Chirps

📐 Definition

A chirp is a sinusoid (or complex exponential) with a time-varying phase:

x(t) = \sin(\phi(t))

Equivalently, an analytic (complex) chirp is $e^{i\phi(t)}$ .

Domain and Codomain

Defined over time (or space) intervals where $\phi$ is differentiable. Outputs are bounded real-valued sinusoids (or unit-magnitude complex phasors for $e^{i\phi(t)}$ ).

⚙️ Key Properties

Instantaneous Frequency

The instantaneous angular frequency is the phase derivative:

\omega(t) = \phi'(t)

Linear Chirp (Quadratic Phase)

A common model is the linear chirp with sweep rate $\alpha$ :

\phi(t) = \omega_0 t + \tfrac{1}{2}\alpha t^2, \qquad \omega(t) = \omega_0 + \alpha t

Higher-order $\phi$ produce nonlinear sweeps.

🎯 Special Cases and Limits

$\alpha=0$ reduces to a constant-frequency sinusoid.
Windowed chirps taper endpoints to control sidelobes and spectral leakage.

Complex exponentials $e^{i\phi(t)}$ provide the analytic counterpart. Gabor functions window oscillations for time–frequency localization, and Fourier methods expose the sweep structure in the time–frequency plane.

Usage in Oakfield

Oakfield includes chirps as a stimulus variant:

stimulus_chirp operator sweeps wavenumber/frequency over time (sinusoidal family), useful for probing response across bands.
Chirps also appear indirectly as spectral phase evolution when using the dispersion operator (frequency-dependent phase advance).

Historical Foundations

📜 Sweep Signals

Frequency-swept signals have long been used in acoustics, radar/sonar, and system identification because they probe a wide band in a single experiment.

🌍 Modern Perspective

Chirps are standard primitives in time–frequency analysis and numerical testing, often paired with windowing and spectrogram diagnostics.

📚 References

Cohen, Time-Frequency Analysis
Oppenheim & Schafer, Discrete-Time Signal Processing