---
kind: concept
slug: discrete-cosine-transform
title: Discrete cosine transform (DCT)
summary: >-
  A fixed transform that rewrites a finite sequence as cosine frequency coefficients, and maps them back. One card: siMLPe applies a 50-point DCT along time to observed human poses, then learns on the coefficients. Trade-off: the matrix and the time-mixing layers are tied to the window length T, and the paper's ablation shows only a small gain over no DCT.
tags: [frequency-domain, discrete-cosine-transform, motion-prediction, time-series]
papers: ["2207.01567"]
related: [motion-prediction, multi-layer-perceptron, frequency-domain-learning, wavelet-transform]
status: draft
updated_at: 2026-11-09
---

## Overview

The discrete cosine transform (DCT) writes a finite sequence as a weighted sum
of cosines at increasing frequencies. For a window of T samples it is a fixed
T × T matrix, built once and applied to the data. A learned model can then work
on the frequency coefficients instead of the raw values, and an inverse
transform returns the result to the original domain. The only card here uses
the DCT along the time axis of human pose sequences; the paper credits an
earlier motion-prediction work with the same use [arxiv:2207.01567 §3.1]. The
matrix entries and their scale are defined in [arxiv:2207.01567 eq. 1].

## Variants

Only one variant has a card in this wiki. Other uses of the DCT, such as the
image-forensics uses listed on [Frequency-domain learning](/concepts/frequency-domain-learning),
are not covered here.

### DCT over the time axis of pose sequences

What it is: the window of observed poses is viewed as C channels, one per
joint coordinate, each a sequence of T = 50 frames. The DCT matrix is applied
along the frame axis, so every channel becomes T frequency coefficients. The
network acts on these coefficients, and the inverse transform maps the output
back to frames [arxiv:2207.01567 §3.1] [arxiv:2207.01567 §4.2]. Card:
[Back to MLP: A Simple Baseline for Human Motion Prediction](/papers/2207.01567).

Key mechanism: the time-mixing layers are T × T, so the transform and the
layers share the same window length. The network produces T frames, and the
prediction horizon is taken as the first few of them [arxiv:2207.01567 §3.2].

Reported effect: the paper's ablation without the DCT reaches 9.9 mm at 80 ms
and 110.5 mm at 1000 ms on Human3.6M, against 9.6 mm and 109.4 mm with it
[arxiv:2207.01567 tab. 6]. The differences, computed from that table, are
0.3 mm and 1.1 mm.

## When to use which

| Situation | Use | Why |
|---|---|---|
| Fixed input window and a model that mixes information across time with dense layers | DCT along time, as in siMLPe | The paper uses it with T × T time-mixing layers and reports the best result with it [arxiv:2207.01567 tab. 6] |
| Gain matters more than the transform itself | Expect a small effect | In the paper's ablation the DCT adds 0.3 to 1.1 mm over a time-domain input [arxiv:2207.01567 tab. 6]; the cards do not show the gain on other tasks |
| Window length may change at inference | Not covered by our cards | The matrix and the time-mixing layers are built for T = 50 [arxiv:2207.01567 §4.2]; a different T needs a new matrix and new layers |
| Localised time-frequency detail is needed | Not covered by our cards | See [Wavelet transform](/concepts/wavelet-transform); no card compares the two |

## Implementation sketch

Our own pseudocode of the time-domain use in siMLPe. The shapes follow the
paper's setting: T = 50 frames and C = 66 channels for 22 joints in 3D
[arxiv:2207.01567 §4.2].

```pseudocode
# Fixed DCT along time  [arxiv:2207.01567 §3.1] [arxiv:2207.01567 eq. 1]
# Input:  window x of shape (T, C), T = 50, C = 66
# Output: a prediction of shape (T, C) in the time domain

# Build the T x T basis once, for this window length  [arxiv:2207.01567 eq. 1]
D = dct_matrix(T)                    # (T, T), fixed, not learned
D_inv = inverse(D)                   # (T, T), maps coefficients back

# Move the window into the frequency domain  [arxiv:2207.01567 §3.1]
coeffs = D @ x                       # (T, C) frequency coefficients

# Learned layers act on the coefficients; T is fixed in their shapes
# [arxiv:2207.01567 §3.2]
coeffs = learned_layers(coeffs)      # (T, C)

# Return to the time domain  [arxiv:2207.01567 §3.1]
y = D_inv @ coeffs                   # (T, C)
```

Where variants would differ: the basis itself (here a cosine basis of size
T), and whether the learned layers see the coefficients or the frames. Only
the first is covered by a card.

## Contradictions

None known. Only one card uses the DCT. Within that paper, the ablation shows
a small gain from the DCT [arxiv:2207.01567 tab. 6], and no other card reports
a different result.

## Papers

- [Back to MLP: A Simple Baseline for Human Motion Prediction](/papers/2207.01567): applies a fixed 50-point DCT along time to observed poses before its time-mixing layers, and reports a small gain over the same model without it.