectors
The DCT Basis Vectors
Another possible basis – these vectors represent sampled
cosines of different frequencies