Understanding an OFDM transmission
Let us try to understand simulation of a typical Orthogonal Frequency Division Multiplexing (OFDM) transmission defined per IEEE 802.11a specification.
Orthogonal pulses
In a previous post (here ), we have understood that the minimum frequency separation for two sinusoidals with arbitrary phases to be orthogonal is , where
is the symbol period.
In Orthogonal Frequency Division Multiplexing, multiple sinusoidals with frequency separation is used. The sinusoidals used in OFDM can be defined as (Refer Sec6.4.2 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]:
, where
correspond to the frequency of the sinusoidal and
is a rectangular window over
.
Using of inverse Fourier Transform
We now have understood that OFDM uses multiple sinusoidals having frequency separation where each sinusoidal gets modulated by independent information. The information
is multiplied by the corresponding carrier
and the sum of such modulated sinusoidals form the transmit signals. Mathematically, the transmit signal is,
The interpretation of the above equation is as follows:
(a) Each information signal multiplies the sinusoidal having frequency of
.
(b) Sum of all such modulated sinusoidals are added and the resultant signal is sent out as .
Update: 29th May 2008
(thanks to the comment by Mr. Mork)
The sampled version of the above equation is,
It is reasonable to understand that above operation corresponds to an inverse Discrete Fourier Transform (IDFT) operation.
Understanding of an OFDM transmission specified per IEEE 802.11a [80211A]
From the Table 79 in [80211A], the symbol duration .
This implies that the used subcarriers are and so on.
The available bandwidth of 20MHz is split into 64 subcarriers. Out of the available 64 subcarriers having indices , the number of used subcarriers is 52. The used subcarriers are having indices from
are used for transmitting information sequence
to
.
To understand in detail the correspondence between the subcarrier index and frequency, one may refer to a previous post (here).
Once the bits in each symbol are assigned to appropriate IDFT inputs (Ref. Figure 109 in [80211A]), the IDFT operation is performed to obtained the time domain signal.
For each symbol, some samples from the end are appended to the beginning of the symbol (referred to as cyclic prefix insertion). We may discuss the pros/cons of cyclic prefix in a future post.
Simulation Model
A simple Octave script where a BPSK modulated signal is transmitted on the 52 used subcarriers, may be used for understanding generation of an OFDM signal. The script is loosely based on 802.11a specification.
Click here to download.
Figure: Spectrum of an OFDM transmission (based on 802.11a specification)
Reference
[DIG-COMM-BARRY-LEE-MESSERSCHMITT]
Hope this helps.
Krishna
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Comments
Sorry for the double-posting, but I just happened to find the answer to my question (by pure chance — I’ve been trying to find out for quite a while already).
The relation between s(t) as you wrote it (and as it is widely used) and the actual (I)DFT implementation is looked at in a paper by Lin and Phoong (http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1223555). Summarized very briefly, they find that for general window functions — g(t) in your example — including the commonly used rectangular windows, it is indeed *not* possible to generate the same s(t) using a DFT followed by a DAC. It is possible to generate a signal with the same values at the times t=nT, but the corresponding signal will still be different (and thus have a different PSD, too).
Thanks for your attention 
- M
@Mork:Thanks for the valuable comments.
Trying to answer the first comment:
(a) Yes, I agree. I was loosely specifying that a continuous time signal can be represented using IDFT. Infact, I should have added one more step, where I defined s(nT) and then claimed that it can be represented using IDFT.
(b) Yes, direct implementation is K oscillators 
Trying to answer the second comment.
Thanks for the link to the paper “OFDM transmitters: analog representation and DFT-based implementation”
(a) From a quick read, the authors are showing that one has to consider the sampled window function g(nT) rather than the contnuous function g(t). I did not follow the full paper, however I was unable to find the reason behind y(t) having more psd than x(t) in region from 0.5 to 0.75 omega_s (ref. figure 5). Any thoughts?
(b) Further, just to note that, in a typical implementation, the output x(nT) will be upsampled followed by a digital filtering, then passed to the DAC. Not of relavence to this discussion, but just to add as FYI.
Thanks again for your comments. You helped me provide a better insight.
Ps. I made a brief update to the post to reflect that s(nT) and not s(t) can be implemented using iDFT.
[...] may refer to post Understanding an OFDM Transmission and the post BPSK BER with OFDM modulation for getting a better understanding of the above [...]
Hi,
After providing the formula for s(t), you wrote
>It is reasonable to understand that above operation corresponds to an inverse Discrete Fourier Transform (IDFT) operation.
Why? s(t) is a continuous-time signal, so wouldn’t it be the Fourier Series (if anything) rather than the DFT?
Another question: I understand the properties of s(t) as it is, but how can this be implemented efficiently for large values of K? A direct implementation of the formula would require K oscillators at the receiver and the transmitter (to generate the individual sinusoidals), right?
Thanks!
– M