Outline Chapter 4: Orthogonal Frequency Division Multiplexing. Orthogonal Frequency Division Multiplexing

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1 Outline Chapter 4: Orthogonal Frequency Division Multiplexing Fading Channel Flat fading channel Frequency selective channel II ingle Carrier Equalization Orthogonal Frequency Division Multiplexing Principle of Multi-carrier systems Inter Carrier Interference Orthogonal Frequency Division Multiplexing: OFDM Cyclic Prefix Equalization Impulse hortening Equalizer OFDM Channel Estimation Digital-to-Analog Interface 1 Fading Channel Transmission channel is in general frequency selective H(jω) B ω If transmit signal is narrow band (small bandwidth B) Frequency response is almost constant in considered band: H(jω) = H 0 Impulse response of channel is given by weighted dirac h[k] =h 0 δ[k] s[k] h 0 n[k] r[k] 1-Path Fading Channel (memoryless channel) 2

2 1-Path-Fading Channel ignal space diagram for QPK transmission Transmitted QPK signal: s[k] Rotation & attenuation by h 0 : h 0 s[k] Receive signal: r[k] = h 0 s[k]+ n[k] For signal detection r[k] is de-rotated s[k] = h 0 h 0 r[k] = h 2 0 h 0 (h 2 0 s[k]+n[k]) = s[k]+ñ[k] s[k] h n[k] 0 r[k] h 0 / h 0 2 ˆd[k] ˆb[`] Transmit signal has long time duration small baud rate 3 (L+1)-Path-Fading Channel H(jω) h(τ) B h 0 h1 h2 ω To increase baud rate, time per symbol has to be reduced hort transmission pulse leads to increased band width B Frequency selectivity of channel becomes effective Discrete impulse response of channel is given by h[k] =h 0 δ[k]+h 1 δ[k 1] h L δ[k L] s[k] z 1 h[k] h 3 h 4 h 5 h 6 h 0 h 1 τ n[k] r[k] L denotes the number of memory elements w.r.t. T s 1 st Nyquist condition violated Intersymbol Interference (II) r[k] =h 0 s[k]+h 1 s[k 1] h L s[k L]+n[k] z 1 h L 4

3 ingle Carrier Equalization ignal space diagram for QPK transmission Transmitted QPK signal s[k] Influence of frequency selective channel P L`=0 h`s[k `] II Disturbance by noise n[k] II-channel plus noise r[k] = P L `=0 h`s[k `]+n[k] Estimation of signals becomes demanding task Equalization schemes Linear equalization (applying linear filter) Decision Feedback Equalizer (DFE) Maximum Likelihood equence Estimation, i.e. Viterbi Algorithm effort increases exponentially with channel memory 5 Outline Chapter 4: Orthogonal Frequency Division Multiplexing Fading Channel Flat fading channel Frequency selective channel II ingle Carrier Equalization Orthogonal Frequency Division Multiplexing Principle of Multi-carrier systems Inter Carrier Interference Orthogonal Frequency Division Multiplexing: OFDM Cyclic Prefix Equalization Impulse hortening Equalizer OFDM Channel Estimation Digital-to-Analog Interface 6

4 Principle of Multi-Carrier ystems H(f) B f T s h(τ) t Intersymbol interference of singlecarrier systems MLE equalization using Viterbi algorithm Complexity increases exponentially with the channel memory H(f) B f N T s τ t Multi-carrier systems experience nonselective sub-channels imple equalization with scalar multiplication 7 ource Multi Carrier Transmission (MC) Transmitter bi ( b) Receiver /P ^ bi ( b) Dest. P/ ld( M) ld( M) Map. Mod. Map. Mod. d Map. Mod. Demap. Demod. Demap. Demod. d0( i) d1 () i N-1( ) g TX(t) gt () g TX(t) gt () i g TX(t) gt () ^ d0 () i g RX(t) ht () ^ d1 () i g RX(t) ht () e j2 πf 0 t e j2 πf 1 t e j2 πf N-1 t e -j2πf 0 t e -j2πf 1 t Σ ^sa () t Channel s a () t channel hc a(t) () t η() t a Demap. Demod. ^ d N-1( ) i g RX(t) ht () e -j2πf N- 1 t 8

5 Inter Carrier Interference (ICI) ICI Problem of MC: If the frequency bands of different subcarriers overlap, Inter Carrier Interference (ICI) appears. olution: A special design of transmit and receive filter leads to orthogonality of the subcarriers. 9 Orthogonal Frequency Division Multiplexing (OFDM) Orthogonality of subcarriers g(t) G(f) 0 T s t 1/T s 0 1/T s f carrier distance: G n (f) G n+1 (f) 1 T s G n 1 (f) G n+2 (f) 0 f n 1 f n f n+1 f n+2 f 10

6 Mathematical Description of the OFDM-Transmitter continuous-time representation of an OFDM transmitter: N 1 n n= 0 st () = d () igt it e j2π fnt g t = rect t T f = n Δ f = n T () /, n / N 1 j2π nt/ T n n= 0 = d () i e, it t i+ 1 T discrete-time N 1 j2 π nkt / T representation of sik (, ) = st ( ) = d ( ) A n ie, k [0,1,2,..., N 1] t= it + kt A an OFDM transmitter: n= 0 N = T / T A N 1 d () j2 nk/ N = n i e π = N IDFT d (), i d (),..., i d ( i) 0 1 N 1 n= 0 11 Mathematical Description of the OFDM-Receiver received signal in time domain received signal after DFT (n = subcarrier index) r(i, k) =s(i, k) +n(i, k) =IDFT{d n (i) +n(i, k)} x n (i) = DFT{r(i, 0),r(i, 1),,r(i, N 1)} = NX 1 k=0 r(i, k)exp(j2πkn/n) Complete ystem: Decided Data: x n (i) = DFT IDFT{d n (i)} + n(i, k) ª = d n (i) +DFT{n(i, k)} ˆdn(i) =Q{x n(i)} 12

7 ymbol Rate Model of an OFDM ystem 13 Outline Chapter 4: Orthogonal Frequency Division Multiplexing Fading Channel Flat fading channel Frequency selective channel II ingle Carrier Equalization Orthogonal Frequency Division Multiplexing Principle of Multi-carrier systems Inter Carrier Interference Orthogonal Frequency Division Multiplexing: OFDM Cyclic Prefix Equalization Impulse hortening Equalizer Channel Estimation Digital-to-Analog Interface 14

8 Inter-ymbol- (II) and Inter-Carrier-Interference (ICI) (- i2 ) (- i1) (+ i 0 ) (+ i 1 ) (+ i 2) OFDM symbol OFDM symbol OFDM symbol OFDM symbol OFDM symbol magnitude of channel impulse response: c h a (t) (t) a t t... symbol( i1 - ) fade out (II) fade in (ICI) symbol i 15 The OFDM Cyclic Prefix / Guard Interval cyclic prefix G (- i1 ) G (+ i 0 ) G (+ i 1) G magnitude of channel impulse response: h c a(t) a t... symbol( i1 - ) fade in symbol i fade out Tg Ts The OFDM cyclic prefix serves for the suppression of II and ICI! 16

9 OFDM Transmitter Mod. Map. d i 0( ) Map. d i 1( ) source CC /P... Map. Map. d i 2( ) dn-1( i) IDFT N P/GI DAC channel coding (convolutional codes with Viterbi decoding) IDFT: discrete realized filter bank (very efficient FFT) cyclic prefix / guard interval (GI) prevents intersymbol interference (II) 17 ADC PE YNC GI -1 N OFDM Receiver DFT x 0 (i) x 1 (i) x 2 (i) x N 1 (i) Demap. Mod. Demap. Demap. ynchronization FFT window position (time domain) sample and modulation frequency correction Pre equalizer (PE) for impulse compression OFDM: Orthogonal Frequency Division Multiplexing separate multiplicative channel correction on each subcarrier e 0 e 1 e 2 e N-1... P/ CC -1 Viterbi decoder dest. equalizer coefficient design: e n = 1 / H n circular convolution 18

10 OFDM Circular Convolution Received signal in time domain (noiseless case) r(i, k) =s(i, k) Circular convolution due to cyclic prefix iff * h(k); * = Circular convolution w.r.t. T > τ g max k Discrete Fourier Transform yields n o n o n DFT (k) N s(i, k) * h(k) =DFT (k) N s(i, k) DFT (k) N {z } {z } x n (i) d n (i) Equalization by simple division separately for each subcarrier y n (i) = 1 H (n) x n(i); e n = 1 H (n) o h(k) {z } H (n) 19 Eye Pattern for QPK Transmission (real part) 20

11 Influence of Guard Interval Bandwidth Efficiency under Guard Interval 1 N symbol rate T s + T G 1 β = = = bandwidth 1 T N 1 + T T NR Loss caused by Guard Interval s G E NR = T g ( t ) g ( t ) dt 0 RX TX T T 2 N 2 0 g ( ) ( ) 0 RX t dt g T TX t dt G 2 E T E T E = = = β N T T + T N T + T N ( ) ( ) 0 G 0 G 0 2 e.g. T = 20% of T + T G G 1dB loss 21 Impulse hortening Equalizer Linear pre-filter; impulse response e pre (k) such that ½ g(k), κ =0, h(k) e pre (k) k=k0 +κ =,`g 1; `g <N g ε 0 (k) else Cost function: P ε 0 (k) 2 min, s.t. (k) MME-solution: ½ g(0) = 1 MME, Kammeyer 1995 P g(k) 2 = 1 Falconer 1973 e pre =[R rr 1 σ 2 R rs R H rs ] 1 r rs where R rr = autocorr.matrix received signal; R rs ; r rs = Cross corr. Matrix/vector 22

12 Examples h(k) N g =16 Equalizer: n =32, `g =8 23 Parameters of an OFDM ystem Meaning ampling frequency Core symbol duration Guard time Total symbol duration ubcarrier spacing Bandwidth Data rate Time continuous - T Time discrete f A = 1 N T = T A N T A N T TG G A T = T + T T = ( N + N ) T G Δ f = 1 Δ f = 1 T N T B N Δ f B N Δ f R = N ld( M) b T + T G A G A A R = N ld( M) b T ( N + N ) G 24

13 Outline Chapter 4: Orthogonal Frequency Division Multiplexing Fading Channel Flat fading channel Frequency selective channel II ingle Carrier Equalization Orthogonal Frequency Division Multiplexing Principle of Multi-carrier systems Inter Carrier Interference Orthogonal Frequency Division Multiplexing: OFDM Cyclic Prefix Equalization Impulse hortening Equalizer OFDM Channel Estimation Digital-to-Analog Interface 25 Pilot ymbol Constellation for WLAN burst structure of HIPERLAN/2 and IEEE802.11a short symbols for AGC and raw synchronization training sequence (T): 2 identical symbols per subcarrier (52) data OFDM symbols with 48 user data and 4 pilot symbols each pilot symbols for fine synchronization (insufficient for channel estimation) n data symbols pilot symbols i 26

14 Nonblind (reference-based) Channel Estimation yk () /P y 0( i) y 1( i) y P-1( i) disc. Prefix & FFT ~ d y0,ref 0 (i) () i ~ d y 1 (i) () i 1,ref ~ yd N 1 () (i) i N-1, ref Ĥ Cn ((n) ) = d n,ref ~ ~ d y n (0) + d y n (1) (1) nref, 2d nref, nref, N Channel estimator N 1 Ĥ Cn ( (n) ) Equalizer d 0( i) d 1( i) d N-1( i) Averaging over only two identical training symbols 2 db loss in NR compared to estimator with ideal channel knowledge Perform additional noise reduction (NR) to increase estimation quality 27 Noise Reduction Algorithm (NR) Background a-priori knowledge: limited channel impulse response in time domain channel impulse response fits into guard interval lowpass filtering in frequency domain NR algorithm (required operations) transform the estimated channel transfer function into time domain (IDFT) truncate the estimated impulse response (rectangular window) re-transform into frequency domain (DFT) noise reduction (NR) CE ĤC IDFT ĥc ~ h c DFT ~ C H N C windowing in time domain 28

15 Noise Reduction Algorithm Example estimated and real channel transfer functions (frequency domain) 0 db -10 db -20 db Ĥn C n H n C n n... in time domain ĥ(k) c(k) h(k) c(k) k 29 Noise Reduction Algorithm - Example smoothed and real transfer functions (in frequency domain) 0 db -10 db -20 db ~ H C n H n C n n time limited (windowed) impulse response ~ h(k) c(k) k 30

16 Noise Reduction Algorithm imulation Result simulation of a HIPERLAN/2 system (27 Mbit/s) time invariant Rayleigh distributed multipath channel 10 0 only CE CE+NR ideal (only CE) E b /N 0 loss: about 1.8 db PER 10-1 (CE+NR) E b /N 0 loss: about 0.5 db 10-2 (ideal) perfectly known channel E /N b 0 31 cattererd Pilot Constellation:Time-Frequency Interpolation n distance of sampling points in time direction i distance of sampling points in frequency direction 32

17 cattererd Pilot Constellation:Time-Frequency Interpolation (2) Estimation of transfer function at Choose a set of neighbored pilot symbols P(n, i) At sampling points the transfer function is simply: H (n Pi,i Pi )= x n Pi (i Pi ) d npi (i Pi ) P (n, i) n i Perform two-dimensional Wiener Interpolation X Ĥ(n, i) = b(n n 0,i i 0 ) H(n 0,i 0 ) {n 0,i 0 } P(n,i) Wiener ansatz: b opt =argmin E{ H (n, i) Ĥ (n, i) 2 } ª b 33 Outline Chapter 4: Orthogonal Frequency Division Multiplexing Fading Channel Flat fading channel Frequency selective channel II ingle Carrier Equalization Orthogonal Frequency Division Multiplexing Principle of Multi-carrier systems Inter Carrier Interference Orthogonal Frequency Division Multiplexing: OFDM Cyclic Prefix Equalization Impulse hortening Equalizer OFDM Channel Estimation Digital-to-Analog Interface 34

18 Digital to Analog Interface Critical sampling f A = N/T, no guard interval For correct analog reconstruction: Ideal low-pass (band-pass) necessary, which is not realistic. olution: Oversampling, overlapping rect-impulses with raised cosine slopes instead of pure rect impulses. 35 Realistic analog transmit filter: 36

19 Intersymbol interference due to overlapping symbols 37 Nonlinearity of the Power Amplifier 38

20 Characteristics of Nonlinear Class A and B Amplifier 39 Input Backoff (IBO) Clipping Class A Class AB IBO: Ratio between maximum output magnitude and root mean square of input signal 40

21 Conclusions OFDM converts frequency selective channel into N non-selective subchannels. Very simple equalization by the use of a guard interval Equivalent structure: ingle Carrier Frequency Domain Equalizer Pre-equalizer for impulse shortening Channel estimation: Preamble for slowly varying channels (noise reduction) cattered pilots with Wiener interpolation for fast varying channels Digital-to-analog interface Time-domain cos-roll-off shaping Nonlinear distortions by power amplifier 41