FORMULA / 004

Convolution theorem: bridge between time and frequency

Interactive derivation module for Convolution theorem: bridge between time and frequency.

Ready

\[y(t)=x(t)*h(t)\Longleftrightarrow Y(\omega)=X(\omega)H(\omega)\]

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

Problem Definition

Problem statement and variable definition

A filter output looks like the input has been smeared by the circuit memory.

Core variable legend

Symbol	Quantity	Unit	Meaning
$x(t)$	Input signal	依系統而定	Signal entering the system.
$h(t)$	Impulse response	依系統而定	System memory fingerprint.
$H(\omega)$	Frequency response	依系統而定	Gain and phase per frequency.

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

y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau

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

Choose the model state or domain.

Apply engineering assumptions

y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau

The model is simplified so the dominant mechanism is visible.

Drop second-order parasitics unless they are the topic.

Rearrange to the working formula

Y(\omega)=X(\omega)H(\omega)

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

Rectangular pulse spectrum

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 lti_output(x, h):
    return np.convolve(x, h, mode='same')