 myself Dr. Rajendra Dubey. Today we are discussing on circular conversion one of the important property of discrete Fourier transform. The learning outcome at the end of this particular session we are the students will be able to explain this concept of circular conversion. The main contents for this important property of discrete Fourier transform we require multiplication of 2 dfts. Then let us see what should be the application of circular conversion as linear filtering and what is the exact use of discrete Fourier transform for linear filtering. Now let us see if x1 of n and x2 of n be the two finite duration sequence of length n, their respective end point dft can be shown. So this is we have to recall what the definition of discrete Fourier transform has been given in terms of summation n is equal to 0 to n minus 1 x of n e raised to minus j 2 pi n k by n where it is has been sampled omega at 2 pi by n multiplied by k where k it is ranging from 0 to n minus 1. So for x1 of n and x2 of n we are representing the two corresponding dfts x1 of k and x2 of k and the multiplication of this 2 dft has been represented as a third dft as x3 of k and its relation between x1 of n and x2 of n has been represented by the inverse discrete Fourier transform definition and we are calling as a property of circular conversion equivalent to multiplication of 2 dft. So let us recall the i dft equation given in terms of capital N 1 by n summation k is equal to 0 to n minus 1 where x3 of k has been modified as the product of the multiplication of 2 dfts x1 of k and x2 of k. So it is given summation k is equal to 0 to n minus 1 x1 of k x2 of k e raised to j 2 pi k m by n where x3 of m will be calculated by rearranging the inter summation that has been given previously in equation 1 and 2 in terms of x1 of n and taking the range for the new variable l that is summation l is equal to 0 to n minus 1 as x2 of l but the exponential parameter has been considered with respect to the three variables that has been defined in terms of m minus n minus m, n and l and the fundamental period n. So let us express that exponential term defined previously e raised to minus j 2 pi k m minus n minus l by n in terms of a as the basic mathematical equation represented summation k is equal to 0 to n minus 1 raised to k for a is equal to 1 it is represented as n and for a not equal to 1 it is represented as 1 minus a n divided by 1 minus a. Now if we can equate these values suppose a is equal to 1 where m is the integer value n is integer value l is also integer value so m minus n minus l is also integer value and if we are considering in multiples of n that is n 2 n 3 n 4 n so on then that difference can be evaluated equal to capital N and if that a is not equal to 1 otherwise it is always equal to 0 since the previous equation 1 minus a raised to n divided by 1 minus a it becomes 0 for a not equal to 1. So if you can substitute this two approximation as the case 1 a is equal to 1 and a not equal to 1 the final equation for the circular convolution can be shown as x 3 of m is equal to summation n is equal to 0 to n minus 1 x 1 of n x 2 of m minus n of n where again m is having a range from 0 to n minus 1. So this is the basic equation of the circular convolution equation and if we can compare this equation of circular convolution with the convolution sum defined then that index m minus n which is periodic in terms of capital N it is called as the circular convolution. So the multiplication of DFT of two sequences which is equivalent to circular convolution of two sequences defined in time domain. Now let us see the application as such for the linear filtering how exactly the discrete Fourier transform and the circular convolution can be combined to give the result which should be the same with respect to the linear convolution. If h of omega is the frequency response y of omega again the response defined with respect to the output signal and x of omega is the spectrum defined for input then always we can express h of omega as y of omega divided by x of omega or y of omega it is given as product of x of omega with h of omega. So all these three functions they are continuous variable they are dependent on the variable omega. So in frequency domain approach this calculation is more efficient than the time domain convolution to the existence of the efficient algorithms for calculation of discrete Fourier transform. The finite if x of n is the finite duration sequence of the length l for given for the FI filter if it is 0 for n less than 0 and always having the minimum condition n greater than or equal to l and n greater than equal to m without loss of any information or loss of generality then the convolution equation can be expressed as y of n is equal to k is equal to 0 to m minus 1 h of k x of n minus k. Now if we can find out if the given sequence is having a length l and the length of the impulse response is m the final output response y of n is having a length is equal to l plus m minus 1. If we are writing equivalently in the frequency domain we have to represent previously it should be a multiplication of the frequency spectrum x of omega and h of omega. So in frequency domain we have to make the size of that DFT at least equal to l plus m minus 1 or that should be greater than l plus m minus 1 otherwise we are I mean the problem of the mixing of the two signals. So equivalently if omega is expressed in terms of 2 pi k bar n then we can show the product of the two spectrum can be equivalently written as y of k as well as is multiplication of DFT can be shown as x of k with h of k equal to y of k. So let us apply this n point circular convolution definition with respect to the linear convolution. So if the linear convolution suppose x of n is having a length l is equal to 5 and the length of impulse response is equal to length equal to 3 in linear convolution it is given as l plus m minus 1 that is 5 plus 3 minus 1 the length of the output sequence it is equal to 7. If we want to get the same result in by using the definition of circular convolution the size of the DFT should be equal or greater than l plus m minus 1 where to increase the length of the sequence either x of n or h of n to make that equal to the length equal to l plus m minus 1 otherwise whatever the length which is higher for x of n or h of n that can be selected by the circular convolution and it is giving a result which is mixing of the two signals and it is very difficult to reconstruct that this signal. So we have to make the number of zeros that way to add in x of n as m minus 1 or h of n as l minus 1. So we are adding number of zeros in x of n as well as h of n to make that length equal to l plus m minus 1 then we can show that the result of the linear circular convolution can be equivalently written as the result of linear circular convolution. So this concept of appending zeros or making the addition of zeros we are calling as zero padding. So this zero padding concept it is always used in discrete Fourier transform to perform the linear filtering approach. The references for this topic it is taken from DeHel signal processing by John Plochis and Manolchis. The fourth edition and publication Prentice Hall. Thank you.