SAR Coherence Length – Using Doppler bandwidth For a C-band Synthetic Aperture Radar (SAR) with Doppler bandwidth Δf_D = 1300 Hz, estimate the coherence length l_coh (use the relation l_coh ≈ c / Δf for coherence length based on spectral width).

Introduction / Context:Coherence length characterizes the path-length over which a wave maintains a predictable phase relationship. In radar/optics, it is inversely related to spectral width. When bandwidth is given in hertz, a useful first-order estimate is l_coh ≈ c / Δf, where c is the speed of light and Δf is the effective spectral width.

Given Data / Assumptions:

• Doppler bandwidth Δf_D = 1300 Hz.
• Speed of light c ≈ 3.0 * 10^8 m/s.
• Using the simple coherence-length estimate l_coh ≈ c / Δf (no factor of 2 for this definition).

Concept / Approach:For a finite spectral width, coherence time τ_coh ≈ 1 / Δf. Coherence length is l_coh ≈ c * τ_coh ≈ c / Δf. Substitute Δf = 1300 Hz to compute l_coh and convert to kilometres.

Step-by-Step Solution:τ_coh = 1 / Δf = 1 / 1300 s ≈ 7.6923 * 10^-4 sl_coh = c * τ_coh ≈ (3.0 * 10^8 m/s) * (7.6923 * 10^-4 s)l_coh ≈ 2.3077 * 10^5 mConvert to km: 2.3077 * 10^5 m ≈ 230.8 kmRounded to the nearest option ≈ 230 km

Verification / Alternative check:Using the compact form l_coh = c / Δf directly: 3 * 10^8 / 1300 ≈ 2.31 * 10^5 m, confirming ≈ 230 km.

Why Other Options Are Wrong:

• 130 km and 250 km: off by large rounding; not supported by the calculation.
• 500 km: roughly double the computed value.
• 65 km: about one-fourth of the correct magnitude.

Common Pitfalls:Confusing range bandwidth (MHz) with Doppler bandwidth (Hz) or inserting an unnecessary factor of 2 (which would correspond to certain coherence definitions) will change the numerical result.

Final Answer:230 km