OFDM modulation works on the principle of converting a serial symbol stream to a parallel symbol stream with each symbol from the parallel set modulating a seperate carrier. The spacing between the carriers is 1/T where T is the duration of the OFDM symbols (without cyclic prefix). This guarantees orthogonality of the carriers i.e. there is no interference between carriers. The addition of orthogonal carriers modulated by parallel symbol streams is equivalent to taking the IFFT of the parallel symbol set. At the receiver the inverse operation of FFT is performed and the parallel symbol streams are converted to serial symbol streams.

The main advantage of this scheme is that one carrier (or set of carriers) may undergo severe fading but other carriers would be able to carry data. Equalization on these narrowband channels is also much easier than equalization of one wideband channel. Intersymbol Interference (ISI) which effects the signal in the time domain is removed by adding a guard period between symbols, called cyclic prefix (which we will discuss later).

We demonstrate the performance of OFDM with QPSK modulation in a simple AWGN channel. Although the true benefits of OFDM are really visible when we have a fading channel but this simple example would serve as a good starting point.

```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FUNCTION TO CALCULATE BER OF OFDM IN AWGN
% seq_len: Input, number of OFDM symbols
% n_carr: Input, number of subcarriers
% EbNo: Input, energy per bit to noise PSD
% ber: Output, bit error rate
% Copyright RAYmaps (www.raymaps.com)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function[ber]=OFDM_err(seq_len,n_carr,EbNo)
for n=1:seq_len
si=2*(round(rand(1,n_carr))-0.5);
sq=2*(round(rand(1,n_carr))-0.5);
s=si+j*sq;
s_ofdm=sqrt(n_carr)*ifft(s,n_carr);
wn=(1/sqrt(2*10^(EbNo/10)))*(randn(1,n_carr)+j*randn(1,n_carr));
r=s_ofdm+wn;
s_est=fft(r,n_carr);
si_est=sign(real(s_est));
sq_est=sign(imag(s_est));
ber1(n)=(n_carr-sum(si==si_est))/n_carr;
ber2(n)=(n_carr-sum(sq==sq_est))/n_carr;
end
ber=mean([ber1 ber2]);
return
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
```

As expected in an AWGN channel the BER performance of OFDM is simply the BER performance of QPSK in AWGN. With cyclic prefix, some of the transmitted energy would be wasted and the BER performance would be a bit worse (to be discussed in future posts).

#### Author: Yasir Ahmed (aka John)

More than 20 years of experience in various organizations in Pakistan, the USA, and Europe. Worked as a Research Assistant within the Mobile and Portable Radio Group (MPRG) of Virginia Tech and was one of the first researchers to propose Space Time Block Codes for eight transmit antennas. The collaboration with MPRG continued even after graduating with an MSEE degree and has resulted in 12 research publications and a book on Wireless Communications. Worked for Qualcomm USA as an Engineer with the key role of performance and conformance testing of UMTS modems. Qualcomm is the inventor of CDMA technology and owns patents critical to the 4G and 5G standards.