Orthogonal Frequency Division Multiplexing (OFDM) Spring'09 ELE 739 - OFDM 1 Frequency Selectivity of the Channel • System bandwidth vs. Coherence Bandwidth • Frequency Non-Selective Channel • Frequency-Flat channel • Channel impulse response is a simple impulse function. • Detecting symbols at the MF output is optimum in the ML sense • No need for an equalizer/MLSD. • Generally low data rate. Spring'09 • Frequency Selective Channel • Channel impulse response has a certain width in time, τmax. • Detecting symbols at the MF output is NOT optimum in the ML sense • An equalizer/MLSD required. • Generally high data rate. ELE 739 - OFDM 2 SC vs. MC • If we use a single carrier modulation scheme, – For relatively low bit rate there is no problem • Channel is flat, • No equalizer is required. – For increased data rate, system bandwidth will increase • • • • May cause ISI Gets worse as data rate (system BW) increases. ISI causes severe error in the detected symbols. An Equalizer/MLSD is required for better reception. • What if we have many independent low bit rate (system BW) transmissions in parallel? • Single Carrier Multi-Channel (Carrier) • no.chnl=1, Rk=R, ΣRk=R no.chnl=N, Rk=R/N, ΣRk=R • W ∝ R >> Bc Wk ∝ R/N << Bc Spring'09 ELE 739 - OFDM 3 FDMA • One way of generating independent multi-channel systems is to divide the frequency range into smaller parts – subcarriers (freq. bins) How can we seperate subchannels in freq. so that they do not interfere – FDMA: Subcarriers must be separated at least by the BW of the xmission – Waste of precious spectrum. • Spring'09 ELE 739 - OFDM 4 OFDM • Instead place the subcarriers at frequencies – Obviously, – Pulse shape is rectangular → spectrum is the sinc function. – Spectra of the above pulses overlap but the sub-carrier frequencies are placed at the spectral nulls of all other pulses. Spring'09 ELE 739 - OFDM 5 have N oscillators at frequencies – Many practical problems (i+1)T ci.N-1 ~ ci.N-1 Spring'09 ELE 739 .0 ~ ci.0 ci.1 ci.1 ~ ci.OFDM • How can we generate these pulses.OFDM 6 . – Analog way. 1 ci.0 ci.N-1 Spring'09 ELE 739 .0 ~ ci.1 ~ ci.OFDM 7 .OFDM • Alternative (digital) way. use the IFFT/FFT pair – Much easier to implement on a digital platform – Overcomes the problems of the analog implementation (i+1)T ci.N-1 ~ ci. n} If N is a power of 2. IFFT N (k) time 8 Spring'09 ELE 739 .g.OFDM • Consider the transmitted signal i: OFDM Symbol index n: Subcarrier index where the normalized (rectangular) basis pulse gn(t) is • Now.l. and sample at instances • • This is the IDFT of the transmit symbols {c0. N (n) freq. can be realized by IFFT. w.OFDM .o. consider only i = 0. subcarriers are orthogonal. the procedure is reversed – – – – Collect N samples in time S/P FFT ~ Obtain the estimates cn regarding the transmitted cn.OFDM Data Packet Delay dispersion time 9 . N data symbols freq.OFDM • • {sk} are time samples → transmitted sequentially in time → P/S At the receiver. Single Freq. • Works fine for the AWGN channel. Guard Interval N data symbols freq. OFDM Spring'09 OFDM Symbol (frame) Delay dispersion time ELE 739 . Spring'09 ELE 739 . – Causes Inter-Carrier Interference (ICI).OFDM 10 . • Can be prevented by adding a Cyclic Prefix (CP) to the OFDM symbol. samples)) ICI is prevented.Cyclic Prefix • Delay dispersion may destroy the orthogonality of the subcarriers. – Number of samples per OFDM symbol increases from N to N+Ncp. • If Ncp ≥ L-1 (delay dispersion of the channel (no. – Copy the last Ncp samples of the OFDM symbol to the beginning. ^ ^ • • Normally. – -Tcp < t < 0 ^ – 0 < t < Ts : cyclic prefix part. the signal arriving from a delay-dispersive channel is the “linear” convolution of the transmitted signal and the channel IR.CP • Define a new basis function where W/N is the carrier spacing and Ts=N/W. CP converts this “linear” convolution to “cyclical” convolution. : data part. ELE 739 .OFDM 11 Spring'09 . – If τmax ≤ Tcp. • OFDM symbol duration is Ts=Ts+Tcp. to eliminate Linear ISI from the previous OFDM symbol (i-1). there is bank of filters matched to the basis functions without the CP: • After removing CP. Assume that Tcp = τmax. In the receiver. We end up with cyclical ISI. simply discard the CP part of the signal received corresponding to OFDM symbol i.OFDM 12 Spring'09 . can be implemented by the FFT operation. ELE 739 . also let i=0.Linear ISI CP Cyclical ISI CP data OFDM symbol i-1 OFDM symbol i OFDM symbol i+1 • • • • At the receiver. totally contained in the i-th symbol. OFDM • The signal at the output of the MF is the convolution of the – Transmit signal (transmit data + IFFT). τ)=h(τ) • Substituting gk(t) into the inner integral: Spring'09 ELE 739 . – The channel impulse response.OFDM 13 . h(t. and – The receive filter: • nn: noise at the MF output. Assume quasi-static channel. since the basis functions are orthogonal during 0 < t < Ts ^ • The OFDM system is respresented by a number of parallel and orthogonal (non-interfering) non-dispersive (flat) fading channels. • Spring'09 ELE 739 . Equalization is very simple: – Divide signal from each subchannel by the transfer function at that freq. each with its own complex attenuation H(nW/N).OFDM • Moreover.OFDM 14 . – Happens when τmax ≥ Tcp • CP does not convey data. only used to prevent ISI/ICI – Decreases useful SNR – Decreases throughput by Tcp/(Ts+Tcp) Spring'09 ELE 739 .OFDM 15 .OFDM • • • A simple structure which employs FFT/IFFT Very simple equalization (1-tap) if subcarriers are orthogonal ICI occurs when subcarriers are not orthogonal. ELE 739 . Consider two consequent data blocks. c0.Another Perspective • • Assume that a data block composed of N symbols. .i i’th data block.i-1 cN-1.i cN-1.i c1.i-1 Spring'09 c0.i-1 c1.OFDM 16 (i-1)th data block. Another Perspective • FFT matrix • • IFFT matrix: Time-domain transmit signal: • Parallel-to-serial convert and trasmit through the channel. ELE 739 .OFDM 17 Spring'09 . i-1 c1. – Causing ISI between OFDM symbols.i c1.OFDM 18 . cN-1. cN-1.i-1 c0. ISI L-1 Spring'09 ELE 739 .i-1 (i-1)th OFDM symbol.i • In an ISI channel with length L taps – Last L-1 samples of the (i-1)th OFDM symbol will interfere the first L-1 samples of the i’th OFDM symbol (in time).i i’th OFDM symbol.Another Perspective • If we transmit both sequentially c0. Place all-zeros in the guard period.Guard Period • • In order to prevent ISI. ISI ISI GP L-1 No ISI • • • No ISI between OFDM symbols ISI only within an OFDM symbol → controllable Received signal is the convolution of the transmit signal and the channel. place a guard period of at least L-1 samples between adjacent OFDM symbols.OFDM 19 Spring'09 . ELE 739 . discard discard Collect these N samples for FFT Collect these N samples for FFT Spring'09 Received Signal ELE 739 .OFDM GP GP 20 . OFDM 21 .Received Signal • This is equivalent to GP • Example: N=8. L=3: Spring'09 HGP H ELE 739 . i and cn. i.e.i • ^ For a one-to-one relation between cn. we obtain the estimates of cn.OFDM 22 . without ICI – must be a diagonal matrix.i. • Q must diagonalize H.FFT/IFFT • Now consider the system after removing CP • Taking FFT of yi. Spring'09 ELE 739 . To diagonalize the channel.OFDM 23 . hence E. Channel estimation is performed at the receiver. we should use this matrix (EH more precisely) at the transmitter and at the receiver. • We need a feedback channel to move the E matrix to the transmitter. and Λ is a diagonal matrix with the eigenvalues of H on the main diagonal. transmitter does not know the channel. Spring'09 ELE 739 . • Not practical in many cases.Diagonalization • Let us consider the Eigendecomposition of the matrix H where E is a unitary matrix containing the eigenvectors of H. • • • For an arbitrary H matrix E will also be arbitrary. ? Spring'09 ELE 739 . E becomes the FFT matrix.Diagonalization • • As a special case.OFDM 24 . For our previous example with N=8. when the H is a circulant matrix. L=3. same Spring'09 ELE 739 .OFDM 25 . repeat the last L-1 samples of si as the Cyclic Prefix at the beginning of and OFDM symbol.Cyclic Prefix • Instead of a Guard Period L-1 samples. hence the eigenvalues correspond to the samples of the transfer function of the channel Spring'09 ELE 739 .OFDM 26 .Diagonalization • The eigenvectors of Hcirc form the FFT matrix. • Cyclic prefix enables circular convolution Spring'09 ELE 739 .Cyclic Prefix • FFT and IFFT (DFT/IDFT) are the pairs for – Circular convolution in the time domain. – Product of the transfer functions will not give what we want. ie. with the zero guard period. we have the linear convolution of the transmit signal and channel at the channel output. and – Product of the transfer functions in the frequency domain. • For regular packet structure.OFDM 27 . i – In total these coef. • Spring'09 ELE 739 . – Eigenvalue-decomposition based methods. – Scattered pilot symbols.s give the Transfer Function of the channel. Hn. There are three approaches: – Pilot OFDM symbols.OFDM 28 . flat) subcarriers – Each subcarrier channel can be represented by a single complex coefficient. we have a number of narrowband (freq.Channel Estimation • We have seen the estimation of the Channel Impulse Response for single-carrier modulation. • • We wish to obtain the estimates of the N samples of the Transfer Function. In OFDM. or the Transfer Function is Spring'09 ELE 739 . – Data on each subcarrier is known.i. • Appropriate for initial acquisition of the channel at the beginning of a transmission burst. then • in the LS sense.OFDM 29 .Pilot-OFDM Symbol Based • We have a dedicated OFDM symbol containing known data. If the known data on subcarrier n for OFDM symbol index i is cn. and is the autocovariance matrix of the LS estimates. channel gains and the LS estimates of channel gains.Pilot-OFDM Symbol Based • Linear MMSE estimator: where is the covariance matrix bw.OFDM 30 . • If the channel noise is AWGN – – where Spring'09 ELE 739 . Very high computational complexity. Time correlation can be exploited: subsequent OFDM symbols are not used as pilots. Less than N subcarrier can be used as pilot tones since neighbouring subcarriers are correlated (in frequency). Suitable for initial channel acquisition. Uses all OFDM frame as pilot: no room for data. Spring'09 ELE 739 .OFDM 31 • • • • • .Pilot-OFDM Symbol Based • Produces very good estimates (the transfer function of the OFDM symbol of concern). Scattered Pilots (in time & frequency) • We could use a grid structure for the pilots. – Pilots scattered in frequency and time – What should be the spacing between the pilots? • Nt = ? • Nf = ? – Sampling Theory: – Interpolate between the pilots to estimate the channel.e.OFDM 32 . i. Spring'09 ELE 739 . OFDM 33 . then where • Less complexity.Scattered Pilots (in time & frequency) • • Interpolation can be done by MMSE estimation. Let be the LS estimates of the pilot tones. Spring'09 ELE 739 . OFDM 34 . Spring'09 ELE 739 . rest can be ignored. approximately Ncp+1 eigenvalues of Λ will have significant values. Consider the LMMSE estimator: • • .Eigenvalue Decomposion Based Methods • • • Length of the Channel Impulse Response < OFDM symbol length Channel can be represented by less coefficients in the time domain. • Then a CP of length at least L-1 is required in an OFDM symbol to eliminate ISI/ICI. If the system allows κ taps for CP. we can have an equalized channel of length κ+1 taps → more degrees of freedom.Channel Shortening • Assume that the length of the communication channel is L taps. ELE 739 . N↑ → η↑ (gets closer to 1) We cannot arbitrarily increase N due to the time-selectivity of the channel. N determines the efficiency. Then we may decrease L ? Equalization effectively decreases L to 1.OFDM 35 Spring'09 . • Spectral Efficiency of OFDM with N subcarriers is • • • • • If L is fixed. f + t[n] Filter. b • Complete Equalization Forced to be these values. • Channel Shortening Equalization Freely determined by the cost function.Channel Shortening η[n] x[n] Effective Channel. w z[n] ^ z[n] + ε[n] Delay.OFDM 36 . {bk} are variable! Spring'09 ELE 739 . δ TIR. Spring'09 ELE 739 .Channel Shortening (MMSE) • Design a receiver filter of nw taps whose output is Toeplitz Matrix ? Same expressions as we have seen before.OFDM 37 . OFDM 38 .Cost Function • Now. the MMSE cost function is For complete equalization we have • We can proceed with the same derivations as complete equalization by substituting Spring'09 ELE 739 . OFDM 39 Spring'09 . first optimize wrt.Cost Function • Expanding the cost function • Using the property that data and noise are uncorrelated E{xη*}=0 • This is a quadratic function of w and b. w ELE 739 . Optimum Equalizer • Optimum equalizer coefficients are: • Substituting back to the MSE term which is minimized by OOPS!!! It says: Do not transmit anything! Spring'09 ELE 739 .OFDM 40 . OFDM 41 . bk’: variable – Unit norm constraint on TIR. wTw=1 – Unit tap constraint on TIR. impose a constraint on the filters: – Unit norm constraint on w.Optimum TIR • To avoid the trivial solution. bTb=1 • Unit norm constraint on TIR gives better performance: Hermitian symmetric • Cost and constraint are convex: Use Lagrangian method This is the eigenvalue problem. bk=1. Spring'09 ELE 739 . ELE 739 . Obviously. Above problem has to be solved for every possible δ. this still a function of delay δ.OFDM 42 Spring'09 .Optimum TIR • defines a square windows in T T= • • • Lagrangian becomes is the minimum eigenvalue of T’ and b is the corresponding eigenvector. OFDM 43 .Peak-to-Average Power Ratio (PAPR) • For frequency-time domain conversion we use the FFT/IFFT matrix: • Let the symbols to be transmitted be • Then the transmitted signal is ( + CP) Spring'09 ELE 739 . They can be linear only in a limited range.OFDM IFFT freq. But. even if the symbols to be transmitted are drawn from a constant modulus constellation (such as M-PSK). There is no problem for a constant modulus signal since the amplitude of the signal does not change.Peak-to-Average Power Ratio (PAPR) • Most practical RF power amplifiers have limited dynamic range. Example: non-constant modulus constant modulus • • • Spring'09 time ELE 739 . the output of the IFFT operator may have different amplitudes for every sample (in time). 44 . due to the central limit theorem. with the IFFT operation. – Amplitude of the signal to be transmitted is proportional to N – Power goes with N2 • Another point of view is – We can consider the symbols on subcarriers as random variables – If the number of subcarriers is large. adding these symbols up will result in a complex Gaussian distribution with a variance of unity (mean power). the symbols on the subcarriers sometimes add constructively and at other times destructively. – Example: Probability that the peak power is 6 dB above the average power Spring'09 ELE 739 .Peak-to-Average Power Ratio (PAPR) • The problem is. 4 times – Absolute amplitude is Rayleigh distributed.OFDM 45 . – Destroys the orthogonality of the subcarriers. increased BER. – Nonlinearity causes spectral regrowth.Impact of the Amplitude Distribution • An amplifier that can amplify nearly up to the possible peak value of the transmit signal is not practical – Requires expensive. • Using a non-linear amplifier will cause distortions in the output signal. inefficient class-A amplifiers. Spring'09 ELE 739 . increased out-of-band emissions interfering systems in the neighbouring frequency bands. causes ICI. • We may use PAPR reduction techniques.OFDM 46 . – There are Mn possible combinations of the symbols.Coding for PAPR Reduction • In error control coding.k) symbols. – Codewords chosen wisely. – But. • We can think of an OFDM symbol as a possible combination of N. M-ary symbols (MN symbols) – Among these combinations. i. For sending k symbols we use n symbols (n ≥ k) • Redundancy of (n . so that the distance between them is maximized. – Significant loss of throughput due to redundancy. Spring'09 ELE 739 .e. – Have some coding gain (but less than a dedicated ECC code). we use a subset of these combinations of dimension Mk (#codewords) – We choose one these Mk codewords of length n. choose appropriate codewords so that PAPR is guaranteed to be below a certain level. for an (n.OFDM 47 .k) code and M-ary modulation. – Completely eliminates PAPR problem. ^ – Receiver undos phase adjustment by using the index l.... {cn}l^is transmitted together with the index l..Phase Adjustments • Define an ensemble of phase adjustment vectors ΦI={φn}l. – Cannot guarantee a certain level of PAPR. – Transmitter multiplies the OFDM symbol to be transmitted Ci by each of these phase vectors to get and then selects: to get the lowest PAPR possible.OFDM 48 ... Spring'09 ELE 739 .. ^ ^ – Instead of the sequence {cn}. – Less overhead..N – Known both at the transmitter and receiver. l = 1. n = 1.L. multiply the signal by a Gaussian function centered at times when the level exceeds the threshold (penalized by the Gauss func. i.OFDM 49 .) • Multiplying by a Gaussian in time is equivalent to convolving with a Gaussian in frequency → spectral regrowth controlled by σt2 • Causes significant ICI → increased BER. – Another example. multiply the signal to be transmitted by A0/sk.Correction by Multiplicative Function • Multiply the OFDM signal by a time-dependent function whenever the peak value is high. Spring'09 ELE 739 . – Simplest example. Clipping (penalize by saturating the output) • If signal attains a level sk>A0.e. Spring'09 ELE 739 . – Correction function acts as pseudo-noise → increased BER.OFDM 50 . There is trade-off between – – – – – PAPR Redundancy/Overhead Guaranteeing a certain PAPR level ICI/BER Out-of-band interference. • No best PAPR reduction technique.Correcting by Additive Function • Instead of multiplicative. we can use an additive correction function. Inter-Carrier Interference • Cyclic prefix completely eliminates Inter-Carrier Interference and InterSymbol Interference caused by the quasi-static frequency selective channel – If the channel delay spread is less than the cyclic prefix • If the channel is time-varying (time-selective) and changes within an OFDM symbol. Spring'09 ELE 739 . Doppler shifted Subcarrier. – Orthogonality of subcarriers is destroyed.OFDM 51 . – Doppler shift of one subcarrier causes ICI in many adjacent subcarriers. to increase spectral efficiency.ICI • Impact of time-selectivity is mostly determined by – Product of maximum Doppler frequency and duration of the OFDM symbol. Shorter CP → ICI. – CP does not have to be chosen to cope with the worst case channel if the loss due to ICI is amenable. • Delay dispersion can also be a source of ICI – if CP is shorter than the maximum excess delay.OFDM 52 . • Length of the channel maybe changing from time to time. – Tradeoff: Large excess delay requires long CP → reduced spectral efficiency. a small Doppler shift can cause considerable ICI. Spring'09 ELE 739 . • Spacing between subcarriers is inversely proportional to symbol duration – Large symbol duration. • For a time-invariant channel (h[q. – h[n. – u[n] : unit step function.OFDM 53 . l] : sampled version of the time-variant channel IR h(t.ICI • Received signal in case of ICI occuring as a result of Doppler shift or insufficient CP. above expression reduces to Spring'09 ELE 739 . l]=h[ l ]δ[q-l]) and sufficiently long CP. τ) – L : maximum excess delay in units of samples L = τmaxN/Ts. Long symbol duration (narrow spacing) is good for satisfactory spectral efficiency – TCP is limited by the maximum excess delay. – CP should be around 10% of the OFDM symbol for high efficiency • Choose Ts (N) to maximize : function of the channel Spring'09 ELE 739 .ICI • Optimum choice of carrier spacing and OFDM symbol length: – Tradeoff between ICI and spectral efficiency (N/(N+Ncp)) • • Short symbol duration (Ts) (large subcarrier spacing) is good for reducing Doppler-caused ICI.OFDM 54 . 10-2.g. Spring'09 ELE 739 . – Subchannels at peaks are good (narrowband) channels – Almost no information can be transmitter through valleys Energy over subcarriers are the same for this example.Adaptive Modulation • Transfer function of a frequency selective channel has peaks and valleys. For satisfying a target BER e. too much energy would be required at these subcarriers.OFDM 55 . Power is limited!. OFDM 56 . Problem: – We want to maximize the capacity of the system by wisely distributing the energy over subchannels – Under the constraint of limited power • Solution is the waterfilling algorithm where λ is the water level chosen to satify Spring'09 ELE 739 .Adaptive Modulation • • We see that fixed power allocation loads bad channels with low SNR (high BER and low capacity) → waste of energy. Adaptive Modulation waterlevel.OFDM 57 . λ Total power Power allocated to subcarrier .1 No power is allocated to this subcarrier Spring'09 ELE 739 . What modulation should we use in the subcarriers to get as close as possible to the assigned capacity? – This means. – For low SNR. – A constellation with Na points has a capacity of log2(Na) bits/channel use – A higher order modulation (64-128-QAM) can be used for a subcarrier with high SNR. the transmitter has to adapt the data rate according to the SNR available for a subcarrier. modulation order has to be decreased.OFDM 58 .Adaptive Modulation • • We have found the power allocation for each subcarrier which maximizes the capacity. Spring'09 ELE 739 .