 Hello everyone, I am Prashant S. Malge, Assistant Professor, Department of Electronics, Walchand Institute of Technology, Solapur. Today we will discuss the structures for FIR system, the learning outcome. At the end of the session, student will be able to draw the realization structure for a given transfer function of FIR system. Now, consider an LTI system characterized by y of n equal to minus k equal to 0 to n a k y of n minus k plus k equal to 0 to m b k x of n minus k. This is the generalized constant coefficient difference equation for an LTI system. For implementation, we can do it either in hardware or software depending on the applications. A structural representation using interconnected basic building blocks is the first step in the implementation. Now, we will consider the factors that influence choice of specific relation. These are computational complexity, memory requirements and finite word length effects. Now, computational complexity refers to the number of operations such as additions, subtractions, divisions and multiplications that are required to compute the output y of n. Memory requirement refers to the number of memory locations required for storing the system parameters, past inputs, past outputs as well as any other intermediate computed values. And now, the finite word length effects refers to quantization effects that are inherent in any digital implementation system either in software or hardware because these parameters need to be represented with finite precision. Computations that are performed in the process of computing the output y of n must be truncated or rounded off to fit within the finite precision constraints of the computer or the hardware used for implementation. Now, let us consider the structures for realization of FIR system. Now, FIR system is described by the difference equation y of n equal to k equal to 0 to m minus 1 b k x of n minus k. Now, by taking z transform, we can write this as y of z equal to summation k equal to 0 to m minus 1 b k z raise to minus k x of z. Therefore, by taking the ratio of y z to x of z, the transfer function is h of z is equal to k equal to 0 to m minus 1 b k z raise to minus k. The unit impulse response of the FIR system is actually identical to the coefficients b k's. Now, the different basic building blocks required for the realization structure. Computational algorithm for LTI system can be represented by block diagram using basic building blocks. The first block is a adder where in which we have multiple inputs and the output is some of the inputs. Second one is a multiplier. Now, here if x n is the input, a is a multiplying factor, y of n will be equal to a into x of n minus 1. Third one is a delay which refers to memory x of n. So, when it is delayed, y of n is equal to x of n minus 1, delay by one unit. And another is a pick off node where from the same input can be taken to multiple points. Now, let us discuss the direct form structure for FIR realization. The direct form structure for FIR system represented by the following equation that is y of n is equal to k equal to 0 to m minus 1 b k x of n minus k. Equivalently, this y of n is written as k equal to 0 to m minus 1 h of k x of n minus k. As I said previously, the coefficients b k's and your impulses forms h of n are same. If we expand this, we can write this as h of 0 into x of n plus h of 1 into x of n minus 1 h of 2 into x of n minus 1 plus dot dot dot. Finally, we can get h of m minus 1 into x of n minus m plus 1. Now, this can be shown by a structure. Now, this is x of n. With this, we get here as an output x of n minus 1. Here it will be x of n minus 2 n minus 3 and so on. So, as per our equation, this will be h of 0 into x of n plus h of 1 into x of n minus 1 plus h of 2 into x of n minus 2 h of 3 into x of n minus 3 and so on. Adding all these terms, we get y of n. So, this is the direct form structure for an FIR system. Now, think about this. What is the computational complexity of the structure and how many memory locations are required for implementation? Pause this video for a minute and write down your answer. Now, I will see. You might have written your answers. Let us see. Here, this particular structure requires m minus 1 memory location for storing m minus 1 previous inputs. That is, the previous inputs are x of n minus 1, x of n minus 2, x of n minus 3 and so on up to x of n minus m plus 1. Also, it requires multiplication by m number of coefficient. That is, we are from h of 0, h of 1 to h of m minus 1. So, it requires m multiplications and as total m terms are to be added, we need m minus 1 additions. Now, we will see this structure for linear phase FIR system. For linear phase FIR system, the impulse response should satisfy the condition h of n is equal to plus or minus h of m minus 1 minus n. That is, your impulse response should be either symmetric or anti-symmetric. So, once again, considering the equation for the output y of n which is equal to k equal to 0 to m minus 1 h of k x of n minus k. If we expand this, we get h of 0 into x of n plus h of 1 into x of n minus 1 plus h of 2 into x of n minus 2 dot dot dot plus h of m minus 2 into x of n minus m plus 2 and the last term will be h of m minus 1 into x of n plus m plus 1. Because of the symmetry here, the term coefficient h of 0 and h of m minus 1 are same. Similarly, this h of 1 and h of m minus 2 are same. And therefore, for implementing the structure, the structure will be now. Here you have x of n, x of n minus 1, x of n minus 2 and so on. And the last term here will be x of n minus m plus 1. As the two coefficients that is h of 0 and these are equal, we can add these two and multiply by h of 0. Same way, here this is x of n minus 1 and this will be x of n minus m plus 2. So, those two are added together and multiplied by h of 1. So, therefore, in this particular structure, the number of multipliers required will get reduced. So, this particular structure is for a linear phase FIR system, where the value of m that is the length of the filter m is r.