THEORY BEHIND GENERATION OF OFDM SUB-CARRIERS

                In my previous post I briefly explained the advantages of OFDM. But you may think how we can generate such closely aligned frequency spectrum. You may think that it would need several precise oscillators to    generate such orthogonal sub carriers. Your worry was true once in the past. That is why the OFDM (theory of which was proved in early 1960's) was out of the picture until late 1990's. But thanks to the rapid development of the Digital Signal Processors (DSP), OFDM became a feasible and practical technique in telecommunication in late 1990's. In this post I am trying to explain you the Discrete Algorithm which is behind this wonderful multiplexing scheme. In other words it is this algorithm which brought the DSPs into the OFDM.

                As you know in case of single carrier modulation (like BPSK, QPSK, ASK..etc),let's consider BPSK for simplicity. Initially the input bit stream is mapped into some value (1 for '1' and -1 for '0'). Then a carrier ( a sin wave in time domain) is multiplied by these mapped values.

eg: let's say we need to transmit '1' using BPSK at rate of 1Mbps,

 you know that frequency spectrum of square pulse is sinc function

 as shown is the figure below,

figure 1: Frequency spectrum of a bit in BPSK

In the above figure,

                                   BT =  2T (T is the symbol period)

                    hence,     BT =  500KHz

               The above mentioned mapping is of vital important in case of OFDM too. Thus the first step of OFDM signal generation is the baseband mapping. Next we need to generate the orthogonal carriers. Here you have to keep in mind that in OFDM we transmit  a frame of bits simultaneously using several orthogonal sub-carriers.

 For example say we need to transmit [1 0 0 0],

then the BPSK mapped samples, X=[1 -1 -1 -1]

we know that a frequency shift of 'fk ' in frequency spectrum can be obtained by multiplying the time domain signal by e^j2π(fk)t 

hence the OFDM signal formula is,

S(t) =ΣXk e^j2π(fk)t       

The resulting time domain OFDM signal must be real,

S(t) =Re{ΣXk e^j2πfkt } 

If this time domain signal is sampled at a rate 1/T, then

S(nT) =Re{ΣXk e^j2πfk(nT) } 

For a square pulse with time period Tf, the required frequency shift fk=k* 1/Tf

S(nT) =Re{ΣXk e^j2π(k/Tf)nT } 

But,        Tf=N*T

                Where, Tf- frame time

                                N- Number of bits in a frame

                                T- Period of a bit in serial data stream

S(nT) =Re{ΣXk e^j2π(kn/N) }  

This is similar to IDFT equation (1/NΣXk e^j2π(kn/N))

it is possible to use IFFT to calculate IDFT of the frame, hence

s(nT)= ΣIFFT(Xk)*N