Discrete cosine transform

The type-II discrete cosine transform of

x_{0}, \dots, x_{N - 1}

is

Z_{k} = \sum_{n = 0}^{N - 1} x_{n} cos (\frac{π}{N} (n + \frac{1}{2}) k) .

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- Discrete cosine transform from an even-reflected discrete Fourier transform Discrete cosine transform

Discrete cosine transform from an even-reflected discrete Fourier transform

Reflect

x

evenly to length

2 N

by

y_{n} = x_{n}

and

y_{2 N - 1 - n} = x_{n}

. If

Y

is the discrete Fourier transform of

y

, then

Z_{k} = \frac{1}{2} e^{- πik / (2 N)} Y_{k} .

This reduction computes the cosine transform with one fast Fourier transform and

O (N)

additional work.

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 57 / 8 / Solution
Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 68 / 6 / Solution
Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 41E / b / Solution
Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 41D / c / Solution

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The Discrete Cosine Transform (DCT) is a mathematical operation that converts a sequence of data points into a sum of cosine functions oscillating at different frequencies. It is widely used in signal processing and image compression techniques because it has properties that are beneficial for representing signals efficiently.

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