这种恒定相移的差异不影响回波的幅度或多普勒频移，可以忽略不计。

This difference inthe constant phase shift does not affect the magnitude or Doppler shift of theecho and can be ignored.

因此，式（2.88）的分析方法在所有重要方面都与先前的结果是一致的。

Thus the analysisapproach of Eq. (2.88) is consistent with the earlier results in all importantrespects.

关于式（2.88）使用的更有趣的例子，请再次回顾图2.25。

For a moreinteresting example of the use of Eq. (2.88), consider Fig. 2.25 again.

假设雷达位于(x, y)坐标的(x_r= 0, y_r = 0)位置，天线指向为+y轴，目标飞机的坐标为(x_t= vt, y = R₀ )。

Let the radar belocated at (x, y) coordinates (x_r = 0, y_r = 0) with itsantenna aimed in the +y direction, and let the coordinates of the targetaircraft be (x_t= vt, y_t = R₀ ).

这意味着目标飞机在时刻t=0时的距离R₀处的雷达视线上，以每秒v米的速度垂直于雷达视线穿过。

This means that thetarget aircraft is on the radar boresight at a range R₀ at time t =0 and is crossing orthogonal to the radar line of sight at a velocity v metersper second.

雷达与飞机之间的距离为：

The range betweenradar and aircraft is

虽然可以直接使用式（2.100），但通常以幂级数的形式展开平方根：

While it is possibleto work with Eq. (2.100) directly, it is common to expand the square root in apower series:

在评估这一表达式时，必须考虑t的范围可能受到几个因素中的任何一个的限制，例如目标在雷达主波束内驻留的时间或相干处理间隔时间，在这段时间内的脉冲将被收集用于后续处理。

In evaluating thisexpression, the range of t that must be considered may be limited by any ofseveral factors, such as the time the target is within the radar main beam orthe coherent processing interval duration over which pulses will be collectedfor subsequent processing.

假设目标在这段时间内运动的距离远小于R₀，因此可以忽略(vt/R₀)中的高阶项：

Assume that thedistance traveled by the target within this time of interest is much less thanthe nominal range R₀ so that higher-order terms in (vt/R₀)can be neglected:

式（2.102）表明，对于图2.25中的目标运动场景，距离变化约为时间的二次函数。

Equation (2.102)shows that the range is approximately a quadratic function of time for thecrossing target scenario of Fig. 2.25.

利用式（2.98）中的截断序列，得出

Using this truncatedseries in Eq. (2.98) gives

除中间指数项以外，所有项均与式（2.99）中的等速运动情况相同。

All of the terms arethe same as in the constant-velocity case of Eq. (2.99) except for the middleexponential.

回想一下，瞬时频率与相位的时间导数成正比。

Recall thatinstantaneous frequency is proportional to the time derivative of phase.

因此，二次相位函数表示由于雷达目标几何结构的变化而随时间线性变化的多普勒频移：

The quadratic phasefunction therefore represents a Doppler frequency shift that varies linearlywith time due to the changing radar target geometry:

——本文译自Mark A. Richards所著的《Fundamentals of Radar Signal Processing（Second edition）》

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