Discrete Cosine Transform

3: Discrete Cosine Transform


• DFT Problems

• DCT +

• Basis Functions

• DCT of sine wave

• DCT Properties

• Energy Conservation

• Energy Compaction

• Frame-based coding

• Lapped Transform +

• MDCT (Modified DCT)

• MDCT Basis Elements

• Summary

• MATLAB routines

DSP and Digital Filters (2017-10120) Transforms: 3 – 1 / 14

DFT Problems


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


For processing 1-D or 2-D signals (especially coding), a common method isto divide the signal into “frames” and then apply an invertible transform toeach frame that compresses the information into few coefficients.

The DFT has some problems when used for this purpose:

DFT Problems


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




• N real x[n]↔N complex X[k] : 2 real, N2 − 1 conjugate pairs

→

DFT Problems


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines





→

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

DFT Problems


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• 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

→

DFT Problems


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• 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.

DCT +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• 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, odd-sample-only sequence.

DCT +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



→

0 12 23

y[r]

Take the DFT of length 4N real, symmetric, odd-sample-only sequence.Result is real, symmetric and anti-periodic:

012

23

Y[k]

DCT +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



→

0 12 23

y[r]

Take the DFT of length 4N real, symmetric, odd-sample-only sequence.Result is real, symmetric and anti-periodic: only need first N values

012

23

Y[k]

÷2−→

DCT +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



→

0 12 23

y[r]


012

23

Y[k]

÷2−→

Forward DCT: XC [k] =∑N−1

n=0 x[n] cos 2π(2n+1)k4N for k = 0 : N − 1

DCT +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



→

0 12 23

y[r]


012

23

Y[k]

÷2−→

Forward DCT: XC [k] =∑N−1

n=0 x[n] cos 2π(2n+1)k4N for k = 0 : N − 1

Inverse DCT: x[n] = 1NX[0] + 2

N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


DFT basis functions: x[n] = 1N

∑N−1k=0 X[k]ej2π

kn

N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N

DCT basis functions: x[n] = 1NX[0] + 2

N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N


N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N


N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N


N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N


N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N


N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

Basis Functions


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




kn

N


N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

DCT of sine wave


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


DCT: XC [k] =∑N−1

n=0 x[n] cos 2π(2n+1)k4N

DCT of sine wave


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




f = mN

x[n]N=20f=0.10

DCT of sine wave


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




f = mN

x[n]N=20f=0.10

|XF [k]|

DCT of sine wave


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




f = mN

f 6= mN

x[n]N=20f=0.10

N=20f=0.08

|XF [k]|

DCT of sine wave


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




f = mN

f 6= mN

x[n]N=20f=0.10

N=20f=0.08

|XF [k]|

DCT of sine wave


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




f = mN

f 6= mN

x[n]N=20f=0.10

N=20f=0.08

|XF [k]|

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

N; |XF [k]| ∝ k−1 for Nf < k ≪ N

2

DCT of sine wave


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




f = mN

f 6= mN

x[n]N=20f=0.10

N=20f=0.08

|XF [k]|

|XC [k]|


N; |XF [k]| ∝ k−1 for Nf < k ≪ N

2

DCT of sine wave


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




f = mN

f 6= mN

x[n]N=20f=0.10

N=20f=0.08

|XF [k]|

|XC [k]|


N; |XF [k]| ∝ k−1 for Nf < k ≪ N

2DCT: Real→Real; Freq range [0, 0.5]; Well localized ∀f ;

|XC [k]| ∝ k−2 for 2Nf < k < N

DCT Properties


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


Definition: X[k] =∑N−1


DCT Properties


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




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

DCT Properties


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines





• “Convolution←→Multiplication” property of DFT does not hold /

DCT Properties


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines






• Symmetric: X[−k] = X[k] since cos−αk = cos+αk

DCT Properties


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines







• Anti-periodic: X[k + 2N ] = −X[k]

DCT Properties


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines







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

◦ cos (θ + π) = − cos θ

DCT Properties


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines








◦ cos (θ + π) = − cos θ

⇒X[N ] = 0 since X[N ] = X[−N ] = −X[−N + 2N ]

DCT Properties


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines








◦ cos (θ + π) = − cos θ

⇒X[N ] = 0 since X[N ] = X[−N ] = −X[−N + 2N ]

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

Energy Conservation


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


DCT: X[k] =∑N−1


IDCT: x[n] = 1NX[0] + 2

N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

rep→

0 12 23

y[r]

DFT→ 0

1223

Y[k]

÷2→

Energy Conservation


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


DCT: X[k] =∑N−1


IDCT: x[n] = 1NX[0] + 2

N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

rep→

0 12 23

y[r]

DFT→ 0

1223

Y[k]

÷2→

Energy: E =∑N−1

n=0 |x[n]|2

Energy Conservation


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


DCT: X[k] =∑N−1


IDCT: x[n] = 1NX[0] + 2

N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

rep→

0 12 23

y[r]

DFT→ 0

1223

Y[k]

÷2→

Energy: E =∑N−1

n=0 |x[n]|2= 1

N|X[0]|

2+ 2

N

∑N−1n=1 |X[n]|

2

Energy Conservation


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


DCT: X[k] =∑N−1


IDCT: x[n] = 1NX[0] + 2

N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

rep→

0 12 23

y[r]

DFT→ 0

1223

Y[k]

÷2→

Energy: E =∑N−1

n=0 |x[n]|2= 1

N|X[0]|

2+ 2

N

∑N−1n=1 |X[n]|

2

In diagram above: E → 2E→ 8NE→≈ 0.5NE

Energy Conservation


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


DCT: X[k] =∑N−1


IDCT: x[n] = 1NX[0] + 2

N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

rep→

0 12 23

y[r]

DFT→ 0

1223

Y[k]

÷2→

Energy: E =∑N−1

n=0 |x[n]|2= 1

N|X[0]|

2+ 2

N

∑N−1n=1 |X[n]|

2


Orthogonal DCT (preserves energy:∑

|x[n]|2 =∑

|X[n]|2)

Note: MATLAB dct() calculates the ODCT

Energy Conservation


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


DCT: X[k] =∑N−1


IDCT: x[n] = 1NX[0] + 2

N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

rep→

0 12 23

y[r]

DFT→ 0

1223

Y[k]

÷2→

Energy: E =∑N−1

n=0 |x[n]|2= 1

N|X[0]|

2+ 2

N

∑N−1n=1 |X[n]|

2



|x[n]|2 =∑

|X[n]|2)

ODCT: X[k] =

√

1N

∑N−1n=0 x[n] k = 0

√

2N

∑N−1n=0 x[n] cos 2π(2n+1)k

4N k 6= 0


Energy Conservation


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


DCT: X[k] =∑N−1


IDCT: x[n] = 1NX[0] + 2

N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N

rep→

0 12 23

y[r]

DFT→ 0

1223

Y[k]

÷2→

Energy: E =∑N−1

n=0 |x[n]|2= 1

N|X[0]|

2+ 2

N

∑N−1n=1 |X[n]|

2



|x[n]|2 =∑

|X[n]|2)

ODCT: X[k] =

√

1N

∑N−1n=0 x[n] k = 0

√

2N

∑N−1n=0 x[n] cos 2π(2n+1)k

4N k 6= 0

IODCT: x[n] =√

1NX[0] +

√

2N

∑N−1k=1 X[k] cos 2π(2n+1)k

4N


Energy Compaction


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


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

Energy Compaction


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• 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|.

Energy Compaction


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• 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⟨

CxxHCH⟩

= CSCH (similarly FSFH )

Diagonal elements give mean coefficient energy.

Energy Compaction


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines





⟨

x2[n]⟩

= 1 and 〈x[n]x[n− 1]〉 = ρ.


xxH⟩

i,j= ρ|i−j|.



CxxHCH⟩



Energy Compaction


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines





⟨

x2[n]⟩

= 1 and 〈x[n]x[n− 1]〉 = ρ.


xxH⟩

i,j= ρ|i−j|.



CxxHCH⟩



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

• Used in speech recognition to decorrelatespectral coeficients: DCT of log spectrum

Energy Compaction


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines





⟨

x2[n]⟩

= 1 and 〈x[n]x[n− 1]〉 = ρ.


xxH⟩

i,j= ρ|i−j|.



CxxHCH⟩



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

• Used in speech recognition to decorrelatespectral coeficients: 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

Frame-based coding


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


• Divide continuous signalinto frames x[n]

X[k] k=30/220

y[n]

y[n]-x[n]

Frame-based coding


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


• Divide continuous signalinto frames

• Apply DCT to each frame

x[n]

X[k] k=30/220

y[n]

y[n]-x[n]

Frame-based coding


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• 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]

Frame-based coding


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




• Encode DCT


• Apply IDCT→ y[n]

x[n]

X[k] k=30/220

y[n]

y[n]-x[n]

Frame-based coding


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




• Encode DCT



x[n]

X[k] k=30/220

y[n]

y[n]-x[n]

Problem: Coding may create discontinuities at frame boundaries

Frame-based coding


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines




• Encode DCT



x[n]

X[k] k=30/220

y[n]

y[n]-x[n]

Problem: Coding may create discontinuities at frame boundariese.g. JPEG, MPEG use 8× 8 pixel blocks

8.3 kB (PNG) 1.6 kB (JPEG) 0.5 kB (JPEG)

Lapped Transform +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


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

x[0 : 2N − 1]

X0[k]

y0[n]

x[n]

0 N

N

2N

2N

3N

3N

4N

4N

Lapped Transform +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X0[0 : N − 1]

X0[k]

y0[n]

x[n]

0 N

N

2N

2N

3N

3N

4N

4N

MDCT: 2N → N coefficients

Lapped Transform +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X0[0 : N − 1]

IMDCT→ y0[0 : 2N − 1]

X0[k]

y0[n]

x[n]

0 N

N

2N

2N

3N

3N

4N

4N

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

Lapped Transform +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X0[0 : N − 1]

IMDCT→ y0[0 : 2N − 1]

x[N : 3N − 1]MDCT→ X1[N : 2N − 1]

X0[k]

y0[n]

X1[k]

y1[n]

x[n]

0 N

N

2N

2N

3N

3N

4N

4N


Lapped Transform +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X0[0 : N − 1]

IMDCT→ y0[0 : 2N − 1]

x[N : 3N − 1]MDCT→ X1[N : 2N − 1]

IMDCT→ y1[N : 3N − 1]

X0[k]

y0[n]

X1[k]

y1[n]

x[n]

0 N

N

2N

2N

3N

3N

4N

4N


Lapped Transform +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X0[0 : N − 1]

IMDCT→ y0[0 : 2N − 1]

x[N : 3N − 1]MDCT→ X1[N : 2N − 1]

IMDCT→ y1[N : 3N − 1]

x[2N : 4N − 1]MDCT→ X2[2N : 3N − 1]

X0[k]

y0[n]

X1[k]

y1[n]

X2[k]

y2[n]

x[n]

0 N

N

2N

2N

3N

3N

4N

4N


Lapped Transform +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X0[0 : N − 1]

IMDCT→ y0[0 : 2N − 1]

x[N : 3N − 1]MDCT→ X1[N : 2N − 1]

IMDCT→ y1[N : 3N − 1]

x[2N : 4N − 1]MDCT→ X2[2N : 3N − 1]

IMDCT→ y2[2N : 4N − 1]

X0[k]

y0[n]

X1[k]

y1[n]

X2[k]

y2[n]

x[n]

0 N

N

2N

2N

3N

3N

4N

4N


Lapped Transform +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X0[0 : N − 1]

IMDCT→ y0[0 : 2N − 1]

x[N : 3N − 1]MDCT→ X1[N : 2N − 1]

IMDCT→ y1[N : 3N − 1]

x[2N : 4N − 1]MDCT→ X2[2N : 3N − 1]

IMDCT→ y2[2N : 4N − 1]

y[n] = y0[n]+ y1[n]+ y2[n]

X0[k]

y0[n]

X1[k]

y1[n]

X2[k]

y2[n]

y[n]

x[n]

0 N

N

2N

2N

3N

3N

4N

4N

MDCT: 2N → N coefficients, IMDCT: N → 2N samplesAdd yi[n] together to get y[n]. Only two non-zero terms far any n.

Lapped Transform +


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



x[0 : 2N − 1]MDCT→ X0[0 : N − 1]

IMDCT→ y0[0 : 2N − 1]

x[N : 3N − 1]MDCT→ X1[N : 2N − 1]

IMDCT→ y1[N : 3N − 1]

x[2N : 4N − 1]MDCT→ X2[2N : 3N − 1]

IMDCT→ y2[2N : 4N − 1]

y[n] = y0[n]+ y1[n]+ y2[n]

X0[k]

y0[n]

X1[k]

y1[n]

X2[k]

y2[n]

y[n]

y[n]-x[n] = error

x[n]

0 N

N

2N

2N

3N

3N

4N

4N

MDCT: 2N → N coefficients, IMDCT: N → 2N samplesAdd yi[n] together to get y[n]. Only two non-zero terms far any n.Errors cancel exactly: Time-domain alias cancellation (TDAC)

MDCT (Modified DCT)


MDCT: X[k] =∑2N−1

n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

MDCT (Modified DCT)



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

If x and X are column vectors, then X = Mx

where M is an N × 2N matrix with mk,n = cos 2π(2n+1+N)(2k+1)8N .

MDCT (Modified DCT)



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N

If x and X are column vectors, then X = Mx


MDCT (Modified DCT)



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N

If x, X and y are column vectors, then X = Mx and y = 1NMTX


MDCT (Modified DCT)



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N

If x, X and y are column vectors, then X = Mx and y = 1NMTX = 1

NMTMx


MDCT (Modified DCT)



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N


NMTMx


Quasi-Orthogonality: The 2N × 2Nmatrix, 1NMTM, is almost the identity:

1NMTM = 1

2

[

I− J 0

0 I+ J

]

with I =

1 · · · 0...

. . ....

0 · · · 1

, J =

0 · · · 1... . .

. ...

1 · · · 0

MDCT (Modified DCT)



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N


NMTMx



1NMTM = 1

2

[

I− J 0

0 I+ J

]

with I =

1 · · · 0...

. . ....

0 · · · 1

, J =

0 · · · 1... . .

. ...

1 · · · 0

When two consective y frames are overlapped by N samples, the second half of thefirst frame has thus been multiplied by 1

2 (I+ J) and the first half of the second frameby 1

2 (I− J). When these y frames are added together, the corresponding x sampleshave been multiplied by 1

2 (I+ J) + 12 (I− J) = I giving perfect reconstruction.

MDCT (Modified DCT)



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N


NMTMx



1NMTM = 1

2

[

I− J 0

0 I+ J

]

with I =

1 · · · 0...

. . ....

0 · · · 1

, J =

0 · · · 1... . .

. ...

1 · · · 0

When two consective y frames are overlapped by N samples, the second half of thefirst frame has thus been multiplied by 1

2 (I+ J) and the first half of the second frameby 1

2 (I− J). When these y frames are added together, the corresponding x sampleshave been multiplied by 1

2 (I+ J) + 12 (I− J) = I giving perfect reconstruction.

Normally the 2N -long x and y frames are windowed before the MDCT and again afterthe IMDCT to avoid any discontinuities; if the window is symmetric and satisfiesw2[i] + w2[i+N ] = 2 the perfect reconstruction property is still true.

MDCT Basis Elements


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N

MDCT Basis Elements


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N

In vector notation: X = Mx and y = 1NMTX = 1

NMTMx

MDCT Basis Elements


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N


NMTMx

The rows of M form theMDCT basis elements.

MDCT Basis Elements


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N


NMTMx


Example (N = 4):

M =

0.56 0.20 −0.20 −0.56 −0.83 −0.98 −0.98 −0.83

−0.98 −0.56 0.56 0.98 0.20 −0.83 −0.83 0.20

0.20 0.83 −0.83 −0.20 0.98 −0.56 −0.56 0.98

0.83 −0.98 0.98 −0.83 0.56 −0.20 −0.20 0.56

MDCT Basis Elements


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N


NMTMx


Example (N = 4):

M =

0.56 0.20 −0.20 −0.56 −0.83 −0.98 −0.98 −0.83

−0.98 −0.56 0.56 0.98 0.20 −0.83 −0.83 0.20

0.20 0.83 −0.83 −0.20 0.98 −0.56 −0.56 0.98

0.83 −0.98 0.98 −0.83 0.56 −0.20 −0.20 0.56

MDCT Basis Elements


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



n=0 x[n] cos 2π(2n+1+N)(2k+1)8N 0 ≤ k < N

IMDCT: y[n] = 1N

∑N−1k=0 X[k] cos 2π(2n+1+N)(2k+1)

8N 0 ≤ n < 2N


NMTMx


Example (N = 4):

M =

0.56 0.20 −0.20 −0.56 −0.83 −0.98 −0.98 −0.83

−0.98 −0.56 0.56 0.98 0.20 −0.83 −0.83 0.20

0.20 0.83 −0.83 −0.20 0.98 −0.56 −0.56 0.98

0.83 −0.98 0.98 −0.83 0.56 −0.20 −0.20 0.56

The basis frequencies are {0.5, 1.5, 2.5, 3.5} times the fundamental.

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


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

[

x ←−x]

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



[

x ←−x]

• Often scaled to make an orthogonal transform (ODCT)

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



[

x ←−x]

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

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



[

x ←−x]


◦ Energy Compaction: Most energy is in only a few coefficients

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



[

x ←−x]


◦ Energy Compaction: Most energy is in only a few coefficients◦ Decorrelation: The coefficients are uncorrelated with each other

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



[

x ←−x]



• Nice convolution property of DFT is lost /

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



[

x ←−x]




MDCT: Modified Discrete Cosine Transform• Lapped transform: 2N → N → 2N

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• 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


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• 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• Similar to DCT for energy compaction and decorrelation ,

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



[

x ←−x]





• Overlapping windowed frames can avoid edge discontinuities ,

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



[

x ←−x]






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

Summary


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines



[

x ←−x]






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

For further details see Mitra: 5.

MATLAB routines


• DFT Problems

• DCT +

• Basis Functions


• DCT Properties







• Summary

• MATLAB routines


dct, idct ODCT with optional zero-padding

Discrete Cosine Transform

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