ELEC 546 Lecture #9 Ortogonal Frequency Division Multiplexing (OFDM): Basic OFDM System
Outline Motivations Diagonalization of Vector Cannels Transmission of one OFDM Symbol Transmission of sequence of OFDM Symbols Sample and Symbol Time Syncronization Carrier Frequency Syncronization Peak-to-Average Power Ratio (PAPR) Issue 2
(f) OFDM: Motivations Realization of Frequency Selective Fading Cannel Eac Sub-cannel is a Flat Fading cannel Treat a Wideband FS fading cannel as Multiple arrowband Flat fading cannels Cange in TX so tat RX does not suffer from ISI Use FEC wit codeword span across all sub-cannels acieve Frequency Diversity, but wit no ISI problem f 3
OFDM : Motivations Motivation: Split a frequency selective fading cannel into multiple, say 24, narrowband flat fading sub-cannels Send te bits over tese sub-cannels in parallel Serial to Parallel modulator modulator Mixer f Mixer Combiner f 24 4
OFDM: Motivation Problems: Multiple transmitter front ends (mixer, modulator, etc) require guard bands Solutions: Do all tese in digital domain using a wide baseband signal Use DFT (discrete Fourier transform) to create te baseband equivalent of te transmit signal and ten up-covert it to te center frequency using one front end As DFT is an ortogonal transformation, no guard band is needed 5
Diagonalization of Vector Cannels Consider a Vector Cannel wit input x and output y s y i x + n j i, j y x + V j x n i Want to diagonalize it suc tat z Uy UVs + Un D ~ s + n zi disi + n~ i i,, n y U z n i s i d i z i 6
Maintaining te same SR during Diagonalization U as to be Unitary to prevent noise enancement Enn ~~ unitary V as to be Unitary to maintain te same transmit power Ex x EUnn Es Es UU V s if U U Vs is if V is unitary 7
Advantage and Issues wit Diagonalization eed to find U and V s.t. UV D is diagonal Decompose te vector cannels into parallel cannels wit different gain (allow adaptive modulation, and TX optimization to be discussed in 2 nd part of OFDM notes) V depends on (TX needs to know te vector cannel) 8
Diagonalization of ISI cannel For frequency selective fading cannel (# of resolvable pats 2), time domain response as ISI is Toeplitz y x+ n, y x + x + n y x + x + n 2 2 2 3 3 2 3 is a circulant matrix if ~ ~ i, j, ~,... ( i j ) ~ i, j j i y x n + y x n Y X 9
Diagonization of Circulant Matrix If is circulant, ten W ~ W D ( D ) nn m ; m e W mn jπ exp ote tat TE W tat diagonalized ~ is independent of ~! ence, TX does not need to know ~!! Use cyclic prefix to create an effective circulant matrix mn mn jπ DFT of te cannel impulse response: Gains of te subcannels
Cyclic Prefix Instead of transmitting xws, transmit Ten, Transmit a lengt + s vector for a lengt- data vector. Efficiency /(+ s ) wit s >L (ISI lengt) For 2 24, s, Efficiency ~ 99%. [ ] T x x x ~ x x ~ ~ x x x x x
X W S IFFT samples Serial to Parallel Parallel to Serial A Frequency Domain Samples OFDM Transmission IFFT FFT B Time Domain Samples Z W Y~ samples FFT Parallel to Serial Serial to Parallel Add Cyclic Prefix & Pulse Saping Matced Filter and Remove Cyclic Prefix X Y ~( + s ) samples C Mixer f c Mixer & Filter Frequency Selective Cannel f c ~( + s ) samples 2
OFDM Transmission W/ -Time-domain modulation -Modulation Pulse overlaps in time -At ideal sampling position, tere is no ISI -Wit timing offset ISI -OFDM -Modulation pulse overlaps in freq -At ideal sampling position (in freq), tere is no ICI -Wit frequency offset ICI 3
Transmission of a sequence of OFDM symbols Using a block of samples to create an OFDM symbol (xws) and te cyclic prefix, ISI between samples witin an OFDM symbol is eliminated Wat appens to te intersymbol interference between OFDM symbols? s s t s s t TX: RX: o ISI 4
Cyclic Prefix Insert a Cyclic Prefix before every OFDM symbol Cycle Prefix lengt > τ max Overead is τ max /(T s ) were T s /B is te sampling period, B is te bandwidt and is te number of sub-carriers or points in te IFFT te larger te, te smaller te overead!! If τ max /T s s. Ten, tere will ave + s sampled points for every OFDM symbol 5
Cyclic Prefix If we just take te last points out of te + s points to do te FFT at te receiver, y x Ten n n n were denotes circular convolution and Y k k X k were Y k, X k, and k are te DFT of y n, x n, and n, resp s T s T s 6
Equivalent Cannel of OFDM Using IFFT, FFT and cyclic prefix, te OFDM transforms a frequency selective fading cannel (in time domain) parallel cannels (in frequency domain). Discussion: Is OFDM optimal in capacity sense (capacity of frequency selective fading cannels)? S d Z S d Z Frequency Domain Inputs Frequency Domain Outputs 7
Advantages of OFDM Wit cyclic prefix, we can eliminate ISI completely Provide frequency diversity Forward error correcting code suc as convolutional code wit interleaver is needed as some sub-carriers will be in deep fade Potential If te transmitter knows te cannel conditions can select only te good sub-carriers to transmit or transmit different numbers of bits based on te sub-carriers gains Power waterfilling in frequency domain. If te transmitter knows te cannel, OFDM wit bit allocation is better tan te best equalizer (e.g. MLSE) 8
Effect of Timing Offsets Sample and Symbol Syncronization Sampling time syncronization Sampling Frequency needs to be correct, but sampling instance offset only leads to linear pase sift in te sub-cannels gains. (wic will be andled by cannel estimation) only X ADC is needed at te receiver OFDM Symbol Syncronization Determine te beginning of te OFDM symbol and te beginning of te cyclic prefix (to avoid Inter-OFDM symbol interference) Use Cyclic Prefix Compute Correlation between two intervals separated by T s 9
Effects of Frequency Offsets Carrier Frequency Syncronization Carrier Frequency offset can cause significant inter-subcarrier interference (Similar to timing offset causing ISI in time domain modulation) As tere is no guard band, very small frequency offset can lead to large inter-subcarrier (or inter-subcannel) interference Very important and performance is sensitive to tis f 2
Effects of Doppler Spread Wen Tc >> OFDM symbol time, cannel fading coefficients are quasi-static slow fading scenario. Wen Tc < OFDM symbol time, cannel fading coefficients are no longer quasi-static. For simplicity, consider flat fading cannel. Effect of fast fading is equivalent to te multiplication of OFDM signal by a time-domain window (t). F yt () xtt () () F Xnsinc( f nδf) Equivalently, te effect of fast fading after FFT is te circular convolution of te (f) wit X(f). te sape of te subcarrier is distorted from te SIC pulse (in freq domain) to a smeared subcarrier sape (as a result of te convolution). Fast fading results in ICI. F n Δ n Δ n n 2 n F ( ) ( ) Y( f) X( f) ( f) X sinc( f n f) ( f) X G f n f
Peak-to-Average Power Ratio Te data symbol, s i, may be QPSK modulated (constant magnitude), but te transmitted samples, x i, is te output of te IFFT and ence takes values over a wide range. Statistically, as s i are independent and as random pase, x i approaces an Gaussian distribution wen is large ence, ig Peak-to-Average Power Ratio PAPR ( max x ) i 2 i Ex i 2 22
Disadvantages of OFDM Overeads Cyclic Prefix: can be reduced by increasing Power to transmit cyclic prefix: can be lower by increasing Implementation issues sensitivity to frequency offsets especially wen is large and sub-carrier spacing is small require igly linear power amp ig peak-to-average-power (PAP) ratio, especially wen is large Poor Adjacent band rejection ~ 2/3dB only. Q: In wireless LA (82.g), te AP can receive and decode packets on an adjacent cannel. Wy? Typical value for is 2 7 to 2 23