DCT基本介绍

《数字信号处理》研究型作业一

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——离散余弦变换

DCT

的基本介绍

学生：

X

X

X

学号：

XXXXXX

指导老师：

二零

一

三

年

11

月

10

日

No.1 The Definition of DCT

A

discrete

cosine

transform

(DCT)

expresses

a

finite

sequence

of

data

points

in

terms

of

a

sum

of

cosine

functions

oscillating

at

different

frequencies.

DCTs

are

important

to

numerous

applications

in

science

and

engineering,The

use

of

cosine

rather than sine functions is critical in these applications: for compression, it turns out

that cosine functions are much more efficient (as described below, fewer functions are

needed to approximate a typical signal), whereas for differential equations the cosines

express a particular choice of boundary conditions.

In particular, a DCT is a Fourier-related transform similar to the discrete Fourier

transform

(DFT),

but

using

only

real

numbers.

DCTs

are

equivalent

to

DFTs

of

roughly

twice

the

length,

operating

on

real

data

with

even

symmetry

(since

the

Fourier transform of a real and even function is real and even), where in some variants

the

input

and/or

output

data

are

shifted

by

half

a

sample.

There

are

eight

standard

DCT variants, of which four are common.

A

real

symmetric

or

antisymmetric

finite-length

sequence

is

a

product

of

a

linear-phase

term

and

a

real

amplitude

the

phase

term

is

known

for

given

length

sequence

,the

amplitude

function

uniquely

describes

the

time-domain

sequence in the transform class of real orthogonal transform is based on

converting

the

real

arbitrary

sequence

into

either

a

symmetric

or

an

antisymmetric

sequence and then extracting the real orthogonal transform coefficients from the DFT

of

the

generated

sequence

with

geometric

transforms

developed

via

this approach are called

the discrete cosine transform,often abbreviated

as

DCT,and

discrete sine transform,often abbreviated as DST.

No.2 The engineering background of DCT

The

DCT,

and

in

particular

the

DCT-II,

is

often

used

in

signal

and

image

processing,

especially

for

lossy

data

compression,

because

it

has

a

strong

compaction

in

a

few

low- frequency

components

of

the

DCT,

approaching

the

Karhunen-Lo

è

ve

transform

(which

is

optimal

in

the

decorrelation

sense)

for

signals

based

on

certain

limits

of

Markov

processes.

As

explained

below,

this

stems

from

the

boundary

conditions implicit in the cosine functions.

DCT-II (bottom) compared to the DFT (middle) of an input signal (top).

A related transform, the modified discrete cosine transform, or MDCT (based on the

DCT-IV), is used in AAC, V

orbis, WMA, and

audio compression.

DCTs

are

also

widely

employed

in

solving

partial

differential

equations

by

spectral

methods,

where

the

different

variants

of

the

DCT

correspond

to

slightly

different

even/odd boundary conditions at the two ends of the array.

DCTs are also closely related to , and fast DCT algorithms (below) are used in

of

arbitrary functions by series of Chebyshev polynomials, for example in .

No.3 The transform signification of DCT

Formally,

the

discrete

cosine

transform

is

a

,

invertible

错误

!

(where

错误

!

denotes the set of ), or equivalently an invertible N

×

N . There

are

several

variants

of

the

DCT

with

slightly

modified

definitions.

The

N

real

numbers x0, ..., xN-1 are transformed into the N real numbers X0, ..., XN-1 according

to one of the formulas:

DCT-I

错误

!

Some

authors

further

multiply

the

x0

and xN-1

terms

by

√

2,

and

correspondingly

multiply

the

X0

and

XN-1

terms

by

1/

√

2.

This

makes

the

DCT-I

matrix

,

if

one

further

multiplies

by

an

overall

scale

factor

of

错误

!

,

but

breaks

the

direct

correspondence with a real-even DFT.

The

DCT-I

is

exactly

equivalent

(up

to

an

overall

scale

factor

of

2),

to

a

DFT

of

错误

!

real numbers with even symmetry. For example, a DCT-I of N=5 real numbers

abcde is exactly equivalent to a DFT of eight real numbers abcdedcb (even symmetry),

divided

by

two.

(In

contrast,

DCT

types

II-IV

involve

a

half- sample

shift

in

the

equivalent DFT.)

Note, however, that the DCT-I is not defined for N less than 2. (All other DCT types

are defined for any positive N.)

Thus, the DCT-I corresponds to the boundary conditions: xn is even around n=0 and

even around n=N-1; similarly for Xk.