Frequency-Wavenumber Domain EECS 800 – Patrick McCormick

2-D Frequency Domain  The 2D FFT across the SAR data after downconversion to baseband produces the Frequency-Doppler domain.  Doppler (Hz) is transformed to wavenumber (rad/m) through the relationship  Because of this proportionality, let’s consider the Frequency (Range) – Doppler (Slow-Time) domain. Spatial Wavenumber Doppler frequency Sensor Velocity

Principle of Stationary Phase (POSP)  Approximate spectrum estimation tool that allows for an analytical solution to the spectral content of an LFM.  Allows for easy calculation of spectral magnitudes and phases without brute force techniques.  The approximation becomes more accurate as the time-bandwidth product (TBP) increases.

Principle of Stationary Phase (POSP)  LFM  Fourier Transform  POSP approximation  Stationary Phase Point

Range Frequency – Slow Time Domain Scalar terms Range spectral envelope Azimuth beampattern 2-D phase Range frequency Slow-time Center frequency Fast-time chirp-rate Range frequency Nearest range to target Single Scatterer Sensor velocity Speed of light

Range Frequency – Doppler Domain Fourier Transform Doppler Frequency

Range Frequency – Doppler Domain  To invoke POSP, must find slow-time variable η in terms of the Doppler frequency f η  Recall: (5.23)  Stationary Phase Point

Range Frequency – Doppler Domain

F-k algorithm  This analytical solution to the spectral content of an LFM is used in the f-k algorithm bulk compression operation or Reference Function Multiply (RFM).  This is essentially a 2-D matched filtering technique that matches to a single reference range R ref and velocity V ref.  The filter is a point-to-point complex multiply in the frequency-Doppler domain using the conjugate of the presumed reference phase response.

Example – single scatterer h=1 km offset=2.265 km L=1 km V r =100 m/s PRF=1 kHz f 0 =1 GHz B=50 MHz T=10 μs f s =300 MHz Radar Parameters: Antenna Parameters: Beamwidth = 34 o Pattern: cosine Purpose: To compare POSP approximation against brute force reference function calculation.

Example – Data Time DomainFrequency Domain

Example – Reference Function Multiply

Example – Point Spread Responses Note: This does not include all of the steps of the f-k algorithm. This example only pertains to a point scatterer at the reference range.

Questions?