# Why the OFDM symbol length is the total number of original modulated symbol, but not one symbol length?

Recently, I am learning the principle of OFDM, and the reference is http://defenseelectronicsmag.com/site-files/defenseelectronicsmag.com/files/archive/rfdesign.com/images/archive/0101Puegel30.pdf.

Everything is reasonable untill page 3 at the left and middle column.

The IFFT takes in N symbols at a time where N is the number of subcarriers in the system. Each of these N input symbols has a symbol period of T seconds.

Then, in expectation, the output of the IFFT, aka a OFDM symbol, should be combined with N symbols that are arranged to N orthogonal sinusoids. That is, a OFDM symbol with period of T seconds. However, at the middle column, the description is out of my expectation.

The block of N output samples from the IFFT make up a single OFDM symbol. The length of the OFDM symbol is NT where T is the IFFT input symbol period mentioned above.

Why the OFDM symbol period is N*T but not just T? A simple example in my understanding will be: If it is, then the OFDM should not have the higher transmission rate or speed, because it costs the transmission time the same with the original modulation method that without the OFDM modulation.

Your understanding of what the IDFT does is simply wrong!

you don't get

symbol1 symbol2 symbol3 … symbolN


one after another in time domain, you get all these symbols modulating the complex sinusoid, added up.

If it is, then the OFDM should not have the higher transmission rate or speed, because it costs the transmission time the same with the original modulation method that without the OFDM modulation.

That however is still true: OFDM doesn't "cheat" physics. You take a channel of bandwidth $B$, divide it into $N$ channels of bandwidth $\Delta f = \frac BN$, and then you send $N$ data symbols at once. So, that's basically $B$ as a symbol rate (not accounting for sidelobes and empty carriers).

In a simple single carrier system with the same pulse shape (sinc), you'd get $1$ symbol with a rate of $B$. The same rate.

So, OFDM doesn't increase the amount of data you can send over one channel inherently.

What it does, however, is split the channel into subchannels, which then are easier to equalize. That's the main thing there is to multicarrier systems like OFDM: You reduce the (very hard!) problem of a wide, frequency-selective channel to $N$ easier problems of flat subcarriers. OFDM systems always pick their $N$ accordingly to make sure fading is flat within one subcarrier.

• Thanks for your response. However, from the tutorial:home.iitj.ac.in/~ramana/ofdm-tutorial.pdf, in page 7 Fig. 11. I find that the four modulated QPSK symbol is moved onto four orthogonal subcarriers, and they are actually added up in time-domain. In this case, I think the OFDM can transmit more data in a symbol period. Also in Fig.14, the time domain and frequency domain view of the OFDM symbol, we can find that a OFDM symbol is actually added up by multiple orthogonal subcarriers original symbols. – desword Sep 7 '17 at 8:24
• However, in many implementation of matlab code and the material mentioned above, they all just connect the IFFT results. I.e., four symbols that are moved onto four orthogonal subcarriers. Then the resulted OFDM symbol is the connection of four symbols, but not be added up. I am confused which one is the correct understanding of the OFDM modulation. – desword Sep 7 '17 at 8:26
• @desword I explained exactly that: you add up the symbols; you transmit $N$ at once, but they take $N$ times as many time each. No, OFDM cannot transport more data in the same bandwidth than single carrier. Please read my answer carefully. I don't care what the Matlab examples do. OFDM and the IFFT aren't up for discussion by some Matlab examples. – Marcus Müller Sep 7 '17 at 9:22