Finite-Length Discrete Transforms
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Transcript of Finite-Length Discrete Transforms
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Finite-Length Discrete
Transforms
1
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Orthogonal Transforms
Let x[n] denote a length-N time-
domain sequence with
denoting the coefficient of its N-
point orthogonal transform. The
general form of the orthogonal
transform pair is of the form
Referred to as the analysis and
synthesis equation respectively.
2
The above condition are said to be
orthogonal to each other
The verification of this equation in
book page 200
x [ k]= n=0
N 1
x [ n ] [ k, n ], 0 k N 1
x [ n ]=1
N n=0
N 1
x [n ] [ k, n ], 0 k N 1
n=0
N 1
[ k, n ] [ l,n] = {1, l=k0 l k}[k]
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The inverse discrete Fouriertransform (IDFT) is given by
As can be seen from the above
expression, the inverse DFT x[n] can
be a complex sequence even when
the DFT X[k] is real sequence.
4
The Discrete Transform
x [ n ]=1
Nk=0
N 1
X[ k] WN kn, 0 n N 1
X[k]
Example - Consider the length-Nsequence defined for 0 n N-1
Where is r is an integer in the range0 n N-1
Using a trigonometric identity, wecan write as
x [n ]= cos(2 rn N), 0 n N 1
x [ n ]= 12
(e j2rn /N+e j2rn /N)
1
2(WN
rn+WN
rn )
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5
The Discrete Transform
The N-point DFT ofg[n] is thus given by
Making use of the identity
X[ k ]= n=0
N 1
g[n ] WNkn
1
2 (n=0N 1
WN ( r k)n+
n=0
N 1
WN( r+k)n)
0 k N 1
n=0
N 1
WN (k )n
={N,fork =rN,ran int eger
0,otherwise }G[ k]={
N/2, fork=r
N/2, fork=N r
0otherwis }0 k N 1
we
get
Matrix Relations
The DFT samples defined by
can be expressed in matrix form as
WhereX is the vector composed of
the N DFT samples and x is thevector ofNinput samples
X[ k]= n=0
N 1
x[ n]WN
k n, 0 k N 1
X=DNx
X=[ X[0 ] X[1 ] X[ N 1 ] ]t
x=[ x [0 ]x [1 ]x [ N 1 ] ]t
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6
The Discrete Transform
and DNis the NNDFT matrix given
by
Likewise, the IDFT relation given by
DN=[1 1 1 1
1 WN1 W
N2 W
N(N 1)
1 WN
2 WN
4 WN
2( N 1)
1 WN
(N 1 ) WN
2(N 1) WN
(N 1)2 ]
can be expressed in matrix form as
where is the NNDFT matrix
Where
Note
DN
1
x=DN 1
X
DN 1
=
[1 1 1 1
1 WN
1 WN
2 WN
(N 1 )
1 WN
2 WN
4 WN
2(N 1)
1 WN
(N 1) WN
2(N 1) WN
(N 1)2
]D
N
1=1
N
DN
x [ n ]=1
Nk=0
N 1
X[K]WN kn, 0 n N 1
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The FT of length-Nsequence x[n]
limit the extent of the summation to
N points and evaluate the
continuous function of frequency at
Nequally spaced points
7
Relation with DTFT
X(ej)= n=
x [ n ]e jn
= n=0
N 1
x [ n ]e jn
X(ej)=
2
Nk
=X(k) =Xk
= n=0
N 1
x [ n ]e j2kn/N
Numerical Computation of the
DTFT using the DFT
A practical approach to the
numerical computation of the DTFT
of a finite-length sequence.
LetX(ej) be the DTFT of a length-N
sequencex[n]
We wish to evaluate X(ej) at a
dense grid of frequencies k=2k/M,0kM-1,where M>> N:
Relation Between the DTFT and the DFT and Their Inverses
ejn
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Define a new sequence
Then
X(ej
k)= n=0
N 1
x [ n ]e j
kn=
n=0
N 1
x [ n ]e j2kn/M
X(ej
k)= n=0
M 1
xe[ n ]e j2kn/M
xe[n ]= {x [n ],0 n N 10, N n M 1 }
Thus is essentially an M-point DFT Xe[k] of the length-Msequencexe[n]
The DFT Xe[k] can be computedvery efficiently using the FFTalgorithm ifMis an integer power of2
The function freqz employs this
approach to evaluate the frequencyresponse at a prescribed set offrequencies of a DTFT expressed asa rational function in e-j
X(ej
k)
Relation Between the DTFT and the DFT and Their Inverses
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DTFT from DFT by Interpolation
The N-point DFT X[k] of a length-N
sequence x[n] is simply the
frequency samples of its DTFT
X(ej) evaluated at N uniformly
spaced frequency points
Given the N-point DFT X[k] of alength-N sequence x[n], its DTFT
X(ej) can be uniquely determined
fromX[k]
=k= 2k/N , 0 k N 1
Thus
X(ej
)= n= 0
N 1
x [n ]e jn
n= 0
N 1
[1
Nk=0
N 1
X[ k]WN kn ]e jn
1N k= 0
N 1
X[ k] n= 0
N 1
e j ( 2k/N)n
S
Relation Between the DTFT and the DFT and Their Inverses
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To develop a compact expressionfor the sum S, let
Then
From the above
Or, equivalently,
r=e j( 2k/N)
S= rn
rS= rn= 1+ rn+rN 1
r
n
+r
N
1=S+rN
1
Hence
Therefore
S=1 rN
1 r=1 e j (N 2k)
1 e j [ (2k/N) ]
sin
(
N 2k
2 )sin(N 2k2N )
e j [ 2k/N)][(N 1)/2]
S rS=(1 r) S=1 rN
5.3 Relation Between the DTFT and the DFT and Their Inverses
X(ej)
1
Nk= 0
N 1
X[ k]
sin(N 2k2 )sin
(
N 2k
2N
)
e j [ 2k/N)][(N 1)/2 ]
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Sampling the Fourier Transform
Consider a sequencex[n] with a DTFT
X(ej)
We sampleX(ej) at Nequally spaced
points k=2k/N, 0kN-1 developingthe Nfrequency samples
These N frequency samples can be
considered as an N-point DFT Y[k]
whose N- point IDFT is a length-N
sequence y[n]
now
{X(ejk)}
Relation Between the DTFT and the DFT and Their Inverses
X(ej
)=
x [ ]e j
Thus
An IDFT ofY[k] yields
Y[ k]=X(ejk)=X(e
j2k/N)
x [ ]e j2k/N=
x [ ]WNk
y [ n ]=1
Nk0
N 1
Y[k]WN kn
y [ n ]=1
Nk0
N 1
=
x[ ]WNkW
N kn
=
x [ ][1N k= 0N 1
WN k( n )]
1
N
n0
N 1
WN k( n r)= (
1, forr=n+mN
0, otherwise
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we arrive at the desired relation
Thus y[n] is obtained from x[n] by
adding an infinite number of shifted
replicas of x[n], with each replicashifted by an integer multiple of N
sampling instants, and observing the
sum only for the interval 0nN-1
y [n ]= m=
x [ n+mN],0 n N 1
To apply
To finite-length sequences, we
assume that the samples outside the
specified range are zeros
y [ n ]= m=
x [ n+mN], 0 n N 1
Relation Between the DTFT and the DFT and Their Inverses
Thus ifx[n] is a length-M sequence
with MN, then y[n]= x[n] , for
0nN-1
Example 5.6
Let {x[n]}={0 1 2 3 4 5}
By sampling its DTFT X(ej) at
k=2k/4, 0k3 and then applying
a 4-point IDFT to these samples, we
arrive at the sequence y[n] given by
y[n]=x[n]+x[n+4]+x[n-4] 0n3
i.e. {y[n]}={4 6 2 3}
{x[n]} cannot be recovered from {y[n]}
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Circular Convolution
This operation is analogous to linear
convolution, but with a subtle
difference
Consider two length-N sequences,
g[n] and h[n], respectively
Their linear convolution results in a
length-(2N-1) sequence yL[n] given
by
yL
[n ]= m=0
N 1
g[ m ] h[ n m], 0 n 2N 2
In computing yL[n] we have assumedthat both length-N sequence havebeen zero-padded to extend theirlengths to 2N-1
The longer form ofyL[n] results fromthe time-reversal of the sequenceh[n] and its linear shift to the right
The first nonzero value ofyL[n] is
yL[0]= g[0]h[0] ,and the last nonzerovalue is
yL[2N-2]= g[N-1]h[N-1]
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Definition:
To develop a convolution-like
operation resulting in a length-N
sequence yC[n] , we need to define a
circular time-reversal, and then apply
a circular time-shift
Resulting operation, called a
circular convolution, is defined by
yC
[ n ]= m=0
N 1g[ m ] h [n m
N],
0 n N 1
Since the operation defined involves
two length-N sequences, it is often
referred to as an N-point circular
convolution, denoted as
The circular convolution is
commutative, i.e.
y[n] =g[n] h[n]
g[n] h[n] = h[n] g[n]
Circular Convolution
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The N-point circular convolution canbe written in matrix form as
Note: The elements of each diagonalof the NNmatrix are equal
Such a matrix is called a circulantmatrix
[yC[0 ]
yC[1 ]
yC[2 ]
yC[N 1 ]]=
[h[ 0 ] h [N 1 ] h[ N 2 ] h [1 ]
h[ 1 ] h[ 0 ] h [N 1] h[ 2 ]
h[ 2 ] h[ 1] h[ 0 ] h [3 ]
h [N 1 ] h[ N 2] h [N 3] h[ 0 ]][g[ 0 ]
g[1 ]
g[ 2 ]
g[N 1 ]]
5.4 Circular Convolution
Tabular Method
Consider the evaluation of y[n]= h[n]
g[n]
where {g[n]} and {h[n]} are length-4
sequences
First, the samples of the two
sequences are multiplied using theconventional multiplication method as
shown on the next slide
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The partial products generated in the 2nd, 3rd, and 4th rows are circularly shiftedto the left as indicated above
Circular Convolution
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Circular Convolution
The modified table after circular
shifting is shown below
The samples of the sequence {yc[n]}
are obtained by adding the 4 partial
products in the column above of eachsample
Thus
yc[0]=g[0]h[0]+ g[3]h[1]+ g[2]h[2]+ g[1]h[3]
yc[1]=g[1]h[0]+ g[0]h[1]+ g[3]h[2]+ g[2]h[3]
yc[2]=g[2]h[0]+ g[1]h[1]+ g[0]h[2]+ g[3]h[3]
yc[3]=g[3]h[0]+ g[2]h[1]+ g[1]h[2]+ g[0]h[3]
The definition of circular conjugate-symmetric sequence and circularconjugate-antisymmetric sequence.
Circular conjugate-symmetry
Circular conjugate-antisymmetry
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Classifications of Finite-Length Sequences
Classifications Based on Conjugate
Symmetry Any complex length-N sequence x[n]can be expressed as:
Where, Xcs[n] is its circularconjugate-symmetric part and Xca[n]is its circular conjugate-anti-symmetric part, defined by:
For a real sequence x[n], it can beexpressed as:
Where, Xev[n] is its circular evenpart and Xca[n] is its circular oddpart, defined by:
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Classifications of Finite-Length Sequences
Classifications Based on Geometric
Symmetry Two types of geometric symmetries
are usually defined:For a real
(1) Symmetric sequence
(2) Antisymmetric sequence
Since the length N of a sequencecan be either even or odd, four typesof geometric symmetry are defined:
Type1: Symmetric impulse responsewith odd length.
Type2: Symmetric impulse responsewith even length
Type1: Anti-symmetric impulseresponse with odd length.
Type1: Anti-symmetric impulseresponse with even length.
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Classifications of Finite-Length Sequences
Type 1 symmetric sequence, withN=9,
is
x[n]=x[0]+x[1]+x[2]+x[3]+x[4]+x[5]+
x[6]+x[7]+x[8]
The Fourier transform is
Type 1 Symmetry with Odd Length
Taking e-j4 as a common factor ineach group of.
Factoring out e-j4 in the right hand
side
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Classifications of Finite-Length Sequences
Notice that the quantity inside the
braces, { }, is a real function of andcan assume positive or negative values
in the range 0
The of the sequence is given by
() = 4 + where is either 0 or
, and hence the phase is a linear
function of
In general, for Type 1 linear-phase
sequence of length-N
Type 2 Symmetry with Even Length
Similarly, the Fourier transform ofType 2 symmetric sequence, with N=8,
can be written.
where the phase is given by
In general, for Type 2 linear-phase
sequence of length-N
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Classifications of Finite-Length Sequences
The Fourier transform of Type 3
antisymmetric sequence, with N=9, is
(notice that x[4]=0)
Now, x[0]=-x[8], x[1]=-x[7], x[2]=-x[6],
x[3]=-x[5] and x[4]=0
Type 3 Antisymmetry with Odd Length
Multiplying by j=ej/2and 2, we obtain
which results in
The phase is now
The antisymmetry introduces a phase
shift of/2
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Classifications of Finite-Length Sequences
In general, the Fourier transform of
Type 3 linear phase antisymmetric
sequence of odd length-N is
Similarly, the Fourier transform of
Type 4 linear phase antisymmetric
sequence of even length-N is
In both cases, j=ej/2 introduces a
phase shift of/2
Type 4 Antisymmetry with Even Length
Multiplying by j=ej/2and 2, we obtain
which results in
The phase is now
The antisymmetry introduces a phase
shift of/2
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DFT Symmetry Relations
In general, the DFT X[k] of a finite
sequence x[n] is a sequence of complex
numbers and can be expressed as
Real and imaginary parts of the DFT
sequence can be found as:
Assuming that the original time-domain
signal is complex
its DFT can be found as:
Therefore, real and imaginary parts ofthe DFT sequence are:
X[ k]= Xre
[ k]+ j Xim [ k]
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DFT Symmetry Relations
Symmetry Properties of the DFT of a
real sequence
Symmetry Properties of the DFT of a
complex sequence
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Discrete Fourier Transform Theorems
The DFT satisfies a number of
properties that are useful in signalprocessing.
All time domain sequences are
assumed to be of length-N with N-point
DFT.
Linearity Theorem
Consider a sequence x[n] obtained by a
linear combination of g[n] and h[n]
Circular Time Shifting Theorem
The DFT of the circularly time shifting
sequence x[n] is given by
Circular Frequency Shifting Theorem
The inverse DFT of the circularly
frequency shifting DFT is given by
Duality Theorem
If the N-point DFT of the length-N
sequence g[n] is G[k], then
Circular Convolution Theorem
The N-point DFT Y[k] of the length N-sequence is given by
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Computation of the DFT of Real Sequence
Modulation Theorem
The N-point DFT Y[k] of the length-Nproduct sequence is given by a
circulation of theirDFTs
Parsevals TheoremThe total energy of a length-N sequence
g[n] can be computed by summing the
square of the absolute values of the
DFT.
N-point DFTs of two Real Sequences
using a single N-point DFT
Let g[n] and h[n] be two length-N realsequences with G[k] and H[k] denoting
their respective N-point DFTs
These two N-point DFTs can be
computed efficiently using a single N-
point DFT
Define a complex length-N sequence
The inverse DFT of the circularly
frequency shifting DFT is given by
LetX[k] denote the N-point DFT of x[n]
Note that for 0 k N1,
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Computation of the DFT of Real Sequence
Thus
Linear Convolution using the DFT
Since a DFT can be efficiently
implemented using FFT algorithms, it is
of interest to develop methods for the
implementation of linear convolution
using the DFT
Let g[n] and h[n] be two finite-length
sequences of length N
and M, respectively
Define two length-L (L = N + M 1)sequences
Thus
2N-point DFT of a Real Sequence
using a single N-point DFTLet v[n] be a length-N real sequence
with a 2N-point DFT V[k]
Define two length-N real sequences
g[n] and h[n] as follows:
Let G[k] and H[k] denote their espective
N-point DFTs
Define a length-N complex sequence
with an N-point DFT X[k]
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Linear Convolution using the DFT
Linear Convolution of a Finite-Length
Sequence with an infinite Length
Sequence
Overlap-Add Method
We first segment x[n], assumed to be a
causal sequence here without any loss
of generality, into a set of contiguous
finite-length subsequences of length N
each:
Where
Thus
where
The corresponding implementation
scheme is illustrated below
We next consider the DFT-basedimplementation of
where h[n] is a finite-length sequence oflength M and x[n] is an infinite length (or
a finite length sequence of length much
greater than M)
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Linear Convolution using the DFT
The desired linear convolution y[n] =
h[n] x[n] is broken up into a sum of
infinite number of short-length linear
convolutions of length N + M 1 each:
ym[n] = h[n] xm[n]
Consider implementing the following
convolutions using the DFT-basedmethod, where now the DFTs (and the
IDFT) are computed on the basis of (N
+ M 1) points
In general, there will be an overlap ofM
1 samples between the samples of the
short convolutions h[n] xr-1[n]and h[n]
xm[n] for (r 1)N n rN + M 2
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Linear Convolution using the DFT
Therefore, y[n] obtained by a linear
convolution of x[n] and h[n] is given by
The above procedure is called the
overlap-add method
since the results of the short linear
convolutions overlap and the
overlapped portions are added to get
the correct final result
Overlap-Save Method
In implementing the overlap-add
method using the DFT, we need to
compute two (N + M1)-point DFTs and
one (N + M 1)-point IDFT for each
short linear convolution
It is possible to implement the overalllinear convolution by performing instead
circular convolution of length shorter
than (N + M 1)
To this end, it is necessary to
segment x[n] intooverlapping blocksxm[n] , keep the terms o f the circular
convolution of h[n] with that
corresponds to the terms obtained by a
linear convolution ofh[n] and xm[n], and
throw away the other parts of the
circular convolution
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Linear Convolution using the DFT
To understand the correspondence
between the linear and circular
convolutions, consider a length-4
sequencex[n] anda length-3 sequence
h[n]
Let yL[n] denote the result of a linear
convolution of x[n] with h[n]
The six samples ofyL[n] are given by
If we append h[n] with a single zero-
valued sample andconvert it into a
length-4 sequence he[n], the 4-point
circularconvolution yC[n] of he[n] and
x[n] is given by
Next form
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Linear Convolution using the DFT
Or equivalently
Computing the above for m = 0, 1, 2, 3,
. . . , and substituting the values of xm[n]we arrive at
Then, we reject the first M 1 samples
of wm[n] and abutthe remaining M M
+ 1 samples of wm[n] to form yL[n], the
linear convolution ofh[n] and x[n]
Ifym[n] denotes the saved portion ofwm[n], i.e.,
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Linear Convolution using the DFT
The approach is called overlap-save
method since the input is segmented
into overlapping sections and parts of
the results of the circular convolutions
are saved and abutted to determine the
linear convolution result
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Discrete Cosine Transform
In general, the N-point DFT X[k] oflength-N real sequence is a complex
sequence satisfying the symmetry
condition
For N even, the DFT samples X[0]
and X[N-2]/2 are real and distinct. The
remaining N-2 DFT samples are
complex and only half of these samples
are distinct
For N odd, the DFT samples x[0] is
real and the remaining N-1 DFT
samples are complex, of which only half
of these samples are distinct
The DFT of a real symmetric and anti-symmetric finite sequence is a product
of a linear phase term and a real
amplitude function.
Orthogonal transform is based onconverting an arbitrary sequence into
either a symmetric or an anti-symmetric
sequence and then extracting the real
orthogonal transform coefficients from
the DFT, the transform develop via this
approach is called discrete cosine
transform (DCT)
kX=X[k]*
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Discrete Cosine Transform
To develop the expression for theDCT for symmetric periodic sequence,
consider type 1 DCT.
then the extracted periodic is given by
To develop the Type-2 DCT, extracty[n] of the symmetric periodic sequence
let x[n] be a length-N sequencedefined for 0nN-1. First, x[n] is
extended to a length-2N sequence by
zero padding.
Next, type-2 symmetric sequence y[n]
of length 2N is formed from xe[n]
according to
{}{ cba=x[n]
{}{ cdcba=y[n]
{}{ bcddcba=y[n]
2,010],[][
nN
Nnnxnxe
][nxe
120
]12[][][
Nn
nNxnxn ee
2],12[
0],[
nNnNx
nnx
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Discrete Cosine Transform
The generated sequence y[n]satisfies the symmetry property.
the 2N-point DFT Y[k] of the length-2N sequence y[n] is thus
rewrite
By making a change of variables, the
DFT Y[k] can be expressed as
The Type-2 N point DCT,
The samples of XDCT[k] are real for
real sequence x[n]
]12[][ nNyn
20,][][12
0
2 nWnykYN
n
kn
N
120
,]12[,][
,][][][
12
0
2
1
0
2
1
0
12
0
22
Nk
WnNxWnx
WnyWnykY
N
n
kn
N
N
n
kn
N
N
n
N
n
kn
N
kn
N
120
,2
)12(cos][2
][
,][][][
1
0
2/
2
1
0
2/
22
2/
222
1
0
1
0
)12(
222
Nk
N
nknxW
WWWWnxW
WWnxWnxkY
N
n
k
N
N
n
k
N
kn
N
k
N
kn
N
kn
N
N
n
N
n
Nk
N
kn
N
kn
N
XDCT
[ k]= n= 0
N 1
2x[n ] cos(k(2n + 1)2N ),0 k 2N 1
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Discrete Cosine Transform
The inverse discrete cosine transform
(IDCT) of an N-point DCT XDCT[k] is
given by
where
a DCT pair may often be denoted as
it can be shown that
In other words, the basis sequence
are orthogonal to each other.
To verify that x[n] is indeed the IDCT
of XDCT[k]
The IDCT of an N-point DCT XDCT[k]
can also be computed using the DFT
120
2
)12(cos][][
1][
1
0
Nk
N
nkkXk
Nnx
N
n
DCT
1102/1][k
kk
x [ k] DCT
XDCT[ k]
,,0
,0,2/1
,0,1
2
)12(cos
2
)12(cos
1 1
0
mk
mk
mk
N
nm
N
nk
N
N
n
Nnk2
)12(cos
10
2
)12(cos
2
)12(cos
1][][2
2
)12(cos
2
)12(cos][][
2][
1
0
1
0
1
0
1
0
Nk
N
nk
N
nl
NlXl
N
nk
N
nllXl
NkX
N
l
N
l
DCT
N
n
N
l
DCTDCT
21],2[
,,0
10][
][
2/
2
2/
2
kNkNXW
Nk
NkkXW
kY
DCT
k
N
DCT
k
N
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Discrete Cosine Transform
The 2N-point IDFT y[n] of Y[k] is
given by
We get
The length-N IDCT x[n] of the N-point
DCT
The DCT satisfies a number ofproperties that are useful in certain
application. Assume all time domain
sequences to be length N with an N-
point DCT.
The DCT XDCT[k] of a sequence
obtained by a linear combination of two
sequences, g[n] and h[n]. Where and arbitrary constants.
20,][2
1][
12
0
2 nWkyNnN
k
kn
N
12
1
(
2
1
0
)2
1(
2 ]2[
2
1][
2
1][
N
Nk
DC
N
k
kn
NDCT kX
N
WkX
N
n
1
1
2)(2
1(
2
1
0
)2
1(
2 ][2
1][
2
1 N
k
n
NDC
N
k
kn
NDCTkX
NWkX
N
1
1
(
2
12
0
)2
1(
2 ][2
1][
2
1 N
k
DC
N
k
kn
NDCTkX
NWkX
N
120
2
)12(cos][
1
2
]0[ 1
1
Nn
nkkX
NN
XN
k
DCT
DCT
10][][ Nnnynx
DCT Properties
][][
][][
kHnh
kGng
DCT
DCT
DCT
DCT
Linearity Property
[][][][ kGnhng DDCTDCT
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The Haar Transform
The DCT of the conjugate sequenceis given by
The energy preservation property of
the DCT is similar to the persevals
relation for the DFT.
Symmetry Properties
Energy Preservation Property
][][ ** kGng DCTDCT
1
0
1
0
2|][|][
2
1|][|
N
k
D
N
n
GkN
ng
The discrete-time Haar transform isderived by sampling the continouse-
time Haar function.
The set of Haar function hl(t) contains
N numbers, with N a power-of-2 positive
integer: that is N=2v+1, where v 0.
In defining the Haar function, the
integer subscript l is uniquely
represented as a function of two non
negative integer variable, r and s.
Where variables r and s have ranges
0 r v; 0 s 2r
The Haar Transform
12 sl r
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The Haar Transform
The NN discrete-time Haar
transform matrix HN is obtained by
discretizing the Haar functions at
discrete values of t, given by t = n/N,
0 n N-1
Definition
To derive the high order Haartransform
Where denotes the kroneckerproduct and IK is the KK identity.
The N-point transform XHaar of length
N sequence x[n] is given by
where l= 2r+s 1
10,1)()( 0,00 tthth
00
22
5.0
2
,2
5.0
2
1,2
)()(
2/
2/
,
tforotherwise
s
t
s
st
s
thth rrr
rr
r
srl
.1,112
11
2
2/
2
12v
I
HH
v
v
vv
,.xHhNHaar
HaHaarHaarHaar XXXh []...1[]0[
Nxxxx ]1[]...1[]0[
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The Haar Transform
The inverse Haar transform is giveby
2 2 normalized Haar ransform
matrix is given by
Higher order normalized Haar
transform is given by
Haar Transform Properties
The Haar transform matrix is
orthogonal and hence
Haar transform expression reduced to
If denote the (k, l)-th element of HN ashN(k,l) then
t
NN HNH
11
10
[),(1
][1
0
Nn
XlkhN
nxN
k
HaN
HaarNXHx1
2
1
2
1
2
1
2
1
x
.0,
2
1
2
1
2
1
2
1
2
,2
21v
I
H
H
v
v
v
n
Orthogonality Property
HaartNXH
Nx1
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The Haar Transform
The expression is similar to theParsevals relation for the DFT also
exists for the Haar transform
If L samples of the transform with
indices in the range R are set to zero
with L < < N and if x(m) [n] denotes the
inverse of the modified transform, then
a measure of the energy compaction
property
Energy conservation property Consider the energy compaction
proper of the DFT, the DCT and theHaar transform.
N-point DFT, the high frequency
indices around
The Discrete Fourier Transform
12
1],[
2
1
2
1
,0
2
10],[
][)(
NkLN
kX
N
k
LN
LNkkX
kX
DFT
DFT
m
DFT
1
0
1
0
2|][|
1|][|
N
k
Ha
N
n
kHN
nx
Energy Compaction Properties
1
0
)(|][][|
1)(N
k
mxnx
NL
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The Energy Compaction Properties
The original time domain sequence
x[n] is obtained by computing the IDFT
The corresponding approximation
error is given by
The high frequency samples havehigh indices and thus the modified DCT
obtained
The original time domain sequence
x[n] is obtained by computing the IDCT
Hence the approximation error is
The modified Haar transform obtained
The x[n] is obtained by computing theIDCT
1
0
)()(][
1][
N
k
m
DFT
m
DFT kXNn
1
0
)(|][][|
1)(N
k
m
DXnx
NL
The Discrete Cosine Transform
10][][)(
NkkXkDCTm
DCT
LN
k
mDCT
mDCT kkXkNn
1
0
)()(
22(co][][1][
1
0
)( |][][|1)(N
k
m
DCxnxNL
The Haar Transform
0
10][][)(
kLN
NkkXkHaarm
Haar
LN
k
H
m
Haar XnkhNnx
1
0
)( [],[1][
1)( |][][|
1)(N
m
HaxnxL