FORMULA / 012

FFT butterfly decomposition

Interactive derivation module for FFT butterfly decomposition.

Ready

\[X[k]=E[k]+W_N^kO[k]\]

Connects the formula to engineering interpretation, limits, and computation.

Problem Definition

Problem statement and variable definition

An embedded DSP can compute a 1024-point spectrum in real time only because FFT reuses smaller DFTs.

Core variable legend

Symbol	Quantity	Unit	Meaning
$N$	FFT length	samples	Transform size.
$W_N^k$	Twiddle factor	無	Complex rotation reused across butterflies.
$E[k],O[k]$	Even/odd DFTs	依資料而定	Half-size subproblems.

Analytical Derivation

Model assumptions and analytical derivation

Model simplifications

Item	Assumption
Assumption 1	Linear time-invariant or locally linear behavior is assumed when using transfer functions.
Assumption 2	Parameters are treated as constant over the analysis interval.
Assumption 3	Numerical plots are educational approximations, not full circuit or field solvers.

Start from the governing relation

X[k]=\sum_n x[n]W_N^{kn}

This is the smallest equation that preserves the physics needed for the formula.

Choose the model state or domain.

Apply engineering assumptions

X[k]=\sum_n x[n]W_N^{kn}

The model is simplified so the dominant mechanism is visible.

Drop second-order parasitics unless they are the topic.

Rearrange to the working formula

\[X[k]=E[k]+W_N^kO[k]\]

The final form is the one used for design, simulation, or measurement review.

Algebraic rearrangement.

parameter $\to0$dominant term remains

The formula reduces to its simplest physical behavior.

parameter $\to\infty$parasitics and implementation limits dominate

The closed-form equation becomes a warning rather than a full implementation model.

Parametric Verification

Parametric simulation and numerical verification

Use the sliders to change the physical parameters and watch the formula expose its behavior. The metric panel keeps sample count and compute latency visible so the visualization remains an engineering instrument, not only a picture.

INTERACTIVE MODULE

DFT versus FFT operation growth

Open interactive view

Engineering Application

Engineering application and implementation notes

Design rules

Check units before trusting numeric output.
Sweep the key parameter through the critical region, not only the nominal value.
Validate the simplified formula against measurement or simulation before committing hardware.

Core algorithm

python

import numpy as np

def fft_reference(x):
    return np.fft.fft(x)