Periodic and Discrete x(n) in Time Domain from IDFT/IFFT of X(k) in Frequency Domain

[engineer_eda]

Hi All,

DTFT of a finite length and discrete time signal x is periodic with 2*pi and continues in frequency domain.

With DFT or FFT we find discrete and periodic X(k) in frequency domain for finite x.

So lets say we have a X(k) in discrete frequency domain and we want to make IDFT(IFFT). My question is that by performing IDFT do we get a periodic and discrete time x signal (with N samples)?

Thanks in advance!

 

[engineer_eda]

I would appreciate if anyone help me on this query.

 

[zorro]

My question is that by performing IDFT do we get a periodic and discrete time x signal (with N samples)?

Yes. Performing IDFT of an N-point X[k] (k=0,...,N-1) you get a sequence of N-point different (in general) values x[n] (n=0,...,N-1).
It can be seen in the formula of IDFT that substituting n by n+N you get the same result, i.e. x[n] is periodic with period N.
DFT and IDFT are (except for a factor N and a sign accompanying "j") symmetric in their properties.
Regards

Z

 

[engineer_eda]

Thank you Zorro. I am also trying to understand the output of the IFFT in OFDM system. The inputs X(k)s are generated by the constellation mapper in frequency domain and then then N parallel X(k) inputs are given into the IFFT block to create N-parallel subcarriers in time-domain.

If I am correct, according to the DFT formula, each X(k) of N-samples in 2*pi period is representing the discrete-time x with N-samples in one period.

For the output of IFFT block in OFDM system, the inputs (N inputs) are in parallel and we get N-parallel discrete time x subcarriers in different orthogonal frequencies. Is each of these N-parallel subcarriers periodic with 2*pi and has N-samples in each period?

Thanks in Advance!