dct

Discrete cosine transform (DCT)

Syntax

y = dct(x)
y = dct(x,n)

Description

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

y(k)=w(k)n=1Nx(n)cos(π2N(2n1)(k1)),k=1,2,,N,

where

w(k)={1N,k=1,2N,2kN,

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.

See Also

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