# Documentation

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# idct

Inverse discrete cosine transform

## Syntax

x = idct(y)
x = idct(y,n)
x = idct(gpuArrayY,n)

## Description

x = idct(y) returns the inverse discrete cosine transform of y. x has the same size as y. If y is a matrix, then idct transforms its columns.

x = idct(y,n) zero-pads or truncates the vector y to length n before transforming.

x = idct(gpuArrayY,n) returns the inverse discrete cosine transform of gpuArray object gpuArrayY. See GPU Computing and GPU System Requirements for details on gpuArray objects.

## Examples

collapse all

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, that is, greater than 1.

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')

## Inverse Discrete Cosine Transform

The inverse discrete cosine transform reconstructs a sequence from its discrete cosine transform (DCT) coefficients. The idct function is the inverse of the dct function.

The inverse discrete cosine transform is defined by

$x\left(n\right)=\sum _{k=1}^{N}w\left(k\right)y\left(k\right)\mathrm{cos}\left(\frac{\pi \left(2n-1\right)\left(k-1\right)}{2N}\right),\text{ }n=1,2,\dots N$

where

$w\left(k\right)=\left\{\begin{array}{ll}\frac{1}{\sqrt{N}},\hfill & k=1\hfill \\ \sqrt{\frac{2}{N}},\hfill & 2\le k\le N\hfill \end{array}$

and N is the length of x. The series is indexed from n = 1 and k = 1 instead of the usual n = 0 and k = 0 because MATLAB® vectors run from 1 to N instead of from 0 to N-1.

## 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.