Hi purna,
As u know that we determine Fourier Transform only for aperiodic signals, the case that Fourier transform of a periodic signal doesnt come into picture.
DFT of an aperiodic Discrete time signal x[n] is the sampled version of Discrete Time Fourier Transform {DTFT} .So here too the periodicity of x[n] doesnt come into picture. For discrete time periodic signals we have something called as Discrete Time Fourier Series {DTFS}.
I hope this convinces u, if u still have doubt plz be free to ask
Happy learning.........
applying DFT doesnt result in periodicity in the time domain,
but, DFT assumes the time domain signal to be periodic so that the
frequency domain representation becomes discrete and hence easy
to compute.
hey i have one more doubt that does DFT of a periodic signal for exaple let the continuous time signal be x(t) = cos(ω0t). So will the DFT of the Discretized version of this signal exists?
hi purnapragna,
We can represent the discretized version of the signal x(t) = cos(ω0t) as
x[n] = cos[ω0n].Now letz calculate the N point DFT of this signal, as it is periodic with 2Π consider 1 period where w0 = (2Π/N)k0.
Now express the cos[ω0n] in terms of exponentials and find out DFT of the sequence.I cant represent all the symbols here, try findin out urself if u still cant find out tell me i'll edit it in MS-Paint and post here.
Anyhow answer should be
X(k) = [{δ(k-k0)} + {δ(k-(N-k0))}]N/2
i hope this will help u.....Happy learning............
i think the frequencies in DT case and CT time cases are not equal. i mean ω0 is the CT frequency of the sinsoid and it is not the same for DT case. Then how can u find the DFT?
My actual doubt is DFT is the discretized version of DTFT i.e. i sample the FT of the sampled sinusoid and and take the strengths to represent the DFT. So i mean to get DFT i have to first multipy the DTFT of the sinusoid with periodic impulse train and take the strengths of the resulting signal. Now DTFT of the sinusoid is two impulses. So how can i get DFT as the multiplication of impulses is not defined? Hope u understood my problem. I want to attack the problem in frequency domain rather than time domain.
Hi purnapragna,
u told " i think the frequencies in DT case and CT time cases are not equal. i mean ω0 is the CT frequency of the sinsoid and it is not the same for DT case. Then how can u find the DFT? "........i dont understand wat u mean by this.w0 is a constant and will be the same for both CT and DT signals at sampled spaces
Consider for a CT...Θ = w0*t1
for DT....φ = w0*n1
Θ = φ at places where t1 = n1,ie at places where u sample the signal...,its as simple as that....
comming to ur next question....." to get DFT i have to first multipy the DTFT of the sinusoid with periodic impulse train and take the strengths of the resulting signal " Its not as u say....To get DFT from DTFT wat we do is sample the DTFT only at certain points ie obtain the value of DTFT only at certain points....It is not equivalent of multiplying the DTFT by a periodic impulse train...
i hope this convinces u....
happy learning..............
first one is ok but i have reservations with the second answer how can u get the samples from DTFT without multiplying it with impulse train? I think there is no ther way thatn this. If there is something like this pls tell me how can we do that mathematically
hi purnapragna,
as i told earlier sampling of a signal is not multiplying it by an impulse train...
If u want to get the sample of a signal by multiplyin it with an impulse train then u have to take care that both the signals start at same time so that if u want to measure the sample value of the signal at some point then multiply it with the unit impulse function at that point.I had forgotten to mention this condition earlier..
i hope this should satisfy u....
happy learning...........
but i have read in DTSP by oppenheim and schafer that sampling is nothing but multiplying the signal with periodic impulse train and if u want to get DT version of the sampled signal u have to take the strengths of the corresponding samples(CT impulses). I got the same idea from multirate signal processing by Rabiner and Crocherie. But now i m getting different idea from u. I dont know which one to follow.!!!!!!!!!!
if i have understood you right you want to first get the DTFT of a signal and then sample that in the frequency domain to get the DFT of the same signal assuming it to be periodic.
please read the following link, this gives very good mathematical analysis on how to obtain DFT from a DTFT.