Fourier Linearity Visualization Tool

Explore the linearity property of Fourier Transform with interactive signals. No coding knowledge required!

This tool demonstrates how Fourier Transform preserves linearity: F(a·f(t) + b·g(t)) = a·F(f(t)) + b·F(g(t))

Fourier Linearity Principle

The Fourier Transform is a linear operator. This means that for any two signals f(t) and g(t), and any constants a and b:

ℱ{a·f(t) + b·g(t)} = a·ℱ{f(t)} + b·ℱ{g(t)}

Where ℱ denotes the Fourier Transform. This property allows us to analyze complex signals by breaking them down into simpler components.

Signal Selection

Sine Wave

f(t) = A·sin(ωt)

Cosine Wave

f(t) = A·cos(ωt)

Square Wave

Periodic on/off signal

Triangle Wave

Linear rise and fall

Signal Parameters

Amplitude (A) 1.0

Frequency (ω) 3.0

Signal 1 Coefficient (a)

1.0

Signal 2 Coefficient (b)

1.0

Linear Combination

Coefficient a 1.0

Coefficient b 1.0

y(t) = 1.0·f(t) + 1.0·g(t)

How to Embed in Blogger

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3. Switch to HTML view (click the "<> HTML" button)

4. Copy ALL the code from this page and paste it into the HTML editor

5. Publish your post/page

That's it! No coding knowledge needed.

Linearity Visualization

Signal 1: f(t)

Signal 2: g(t)

Linear Combination: a·f(t) + b·g(t)

Fourier Transform of f(t)

ℱ{f(t)} = δ(ω - ω₀)

Fourier Transform of g(t)

ℱ{g(t)} = δ(ω - ω₀)

Fourier Transform of a·f(t) + b·g(t)

a·ℱ{f(t)} + b·ℱ{g(t)}

Linearity Demonstration

The graphs show that the Fourier Transform of the linear combination (bottom right) equals the same linear combination of the individual Fourier Transforms. This confirms the linearity property:

ℱ{a·f(t) + b·g(t)} = a·ℱ{f(t)} + b·ℱ{g(t)}

This property is fundamental to signal processing, allowing complex signals to be analyzed as sums of simpler sinusoidal components.

2

Signal Components

3.0 Hz

Base Frequency

1

Harmonics

✓

Linearity Verified

Fourier Linearity Visualization Tool | Designed for easy embedding in Blogger | Educational Use

Explore the fundamental linearity property of Fourier Transform in signal processing