Elements of a Digital Communication System • Block diagram of a communication system: Information source and input transducer Source encoder Channel encoder Digital modulator Channel Output transducer Source decoder Channel decoder Digital demodulator 1 Mathematical Models for Communication Channels • Additive Noise Channel: – In presence of attenuation: r (t ) (t ) s(t ) n(t ) channel S(t) + r(t) = s(t) + n(t) n(t) 2 Mathematical Models for Communication Channels • The Linear filter channel: r (t ) s(t ) * c(t ) n(t ) c( )s(t )d n(t ) s(t) + Linear filter c(t) r(t) = s(t)*c(t)+n(t) channel n(t) 3 Mathematical Models for Communication Channels • Linear Time-Variant Filter Channel: – Are charachterized by a time-variant channel impulse response c( ; t ) s(t) r(t) + Linear time_variant filter channel n(t) 4 Representation of Band-Pass Signals and Systems • Representation of Band-Pass Signals: s(t ) x(t ) cos(2 f c t ) y(t ) sin(2 f c t ) Energy of the signal: 1 2 sl (t ) dt 2 • Representation of Linear Band-Pass Systems: h(t ) hl (t )e j2 fc t hl (t )e j2 fc t • Response of a Band-Pass System to a Band-Pass Signal: r(t ) Re rl (t )e j2 fc t 5 Orthogonal Expansion of Signals • We can express M orthonormal signals sn (t ) as a Linear combination of basis functions f n (t ) and hence can be defined N as sk (t ) skn f n (t ), k 1,2,.., M n 1 • Linear digitally modulated signals can be expanded in terms of two orthonormal basis functions given by: f1 (t ) 2 cos 2 f c t T and f 2 (t ) 2 sin 2 f c t T 6 Representation of Digitally Modulated Signals • Pulse-amplitude-modulated Signals (PAM): sm (t ) Re Am g (t )e j2 fc t m=1,2,…M • Phase-modulated signals (PSK): s m (t ) g (t ) cos 2 2 (m 1) cos 2 f c t g (t ) sin (m 1) sin 2 f c t M M • Quadrature amplitude modulation (QAM): sm (t ) Re ( Amc jAms ) g (t )e j 2 fct m=1,2,..,M, Amc g (t ) cos2 f c t Ams g (t ) sin 2 f c t 7 Representation of Digitally Modulated Signals • Orthogonal multidimensional signals: Cm cm1cm2 ...cmN • Biorthogonal signals: • Simplex signals: sm ' sm s, • m=1, 2,…, M.. • Signal waveforms from binary codes: Cm cm1....cmN c mj 1 S mj (t ) cmj 0 S mj (t ) 2 c cos 2 f c t Tc 2 c cos2 f ct Tc 0 t Tc 0 t Tc 8 Optimum Receivers Corrupted by additive White Gaussian Noise- I • General Receiver: channel Sm(t) + r(t) = sm(t) + n(t) n(t) r (t ) sm (t ) n(t ), (0 t T ) Receiver is subdivided into: – 1. Demodulator. • (a) Correlation Demodulator. • (b) Matched Filter Demodulator. – 2. Detector. 9 Optimum Receivers Corrupted by additive White Gaussian Noise- II • Correlation Demodulator: – Decomposes the received signal and noise into a series of linearly weighted orthonormal basis functions. • Equations for correlation demodulator: rk 0 r (t ) f k (t )dt 0 s m (t ) n(t ) f k (t )dt T T s mk s T 0 m k 1,2,...N (t ) f k (t )dt, nkm 0n(t ) f k (t )dt, T 10 Optimum Receivers Corrupted by additive White Gaussian Noise- III • Matched Filter Demodulator: – Equation of a matched filter: hk (t ) f k (T t ), 0t T • Output of the matched filter is given by: y k (t ) 0r (t )hk (t )d T T r (t ) f k (T t )d k=1,2, …N 0 11 Optimum Receivers Corrupted by additive White Gaussian Noise- IV • Optimum Detector: – The optimum detector should make a decision on the transmitted signal in each signal interval based on the observed vector. • Optimum detector is defined by: N N N n 1 n 1 2 D(r , s m ) r 2 rn s mn s mn n 1 2 n 2 r 2r s m s m , 2 or m=1,2,… M D(r, sm ) 2r s m s m 2 12 OFDM • It is a block modulation scheme where data symbols are transmitted in parallel by employing a large number of orthogonal sub-carriers. • Equation of complex envelope of the OFDM signal: s (t ) A b(t nT, xn ) n • where N 1 2 ( k )t N 1 2 b(t , xn ) ha (t ) xnk exp j T k 0 13 General FFT based OFDM system-I • Block diagram of FFT based OFDM transmitter : X k X0 X1 ... XN-2 XN-1 IFFT X ~ sl (t ) g In D/A X n X0 X1 ... XN-2 XN-1 Insert Cyclic prefix X g n ~ sQ (t ) X D/A g Qn • Equations at the transmmitter end: j 2kt s(t ) A xk exp u r (t ) NT k 0 s N 1 j 2kn X n s(nTs ) A xk exp , N k 0 s N 1 14 General FFT based OFDM system-II • Block diagram of FFT based OFDM receiver: R ~ rl (t ) k ,n A/D ~ rQ (t ) A/D R Remove cyclic prefix R0 R1 ... RN-2 RN-1 Qn FFT Z n • Z0 Z1 ... ZN-2 ZN-1 Serial metric computer (vm ) At the demodulator: 1 N 1 Z i Rn e N n 0 j 2ni N Z i i Axi L i g m e m 0 j 2mi N 15 General FFT based OFDM system-II • Merits of OFDM: – 1. the modulation and the demodulation can be achieved in the frequency-domain by using a DFT. – 2. the effects of ISI can be eliminated with the introduction of the guard interval. 16 IMPLEMENTATION OF OFDM SYSTEM-I • Basic implementation of OFDM system: 1 N 1 Z i Rn e N n 0 L i g m e j 2ni N j 2mi N m 0 Zi i Axi n0 X0 BPSK . . . BPSK . . . X1 . . . BPSK X127 n1 1 127 R0 + . . . . . . Bits Serial To Parallel Converter 0 . . . n127 R1 Detector + . . . . . . . . . . . . R127 + 17 SIMULATION RESULTS. • Perfomance charachteristics were obtained for the simulated OFDM system. -1 10 Non Fading Channel Fading Channel -2 10 -3 10 -4 10 -5 10 -6 10 10 15 20 25 30 35 40 45 50 18 Conclusion. • 1. OFDM communication system exhibits better Pe Vs SNR curves in case of Non-Fading channel as compared to the Fading channel. • 2. As the value of the SNR is increased the value of Pe gradually decreases. • 3. Perfomance charachteristics of simulated OFDM communication system are consistent with the performance charachteristics of the general OFDM communication system. 19