# OFDM/DFT/Sampling question.

Started by January 6, 2009
```Hello,

I am pretty confused with some DFT/Sampling techniques and how these apply
to OFDM.

Fundamentally, if a (baseband) signal has a highest frequency component of
f Hz, that signal needs to be sampled at 2f Hz for aliasing-free data
reconstruction.

My confusion begins with OFDM, which does the process in reverse. That is
IDFT is performed first.
So, in OFDM, the output of the IDFT block (the time-domain signal) will
consist of frequency components upto N*f_s Hz. Here I assume the frequency
spacing is f_s and N subcarriers are used. So fundamentally, this signal
needs to be sampled at 2*N*f_s Hz for proper reconstruction.

But, in OFDM, the output samples of the IDFT are arranged with a spacing
of t_s=1/f_s.

So my question are,
1) Does sampling theorem not taken into consideration here? how does
sampling theorem/Nyquist frequency come in to play here with f_s and N?
2) What is the signalling bandwidth of this system?
Generally (like QAM transmission), the BW of the system should be the
reciprocal of symbol rate, BW=1/(N*t_s). But in an OFDM system, BW=1/t_s.

I would really appreciate if someone clarify me this confusion. I know I
am missing something very basic here.. but can;t figure out what.

Thanks a lot.

```
```On Jan 6, 1:33&#2013266080;pm, "m26k9" <maduranga.liyan...@gmail.com> wrote:
> Hello,
>
> I am pretty confused with some DFT/Sampling techniques and how these apply
> to OFDM.
>
> Fundamentally, if a (baseband) signal has a highest frequency component of
> f Hz, that signal needs to be sampled at 2f Hz for aliasing-free data
> reconstruction.
>
> My confusion begins with OFDM, which does the process in reverse. That is
> IDFT is performed first.
> So, in OFDM, the output of the IDFT block (the time-domain signal) will
> consist of frequency components upto N*f_s Hz. Here I assume the frequency
> spacing is f_s and N subcarriers are used. So fundamentally, this signal
> needs to be sampled at 2*N*f_s Hz for proper reconstruction.
>
> But, in OFDM, the output samples of the IDFT are arranged with a spacing
> of t_s=1/f_s.
>
> So my question are,
> 1) Does sampling theorem not taken into consideration here? how does
> sampling theorem/Nyquist frequency come in to play here with f_s and N?
> 2) What is the signalling bandwidth of this system?
> Generally (like QAM transmission), the BW of the system should be the
> reciprocal of symbol rate, BW=1/(N*t_s). But in an OFDM system, BW=1/t_s.
>
> I would really appreciate if someone clarify me this confusion. I know I
> am missing something very basic here.. but can;t figure out what.
>
> Thanks a lot.
as per your post, the statement
"in OFDM, the output of the IDFT block (the time-domain signal) will
consist of frequency components upto N*f_s Hz" is wrong.

toe output of IDFT block's frequency range is from -N*f_s/2 to N*f_s/
2.
note that here f_s is subcarrier spacing. sampling frequency is N*f_s.
as per the Nyquist theorem the maximum freq component of the IDFT
block is, N*f_s/2.
the frequency components exceeding this N*f_s/2 will folded to
negative frequency side in IDFT output.
this is the basic of FFT. this may be confusing.
gothrough this tutorials 4,5,6 and 22 in the link
http://www.complextoreal.com/tutorial.htm

and read understanding digital signal processing by rick lyons

```
```>as per your post, the statement
>"in OFDM, the output of the IDFT block (the time-domain signal) will
>consist of frequency components upto N*f_s Hz" is wrong.
>
>toe output of IDFT block's frequency range is from -N*f_s/2 to N*f_s/
>2.
>note that here f_s is subcarrier spacing. sampling frequency is N*f_s.
>as per the Nyquist theorem the maximum freq component of the IDFT
>block is, N*f_s/2.

Thanks a lot charanchar. You cleared that up.
I am actually reading Rick Lyons book, which by the way I think is
awesome. I just have some cofusions when connecting the dots with OFDM.

I just one question that I cannot seem to figure out.

The output of the IFFT block is a 'discrete' time-domain signal, which has
frequency components up to N*f_s/2. So from sampling  theorem, we need to
sample this signal at N*f_s Hz = 1/t_s.

What happens (to my understanding) in an OFDM transmitter is that, the
discrete 'parallel' outputs of the IFFT block are parallel-to-serial
converted, with the spacing between the points as t_s. And then these
discrete points are sent to a DAC and then transmitted.

Now, assuming there is no multipath fading or any attenuation during
transmission, the signal arrives at the receiver, where the signal is
sampled at N*f_s Hz = 1/t_s. So, when these points are sampled,  the
resulting discrete-points are exactly the output points of the IFFT block
at the transmitter. And feeding these points to the FFT will result in
desired frequency domain points.

My problem now is, WHAT IF, at the transmitter, the serial-to-parallel
conversion changed the spacing between the IFFT output sampled to 2*t_s,
instead of t_s? Ofcourse there will be reduced throughput but that is not
what I am concerned here. Now the transmitting signal will be 2 times the
original length. At the receiver, instead of sampling at 1/t_s as before,
now it samples at 1/2*t_s. Same spacing used at the transmitter. The
results of this sampling, will again be the 'same' output points of the
IFFT block at the transmitter. And feeding these N points to FFT will
result in required result same as before, because the FFT does not
care/know how/what rate these are sampled.

So my question is, even though the IFFT output contains frequency
components upto N*f_s/2, sampling at N*f_s/2=1/2*t_s results in the same
desired result. Because the FFT block only cares about the discrete values
entered to it. It hass no idea about the sampling rate, as long as the
input to the FFT block are the same as the output of the IFFT block.

I am very sorry for the long description but I wanted to make it clear.

Thank heaps again.
```