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The discrete cosine transform (DCT) represents a finite sequence of real-valued samples as a weighted combination of cosine patterns. Its output coefficients describe how much of each pattern is present. Because several DCT types and scaling conventions exist, a technical definition or software result should specify both.
What does DCT mean?
DCT stands for discrete cosine transform. It is a transform that takes a finite sequence of samples and expresses it in a cosine basis. Each output value is a coefficient: together, the coefficients describe the input sequence using cosine patterns associated with different frequencies.
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The phrase “the DCT” commonly refers to DCT type II, but it is not a complete specification in every technical context. Types I, II, III and IV use different cosine indexing and endpoint conventions, so their coefficients are not interchangeable.
How is DCT-II defined?
For a length-N input sequence x[n], SciPy defines its unnormalized type-II transform as:
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y[k] = 2 Σ(n=0 to N−1) x[n] cos(π(2n+1)k/(2N)), for 0 ≤ k < N
The cosine is evaluated at half-sample offsets, reflected in the factor 2n+1. The index k selects an output coefficient, while n runs over the input samples. This is SciPy’s documented type-II convention; other scaling conventions can change the coefficient values.
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How do DCT types differ?
The type determines the cosine basis and its sample or endpoint conventions. SciPy’s dct reference implements types I through IV and uses type II by default.
| Type | Convention described in the documentation | Practical distinction |
|---|---|---|
| DCT-I | Uses endpoints, with a denominator of N−1 in the cosine argument. | Its endpoint convention differs from the half-sample indexing of type II. In SciPy, the input length must exceed one. |
| DCT-II | Uses half-sample offsets in the input index. | The usual meaning of “DCT”; SciPy’s default type. |
| DCT-III | Uses a different cosine indexing pattern from type II. | Commonly used as the inverse paired with type II, subject to compatible scaling. |
| DCT-IV | Uses half-sample offsets for both index factors. | Its indexing convention differs from types II and III. |
The table summarizes the distinctions provided by SciPy’s documentation; it is not a substitute for the exact formula when implementing or comparing a transform.
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Why do DCT formulas use different normalization?
Normalization specifies how the transform scales its coefficients. It matters when comparing outputs from different implementations and when reconstructing the original sequence. A formula’s type alone does not determine the coefficient scale.
In SciPy, the norm setting can be "backward", "forward" or "ortho". With the default "backward" convention, the forward transform is unscaled and the inverse is scaled by 1/N, using the logical transform size. With "forward", the 1/N factor is applied to the forward transform instead. "ortho" applies scaling intended to make the transform orthonormal. For types I–III, SciPy’s default orthogonalization further adjusts the formula to make the coefficient matrix orthonormal.
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Is DCT-III the inverse of DCT-II?
Type III is commonly the inverse counterpart of type II, but the scale factor depends on normalization. Under SciPy’s unnormalized convention, DCT-III inverts DCT-II only after accounting for a factor of 2N. With orthonormal scaling, type III is the inverse of type II. Pair compatible type and normalization settings rather than assuming two libraries use identical conventions.
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SciPy documents the function as scipy.fft.dct(x, type=2, norm=...). Its default type is II, it transforms along the last axis by default, and it also accepts an optional length and axis. The documented API is for SciPy v1.18.0; software behavior and defaults should be checked for the particular library and version being used.
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For a result that others can reproduce, record the library and version, DCT type, normalization, and orthogonalization setting. Those choices determine which coefficients the reported transform produces.
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