This example demonstrates how to model and simulate Linear and Logarithmic chirp signals. Chirp signals are widely used in radar pulse compression, audio equipment testing, and system identification to measure frequency response over a wide band in a single sweep.

1. Theoretical Basis

1.1 Linear Chirp

The instantaneous frequency increases linearly with time.

Frequency: $f_{\text{lin}}(t) = f_0 + k \cdot t$ Where $k$ is the sweep rate (Hz/s) and $f_0$ is the start frequency.

Time Domain Signal: $x_{\text{lin}}(t) = A \cdot \sin\left(2\pi \left(f_0 t + \frac{k}{2}t^2\right) + \varphi_0\right)$

1.2 Logarithmic Chirp

The instantaneous frequency increases exponentially, which matches human pitch perception (octaves).

Frequency: $f_{\text{log}}(t) = f_0 \cdot a^t$

Time Domain Signal: $x_{\text{log}}(t) = A \cdot \sin\left(2\pi \cdot f_0 \cdot \frac{a^t - 1}{\ln a} + \varphi_0\right)$

2. Simulation Models

2.1 Linear Chirp Model

To build this in Simit:

1. Source: Use a Ramp block to generate time $t$ (or use simulation time).
2. Math: Implement the phase equation $\Phi(t) = 2\pi (f_0 t + \frac{k}{2}t^2)$.

Parameters:

2.2 Logarithmic Chirp Model

Parameters:

3. Results Analysis

Linear Sweep

• Uniform Compression: The waveform cycles get shorter at a constant rate.
• Verification: At $t=5s$, frequency $f = 9 + 10(5) = 59 \text{Hz}$. Period $T \approx 17 \text{ms}$.

Logarithmic Sweep

• Exponential Compression: Cycles are sparse at first but compress rapidly later.
• Dynamic Range: Covers a wide frequency range quickly, ideal for Bode plot analysis.

4. Applications

• Radar: Pulse compression for high range resolution.
• Audio: Measuring speaker frequency response (Log Chirp matches hearing).
• Vibration: Modal analysis of mechanical structures.