Accelerating the pace of engineering and science

dct

Discrete cosine transform (DCT)

y = dct(x)
y = dct(x,n)

Description

y = dct(x) returns the unitary discrete cosine transform of x,

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

where

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

N is the length of x, and x and y are the same size. If x is a matrix, dct transforms its columns. 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.

y = dct(x,n) pads or truncates x to length n before transforming.

The DCT is closely related to the discrete Fourier transform. You can often reconstruct a sequence very accurately from only a few DCT coefficients, a useful property for applications requiring data reduction.

Examples

expand all

Energy Stored in DCT Coefficients

Find how many DCT coefficients represent 99% of the energy in a sequence.

```x = (1:100) + 50*cos((1:100)*2*pi/40);
X = dct(x);
[XX,ind] = sort(abs(X),'descend');
i = 1;
while norm(X(ind(1:i)))/norm(X)<0.99
i = i + 1;
end
Needed = i;
```

Reconstruct the signal and compare to the original.

```X(ind(Needed+1:end)) = 0;
xx = idct(X);

plot([x;xx]')
legend('Original',['Reconstructed, N = ' int2str(Needed)], ...
'Location','SouthEast')
```

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, chap. 4.