 myself Piyusha Shedgarh. In today's session, we will discuss the topic multiplication of pattern. These are the learning outcomes for today's session. At the end of this session, students will be able to define pattern multiplication. They will be able to draw the radiation pattern for 4 and 8 point sources using pattern multiplication. These are the contents. Before going to start the pattern multiplication, you can pause video here for a second and recall that what is the radiation pattern of an antenna. Yes, the radiation pattern of an antenna is nothing but the directional characteristics. Directional characteristics can be defined with the radiation pattern of an antenna which having major lobes, minor lobes along with some of the side lobes which indicates the distribution of radiation power in free space in different angular regions. It is nothing but a graphical representation of radiation as a function of direction. Now, let us discuss what is the principle of the pattern multiplication. The principle of pattern multiplication states that the radiation pattern of an array is the product of pattern of the individual antenna with the array pattern of isotropic point sources each located at the phase center of individual source. Here we can take the product of pattern of the individual antenna and the array pattern of the isotropic different point sources. What is isotropic radiation? The radiation is considered for along the all direction then it is called as the isotropic radiation. What is array pattern? Because by considering the number of antenna elements that forms the array pattern which can be considered for the pattern multiplication. So, the array pattern is nothing but the function of the location of the antennas in the array and their relative complex excitations amplitudes. The phase center of the array is nothing but the reference point for the total phase pattern. It helps to sketch the radiation pattern of array antennas rapidly for the simple product of element pattern and the array pattern. So, you should know the separate individual pattern for the antenna element and also you should know the group pattern by using the array elements. You can take the product between these two to calculate the pattern multiplication or the resultant radiation pattern. So, the pattern multiplication principle is only applicable for arrays which containing the identical elements. The principle of the pattern multiplication is true for only any number of similar sources. Total phase pattern is nothing but the addition of the phase pattern of the individual sources and that of the array of isotropic point sources. Now, if you are taking the product between these two the resultant pattern will be formed. The resultant pattern of an array of now non isotropic identical radiators which is given by this equation E is equal to f of the function of theta and phi multiplied by capital F of theta phi. Again it is multiplied by f p theta phi plus capital F p theta phi whereas the theta and phi are the coordinates used in spherical coordinate system where if you see the first parameter that is small f of theta phi is nothing but consider as the unit pattern or it is the field pattern of the individual source whereas f p of theta phi is nothing but the phase pattern of the individual source. Capital F theta phi is the group pattern or it is also known as the array factor or field pattern of isotropic array and f p of theta phi denotes the phase pattern of array of isotropic sources. Now, let us consider the four elements given in this figure and we can calculate the pattern multiplication for this number of four array elements. Now, consider this these are the four elements separated by the distance lambda by 2 each element is separated by lambda by 2 distance. These elements are the isotropic elements or it is also known as the non-directional elements. So, if you are considering these all elements as 1, 2, 3 and 4 and if you are taking the center point between the 1 and 2 and 3 and 4 thus you are getting the points A and B. Now, if you calculate the distance between this A and B you getting that distance or these A and B points are separated by the lambda distance. Now, from the above figure the descriptions are given that is elements are spaced at lambda by 2 distance elements 1 and 2 are considered as one of the unit. So, these two elements 1 and 2 grouping of these two elements consider as 1 unit 3 and 4 is considered as the second unit and now you can calculate the center point for this 1, 2 and 3, 4 and you can consider the distance between them is lambda. Elements 3 and 4 are considered as the another unit. Since the elements are identical both the units have the same radiation pattern. Now, consider the individual pattern first instead of first considering the array pattern. So, the unit pattern is the pattern of the two elements which are spaced at lambda by 2 is given by this pattern. So, just you are getting the two major lobes here at it is for the two elements spaced at lambda by 2 distance. So, D is equal to lambda by 2 whereas, the alpha E is equal to 0 degree. This is the pattern shown for the two isotropic elements which are separated by lambda distance. Now, consider the array. So, in the above figure we have located the points A and B by taking the center point between the elements 1, 2 and 3, 4. So, consider the A and B are now separated by the lambda distance. These two units are considered to be one unit whose radiation pattern is shown in this figure. Now, these are the two points and this is the radiation pattern with these are the major lobes and these are the lobes which are perpendicular to these lobes and the spacing between this A and B is nothing, but the lambda. So, this is nothing, but the group pattern. Now, to calculate the resultant pattern use the concept of pattern multiplication. So, therefore, the resultant pattern is nothing, but calculated by taking the product between the unit pattern which is denoted for the single unit pattern is denoted with the two elements which are placed by lambda by 2 distance whereas, the group pattern is nothing, but the group pattern is used for the array which are placed by lambda spacing. So, the product of the unit pattern of 1 and 2 elements or you can consider the 3 and 4 elements because these elements are identical. And a group pattern of A and B these two are multiplied to each other to get the resultant pattern for the array of 4 elements. Now, if you are taking this is the unit pattern which is multiplied to the group pattern group pattern is shown in this figure like this one. Now, the major lobe when multiplied to the major lobe you are getting the major lobe, minor lobes multiplied to the major lobe you are getting the minor lobe and wherever the null point is there you are getting that is nothing, but the zero signal that is the null point. So, the resultant pattern is shown in this figure by taking the product between these two patterns. Now, similarly you can calculate the radiation or the resultant pattern for the 8 elements. Now, consider these are the different 8, 1, 2, 3, 4, till 8 these are the elements spaced by the distance lambda by 2. Again these elements are isotropic or it is also known as the non-directional. Now, if you are taking the center point between the 4 element from the left hand side and the right hand side these two the center points are getting A and B which can be separated by twice of lambda value. Now, center of the first 4 elements and the last 4 elements are marked as A and B and the unit pattern is nothing, but the pattern will be consider for the first 4 elements and the group pattern is the pattern for the 2 elements spaced at 2 lambda distance. Now, it is the unit pattern when it is multiplied to the group pattern of the 4 elements the resultant pattern will be observed. So, it is again the product of the unit pattern multiplied to the group or the array pattern for the 4 elements. So, again by using the pattern multiplication you can calculate the resultant pattern for the any number of elements. So, here the examples are given for the 4 elements and the 8 elements. Thus, you can calculate the number of you can calculate the radiation pattern or the resultant pattern for the n number of elements. These are the references used for today's session. Thank you.