DEFINITION OF DCT

Due to its computational efficiency the discrete

Fourier transform (DFT) is very popular.

But it is very complex.

It has very poor energy compaction. Energy

compaction is the ability to pack the energy of the

spatial sequence into as few freq. coefficients as

possible. It helps in image compression.

When the input data contains only real nos from an

even function, the Sin component of DFT is ’0’, and

the DFT becomes DCT.

DCT is simply a computationally efficient version of

the DFT for the signal that are real and even.

In comparison, (DCT) is a real transform that

transforms a sequence of real data points into its

real spectrum and therefore avoids the problem of

redundancy

Also, as DCT is derived from DFT, all the

desirable properties of DFT (such as the fast

algorithm) are preserved.

One-dimensional DCT

Definition: Let n be a positive integer. The one-

dimensional DCT of order n is defined by an n

x n matrix C whose entries are

n

jiaC iij

2

)12(cos

To derive the DCT of an N-point real signal

sequence we first construct a new

sequence of points:

DFT and DCT basics

DFT AND DCT BASICS

1D-DFT1D-DCT

DCT (1D)

Discrete cosine transform

The strength of the ‘u’ sinusoid is given by C(u) Project f onto the basis function

All samples of f contribute the coefficient

C(0) is the zero-frequency component – the average value!

6

DCT (1D) Consider a digital image such that one row has the following

samples

There are 8 samples so N=8

u is in [0, N-1] or [0, 7]

Must compute 8 DCT coefficients: C(0), C(1), …, C(7)

Start with C(0)

7

Index 0 1 2 3 4 5 6 7

Value 20 12 18 56 83 10 104 114

DCT (1D)

8

DCT (1D) Repeating the computation for all u we obtain

the following coefficients

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DCT MATRIX IS ORTHOGONAL:

DCT matrix is orthogonal i.e

and it is real

Now the DCT can be expressed in matrix form as:

Left multiplying both sides by C we will get

this is the inverse DCT: or in component form:

Example: When N=2 , we have ,

m,n=0,1.

and

For N=4,

Assume the signal is then its DCT transform is:

And the inverse transform is

DCT (2D)

The 2D DCT is given below where the definition for alpha is the same as before

For an MxN image there are MxN coefficients

Each image sample contributes to each coefficient

Each (u,v) pair corresponds to a ‘pattern’ or ‘basis function’

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SEPARABILITY

The DCT is separable The coefficients can be obtained by computing the 1D coefficients for

each row

Using the row-coefficients to compute the coefficients of each column (using the 1D forward transform)

13

INVERTABILITY

The DCT is invertible Spatial samples can be recovered from the DCT coefficients

14

SUMMARY OF DCT

The DCT provides energy compaction

Low frequency coefficients have larger magnitude (typically)

High frequency coefficients have smaller magnitude (typically)

Most information is compacted into the lower frequency coefficients (those coefficients at the ‘upper-left’)

Compaction can be leveraged for compression

Use the DCT coefficients to store image data but discard a certain percentage of the high-frequency coefficients!

JPEG does this15

COMPARED WITH DFT, DCT HAS TWO MAIN

ADVANTAGES:

It is a real transform with better computational efficiency than DFT which by definition is a complex transform.

It does not introduce discontinuity while imposing periodicity in the time signal. In DFT, as the time signal is truncated and assumed periodic, discontinuity is introduced in time domain and some corresponding artifacts is introduced in frequency domain. But as even symmetry is assumed while truncating the time signal, no discontinuity and related artifacts are introduced in DCT.

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INTRODUCTION TO ADAPTIVE FILTERS

Linear filters :

the filter output is a linear function of the filter input

Design methods:

1 The classical approach

frequency-selective filters such as

lowpass / bandpass / notch filters etc

2 Optimal filter design

Mostly based on minimizing the mean-square value

of the error signal

19

THE FILTERING PROBLEM

Filters may be used for three information-processing tasks

Filtering

Smoothing

Prediction

Adaptive filters are self-designing using a recursive

algorithm

Useful if complete knowledge of environment is not

available a priori

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WIENER FILTER

work of Wiener in 1942 and Kolmogorov in 1939

it is based on a priori

statistical information

when such a priori

information is not available,

which is usually the case,

it is not possible to design

a Wiener filter in the first

place

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ADAPTIVE FILTER

The signal and/or noise characteristics are often

nonstationary and the statistical parameters vary

with time

An adaptive filter has an adaptation algorithm, that is

meant to monitor the environment and vary the filter

transfer function accordingly

Based in the actual signals received, attempts to find

the optimum filter design

ADAPTIVE FILTER

The coefficients of an adaptive filter change

in time

Output

signal

Input

signal

Adaptive

algorithm

Criterion of

performance

Filter

structure

23

ADAPTIVE FILTER

In a stationary environment, the filter is expected to

converge, to the Wiener filter

In a nonstationary

environment,

the filter is expected to

track time variations and

vary its filter coefficients

accordingly

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ADAPTIVE FILTER

The basic operation now involves two

processes :

1. a filtering process, which produces an output signal

in response to a given input signal.

2. an adaptation process, which aims to adjust the

filter parameters (filter transfer function) to the

(possibly time-varying) environment

Often, the (avarage) square value of the error signal

is used as the optimization criterion

BLOCK DIAGRAM OF ADAPTIVE SYSTEM

S(n)+No(n)No(n)

+

-

?

Primary

signal

d(n)

N1(n)

Reference

signaly(n)

output

e(n)

adaptive

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ADAPTIVE FILTER

• Because of complexity of the optimizing algorithms most adaptive filters are digital filters that perform digital signal processing

When processing

analog signals,

the adaptive filter

is then preceded

by A/D and D/A

convertors.

[1]

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ADAPTIVE FILTER

• The generalization to adaptive IIR filters leads to stability problems

• It’s common to use

a FIR digital filter

with adjustable

coefficients.

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APPLICATIONS OF ADAPTIVE FILTERS: SYSTEM IDENTIFICATION

Used to provide a linear model of an unknown plant

Parameters u=input of adaptive filter=input to plant y=output of adaptive filter d=desired response=output of plant e=d-y=estimation error

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APPLICATIONS OF ADAPTIVE FILTERS: INVERSE MODELING (ADAPTIVE

CHANNEL EQUALIZATION)

Used to provide an inverse model of an unknown plant

Parameters u=input of adaptive filter=output to plant y=output of adaptive filter d=desired response=delayed system input e=d-y=estimation error

Applications: Equalization

ADAPTIVE ARRAY ANTENNA

Adaptive Arrays

Linear Combiner

Interference

SMART ANTENNAS

Adaptive Array Antenna

APPLICATIONS OF ADAPTIVE FILTERS: INTERFERENCE CANCELLATION

Used to cancel unknown interference from a primary signal

Applications:

Echo / Noise cancellation

hands-free car phone, aircraft headphones etc

Parameters

u=input of adaptive filter=reference signal

y=output of adaptive filter

d=desired response=primary signal

e=d-y=estimation error=system output

EXAMPLE: ACOUSTIC ECHO CANCELLATION

Applications are manyDigital Communications (OFDM , MIMO , CDMA, and RFID)Channel EqualisationAdaptive noise cancellationAdaptive echo cancellationSystem identificationSmart antenna systemsBlind system equalisationAnd many, many others

ADAPTIVE ALGORITHM

An adaptive algorithm is used to estimate a time varying signal.

By adjusting the filter coefficients so as to minimize the error.

There are many adaptive algorithms like Recursive

Recursive Least Square (RLS), Kalmanfilter,

but the most commonly used is the Least Mean Square (LMS) algorithm.

LMS ALGORITHM Estimates the

solution to the Weiner-Hopfequations using gradient descent method which finds minima by estimating the gradient.

is the step size

X(n)

C(n)

Transversal

Filter

LMS

Y(n)

e(n)

d(n)

CONT..

Update the coefficients using

the following computation.

Filtering operation with the

previous version of the

coefficients. Compare the computed output

with the expected output.

e(n)

X(n) y(n)

d(n)

Adaptive

filter

Unknown

system

LMS ALGORITHM

• Most popular adaptation algorithm is LMS

Define cost function as mean-squared error

• Based on the method of steepest descent

Move towards the minimum on the error surface to get

to minimum gradient of the error surface estimated at

every iteration

LMS ADAPTIVE ALGORITHM

• Introduced by Widrow & Hoff in 1959

• Simple, no matrices calculation involved in the adaptation

• In the family of stochastic gradient algorithms

• Approximation of the steepest – descent method

• Based on the MMSE criterion.(Minimum Mean square Error)

• Adaptive process containing two input signals:

• 1.) Filtering process, producing output signal.

• 2.) Desired signal (Training sequence)

• Adaptive process: recursive adjustment of filter tap weights

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The LMS Algorithm consists of two basic processes

Filtering process

Calculate the output of FIR filter by convolving

input&taps

Calculate estimation error by comparing the output to

desired signal

Adaptation process

Adjust tap weights based on the estimation error

LMS ALGORITHM STEPS

Filter output

Estimation error

Tap-weight adaptation

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1

0

*M

k

k nwknuny

nyndne

neknunw1nw *kk

signal

error

vector

input

tap

parameter

rate

-learning

vector

weight-tap of

value old

vector

weigth-tap of

value update

STABILITY OF LMS

The LMS algorithm is convergent in the mean square

if and only if the step-size parameter satisfy

Here max is the largest eigenvalue of the correlation

matrix of the input data

More practical test for stability is

Larger values for step size

Increases adaptation rate (faster

adaptation)

Increases residual mean-squared error