Principles of Modern Radar - Volume 2 1891121537

6.6 Image Metrics 243Along-track ambiguities arise because the synthetic aperture is not sampled finelyenough to avoid aliasing all spatial frequencies. The pulses would have to be spaced at orless than λ/4 to avoid spatial aliasing altogether. Aliased returns manifest as multiplicativenoise in the image, and the typical effect on the image is simply reduced contrast.However, highly-reflective discrete objects can cause visible ghosting in the imagery (e.g.,Figure 7-28 in [3]).Range ambiguities are echoes from earlier or later transmissions superimposed onthe return from the current pulse. Whereas the along-track ambiguities are attenuatedprimarily by the transmit/receive beampattern, the range ambiguities are attenuated byspherical spreading loss, the beampattern of the radar, and the fact that the grazing angle(and thus terrain reflectivity) becomes very small at ranges far beyond the imaged scene.Range ambiguities are also terminated by the radar horizon caused by the earth’s curvature.The total ASR is given by the integral in equation (6.32), where G = G TX G RX is thecomposite two-way gain for a particular spatial frequency k x (proportional to the anglefrom boresight for a broadside-looking sensor), σ 0 is the backscatter coefficient of theterrain, B p is the processed spatial bandwidth, k xs is the spatial sampling frequency inrad/m (k xs = 2π/δx), δx is the distance between along-track samples, and f p is the PRFin Hz. The ASR is expressed as a function of time τ (or equivalently, range R = cτ/2):∞∑∫ −Bp /2ASR(τ) =m,n=−∞m,n≠0−B p /2G(k x + mk xs ,τ+ n/f p ) · σ 0 (k x + mk xs ,τ+ n/f p )dk x∫ −Bp /2−B p /2G(k x ,τ) · σ 0 (k x ,τ)dk x(6.32)The literature frequently describes the along-track ambiguities in terms of Doppler frequency,rather than spatial frequency, as is done here. The formulations are interchangeable,and the two are related by f D = vk x /2π, where v is the forward speed of the sensor.For a fixed temporal frequency, the spatial frequency is proportional to the angle of arrivalθ of the reflected signal: k x = 2k sin θ [17,24].The processed bandwidth is the extent of the spectral region of support of the wavenumberk x (recall Figures 6-13 and 6-14). Thus, the processed bandwidth is equivalent to aneffective beamwidth. It is bounded above by the composite transmit/receive beamwidth,and it may be intentionally reduced to achieve better ASR performance at the expense of asmaller scene size. Interestingly, the processed spatial bandwidth has different meaningsfor spotlight and stripmap imagery. The common thread is that reducing processed bandwidthequates to narrowing the effective beamwidth. For a fixed wavelength λ, narrowerbeams reduce the length of the synthetic aperture (thus coarsening resolution) in stripmapmode while they reduce the size of the illuminated scene in spotlight mode.6.6.1.2 Integrated Sidelobe RatioThe integrated sidelobe ratio is the ratio of the energy in the sidelobes of the imageimpulse response to the energy in its main lobe. Specifically the quantity of interest is thetwo-dimensional ISLR. Like the ASR, it is a scale factor and has no reference quantityassociated with its decibel representation. Recalling that multiplication in one Fourierdomain is convolution in the other, we see that the spectral trimming and windowingapplied as part of the PFA reconstruction image determines the impulse response. TheISLR values for several common windows are given in Table 6-4.