Interactive Visualizations

🌊 Sinusoidal Signals

Explore: Amplitude, frequency, phase, and DC offset

Visualize how changing parameters affects the shape and characteristics of sine and cosine waves. Perfect for understanding periodic signals and their properties.

Interactive Features: Sliders for A, ω, φ, and DC components | Real-time equation display | Multiple signal comparison

📶 Unit Step and Impulse Functions

Explore: Discontinuous signals and their properties

Understand the fundamental building blocks of signal processing: unit step function u(t) and impulse function δ(t). See how these signals behave and how they're used to construct more complex signals.

Interactive Features: Time-shifted versions | Signal composition | Pulse width variation

📉 Exponential Signals

Explore: Growing and decaying exponentials

Visualize exponential functions e^(at) with different time constants. Essential for understanding system transient responses and RC/RL circuits.

⏰ Time Shifting

Explore: x(t - t₀) - Delaying and advancing signals

Understand how time shifts affect signal position. Positive shifts delay the signal (shift right), negative shifts advance it (shift left). Critical for understanding system delay and causality.

Key Concepts:

• x(t - 2) → Delay by 2 seconds (shift right)
• x(t + 2) → Advance by 2 seconds (shift left)
• Compare original and shifted signals side-by-side

🔍 Time Scaling

Explore: x(at) - Compression and expansion

See how time scaling changes signal speed. Time compression (a > 1) makes signals faster, time expansion (0 < a < 1) makes them slower.

➕ Signal Addition and Multiplication

Explore: x₁(t) + x₂(t) and x₁(t) × x₂(t)

Visualize how signals combine through addition and multiplication. Essential for understanding modulation, mixing, and superposition.

Interactive Features: Select different signal types | Adjust individual signal parameters | See the resulting combined signal

🔄 Even and Odd Decomposition

Explore: x(t) = x_e(t) + x_o(t)

Any signal can be decomposed into even and odd components. This visualization shows the original signal alongside its even part [x(t) + x(-t)]/2 and odd part [x(t) - x(-t)]/2.

Key Properties:

• Even part: x_e(t) = x_e(-t) - Symmetric about t=0
• Odd part: x_o(t) = -x_o(-t) - Anti-symmetric about t=0
• Verification: x(t) = x_e(t) + x_o(t)

🔀 Convolution Visualization

Explore: y(t) = x(t) * h(t) - The convolution integral

See convolution in action! This interactive tool shows how the output signal is formed by sliding, flipping, multiplying, and integrating. Watch the h(τ) function flip to h(t-τ) and slide across x(τ).

Interactive Features: Step-by-step animation | Adjustable time slider | Product visualization | Area under curve display

📊 First-Order System Response

Explore: dy/dt + ay = bx(t)

Visualize first-order system behavior including natural response, forced response, and total response. Adjust time constant τ = 1/a and see how it affects settling time.

🎯 Second-Order System Response

Explore: d²y/dt² + 2ζω_n(dy/dt) + ω_n²y = ω_n²x(t)

Understand how damping ratio ζ affects system behavior. See underdamped, critically damped, and overdamped responses side by side.

Key Parameters: Natural frequency ω_n, damping ratio ζ, overshoot, settling time

🎛️ RC Circuit Analysis

Explore: First-order RC low-pass filter

Interactive RC circuit showing both time-domain (step response) and frequency-domain (Bode plot) behavior. Adjust R and C values to see how cutoff frequency changes.

Displays: Time constant τ = RC | Cutoff frequency f_c = 1/(2πRC) | -3dB point | Phase response

📐 Fourier Series Synthesis

Explore: Building periodic signals from harmonics

Watch as a complex periodic waveform is constructed by adding harmonics one by one. Start with the fundamental frequency and add harmonics to see the signal take shape. Demonstrates Gibbs phenomenon at discontinuities.

Waveforms Available: Square wave, Sawtooth, Triangle, Pulse train

🌈 Frequency Spectrum Visualization

Explore: Magnitude and phase spectra

See both time-domain and frequency-domain representations simultaneously. Adjust signal parameters and watch how the spectrum changes in real-time.

Interactive Features: Magnitude spectrum |X(f)| | Phase spectrum ∠X(f) | Time-frequency relationship | Bandwidth visualization

🔄 Fourier Transform Properties

Explore: Duality, time-shift, frequency-shift, scaling

Interactive demonstration of key Fourier transform properties. See how operations in the time domain affect the frequency domain and vice versa.

🎵 Complex Exponentials and Phasors

Explore: e^(jωt) representation and rotating phasors

Visualize complex exponentials as rotating vectors in the complex plane. Understand how real sinusoids relate to complex exponentials through Euler's formula.

Key Concepts: e^(jωt) = cos(ωt) + j·sin(ωt) | Phasor rotation | Real and imaginary components | 3D visualization option

📡 Amplitude Modulation (AM)

Explore: AM modulation and demodulation

See how amplitude modulation works in communications. Adjust carrier frequency, message frequency, and modulation index to understand the modulation process.

Key Parameter: Modulation index μ determines envelope shape and bandwidth

🎚️ Ideal Filter Design

Explore: Low-pass, high-pass, and band-pass filters

Interactive filter design tool showing frequency response and filtered output. Apply different filters to input signals and observe the results.

📊 Sampling and Aliasing

Explore: Nyquist theorem and aliasing effects

Understand the sampling theorem by seeing what happens when you sample below, at, and above the Nyquist rate. Watch aliasing occur when sampling rate is too low.

Interactive Features: Adjustable sampling rate | Original vs sampled signal | Reconstructed signal | Frequency domain view | Aliasing demonstration

🌐 Bode Plot Interactive Tool

Explore: Frequency response of LTI systems

Design transfer functions and see their Bode plots (magnitude and phase). Adjust poles and zeros and watch how they affect the frequency response.

Features: Pole-zero placement | Gain adjustment | Magnitude response (dB) | Phase response (degrees) | Cutoff frequency marking