Single carrier  Multicarrier  OFDM Single Carrier - ISI, Receiver complexity  ISI, Bit rate limitation Multi-carrier - Negligible ISI, Approximately flat subchannels, Less receiver complexity - Adaptive allocation of power to each subbands OFDM - Overlapping spectra still separable at the receiver - Maximum data rate in band-limited channel - Digital implementation is possible - No need of steep bandpass filters

Need of orthogonality in time functions

Overlapping spectra

Synthesis of OFDM signals for multichannel data transmission No need of perpendicular cutoffs and linear phases Overall data rate  2  Overall baseband bandwidth, as N  , where N are no. of subchannels For transmitting filters designed for arbitrary amplitude characteristics, the received signals remain orthogonal for all phase characteristics of the transmission medium The distances in signal space are independent of the phase characteristics of the transmitting filters and the transmitting medium

OFDM of N-data channels over one transmission medium

To eliminate ISI and ICI  Orthogonality    A i 2 (f) H 2 (f) Cos 2  fkT df = 0, k = 1, 2,… i=1,2, …,N, (1) 0   A i (f) A j (f) H 2 (f) Cos[  i (f) -  j (f)]. Cos 2  fkT df = 0 and 0   A i (f) A j (f) H 2 (f) Sin[  i (f) -  j (f)]. Sin 2  fkT df = 0 0 for k = 0,1,2, … i,j =1,2, …,N, i  j (2) f 1 = (h + ½)f s & f i = f 1 + (i-1)f s = (h + i - ½)f s, h is any +ve integer (3) T = 1 / 2f s seconds (4)

Designing transmitter filter For given H(f), A i 2 (f)H 2 (f) = C i + Q i (f) > 0, f i - f s  f  f i + f s = 0 f  f i -fs, f > f i + f s (5) where C i is an arbitrary constant and Q i (f) is a shaping function having odd symmetries about f i + (f s /2) and f i - (f s /2). i.e. Q i [(f i +f s /2) + f] = -Q i [(f i +f s /2) - f], 0  f  f s /2 Q i [(f i -f s /2) + f] = -Q i [(f i -f s /2) - f], 0  f  f s /2 (6) Furthermore, the function [C i + Q i (f)]  [C i+1 + Q i+1 (f)] is an even function about f i + (f s /2). i.e. [C i + Q i (f i + f s /2+f)]  [C i+1 + Q i+1 (f i + f s /2+f)] = [C i + Q i (f i + f s /2-f)]  [C i+1 + Q i+1 (f i + f s /2-f)] 0  f  f s /2 i = 1,2, ….N-1 (7) The phase characteristic  i (f), i = 1,2, … N, be shaped such that –  i (f) -  i+1 (f) =   /2 +  i (f), f i  f  f i + f s, i = 1,2, …, N-1 (8) where  i (f) is an arbitrary phase function with odd symmetry about f i + (f s /2)

Examples of required filter characteristics (i) C i is same for all i (e.g. ½) (ii) Q i (f), i = 1,2, …, N, is identically shaped, i.e., Q i+1 (f) = Q i (f-f s ), i = 1, 2, …, N-1, e.g. Q i (f) = ½·Cos( .(f - f i )/f s ), f i – f s  f  f i + f s, i=1,2, …,N,  A i 2 (f)H 2 (f) = C i + Q i (f) = ½ + ½  Cos(( .(f - f i )/f s )  A i (f)H(f) = Cos(( .(f - f i )/(2f s )), f i – f s  f  f i + f s, i=1,2, …,N

Examples … (continued)

Shaping of phase characteristics  i (f), i = 1,2, …,N, are identically shaped, i.e. –  i+1 (f) =  i (f-f s ) i = 1,2, …,N-1 equation (8) holds when f - f i f – f i f - f i  i (f) = h   +  0 +   m Cos m   +   n Sin n   2f s m f s n f s m=1,2,3,4,5, … n=2,4,6, … f i – f s  f  f i +f s An example with all coefficients zero, except  2 = 0.3 and h is set to –1

Shaping of phase…. (continued)

Satisfaction of 1 st and 2 nd requirements - No perpendicular cutoffs and linear phase characteristics are not required Overall baseband bandwidth = (N+1)f s As data rate/channel is 2f s, Overall data rate = 2N f s = [2N/(N+1)]. Overall baseband b/w = [N/(N+1)]. R max Where R max is 2 times overall baseband b/w, is the Nyquist rate. So, for large N, overall data rate approaches R max

Satisfaction of 3 rd and 4 th requirements - Since phase chracteristic  (f) of the transmission medium does not enter into equatios (1) and (2), the received will remain orthogonal for all  (f). In the case of the fourth requirement, let b k i, k = 0, 1, 2, …; i = 1, 2, …,N, and c k i, k = 0, 1, 2, …; i = 1, 2, …,N be two arbitrary distinct sets of m-ary signal digits to be transmitted by N channels. The distance in signal space between these two received signal sets  d = [  [   b k i u i (t - kT) -   c k i u i (t - kT) ] 2 dt ] ½ -  i k i k With no ISI and ICI and applying transform domain identity –  d ideal = [   (b k i - c k i ) 2  A i 2 (f) H 2 (f) df ] ½ i k -  Thus d ideal is independent of the phase characteristics  i (f) and  (f).

Receiver structure

FFT based modulation and demodulation

Modulation using IDFT Create N = 2N information symbols by defining X N-k = X k *, k=1, …….,N-1 and X 0 = Re{X 0 }, and X N = Im(X 0 ) Then N - point IDFT yields the real-valued sequence 1 N-1 x n =   X k e J2  nk / N, n = 0,1,2,…N-1  N k=0 where 1/  N is simply a scale factor. The resulting baseband signal is then converted back into serial data and undergoes the addition of the cyclic prefix (which will be explained in the next section). In practice, the signal samples {x n } are passed through a digital-to-analog (D/A) converter at time intervals T/N. Next, the signal is passed through a low-pass filter to remove any unwanted high-frequency noise. The resulting signal closely approximates the frequency division multiplexed signal.

Cyclic prefix and demodulation using DFT Cyclic Prefix: -acts as a guard space, -as cyclic convolution is performed with channel impulse response, orthogonality of subcarriers is maintained. Demodulation using DFT: Demodulated sequence will be- X k = H k X k +  k, k = 0,1, …, N-1 where {X k } is the output of the N-point DFT demodulator, and  k is the additive noise corrupting the signal.

Downsides of OFDM Cyclic Prefix Overhead Frequency Control Requirement of coded or adaptive OFDM Latency and block based processing Synchronization Peak-to-average power ratio (PAR)

Future research scope Goals: - Increase capacity, high data rate, minimum bit error rate (BER), spectral efficiency, minimum power requirements Problems: - Spectral limitations, channel delay/doppler shifts, limitation in transmission power, real time, PAR Issues: - coding, diversity, frame overlapping, synchronisation techniques, adaptive estimation, w/f shaping, combined approaches, OFDM application specific DSP architecture AND Any of the combination of above issues to achieve ‘Goals’ in presence of ‘Problems’.