Principles of Modern Radar - Volume 2 1891121537

2.5 Stepped Frequency Waveforms 61The first term to the right of the equal sign is a constant and is associated with the roundtripdelay and the initial transmit frequency. The second term is a complex phasor rotatingat a rate defined by the frequency step size and the range to the target. The phase rotationis similar to that observed when employing a pulsed Doppler waveform. To simply theexpressions, the amplitude of the return is set to unity.Fourier analysis is applied to the complex samples in equation (2.112) to extract thelocation of the return in range. Consider the DTFT defined byX (ω) =N−1∑n=0x (n) exp (− jωn) (2.113)where x (n) are the measured returns collected from N pulses, and ω is the digital frequencywith units of radians/sample. The DTFT represents a filter bank tuned over a continuumof frequencies (or rotation rates), which in this case correspond to different ranges.The samples are often viewed as being collected in the frequency domain, and thereturns are then transformed to the time (or range) domain. The frequency-domain interpretationis based on the assertion that each pulse is measuring the target’s response(amplitude and phase) at a different frequency. From this perspective, an inverse DTFTwould naturally be applied; however, it is the rotating phase induced by the change infrequency that is important. Either a forward or inverse DTFT may be applied as long asthe location of the scatterers (either up- or down-range) is correctly interpreted within theprofile, and the return is scaled to account for the DTFT integration gain.For a point target located at range R 0 , the output of the DTFT is a digital sinc definedby( N ( ) )sin ω − ωR0 |X (ω)| =2(( ))(2.114)ω − ωR0 ∣sin2 ∣where ω R0 = 2π 2R 0c f , ω = 2π 2R f , and R is a continuous variable representingcrange. Equation (2.114) represents the range compressed response. The term ω R0 centersthe response at a particular range or frequency. The shape of the compressed response isexamined by setting R 0 = 0or( )N sin|X (ω)| =2 ω ( )ω (2.115)∣ sin∣2A plot of the compressed response is shown in Figure 2-25. The response consists of a mainlobe and sidelobe structure with peak sidelobes 13.2 dB below the peak of the main lobe.The DTFT is periodic and repeats at multiples of 2π in ω; therefore, the range compressedresponse is also periodic with periodicities spaced by c/2f . An implication is that rangemeasured at the output of the DTFT is relative to the range gate and not absolute range.Absolute range is defined by the time delay to the range gate and the relative range offsetwithin the profile.