slide 21

This information explains the process of deriving the Doppler shift from observed phase change in a SAR (Synthetic Aperture Radar) system. The phase change is converted to velocity, and the Doppler shift is calculated using the time derivative of the phase change. The Doppler shift is used to generate a curved signal with a bandwidth equal to 2 times the platform velocity divided by the wavelength. The azimuth resolution is independent of the platform's attitude and depends only on the size of the antenna. Smaller antennas result in better azimuth resolution. First, we look at the figure. LS means the total length that an object is observed by several footprints. X-coordinate is used to know the azimuth positions of an object in different footprints. S0 is the center of the azimuth position. We take it as the origin. The range between the satellite and the X0 is R0. And the range difference delta R is the difference between the range of the central azimuth position and that at the other azimuth positions. The range difference delta R can be converted to the phase difference, or we call it phase change. The SAR sensor is more sensitive to the phase change. We then try to derive the Doppler shift from the phase change. The delta R is equal to the X squared over 2R0. We can convert the position X to the velocity of the platform times the time. Here the time means the time interval from the position X to the X0. Afterward, we can convert from the delta R to phase change phi. Phi is equal to minus 2 times delta R times 2 pi over lambda. So, phase change phi is equal to minus 2 pi times the square of the velocity of the platform times T over R0 lambda. We then try to relate the phase change phi to the frequency f, which is Doppler shift. f is equal to the time derivative of phi. So, we can do the time derivative of phi from the second equation. We get the derivative is minus 4 pi times velocity of platform squared times T over R0 lambda. Next, we need to know what is T. We calculate the cat at the farthest azimuth position. Because the Doppler shift at the farthest azimuth position determines the bandwidth of the curved signal. The velocity of the platform times T is equal to the half of Ls. The half of Ls can be approximated by R0 times delta R, theta R, theta A. And theta R is equal to lambda over 2L. So, we can know that velocity of the platform times T is equal to the half of R0 times lambda over L. We can then get T. T is equal to R0 times lambda over 2 velocity of the platform. Then, we substitute T into time derivative of phi. We can get time derivative of phi is equal to minus 2 pi times velocity of platform over L. Converting it to the frequency f, it is Doppler shift, f is minus velocity of the platform over L. Note that velocity of platform is negative when approaching x0. The reason for the negative is that the azimuth position for this case of approaching x0 is negative. On the other hand, the velocity of platform is positive when moving away from x0. We have already derived the equation to obtain Doppler shift from the observed phase chain phi. The Doppler shift can generate a curved signal. The bandwidth of the curved signal, we call it Doppler bandwidth, is 2 velocity of platform over L. We can then know the azimuth resolution is velocity of the platform over bandwidth, which is the half of antenna length. Up to now, we can see the azimuth resolution becomes independent to the attitude of the platform. It is only related to the size of the antenna. And the smaller the antenna length, the better the azimuth resolution. We don't need to make a large antenna. In contrast, we should make a small antenna.