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DFT Problems

3: Discrete Cosine Transform

• DFT Problems

• DCT

• DCT of sine wave

• DCT/DFT Equivalence

• DCT Properties

• IDCT

• Energy Conservation

• Energy Compaction

• Frame-based coding

• Lapped Transform

• MDCT

• Summary

• MATLAB routines

DSP and Digital Filters (2013-3622) Transforms: 3 – 2 / 14

For processing 1-D or 2-D signals (especially coding), a common method is

to divide the signal into “frames” and then apply an invertible transform to

each frame that compresses the information into few coefficients.

The DFT has some problems when used for this purpose:

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DFT Problems


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines






• N real x[n] ↔ N complex X [k] : 2 real, N 2 − 1 conjugate pairs

→

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DFT Problems


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines







→

• DFT ∝ the DTFT of a periodic signal formed by replicating x[n] .

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DFT Problems


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines







→

• DFT ∝ the DTFT of a periodic signal formed by replicating x[n] .⇒ Spurious frequency components from boundary discontinuity.

N=20f=0.08

→

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DFT Problems


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines







→

• DFT ∝ the DTFT of a periodic signal formed by replicating x[n] .⇒ Spurious frequency components from boundary discontinuity.

N=20f=0.08

→

The Discrete Cosine Transform (DCT) overcomes these problems.

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DCT


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


To form the Discrete Cosine Transform (DCT), replicate x[0 : N − 1] but inreverse order and insert a zero between each pair of samples:

→

0 12 23

y[r]

Take the DFT of length 4N real symmetric sequence

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DCT


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



→

0 12 23

y[r]

Take the DFT of length 4N real symmetric sequenceResult is real, symmetric and anti-periodic:

012

23

Y[k]

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DCT


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



→

0 12 23

y[r]

Take the DFT of length 4N real symmetric sequenceResult is real, symmetric and anti-periodic: only need first N values

012

23

Y[k]

→

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DCT


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



→

0 12 23

y[r]


012

23

Y[k]

→

Forward DCT: X C [k] =N −1

n=0 x[n]cos 2π(2n+1)k

4N for k = 0 : N − 1

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DCT


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



→

0 12 23

y[r]


012

23

Y[k]

→

Forward DCT: X C [k] =N −1


4N for k = 0 : N − 1

Compare DFT: X F [k] =

N −1n=0 x[n]exp −j2π(4n+0)k4N

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DCT of sine wave


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


DCT: X C [k] =N −1


4N

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DCT of sine wave


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


DCT: X C [k] =N −1


4N

f = mN

x[n]N=20f=0.10

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DCT of sine wave


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


DCT: X C [k] =N −1


4N

f = mN

x[n]N=20f=0.10

|X F [k]|

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DCT of sine wave


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


DCT: X C [k] =N −1


4N

f = mN

f = mN

x[n] N=20f=0.10

N=20f=0.08

|X F [k]|

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DCT of sine wave


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


DCT: X C [k] =N −1


4N

f = mN

f = mN

x[n] N=20f=0.10

N=20f=0.08

|X F [k]|

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DCT of sine wave


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


DCT: X C [k] =N −1


4N

f = mN

f = mN

x[n] N=20f=0.10

N=20f=0.08

|X F [k]|

DFT: Real→Complex; Freq range [0, 1]; Poorly localized unlessf = m

N ; |X F [k]| ∝ k

−1 for Nf < k N 2

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DCT of sine wave


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


DCT: X C [k] =N −1


4N

f = mN

f = mN

x[n] N=20f=0.10

N=20f=0.08

|X F [k]|

|X C [k]|


N ; |X F [k]| ∝ k

−1 for Nf < k N 2

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DCT of sine wave


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


DCT: X C [k] =N −1


4N

f = mN

f = mN

x[n] N=20f=0.10

N=20f=0.08

|X F [k]|

|X C [k]|


N ; |X F [k]| ∝ k

−1 for Nf < k N 2DCT: Real→Real; Freq range [0, 0.5]; Well localized ∀f ;

|X C [k]| ∝ k−2 for 2Nf < k < N

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DCT/DFT Equivalence


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


X C [k] =N −1


4N 0 12 23

y[r]

Define y[r] =

0 r even

xr−1

2

r = 1 : 2 : 2N − 1x4N −1−r

2

r = 2N + 1 : 2 : 4N − 1

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DCT/DFT Equivalence


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


X C [k] =N −1


4N 0 12 23

y[r]

Define y[r] =

0 r even

xr−1

2

r = 1 : 2 : 2N − 1x4N −1−r

2

r = 2N + 1 : 2 : 4N − 1

Y F [k] =4N −1

r=0 y[r]W kr4N where W M = e

−j2πM

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DCT/DFT Equivalence


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


X C [k] =N −1


4N 0 12 23

y[r]

Define y[r] =

0 r even

xr−1

2

r = 1 : 2 : 2N − 1x4N −1−r

2

r = 2N + 1 : 2 : 4N − 1

Y F [k] =4N −1


−j2πM

(i)

=2N −1

n=0 y[2n + 1]W (2n+1)k

4N

(i) odd r only: r = 2n + 1

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DCT/DFT Equivalence


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


X C [k] =N −1


4N 0 12 23

y[r]

Define y[r] =

0 r even

xr−1

2

r = 1 : 2 : 2N − 1x4N −1−r

2

r = 2N + 1 : 2 : 4N − 1

Y F [k] =4N −1


−j2πM

(i)

=2N −1

n=0 y[2n + 1]W (2n+1)k

4N (ii)=N −1

n=0 y[2n + 1]W (2n+1)k4N

+N −1

m=0 y[4N − 2m − 1]W (4N −2m−1)k4N

(i) odd r only: r = 2n + 1(ii) reverse order for n ≥ N : m = 2N − 1 − n

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DCT/DFT Equivalence


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


X C [k] =N −1


4N 0 12 23

y[r]

Define y[r] =

0 r even

xr−1

2

r = 1 : 2 : 2N − 1x4N −1−r

2

r = 2N + 1 : 2 : 4N − 1

Y F [k] =4N −1


−j2πM

(i)

=2N −1

n=0 y[2n + 1]W (2n+1)k

4N (ii)=N −1

n=0 y[2n + 1]W (2n+1)k4N

+N −1

m=0 y[4N − 2m − 1]W (4N −2m−1)k4N

(iii)= x[n]W

(2n+1)k4N +

N −1m=0 x[m]W

−(2m+1)k4N

(i) odd r only: r = 2n + 1(ii) reverse order for n ≥ N : m = 2N − 1 − n(iii) substitute y definition

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DCT/DFT Equivalence


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


X C [k] =N −1


4N 0 12 23

y[r]

Define y[r] =

0 r even

xr−1

2

r = 1 : 2 : 2N − 1x4N −1−r

2

r = 2N + 1 : 2 : 4N − 1

Y F [k] =4N −1


−j2πM

(i)

=2N −1

n=0 y[2n + 1]W (2n+1)k

4N (ii)=N −1

n=0 y[2n + 1]W (2n+1)k4N

+N −1

m=0 y[4N − 2m − 1]W (4N −2m−1)k4N

(iii)= x[n]W

(2n+1)k4N +

N −1m=0 x[m]W

−(2m+1)k4N

= 2

N −1n=0 x[n]cos

2π(2n+1)k4N = 2X C [k]

(i) odd r only: r = 2n + 1(ii) reverse order for n ≥ N : m = 2N − 1 − n(iii) substitute y definition & W 4Nk

4N = e−j2π

4Nk4N ≡ 1

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


Definition: X [k] =N −1


4N

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines




4N • Linear: αx[n] + βy[n] → αX [k] + βY [k]

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines




4N • Linear: αx[n] + βy[n] → αX [k] + βY [k]• DFT “Convolution←→Multiplication” property does not hold

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines




4N • Linear: αx[n] + βy[n] → αX [k] + βY [k]• DFT “Convolution←→Multiplication” property does not hold• Symmetric: X [−k] = X [k] since cos−αk = cos +αk

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines





• Anti-periodic: X [k + 2N ] = −X [k]

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines





• Anti-periodic: X [k + 2N ] = −X [k] because:◦ cos(θ + π) = − cos θ◦ 2π(2n + 1)(k + 2N ) = 2π(2n + 1)k + 8πN n + 4N π

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines





• Anti-periodic: X [k + 2N ] = −X [k] because:◦ cos(θ + π) = − cos θ◦ 2π(2n + 1)(k + 2N ) = 2π(2n + 1)k + 8πN n + 4N π⇒X [N ] = 0 since X [N ] = X [−N ] = −X [−N + 2N ]

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines






• Periodic: X [k + 4N ] = −X [k + 2N ] = X [k]

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


Definition: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N

• Linear: αx[n] + βy[n] → αX [k] + βY [k]• DFT “Convolution←→Multiplication” property does not hold• Symmetric: X [−k] = X [k] since cos−αk = cos +αk


• Periodic: X [k + 4N ] = −X [k + 2N ] = X [k]DCT basis functions:

0:

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


Definition: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N




0:

1:

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


Definition: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N




0:

1:

2:

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


Definition: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N




0:

3:

1:

2:

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


Definition: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N




0:

3:

1: 4:

2:

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DCT Properties


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


Definition: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N




0:

3:

1: 4:

2: 5:

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IDCT


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


y[r] = 14N

4N −1k=0 Y [k]W

−rk4N

0

12

23

Y[k]

W ba = e−j 2πb

a

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IDCT


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


y[r] = 14N

4N −1k=0 Y [k]W

−rk4N =

12N

4N −1k=0 X [k]W

−rk4N

[Y [k] = 2X [k]]

0

12

23

Y[k]

W ba = e−j 2πb

a

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IDCT


• DFT Problems

• DCT



• DCT Properties

• IDCT

• Energy Conservation• Energy Compaction



• MDCT

• Summary

• MATLAB routines


y[r] = 14N

4N −1k=0 Y [k]W

−rk4N =

12N

4N −1k=0 X [k]W

−rk4N

x[n] = y[2n + 1] [Y [k] = 2X [k]]

0

12

23

Y[k]

W ba = e−j 2πb

a

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IDCT


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


y[r] = 14N

4N −1k=0 Y [k]W

−rk4N =

12N

4N −1k=0 X [k]W

−rk4N

x[n] = y[2n + 1] = 12N 4N −1

k=0 X [k]W −(2n+1)k4N [Y [k] = 2X [k]]

0

12

23

Y[k]

W ba = e−j 2πb

a

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IDCT


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


y[r] = 14N

4N −1k=0 Y [k]W

−rk4N =

12N

4N −1k=0 X [k]W

−rk4N

x[n] = y[2n + 1] = 12N 4N −1

k=0 X [k]W −(2n+1)k4N [Y [k] = 2X [k]]

(i)= 1

2N

2N −1k=0 X [k]W

−(2n+1)k4N

− 12N 2N −1

l=0 X [l]W −(2n+1)(l+2N )4N

0

12

23

Y[k]

W ba = e−j 2πb

a

(i) k = l + 2N for k ≥ 2N and X [k + 2N ] = −X [k]

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IDCT


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


y[r] = 14N

4N −1k=0 Y [k]W

−rk4N =

12N

4N −1k=0 X [k]W

−rk4N

x[n] = y[2n + 1] = 12N 4N −1

k=0 X [k]W −(2n+1)k4N [Y [k] = 2X [k]]

(i)= 1

2N

2N −1k=0 X [k]W

−(2n+1)k4N

− 12N 2N −1

l=0 X [l]W −(2n+1)(l+2N )4N

0

12

23

Y[k]

(ii)= 1

N

2N −1k=0 X [k]W

−(2n+1)k4N W

ba = e

−j 2πba


(ii) (2n+1)(l+2N )

4N = (2n+1)l

4N + n + 12

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IDCT


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


y[r] = 14N

4N −1k=0 Y [k]W

−rk4N =

12N

4N −1k=0 X [k]W

−rk4N

x[n] = y[2n + 1] = 12N 4N −1

k=0 X [k]W −(2n+1)k4N [Y [k] = 2X [k]]

(i)= 1

2N

2N −1k=0 X [k]W

−(2n+1)k4N

− 12N 2N −1

l=0 X [l]W −(2n+1)(l+2N )4N

0

12

23

Y[k]

(ii)= 1

N

2N −1k=0 X [k]W

−(2n+1)k4N W

ba = e

−j 2πba

(iii)

=

1

N X

[0] +

1

N N −1

k=1 X

[k

]W

−(2n+1)k

4N +

1

N X

[N

]W

−(2n+1)N

4N + 1N

N −1r=1 X [2N − r]W

−(2n+1)(2N −r)4N


(ii) (2n+1)(l+2N )

4N = (2n+1)l

4N + n + 12

(iii) k = 2N − r for k > N

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IDCT


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


y[r] = 14N

4N −1k=0 Y [k]W

−rk4N =

12N

4N −1k=0 X [k]W

−rk4N

x[n] = y[2n + 1] = 12N 4N −1

k=0 X [k]W −(2n+1)k4N [Y [k] = 2X [k]]

(i)= 12N

2N −1k=0 X [k]W

−(2n+1)k4N

− 12N 2N −1

l=0 X [l]W −(2n+1)(l+2N )4N

0

12

23

Y[k]

(ii)= 1

N

2N −1k=0 X [k]W

−(2n+1)k4N W

ba = e

−j 2πba

(iii)= 1

N X [0] + 1

N N −1

k=1 X [k]W

−(2n+1)k

4N + 1

N X [N ]W

−(2n+1)N

4N + 1N

N −1r=1 X [2N − r]W

−(2n+1)(2N −r)4N

(iv)= 1

N X [0] + 1

N

N −1k=1 X [k]W

−(2n+1)k4N

+ 1N

N −1r=1 −X [r]W

(2n+1)r+2N 4N


(ii) (2n+1)(l+2N )

4N = (2n+1)l

4N + n + 12

(iii) k = 2N − r for k > N (iv) X [N ] = 0 and X [2N − r] = −X [r]

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IDCT


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


y[r] = 14N

4N −1k=0 Y [k]W

−rk4N =

12N

4N −1k=0 X [k]W

−rk4N

x[n] = y[2n + 1] = 12N 4N −1

k=0 X [k]W −(2n+1)k4N [Y [k] = 2X [k]]

(i)= 12N

2N −1k=0 X [k]W

−(2n+1)k4N

− 12N 2N −1

l=0 X [l]W −(2n+1)(l+2N )4N

012

23

Y[k]

(ii)= 1

N

2N −1k=0 X [k]W

−(2n+1)k4N W

ba = e

−j 2πba

(iii)= 1

N X [0] + 1

N N −1

k=1 X [k]W

−(2n+1)k

4N + 1

N X [N ]W

−(2n+1)N

4N + 1N

N −1r=1 X [2N − r]W

−(2n+1)(2N −r)4N

(iv)= 1

N X [0] + 1

N

N −1k=1 X [k]W

−(2n+1)k4N

+ 1N

N −1r=1 −X [r]W

(2n+1)r+2N 4N

x[n] = 1N X [0] + 2N

N −1k=1 X [k]cos 2π(2n+1)k4N


(ii) (2n+1)(l+2N )

4N = (2n+1)l

4N + n + 12


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IDCT


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


y[r] = 14N

4N −1k=0 Y [k]W

−rk4N =

12N

4N −1k=0 X [k]W

−rk4N

x[n] = y[2n + 1] = 12N 4N −1

k=0 X [k]W −(2n+1)k4N [Y [k] = 2X [k]]

(i)= 12N

2N −1k=0 X [k]W

−(2n+1)k4N

− 12N 2N −1

l=0 X [l]W −(2n+1)(l+2N )4N

012

23

Y[k]

(ii)= 1

N

2N −1k=0 X [k]W

−(2n+1)k4N W

ba = e

−j 2πba

(iii)= 1

N X [0] + 1

N N −1

k=1 X [k]W

−(2n+1)k

4N + 1

N X [N ]W

−(2n+1)N

4N + 1N

N −1r=1 X [2N − r]W

−(2n+1)(2N −r)4N

(iv)= 1

N X [0] + 1

N

N −1k=1 X [k]W

−(2n+1)k4N

+ 1N

N −1r=1 −X [r]W

(2n+1)r+2N 4N

x[n] = 1N X [0] + 2N

N −1k=1 X [k]cos 2π(2n+1)k4N = Inverse DCT


(ii) (2n+1)(l+2N )

4N = (2n+1)l

4N + n + 12


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Energy Conservation


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


DCT: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N

IDCT: x[n] = 1N

X [0] + 2N

N −1k=1 X [k]cos

2π(2n+1)k4N

rep

→0 12 23

y[r]

DFT

→ 012

23

Y[k]

÷2

→

Energy: E =N −1

n=0 |x[n]|2

: E → 2E → 8N E →≈ 0.5N E

E = 1N |X |2

[0] + 2N N −1

n=1 |X |2

[n]

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Energy Conservation


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


DCT: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N

IDCT: x[n] = 1N

X [0] + 2N

N −1k=1 X [k]cos

2π(2n+1)k4N

rep

→0 12 23

y[r]

DFT

→ 012

23

Y[k]

÷2

→

Energy: E =N −1

n=0 |x[n]|2

: E → 2E → 8N E →≈ 0.5N E

E = 1N |X |2

[0] + 2N N −1

n=1 |X |2

[n]

Orthogonal DCT (preserves energy)

Define: c[k] =

2−δkN ⇒ c[0] =

1N

, c[k = 0] =

2N

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Energy Conservation


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


DCT: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N

IDCT: x[n] = 1N

X [0] + 2N

N −1k=1 X [k]cos

2π(2n+1)k4N

rep

→0 12 23

y[r]

DFT

→ 012

23

Y[k]

÷2

→

Energy: E =N −1

n=0 |x[n]|2

: E → 2E → 8N E →≈ 0.5N E

E = 1N |X |2

[0] + 2N N −1

n=1 |X |2

[n]


Define: c[k] =

2−δkN ⇒ c[0] =

1N

, c[k = 0] =

2N

ODCT: X [k] = c[k]

N −1n=0 x[n]cos 2

π(2n+1)k4N

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Energy Conservation


• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


DCT: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N

IDCT: x[n] = 1N

X [0] + 2N

N −1k=1 X [k]cos

2π(2n+1)k4N

rep

→0 12 23

y[r]

DFT

→ 012

23

Y[k]

÷2

→

Energy: E =N −1

n=0 |x[n]|2

: E → 2E → 8N E →≈ 0.5N E

E = 1N |X |2

[0] + 2N N −1

n=1 |X |2

[n]


Define: c[k] =

2−δkN ⇒ c[0] =

1N

, c[k = 0] =

2N

ODCT: X [k] = c[k]

N −1n=0 x[n]cos 2

π(2n+1)k4N

IODCT: x[n] = N −1

k=0 c[k]X [k]cos 2π(2n+1)k

4N

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Energy Conservation

3: Discrete Cosine Transform• DFT Problems

• DCT



• DCT Properties

• IDCT




• MDCT

• Summary

• MATLAB routines


DCT: X [k] =

N −1n=0 x[n]cos

2π(2n+1)k4N

IDCT: x[n] = 1N

X [0] + 2N

N −1k=1 X [k]cos

2π(2n+1)k4N

rep

→0 12 23

y[r]

DFT

→ 012

23

Y[k]

÷2

→

Energy: E =N −1

n=0 |x[n]|2

: E → 2E → 8N E →≈ 0.5N E

E = 1N |X |2

[0] + 2N N −1

n=1 |X |2

[n]


Define: c[k] =

2−δkN ⇒ c[0] =

1N

, c[k = 0] =

2N

ODCT: X [k] = c[k]

N −1n=0 x[n]cos 2π(2n+1)k4N

IODCT: x[n] = N −1

k=0 c[k]X [k]cos 2π(2n+1)k

4N

Note: MATLAB dct() calculates the ODCT

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Energy Compaction


• DCT



• DCT Properties

• IDCT

•

Energy Conservation• Energy Compaction



• MDCT

• Summary

• MATLAB routines


If consecutive x[n] are positively correlated, DCT concentrates energy in afew X [k] and decorrelates them.

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Energy Compaction


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines



Example: Markov Process: x[n] = ρx[n − 1] +

1 − ρ2u[n]where u[n] is i.i.d. unit Gaussian.Then

x2[n]

= 1 and x[n]x[n − 1] = ρ.

Covariance of vector x is Si,j =

xxH i,j

= ρ|i−j|.

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Energy Compaction


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines





x2[n]

= 1 and x[n]x[n − 1] = ρ.


xxH i,j

= ρ|i−j|.

Suppose ODCT of x is Cx and DFT is Fx.

Covariance of Cx is

CxxH

CH

= CSCH

(similarly FSFH

)Diagonal elements give mean coefficient energy.

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Energy Compaction


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines





x2[n]

= 1 and x[n]x[n − 1] = ρ.


xxH i,j

= ρ|i−j|.


Covariance of Cx is

CxxH

CH

= CSCH

(similarly FSFH


5 10 15 20 25 30

50

60

70

80

90

100

DCT

DFT

ρ =0.9N=32

No of coefficients

C u m u l a t i v e

e n e r g y ( % )

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Energy Compaction


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines





x2[n]

= 1 and x[n]x[n − 1] = ρ.


xxH i,j

= ρ|i−j|.


Covariance of Cx is

CxxH

CH

= CSCH

(similarly FSFH


5 10 15 20 25 30

50

60

70

80

90

100

DCT

DFT

ρ =0.9N=32

No of coefficients

C u m u l a t i v e e n e r g y ( % )

• Used in MPEG and JPEG (superseded byJPEG2000 using wavelets)

• Used in speech recognition to decorrelate:DCT of log spectrum

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Energy Compaction


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines





x2[n]

= 1 and x[n]x[n − 1] = ρ.


xxH i,j

= ρ|i−j|.


Covariance of Cx is

CxxH

CH

= CSCH

(similarly FSFH


5 10 15 20 25 30

50

60

70

80

90

100

DCT

DFT

ρ =0.9N=32

No of coefficients

C u m u l a t i v e e n e r g y ( % )

• Used in MPEG and JPEG (superseded byJPEG2000 using wavelets)

• Used in speech recognition to decorrelate:DCT of log spectrum

Energy compaction good for coding (low-valued coefficients can be set to 0)

Decorrelation good for coding and for probability modelling

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Frame-based coding


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines


• Divide continuous signalinto frames

x[n]

X[k] k=30/220

y[n]

y[n]-x[n]

F b d di

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Frame-based coding


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines



• Apply DCT to each frame

x[n]

X[k] k=30/220

y[n]

y[n]-x[n]

F b d di

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Frame-based coding


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines




• Encode DCT◦ e.g. keep only 30 X [k]

x[n]

X[k] k=30/220

y[n]

y[n]-x[n]

F b d di

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Frame-based coding


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines





• Apply IDCT → y[n]

x[n]

X[k] k=30/220

y[n]

y[n]-x[n]

F b d di

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Frame-based coding


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines





• Apply IDCT → y[n]

x[n]

X[k] k=30/220

y[n]

y[n]-x[n]

Problem: Coding may create discontinuities at frame boundaries

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Lapped Transform

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Lapped Transform


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines


Modified Discrete Cosine Transform (MDCT): overlapping frames 2N long

x[0 : 2N − 1]x[n]

X[k]

yA[n]

yB[n]

y[n]

y[n]-x[n]

0 N

N

2N

2N

3N

3N

4N

4N

Lapped Transform

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Lapped Transform


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X [0 : N − 1] x[n]

X[k]

yA[n]

yB[n]

y[n]

y[n]-x[n]

0 N

N

2N

2N

3N

3N

4N

4N

MDCT: 2N → N coefficients

Lapped Transform

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Lapped Transform


• DCT



• DCT Properties

• IDCT

•




• MDCT

• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X [0 : N − 1]

IMDCT

→ yA[0 : 2N − 1]

x[n]

X[k]

yA[n]

yB[n]

y[n]

y[n]-x[n]

0 N

N

2N

2N

3N

3N

4N

4N

MDCT: 2N → N coefficients, IMDCT: N → 2N samples

Lapped Transform

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Lapped Transform


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X [0 : N − 1]

IMDCT

→ yA[0 : 2N − 1]x[N : 3N − 1]

MDCT→ X [N : 2N − 1]

x[n]

X[k]

yA[n]

yB[n]

y[n]

y[n]-x[n]

0 N

N

2N

2N

3N

3N

4N

4N


Lapped Transform

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Lapped Transform


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X [0 : N − 1]

IMDCT

→ yA[0 : 2N − 1]x[N : 3N − 1]

MDCT→ X [N : 2N − 1]

IMDCT→ yB [N : 3N − 1]

x[n]

X[k]

yA[n]

yB[n]

y[n]

y[n]-x[n]

0 N

N

2N

2N

3N

3N

4N

4N


Lapped Transform

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Lapped Transform


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X [0 : N − 1]

IMDCT

→ yA[0 : 2N − 1]x[N : 3N − 1]

MDCT→ X [N : 2N − 1]


x[2N : 4N − 1]MDCT→ X [2N : 3N − 1]

x[n]

X[k]

yA[n]

yB[n]

y[n]

y[n]-x[n]

0 N

N

2N

2N

3N

3N

4N

4N


Lapped Transform

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Lapped Transform


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X [0 : N − 1]

IMDCT

→ yA[0 : 2N − 1]x[N : 3N − 1]

MDCT→ X [N : 2N − 1]


x[2N : 4N − 1]MDCT→ X [2N : 3N − 1]

IMDCT→ yA[2N : 3N − 1]

x[n]

X[k]

yA[n]

yB[n]

y[n]

y[n]-x[n]

0 N

N

2N

2N

3N

3N

4N

4N


Lapped Transform

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Lapped Transform


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X [0 : N − 1]

IMDCT

→ yA[0 : 2N − 1]x[N : 3N − 1]

MDCT→ X [N : 2N − 1]


x[2N : 4N − 1]MDCT→ X [2N : 3N − 1]


y[∗] = yA[∗] + yB [∗]

x[n]

X[k]

yA[n]

yB[n]

y[n]

y[n]-x[n]

0 N

N

2N

2N

3N

3N

4N

4N

MDCT: 2N → N coefficients, IMDCT: N → 2N samplesEven frames → yA, Odd frames → yB then y = yA + yB

Lapped Transform

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Lapped Transform


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X [0 : N − 1]

IMDCT

→ yA[0 : 2N − 1]x[N : 3N − 1]

MDCT→ X [N : 2N − 1]


x[2N : 4N − 1]MDCT→ X [2N : 3N − 1]


y[∗] = yA[∗] + yB [∗]

x[n]

X[k]

yA[n]

yB[n]

y[n]

y[n]-x[n]

0 N

N

2N

2N

3N

3N

4N

4N

MDCT: 2N → N coefficients, IMDCT: N → 2N samplesEven frames → yA, Odd frames → yB then y = yA + yBErrors cancel exactly: Time-domain alias cancellation (TDAC)

MDCT

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MDCT


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N

IMDCT: yA,B [n] = 1N

N −1k=0 X M [k]cos

2π(2n+1+N )(2k+1)8N

MDCT

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MDCT


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N



2π(2n+1+N )(2k+1)8N

MDCT can be made from a DCT of length 2N

MDCT

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MDCT


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N



2π(2n+1+N )(2k+1)8N


Split x[n] into four N

2 -vectors: x =

a b c d

MDCT

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MDCT


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N



2π(2n+1+N )(2k+1)8N



2 -vectors: x =

a b c d

Now form the N -vector u = −d −←−c a −

←−b

where←−c is the vector c but in reverse order

MDCT

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MDCT


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N



2π(2n+1+N )(2k+1)8N



2 -vectors: x =

a b c d


←−b


Now form v =

u −←−u

and take 2N DCT

MDCT

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MDCT


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N



2π(2n+1+N )(2k+1)8N



2 -vectors: x =

a b c d


←−b


Now form v =

u −←−u

and take 2N DCT

V C [2k] = 0 because of symmetry, so set X M [k] = V C [2k + 1]

MDCT

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C


• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N



2π(2n+1+N )(2k+1)8N



2 -vectors: x =

a b c d


←−b


Now form v =

u −←−u

and take 2N DCT

V C [2k] = 0 because of symmetry, so set X M [k] = V C [2k + 1]IMDCT

Undo above to get X M [k] → u = −d −←−c a −

←−b

MDCT

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• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N



2π(2n+1+N )(2k+1)8N



2 -vectors: x =

a b c d


←−b


Now form v =

u −←−u

and take 2N DCT



←−b

Create yA[n] =

a −

←−

b b −

←−a c +

←−

d d +

←−c

MDCT

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• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N



2π(2n+1+N )(2k+1)8N



2 -vectors: x = a b c d


←−b


Now form v =

u −←−u

and take 2N DCT



←−b

CreateyA[n] =

a −

←−

b b −

←−a c +

←−

d d +

←−c

Next frame: yB[n] =

c −←−d d −←−c e +

←−f f + ←−e

MDCT

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• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N



2π(2n+1+N )(2k+1)8N

MDCT can be made from a DCT of length 2N Split

x[n] into four N

2 -vectors:

x = a b c d


←−b


Now form v =

u −←−u

and take 2N DCT



←−b

Create yA[n] =

a −

←−

b b −

←−a c +

←−

d d +

←−c

Next frame: yB[n] =

c −←−d d −←−c e +

←−f f + ←−e

Adding together, the←−d and ←−c terms cancel but c and d add.

MDCT

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• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


MDCT: X M [k] =

2N −1n=0 x[n]cos 2π(2n+1+N )(2k+1)8N



2π(2n+1+N )(2k+1)8N

MDCT can be made from a DCT of length 2N Split

x[n] into four N

2 -vectors: x

= a b c d


←−b


Now form v =

u −←−u

and take 2N DCT



←−b

Create yA[n] =

a −

←−

b b −

←−

a c +

←−

d d +

←−

c

Next frame: yB[n] =

c −←−d d −←−c e +

←−f f + ←−e

Adding together, the←−d and ←−c terms cancel but c and d add.

Used in audio coding: MP3, WMA, AC-3, AAC, Vorbis, ATRAC

Summary

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88/97

y


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines


DCT: Discrete Cosine Transform• Equivalent to a DFT of time-shifted double-length

x ←−x

Summary

8/13/2019 00300 Transforms

89/97


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x ←−x

• Often scaled to make an orthogonal transform (ODCT)

Summary

8/13/2019 00300 Transforms

90/97


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x ←−x

• Often scaled to make an orthogonal transform (ODCT)• Better than DFT for energy compaction and decorrelation

Summary

8/13/2019 00300 Transforms

91/97


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x ←−x

• Often scaled to make an orthogonal transform (ODCT)• Better than DFT for energy compaction and decorrelation• Nice convolution property of DFT is lost

Summary

8/13/2019 00300 Transforms

92/97


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x ←−x


MDCT: Modified Discrete Cosine Transform

• Lapped transform: 2N → N → 2N

Summary

8/13/2019 00300 Transforms

93/97


• DFT Problems

• DCT



• DCT Properties

• IDCT





• MDCT

• Summary

• MATLAB routines



x ←−x


MDCT: Modified Discrete Cosine Transform

• Lapped transform: 2N → N → 2N • Aliasing errors cancel out when overlapping output frames are added

Summary

8/13/2019 00300 Transforms

94/97


• DFT Problems

• DCT



• DCT Properties

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Documents

00300 Transforms