 Good afternoon. In today's session, we are discussing the properties of discrete Fourier transform. Myself, Dr. Rajanadubai, Professor of Electronics and Telecommunication Department, WIT, Sallapur. The learning outcome after the end of this session, the student will be able to explain the concept of properties of DOT. The contents includes the introduction, the periodicity, linearity, circular symmetries of a sequence, and symmetry properties of discrete Fourier transform. Let us recall some of the concepts that we required to discuss the properties of discrete Fourier transform. The definition or the equation of discrete Fourier transform can be written in terms of summation n is equal to 0 to n minus 1, x of n, w n raise to k n, w n is the Twiddle factor, and the value of k is ranging from 0 to n minus 1. And here, n is at least equal to the length of the sequence or greater than length of the sequence, n should be greater than or equal to l. The IDFT equation is given as 1 by n summation k is equal to 0 to n minus 1, x of k, w n raise to minus k n, where again n is ranging from 0 to n minus 1. With this introduction of recalling the concept, the Twiddle factor e raise to minus j 2 pi n, which is again holding the property of periodicity, where the definition of DFT can be calculated with respect to the frequency domain representation, where it is sampled at omega is equal to 2 pi k by n. Now, let us start to see the property of a discrete Fourier transform as the periodicity. If the given sequence is x of n, which is a finite duration sequence, and if we are writing its corresponding DFT as x of k, then if capital N is a fundamental period written as x of n plus n is equal to x of n for all the values of n, then the property of periodicity can be applicable to all the DFT values for all the values of k as x of k plus n is equal to x of k. This periodicity property, it is always explained in terms of the fundamental period, which is satisfied or gives the reconstruction of the original signal as x of n without having any mixing or without having any alising problem. The second property is the linearity or the additive property or the superposition property, if the two sequence has been given x 1 of n, x 2 of n, and its corresponding DFT x 1 of k and x 2 of k, then the sequence, which is if it is written as the algebraic sum with the coefficient a 1, a 2 as a 1 x 1 of n plus a 2 x 2 of n, then the n point DFT again can be explained in terms of its corresponding coefficient a 1 x 1 of k, where x 1 of k is the DFT of x 1 of n plus a 2 of x 2 of k, where x 2 of k is the DFT of x 2 of n. So, this superposition or addition principle is applicable to DFT also. The circular symmetries of a sequence can be explained by considering the basic condition of discrete Fourier transform, if n point of DFT of a periodic sequence x p of n, where x p of n is periodic repetition in terms of small l can be shown as summation l is equal to minus infinity to infinity x of n minus l n, then we can make this new version as x dash p of n by shifting that x p of n, the periodic sequence by k sample, that is x of n minus l n can be shown as x p of n, a new sequence, finite duration sequence x dash p of n, which is the shifted version of k samples can be shown as l is equal to again minus infinity to infinity x of n minus k minus l n, that is the finite duration sequence, a new sequence x dash of n is always written as x dash p of n, where n it is defined in between the range 0 to n minus 1, otherwise it is value 0. The corresponding circular shift can be shown with the previous equation as x of n minus as if x of n is the given sequence and if we can show is periodic sequence, x of 0 is value equal to 1, x of 1 is value equal to 2, likewise we can show x of 3 as a value 4, the corresponding periodic sequence x p of n we can show as if n is equal to l as the shifted, not shifted but a periodic repetition of the same signal x of n and if k is equal to 2, then the x p of n minus 2, that is we are shifting the sequence to the right by k is equal to 2 and the final or the new sequence x dash p or x dash of n can be represented with the values 3 4 1 2. So, shifting to the right by k sample can be shown as x p of n minus 2 and this shifting to the right or to the left or folding the sequence can be represented in circular shift fashion, where x of n or x dash of n can be considered clockwise or anticlockwise and then we can represent both the sequence in circular fashion. So, in general the circular shift of a sequence can be represented as the index module n, this index module n is a divisional factor, where we are dividing x of n minus k with the modulo n the n the factor and then we are writing the remainder and then the equivalent value we can show as x of n minus k of n, where n is the fundamental period. So, if n is equal to 4 and k is equal to 2, so this shifting of the signal x dash of n can be shown as x of n minus 2 4. Likewise, we can show the remaining values in terms of n 0 to 3 as x dash of 0, x dash of 1, x dash of 2 and x dash of 3, which is equality written in terms of x of n minus 2 of 4 as x of 2, x of 3, x of 0 and x of 1. During the circular shift properties again if we are considering n point sequence circularly even or n point sequence circularly odd, it is symmetric about the 0 on the circle we can define x of n minus n equal to x of n, but here the sequence is starting from 1, it is not starting from 0, as well as if the n point sequence is called as the odd then x of n minus n can be shown as minus x of n again it is starting from 1, not it is starting from 0. The time reversal of an n point sequence can be attained by reversing the samples that is by taking the reverse with respect to 0 on the circle that sequence can be represented as a folding sequence, folding the sequence with respect to the initial point on the circle as x of n minus n of n can be equal to x of n minus n where again n is ranging from 0 to n minus 1. Likewise, we can make the use of definition of even sequence and odd sequence and combinedly we can represent the sequence x of n which is again the periodic sequence here this x p of n is always represented as x p of in terms of even sequence x p of n and in terms of odd as x p of n this can be represented as x p of n minus n and minus x p of n minus n respectively. Then if the periodic sequence is complex value up till now we have discussed with the real value if the periodic sequence is complex value then we have to divide its even value and odd value as a conjugate and that conjugate can be written as x p of conjugate of n minus n for even and minus of x p of conjugate of n minus n for odd values. Then the decomposition we can represent in terms of even values in terms of odd values and this decomposition can be shown as x p of n is equal to x p e of n plus x p o of n. Then the symmetric properties of discrete Fourier transform can be written as by considering the complex value of x of n as x i of n plus j x i of n consist of real part and imaginary part. The corresponding part of the DFT is also can be shown as real part x r of k plus x i of k. So if we are putting x of n in the basic equation of the DFT then by removing the real part and by removing the imaginary part we can find out x r of k and x i of k. Similarly we can apply this to show the real part of as small x r of n and x i of n defined for the IDFT equation where again from the IDFT equation we can calculate is real part and imaginary part and that can be shown in terms of x r of n and x i of n. For this we are referring the book digital signal processing by John Corkis and Manu Corkis that is fourth edition. Thank you.