Larsen Effect (Feedback Loop)
The **Larsen effect** (acoustic feedback) occurs when a microphone picks up a loudspeaker that is amplifying the same signal: the closed loop “microphone → amplifier → speaker → propagation → microphone” adds a delayed copy of the output to the input. If at some frequency the round-trip gain exceeds unity and the phase is an integer multiple of **2π** (positive feedback), the amplitude **grows exponentially** until nonlinearity—amplifier saturation, supply limits, driver compression, or clipping—limits it. The simulator uses a **discrete** teaching model y[n] = tanh(G·y[n−D] + A sin(ωn)): **G** is effective loop gain, **D** is delay in samples (standing in for acoustic plus processing delay), and **tanh** provides soft limiting. It is not a substitute for measuring a venue’s transfer function, but it isolates how **G**, **D**, and saturation interact.
Who it's for: Students of acoustics, audio engineering, and control theory; instructors explaining instability conditions in feedback loops.
Key terms
- Acoustic feedback
- Delay
- Loop gain
- Self-oscillation
- Saturation
- Phase coherence
- Whistle frequency
How it works
Microphone, amplifier, and loudspeaker in the same room form a loop: sound returns after a delay; if gain around the loop exceeds one at some frequency, level grows until something saturates — the Larsen effect.
Frequently asked questions
- Why does the whistle pick a particular pitch?
- Real loops have **frequency-dependent** gain and phase. The audible tone corresponds to a frequency where **|G(ω)|** is large and the total phase shift is a multiple of **2π** (constructive feedback). The toy model injects a **test sine**; in practice the room, mic, and EQ set the spectrum.
- Does moving the mic or turning down the volume help?
- Yes: lowering **G** (gain, sensitivity, distance, orientation) or changing **D** and phase (latency, EQ, monitor equalization) can move the system out of the self-oscillation condition—basic concert feedback control.
- Why use tanh instead of hard clipping?
- **tanh** gives smooth saturation and bounded stable trajectories without harsh numerical issues; it qualitatively mimics how real amplifiers behave nonlinearly at large levels.
- Does one discrete delay represent a real room?
- Only **qualitatively**: reflections create a **distribution** of delays and frequency-dependent phases. A single **D** is the minimal model explaining why delayed feedback can go unstable.
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