 Good afternoon. Today we are discussing in this session the discrete Fourier transform. Myself Dr. Rajendra Dube as working as a professor Electronics and Communication Engineering Varshan Institute of Technology, Solapur. The learning outcome of the discrete Fourier transform. At the end of this session student will be able to explain the concept of discrete Fourier transform. The content of DFT includes the frequency domain sampling which is very important and the reconstruction of discrete time signal. After doing the sampling the reconstruction is very essential to recover the original discrete time signal and after sampling and reconstruction how we can get the equation of discrete Fourier transform. To learn about the frequency domain sampling we have to see the a periodic discrete time signal concept of with respect to the Fourier transform. If the frequency spectrum with the a periodic discrete time signal can be represented as the summation n is equal to minus infinity to infinity with the discrete time signal x of n e raised to minus j omega n. If we can evaluate this omega at 2 pi k by n then the modified equation is represented with respect to this frequency as with respect to the k 0 to n minus 1. Let us see how we can represent this frequency spectrum. This frequency spectrum is the periodic repetition of x of n with the n sample number of samples has been shown. If we are selecting the range in between 0 to 2 pi and if the spacing between 2 sample has been considered as delta omega then at each point we are representing this repetition with respect to the fundamental period as represented as x of k delta omega n. So, this is the frequency domain sampling defined with the frequency range. The frequency range generally we are selecting as in between 0 to 2 pi equal spacing and this equal spacing will give the correct adjustment of the sampling concept with respect to the Nyquist criteria. If we are dividing the inner sum in between the range minus n to minus 1 0 to n minus 1 to n to minus infinity that is if we are representing the frequency sample at 2 pi k by n with the range minus infinity to infinity. So, this frequency domain sampling if we are representing as the basic equation with respect to l as l is equal to minus infinity to infinity and if we are considering l as with 0 then the range we can have summation n is equal to 0 to n minus 1 x of n e raise to minus j 2 pi k n by n. If n is equal to minus 1 then we are having a range minus n to minus 1 so on. Generally, we are representing in such a way that it show the periodic repetition of the inner sum ranging from the value n to n minus l n and interchanging the order of the summation will give the periodic repetition of the signal that has been represented as x p of n in terms of x of n minus l n with the range of l generally we are mentioning previously also it is minus infinity to infinity. In this case the capital N is the fundamental period and the frequency domain sampling will give the measurement in terms of the linear combination of exponential signals in terms of the Fourier series and this Fourier series is represented as x p of n is equal to summation k is equal to 0 to n minus 1 with the Fourier coefficient c k e raise to j 2 pi k n by n and this c k will give the actual concept of the frequency domain sampling and we are showing in terms of the range k is equal to 0 to n minus 1 with the periodic repetition n is equal to 0 to n minus 1 in terms of the fundamental period. Then if you can compare this with the previous equation with this equation number 6 then the value of c k can return in terms of the frequency spectrum as 1 by n x of 2 pi n into k. So, if we are showing this x p of n as the periodic repetition as x of 2 pi k by n and if we are considering the number of samples in between 0 to n suppose if you are in the sample as l if the the fundamental period if it is less than l what will be the situation if the fundamental period is higher than l what will be the situation and then what will be the condition or result when n is equal to l. When fundamental period is equal to the length of the sequence l then there is no problem of reconstruction of the original signal we can recover that signal x of n with the periodic repetition x p of n. If n is greater than l then again the there should not be any mixing or the allizing problem since the fundamental period is higher than the length of the sequence. So, in this case the reconstruction is very good and we are recovering the original signal, but if n is less than l then there is always a chance of mixing of two signal and this mixing of two signal will not give the recovery or the reconstruction of the discrete time signal. This is the fundamental point that has to be remember for defining the equation of discrete Fourier transform as x of n is equal to x p of n if and only if the condition is n is at least equal to l or n should be greater than l. So, this will give the calculation of x x of n equal to periodic repetition x p of n in between the range 0 to n minus 1 otherwise this x p of n has been represented equal to 0. Now, let us recall the equation that is x p of n is an allies version of x of n and as represented with the basic equation summation l is equal to minus infinity to infinity in terms of n minus l n the fundamental period then we are defining a sequence x of n in such a way that it should represent x p of n with the range 0 to l minus 1 otherwise if you are considering less than l then it will give the value equal to 0. So, that will not show any periodic repetition of x of n with respect to the fundamental period n. So, in this case either we can represent in terms of capital N or in terms of capital L to give the correct frequency spectrum representation as x of omega ranging from 0 n is equal to 0 to l minus 1 x of n e h 2 minus j omega n. If you are sampling in between 0 to 2 pi we can represent this as the frequency samples which are equally spaced ranging from k is equal to 0 to n minus 1 satisfying the condition of n at least equal to l or greater than l then the definition will represent as the discrete Fourier transform. This Fourier transform will give the frequency analysis of the Fourier transform representation of a DT signal in digital signal processing this Fourier transform we are mentioning as x of k which is sampled at the frequency 2 pi k by n will give the basic definition of DFT. The range of this k is again can be given in terms of the fundamental period from 0 to n minus 1. If it is satisfying the condition n equal to or greater than l we are getting to the definition of discrete Fourier transform correctly. And if we are achieving the DFT there is always 100 percent chance to recover the sequence x of n from frequency samples and this recovery of the signal all the sequence x of n we are calling as inverse discrete Fourier transform. Again this inverse discrete Fourier transform it is represented in terms of the fundamental period with the number of samples that can be considered n is equal to 0 to n minus 1. Condition again here in IDFT it is n should be at least equal to l or greater than l then and then the recover of that sequence x of n is always possible and then we can make the representation as IDFT. Now in most of the applications this definition discrete Fourier transform as well as IDFT both are very useful since it depends upon the frequency domain sampling concept and satisfying the property of the periodicity if we are representing this e ratio minus j 2 pi k n by n in terms of the twiddle factor then this twiddle factor or the cyclic factor always give the periodic repetition. So, after every n sample they are having the repetition of same values where it will show the correctness of the definition what we are getting from frequency domain sampling as x of k. As well as the IDFT we are getting the probability x of n it is also will represent in terms of the fundamental period again this will again we can represent in terms of the twiddle factor. If we are writing in terms of matrix form then the cyclic or the periodicity property of twiddle factor represented in inverse matrix form always it is possibility to have the result of x of n. So, for this we are having the references we are taken from the Bible of the DSP that is John Plochis and Mancholis there is fourth edition and in most of the DSP applications we are using DFT where the use of DFT always give the representation with various properties representation with respect to the circular convolution representation with respect to the fast Fourier transform which is very useful for various applications in signal processing. Thank you.