ECE6272 Fundamentals of Radar Signal Processing Module #18 Doppler Shift ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18 Relativistic Doppler Shift • Consider a monostatic radar transmitting at frequency Ft • Scatterer moves towards the radar with velocity v ((“closing” closing target) • Special relativity predicts the received frequency will be 1+ v c Fr = Ft 1− v c ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18 Simplifying for Low Velocity - 1 • v/c is always small – Mach 2 ((~680 680 m/s) v/c = 2.3x10 2 3x10-6 – LEO (“low earth orbit”) satellite (~7.6 km/s) v/c = 2.5x10-5 • Expand denominator in binomial series: −1 Fr = (1 + v c )(1 − v c ) Ft 2 = (1 + v c ) 1 + ( v c ) + ( v c ) + Ft 2 = 1 + 2 ( v c ) + 2 ( v c ) + Ft ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18 Simplifying for Low Velocity - 2 • Discard quadratic and higher-order terms in (v/c): Fr = 1 + 2 ( v c ) Ft • Doppler shift is the change in frequency: 2v 2v FD = + Ft = + λt c ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18 Values of Doppler Shift • Mach 2 aircraft (~680 m/s) causes 4.5 kHz shift at L band ((1 GHz)) • LEO satellite causes 507 kHz shift at X band B d Band Frequency (GHz) Doppler shift (Hz) for v = 1 m/s L 1 6.67 C 6 40.0 X 10 66.7 Ka 35 233 W 95 633 ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18 Radial Velocity y v φ = 90º FD = 0 v φ FD = + boresig ght directio on • Doppler shift is determined b th by the radial di l component of velocity for a monostatic radar 2v λ cos φ x radar antenna ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18 Envelope and Bandwidth Effects • Radar waveforms are not monochromatic sinusoids, but have finite bandwidth – almost always 10% or less (usually much less) of carrier frequency • Effect of Doppler on bandwidth usually completely l t l iinsignificant i ifi t Br = 1 + 2 ( v c ) Bt • Time dilation or contraction of pulse envelope similarly insignificant ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18 Non-Relativistic Approach • Because v/c is small, relativistic effects are insignificant – simpler approach will give correct expressions for Doppler shift and other motion-induced phase modulations • Consider a general time-varying range from radar to target of R(t); then 2R ( t ) y (t ) = x t − c ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18 Doppler Shift Example - 1 • Target moves toward radar at constant velocity v, implying R ( t ) = R0 − vt – R0 is range at time t = 0 • Constant Constant-frequency frequency complex exponential is transmitted: x ( t ) = a ( t ) exp [ j 2π Ft t ] ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18 Doppler Shift Example - 2 • Received signal becomes 2 ( R0 − vt ) 2 ( R0 − vt ) y (t ) = b t − exp j 2π Ft t − c c 4π Round-trip 2 R0 R0 ≈ b t − exp − j phase c λ t Carrier 2v Delayed exp + j 2π t exp [ j 2π Ft t ] Envelope λt Doppler term ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18 End of Module #18 ECE 6272 Mark A. Richards mark.richards@ece.gatech.edu © Copyright 2006. All Rights Reserved. Module #18