 Welcome to this session, so in today's class we shall be focusing on basics as well as we shall try to cover a few problems to understand the concepts in a better manner. You know I always like to compare this to listening to your favorite song, you know. If you like it, you tend to listen to it repeatedly back to back and it is not going to cost you any boredom, isn't it? And automatically the lines and verses become imprinted in your mind. So similarly the aim of today's lecture is that we shall be revisiting few of the concepts that we discussed earlier. Consider this as a quick recap so that it will be easier for you to solve numericals. So we are going to discuss quickly about real aperture radar and synthetic aperture radar. Remember when we discussed about real aperture radar we mentioned that the resolution of a real aperture radar is primarily dependent on the length of antenna. The larger the antenna, better is the spatial resolution but in the case of real aperture radar it is not practically possible for us to have a very long antenna. Now shown here is the flight direction. By now I assume that you are familiar with these terminologies that is the nadir direction, perpendicular to the platform, the azimuth direction is shown here, the ground range direction is shown here and swath is shown here. So in real aperture radar or side looking airborne radar the amplitude of each return echo is measured and processed amplitude. Shown here are also the expressions for azimuth resolution which we have covered as part of earlier lecture. It is nothing but lambda into R by D where lambda is the wavelength, R is the range and D is the length of antenna and over here we have range resolution in brackets I have written slant range to differentiate it from ground range resolution. So here it is range resolution which is nothing but C tau P by 2, C is the speed of flight. I will give it a moment to sink in. Now remember in the case of real aperture radar there are other determining factors also such as the antenna beam width and the pulse duration that affect the spatial resolution. Now moving on to synthetic aperture radar or SAR. See when it comes to synthetic aperture radar it is exploiting the history of echoes, exploiting the history of echoes. To be more specific it exploits the Doppler history of echoes that are caused by the forward movement of the antenna to synthesize a large antenna. And we also discussed about chirped pulses, isn't it? Chirped pulses are used in the range direction and motion of platform is inducing frequency modulation in the azimuth direction. Once again let me re-iterate. So in the case of synthetic aperture radar chirped pulses are used in the range direction we know what is range direction. This is the movement of flight we know what is range direction. Chirped pulses are used in the range direction and the motion of platform induces frequency modulation in the azimuth direction. So what is the net effect? Net effect is that in both the directions that is in the azimuth as well as the range direction the signals are getting frequency modulated. And it is this frequency modulation which is sensed in the Doppler shift. So shown here are the relationships for azimuthal spatial resolution which is nothing but length of antenna by 2 and given here is the range resolution. The ground range resolution is given as c tau p by 2 into sin theta i where theta i is the incidence angle theta i. Again let us re-iterate the radar equation which gives the fundamental relationship between the power transmitted and power received which is nothing but P r equal to P t into G square into lambda square into sigma that is radar cross section by 4 pi cube into r raised to 4. I am repeating it because it is the fundamental relationship of transmitting power and received power transmitted power by radar received power by radar how they are related the fundamental relationship is given by a radar equation. Now remember SAR we are understanding it as an instrument that is both transmitting and receiving. There is no separate transmitter, there is no separate receiver we till now we are understanding SAR as a transmitting and receiving instrument. At this point it is worthwhile for us to recollect the chirped pulses where chirp it stands for compressed high intensity radar pulse that is chirped pulse. So just to re-iterate the radar engineers use a long pulse with a linearly modulated frequency called a chirp and with this technique it is possible to increase the range resolution. Once again the aim of a chirped pulse is to increase the range resolution. So given here in front of you is the expression for range resolution which is nothing but C by 2B where C is nothing but speed of light and B is the bandwidth. We have seen the relationship as part of earlier lectures is not it? So this was a quick recap of the concepts which we have been covering as part of module 2. Of course this is not giving us a complete summary moving on. Now remember that in one of the earlier lectures we were trying to understand about different formats of synthetic aperture radar data and that is when we discussed that the very first type of data is called as raw data. And in SAR data by now we know what is range direction and what is azimuth direction. SAR system is moving in the azimuth direction or along track direction and the SAR system is collecting data in the range direction or across track direction along track across track azimuth range. Now please note that radar system can receive even 4000 to 5000 responses for one individual target which means in the first type of data that is the raw data the energy scattered from a target will be spread in both the azimuth as well as the range direction. Let me repeat the energy appear to be scattered let me put it that way. In raw data the energy can appear to be scattered in both directions that is azimuth direction as well as range direction which means if we have a point target its echo will be distributed in both the azimuth direction as well as the range direction. So to explain this let me show you a small schematic on the x axis we have samples within echoes on the y axis assume we have individual pulses and whatever shaded area in red that you see assume it is a field within the SAR raw data that contains power reflected from a ground cell R1A1 range azimuth instead of using x y I am using R1A1. So this is one ground cell which is having scattered power in both the azimuth as well as the range direction which means there is a requirement of SAR data focusing. Now by focusing we mean we want to collect all the energy into one single pixel of the output image because right now in the raw data the it is getting scattered in both the direction azimuth direction as well as range direction. So we need to use SAR data focusing and briefly I will try to explain what is meant by SAR data focusing that is there are many arithmetic computations which are involved in SAR data focusing and it typically involves something known as data compression okay data compression in both the range direction as well as the azimuth direction and data compression in range as the name suggests the transmitted pulse is being compressed in one range bin while the echoes are spread along the azimuth direction okay. Let me try to explain with a schematic. So now what you see in front of you is towards the left side you have the raw SAR data that we already discussed and in the middle assume it is the same image but after range compression okay data compression in range direction as the name suggests here the transmitted pulse is being compressed in one range bin while their echoes are spread along the azimuth direction okay that is why you see the red shaded region compressed in range but with information from a single azimuth spread over many pulses. Now which means we need to go for azimuth compression as well okay which means once we complete the data compression in the range direction the data set is now distributed in the azimuth direction which means we need to perform data compression in the azimuth direction as well which is when you get resolution cell represented like this R1A1. So this is the final representation of a synthetic aperture radar image. So you have the range cells as the x axis the azimuth cells as the y axis okay. Now when you go in detail into the mathematics fast Fourier transforms or FFT is used for data compression in both range and azimuth directions while we may not be going into the mathematical details right now I wanted you to be aware that to convert a raw SAR data we need to compress it in both the range as well as azimuth direction so that we have the information correctly represented okay without echoes distributed in both the directions alright. So with this background let us now try to solve a few numericals. We will try to focus on the radar equation and the Doppler effect. I hope you can recollect it as part of the previous lectures alright. Now moving on to the first question. The question in front of you is a very simple question for you to check whether you have understood the Doppler effect correctly. It reads an airborne radar is operating in the X band at 10 gigahertz the wavelength lambda is given as 3 centimeters and is traveling at 700 miles per hour. It is tracking a target ahead which is moving at 800 miles per hour in the same direction. So there are two parts of the question. First question is what is the differential speed okay given 700 miles per hour and it is trying to track the target which is moving at 800 miles per hour. The first part of the question is what is the differential speed. Let us write it down differential speed is nothing but 700 minus 800 is the straightforward isn't it minus 100 miles per hour but then we are more familiar with meter per second isn't it. One mile per hour is 0.44 meter per second. So I can as well write it as minus 44.704 meter per second okay differential speed. Now the second question reads what is the Doppler shift of the target. Second part of the question what is the Doppler shift of the target. Now for that let us try to recollect the expression we had come across to quantify the first target Doppler shift. Let us write it down first target Doppler shift is nothing but f Doppler if you remember the notation which was covered in class which is nothing but 2 into relative velocity by lambda which means it is nothing but 2 into 44.704 wavelength is given as 3 centimeter 0.03 meter. So the answer comes to 2.9 kilohertz first target Doppler shift okay easy. Now go to the second question now okay here the question reads that for a single look complex image that is SLC image the value of one single pixel is given as 5 plus 2.7i and the value of amplitude and phase information is asked okay some sort of a fill in the blanks what is the value of amplitude and phase information if the value of a single pixel is 5 plus 2.7i. Now remember we discussed that SAR image can be considered as composed of complex numbers. So here the value of a single pixel is given as 5 plus 2.7i where we know that i is nothing but root of minus 1 isn't it. Now the SAR amplitude image can be computed on a pixel by pixel basis using the expression amplitude equals root of a square plus b square while the phase can be computed by the expression tan inverse plus a by b. Let us try to solve the simple problem. So here the complex number is given as 5 plus 2.7i okay which means I can compute amplitude using the expression which was shown earlier which will come out to be root of 5 square plus 2.7 square it will come out to be 5.68 amplitude. Now phase value for the same pixel it can be computed as tan inverse of 2.7 by 5 you will get the value as 28.3 degrees remember I have given you just one pixel value and when we are discussing about a single look complex image SLC image there are multiple pixels in a SAR image which means there are we are dealing with complex numbers matrix of complex numbers. This question dealt with just one single pixel value alright. So let us move forward to the third question. So this question reads an X band side looking airborne radar that is SLAR system with a 2 meter antenna has an azimuth antenna beam width of what okay? X band which means wavelength is given what else is given? 2 meter antenna is given. So we know that for an X band SLAR side looking airborne radar system with a 2 meter antenna it is going to have an azimuth antenna beam width of lambda by d 0.03 meters by 2 okay. What is 2? Length of antenna. So you will get the value as 0.015 radian okay 0.015 radian. This question what have we done? We have written down the given details that is lambda equal to 0.03 meters. We know that for an X band SLAR system that a side looking airborne system with a 2 meter antenna it is going to have an azimuth antenna beam width of we can write it as theta A equal to 0.03 by 2. What is 2? It is the antenna length which will come out to be 0.015 radian okay. Straight forward questions what you need to do is you just need to understand what is given and relate it to the relationships that we have been learning over the course of lectures okay. So moving forward the next question reads a radar system is having an unambiguous range of 300 kilometers and a bandwidth of 0.5 megahertz. It has two parts. The first part reads what is its range resolution and the second part reads calculate the pulse width okay. So first part of the question what is the range resolution? We can use the expression delta R is C by 2B which is nothing but C we know speed of light by 2 into bandwidth B is given 0.5 into 10 power 6 to make the units consistent. You can get the values 300 meters range resolution okay. So moving on the second part of the question is calculate the pulse width okay. We know the expression for pulse width which is nothing but by C have just rearranged the expression okay. So we can write it as 2 into 300 by 3 into 10 to the power 8. You will get the value as 2 new seconds pulse width you know simple questions simple numericals to help us understand the concepts that we have been covering so far. So let me hope that you understood this part of the lecture and I will meet you again in the next class. Thank you.