idct

Inverse discrete cosine transform

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Syntax

x = idct(y)

x = idct(y,n)

x = idct(y,n,dim)

y = idct(___,'Type',dcttype)

Description

example

x= idct(y)returns the inverse discrete cosine transform of input arrayy. The outputxhas the same size asy. Ifyhas more than one dimension, thenidctoperates along the first array dimension with size greater than 1.

x= idct(y,n)zero-pads or truncates the relevant dimension ofyto lengthnbefore transforming.

x= idct(y,n,dim)computes the transform along dimensiondim. To input a dimension and use the default value ofn, specify the second argument as empty,[].

example

y= idct(___,'Type',dcttype)specifies the type of inverse discrete cosine transform to compute. SeeInverse Discrete Cosine Transformfor details. This option can be combined with any of the previous syntaxes.

例子

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Signal Reconstruction Using Inverse Discrete Cosine Transform

Open Live Script

Generate a signal that consists of a 25 Hz sinusoid sampled at 1000 Hz for 1 second. The sinusoid is embedded in white Gaussian noise with variance 0.01.

rng('default') Fs = 1000; t = 0:1/Fs:1-1/Fs; x = sin(2*pi*25*t) + randn(size(t))/10;

Compute the discrete cosine transform of the sequence. Determine how many of the 1000 DCT coefficients are significant. Choose 1 as the threshold for significance.

y = dct(x); sigcoeff = abs(y) >= 1; howmany = sum(sigcoeff)

howmany = 17

Reconstruct the signal using only the significant components.

y (~ sigcoeff) = 0;z = idct (y);

Plot the original and reconstructed signals.

subplot(2,1,1) plot(t,x) yl = ylim; title('Original') subplot(2,1,2) plot(t,z) ylim(yl) title('Reconstructed')

DCT Orthogonality

Open Live Script

Verify that the different variants of the discrete cosine transform are orthogonal, using a random signal as a benchmark.

Start by generating the signal.

s = randn(1000,1);

Verify that DCT-1 and DCT-4 are their own inverses.

dct1 = dct(s,'Type',1); idt1 = idct(s,'Type',1); max(abs(dct1-idt1))

ans = 1.3323e-15

dct4 = dct(s,'Type',4); idt4 = idct(s,'Type',4); max(abs(dct4-idt4))

ans = 1.3323e-15

Verify that DCT-2 and DCT-3 are inverses of each other.

dct2 = dct(s,'Type',2); idt2 = idct(s,'Type',3); max(abs(dct2-idt2))

ans = 4.4409e-16

dct3 = dct(s,'Type',3); idt3 = idct(s,'Type',2); max(abs(dct3-idt3))

ans = 1.1102e-15

Input Arguments

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`y`—Input discrete cosine transform
vector|matrix|N-D array|`gpuArray`object

Input discrete cosine transform, specified as a real-valued or complex-valued vector, matrix, N-D array, orgpuArrayobject.

SeeRun MATLAB Functions on a GPU(Parallel Computing Toolbox) andGPU Support by Release(Parallel Computing Toolbox) for details ongpuArrayobjects.

Example:dct(sin(2*pi*(0:255)/4))specifies the discrete cosine transform of a sinusoid.

Example:dct(sin(2*pi*[0.1;0.3]*(0:39))')specifies the discrete cosine transform of a two-channel sinusoid.

Data Types:single|double
Complex Number Support:Yes

`n`—Inverse transform length
positive integer scalar

Inverse transform length, specified as a positive integer scalar.

Data Types:single|double

`dim`—Dimension to operate along
positive integer scalar

Dimension to operate along, specified as a positive integer scalar.

Data Types:single|double

`dcttype`—Inverse discrete cosine transform type
`2`(default) |`1`|`3`|`4`

Inverse discrete cosine transform type, specified as a positive integer scalar from 1 to 4.

Data Types:single|double

Output Arguments

collapse all

`x`— Inverse discrete cosine transform
vector | matrix | N-D array |`gpuArray`object

Inverse discrete cosine transform, returned as a real-valued or complex-valued vector, matrix, N-D array, orgpuArrayobject.

More About

collapse all

Inverse Discrete Cosine Transform

The inverse discrete cosine transform reconstructs a sequence from its discrete cosine transform (DCT) coefficients. Theidctfunction is the inverse of thedctfunction.

The DCT has four standard variants. For a transformed signalyof lengthN, and withδ_kℓthe Kronecker delta, the inverses are defined by:

Inverse of DCT-1:

$x (n) = \sqrt{\frac{2}{N - 1}} \sum_{k = 1}^{N} y (k) \frac{1}{\sqrt{1 + δ_{k 1} + δ_{k N}}} \frac{1}{\sqrt{1 + δ_{n 1} + δ_{n N}}} \cos (\frac{π}{N - 1} (k - 1) (n - 1))$
Inverse of DCT-2:

$x (n) = \sqrt{\frac{2}{N}} \sum_{k = 1}^{N} y (k) \frac{1}{\sqrt{1 + δ_{k 1}}} \cos (\frac{π}{2 N} (k - 1) (2 n - 1))$
Inverse of DCT-3:

$x (n) = \sqrt{\frac{2}{N}} \sum_{k = 1}^{N} y (k) \frac{1}{\sqrt{1 + δ_{n 1}}} \cos (\frac{π}{2 N} (2 k - 1) (n - 1))$
Inverse of DCT-4:

$x (n) = \sqrt{\frac{2}{N}} \sum_{k = 1}^{N} y (k) \cos (\frac{π}{4 N} (2 k - 1) (2 n - 1))$

The series are indexed fromn= 1andk= 1instead of the usualn= 0andk= 0, because MATLAB^®vectors run from 1 toNinstead of from 0 toN– 1.

All variants of the DCT areunitary(or, equivalently,orthogonal): To find the forward transforms, switchkandnin each definition. In particular, DCT-1 and DCT-4 are their own inverses, and DCT-2 and DCT-3 are inverses of each other.

References

[1] Jain, A. K.Fundamentals of Digital Image Processing. Englewood Cliffs, NJ: Prentice-Hall, 1989.

[2] Pennebaker, W. B., and J. L. Mitchell.JPEG Still Image Data Compression Standard. New York: Van Nostrand Reinhold, 1993.

[3] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck.Discrete-Time Signal Processing. 2nd Ed. Upper Saddle River, NJ: Prentice Hall, 1999.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

C and C++ code generation fordctrequires DSP System Toolbox™ software.
The length of the transform dimension must be a power of two. If specified, the pad or truncation value must be constant. Expressions or variables are allowed if their values do not change.
Inputs must be double precision.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, seeRun MATLAB Functions on a GPU(Parallel Computing Toolbox).

idct

Syntax

Description

例子

Signal Reconstruction Using Inverse Discrete Cosine Transform

DCT Orthogonality

Input Arguments

`y`—Input discrete cosine transform
vector|matrix|N-D array|`gpuArray`object

`n`—Inverse transform length
positive integer scalar

`dim`—Dimension to operate along
positive integer scalar

`dcttype`—Inverse discrete cosine transform type
`2`(default) |`1`|`3`|`4`

Output Arguments

`x`— Inverse discrete cosine transform
vector | matrix | N-D array |`gpuArray`object

More About

Inverse Discrete Cosine Transform

References

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

See Also

Topics

Introduced before R2006a

Signal Processing Toolbox Documentation

金宝app

Try MATLAB, Simulink, and Other Products

idct

Syntax

Description

例子

Signal Reconstruction Using Inverse Discrete Cosine Transform

DCT Orthogonality

Input Arguments

y—Input discrete cosine transformvector|matrix|N-D array|gpuArrayobject

n—Inverse transform lengthpositive integer scalar

dim—Dimension to operate alongpositive integer scalar

dcttype—Inverse discrete cosine transform type2(default) |1|3|4

Output Arguments

x— Inverse discrete cosine transformvector | matrix | N-D array |gpuArrayobject

More About

Inverse Discrete Cosine Transform

References

Extended Capabilities

C/C++ Code GenerationGenerate C and C++ code using MATLAB® Coder™.

GPU ArraysAccelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

See Also

Topics

Introduced before R2006a

Signal Processing Toolbox Documentation

金宝app

Try MATLAB, Simulink, and Other Products

`y`—Input discrete cosine transform
vector|matrix|N-D array|`gpuArray`object

`n`—Inverse transform length
positive integer scalar

`dim`—Dimension to operate along
positive integer scalar

`dcttype`—Inverse discrete cosine transform type
`2`(default) |`1`|`3`|`4`

`x`— Inverse discrete cosine transform
vector | matrix | N-D array |`gpuArray`object

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.