 Hello, welcome to video lecture on digital signal processing. In this video lecture we are going to discuss the third part of lattice structure for FIR filter. At the end of this session students will be able to draw the lattice realization structure for a given FIR system. So, in this video lecture we are going to derive an expression which converts the lattice structure coefficients, lattice structure coefficients into direct form coefficients. So, first before deriving the expressions for the general case, let us consider m is equal to 3 which means that we are considering the 3 stage lattice structure here. For m equal to 3, y of n is equal to x of n plus summation k equal to 1 to m alpha m of k x of n minus k. Let us give this one as equation number 1. This can be written as y of n is equal to x of n plus alpha 3 of 1 x of n minus 1 plus alpha 3 of 2 x of n minus 2 plus alpha 3 of 3 x of n minus 3. So, this is the 3 stage output of lattice structure. Now we have y of n is equal to f 3 of n which is equal to f 2 of n plus k 3 g 2 of n minus 1. So, we know we all know that this is the output of 3 stage lattice structure. So, this can be here we know that f 3 of n can be found by the values of f 2 of n and g 2 of n minus 1. So, f 2 of 1 sorry f 2 of n is equal to f 1 of n plus k 2 g 1 of n minus 1. Now from this expression we know that f 2 of n can be found by the values of f 1 of n and g 1 of n here. Similarly, g 2 of n is equal to k 2 f 1 of n plus g 1 of n minus 1. So, these two are the output of stage 2. Similarly, f 1 of n is equal to similarly f 1 of n is equal to f naught of n plus k 1 g naught of n minus 1. Similarly, g 1 of n is equal to k 1 f naught of n plus g naught of n minus 1. So, this is the output of stage 1 stage 1 lattice structure. So, earlier here we have written the output of 2 stage second stage lattice structure here. So, this is the output of second stage lattice structure. So, we will substitute the output of first stage and output of second stage in the above expression in this expression here. So, let us give this one as equation number 3 here. Now after substituting the output of first stage and output of second stage in the equation 3 we get this one as f 3 of n which is nothing but the y of n which is equal to x of n plus alpha 2 of 1 x of n minus 1 plus alpha 2 of 2 x of n minus 2 plus k 3 alpha 2 of 2 x of n minus 1 plus k 3 alpha 2 of 1 x of n minus 2 plus k 3 x of n minus 3 ok. So, after simplifying this one y of n can be written as x of n plus alpha 2 of 1 plus k 3 alpha 2 of 1 which is nothing but x of n minus 1 into x of n minus 1 plus alpha 2 of 2 plus k 3 alpha 2 of 1 into x of n minus 2 plus k 3 x of n minus 3 here. So, let us call this one as equation number 4. Now pause the video for some time and find out the values for alpha 3 of 0 alpha 3 of 1 alpha 3 of 2 and alpha 3 of 3 by comparing equation 2 and equation 4. I assume that I assume that you have written the values for alpha 3 of 0 alpha 3 of 1 alpha 3 of 2 and alpha 3 of 3 ok. So, comparing equation 2 and 4 we get alpha 3 of 0 is equal to 1 and alpha 3 of 1 is equal to alpha 2 of 1 plus k 3 alpha 2 of 2 which is equal to alpha 2 of 1 plus alpha 3 of 3 into alpha 2 of 2 ok. Similarly, alpha 3 of 2 is equal to alpha 2 of 2 plus k 3 alpha 2 of 1 which is equal to alpha 2 of 2 plus alpha 3 of 3 alpha 2 of 1 ok. So, similarly alpha 3 of 3 is equal to k 3 ok. So, let us call this one as equation number 5 ok. Now for general case alpha m of 0 is equal to 1 and alpha m of m is equal to k suffix m ok and alpha m of k is equal to alpha m minus 1 of k plus alpha m of m into alpha m minus 1 of m minus k. So, let us put this one as equation number ok. So, let us call this one as equation number 6 ok. And these are the expressions to convert the m stage coefficients of lattice structure into m stage direct form filter coefficients. So, these are the reference books. Thank you.