# FDM vs OFDM

1. May 12, 2012

### quantumlight

I was working on my thesis when this question popped into my head, why can't there be overlap in FDM wihout ICI but you are allowed to overlap in OFDM without ICI?

Also why is the maximum overlap of subchannels 50% in OFDM before ICI occurs?

From math point of view as long as the carriers are orthogonal and you take the integral from 0 to T it always reduces to a delta function so it doesn't seem to matter how much bandwidth is on each side of the center frequency.

Last edited: May 12, 2012
2. May 13, 2012

### the_emi_guy

FDM and OFDM both have the same overlap!. In FDM the overlap is in the time domain. In OFDM the overlap is in the frequency domain.
First (you may already know this) the relationship between the rectangular pulse and the sin(x)/x (sinc) function:
A rectangular pulse in the time domain transforms to and from the sinc function in the frequency domain. A sinc function in the time domain transforms to and from a rectangular "brickwall" function in the frequency domain. In other words these two functions transform to each other by either FFT or IFFT.

In both FDM and OFDM we are taking multiple carrier frequencies, modulating them, then combining them for transmission.
For simplicity lets assume each carrier is on/off modulated. In idealized FDM, we modulate each carrier then
send each though a brickwall filter before combining to the antenna. Say the carriers are separated by 500KHz, (say at 1GHz + 500KHz, 1GHz + 1MHz ...)
Each carrier's 500KHz brickwall filter in the frequency domain cause a time domain spreading of its on/off pulses into time domain
sinc function with zero crossings every 1us.
Now, if we make the baud rate 1Mbps, each bit's ideal sampling point (center if eye) occurs at the zero crossing point of
all of the potentially interfering sinc functions from previously received bits. In other words there is lots of ISI, but none at the critical moment when the bits are sampled. This is called "signalling at the Nyquist rate" and is related to but not the same as
Nyquist sampling which you hear a lot about. (see en.wikipedia.org/wiki/Nyquist_ISI_criterion, apparently I am not allowed to include link because I am new to physicsforums).
Of course brick wall filters are hard to make, so we use things like raised cosine filters that create the same beneficial sinc zero crossings.

OFDM is analogous to FDM but with time and frequency domain reversed.
We on/off modulate our carriers, but they are combined as unfiltered rectangular pulses and sent straight to the antenna (simplification of course).
These time domain rectangular pulses become spread in the *frequency* domain as sinc functions.
If we on/off modulate each carrier at 1Mbps (1us symbol time), and simultaneously maintain 1MHz carrier spacing ("orthogonal"),
then the zero crossings of the sinc functions occur every 1MHz. Their positions are such that at each carrier frequency, all other carrier's smearing sinc functions have zero crossings. Thus each carrier frequency is free from interference.
Again there is plenty of interference between these signals, but none at the critical frequencies where the carriers are located.
Note the factor of 2 difference between the OFDM bandwidth and the FDM bandwith in my example. This is due to the convention of including negative frequency in the bandwidth in the OFDM case.
Naturally, there is much more to it than this, but this is the basics.
Hopefully you can figure out from this where the 50% comes from (look at the superimposed sinc functions).

Hope this helps,
Cheers.

Last edited: May 13, 2012
3. May 13, 2012

### quantumlight

Thanks. I get it from that perspective but how does it work when you demodulate with cos(2πfct) where fc matches the carrier. when you take the combined signal and integrate from 0 to T, you theoretically could obtain the original signal with just that carrier back perfectly because integral from 0 to T of cos(2πf1t)*cos(2πf2t) is zero if f1≠ f2. What is limitations there?

4. May 13, 2012

### the_emi_guy

We recover the signal from carrier f1, as you mentioned, by coorelating (multiplying then averaging) the full signal with a tone at the carrier f1
frequency over one symbol time.
The carrier f2 signal (and all others) will correlate to 0 over one symbol time only when its sinc function has a zero crossing at f1.
This requires f1 and f2 to have a specific relationship to each other.
Specifically the carriers must each have an integer number of cycles within the symbol time.
Of course the magic of OFDM is that all of the coorelation over all carriers is done by a simple FFT. Also the correlation is not just done
with cos(wt) but with sin(wt) as well so that we can extract both dimensions (I & Q) of modulation.