ECE5984 Orthogonal Frequency Division Multiplexing and Related Technologies Fall 2007 Mohamed Essam Khedr Channel Estimation
Matlab Assignment # Thursday 4 October 2007 Develop an OFDM system with the ollowing components S/P Mapping model (modulation techniques) Coding model (conv, turbo) IFFT CP Channel (Gaussian, SFFF channel) Mapping decoding Decoding model FFT CP removal Channel Estimation (later) Input :pulse shaping, Number o subcarriers, symbol rate, BW, CP ratio Output: Signal in time, spectrum, BER, ICI (later), ISI (later)
HW 2 /2 Due Thursday 4 December
HW 2/2 Comment on ISI Comment on ISI
Major Learning Objectives Upon successul completion o the course the student will be able to: Describe the complete architecture o an OFDM system, ( serial to parallel, FFT/IFFT, Cyclic preix, Modulation techniques, coding techniques) Evaluate the response o OFDM in Gaussian channels and ading channels. Design and analyze standards using OFDM such as IEEE 802.a,g and IEEE 802.6 Deine the problems associated o using multi-carrier in time varying channels and how to mitigate these problems. Describe the principle mechanisms by which multiple access techniques are supported using OFDM. Able to categorize the dierent type o MC-CDMA and the degree o lexibility provided by each type. Able to simulate the basic and advanced techniques used in OFDM systems
Syllabus Analysis o OFDM systems (5%) 2 RF subsystems, ampliier classiication and distortion Crest actor (PAPR) reduction techniques Pre-distortion & adaptive pre-distortion techniques clipping coding techniques partial transmit sequences (PTS) & modiied PTS v. selective mapping nonlinear quantization (companding) Phase noise and I&Q imbalance or QAM Perormance o OFDM in Gaussian channels Perormance o OFDM in Wide-band channels Synchronization and Estimation (5%) 2 ICI and OISI problems Timing estimation Frequency synchronization Frequency error estimation algorithms Carrier phase tracking Frequency domain and time domain approaches or channel estimation coherent detection dierential detection
Our OFDM System Assumptions Usage o cyclic Preix Impulse response o the channel shorter than Cyclic Preix. Slow ading eects so that the channel is time-invariant over the symbol interval. Rectangular Windowing o the transmitted pulses Perect Synchronization o transmitter and receiver Additive, white, Gaussian channel noise
The Mobile Multipath Channel Delay spread Doppler spread Time FT FT Frequency Time Frequency Channel Estimation Frequency Interpolation, or iltering. Assume that known symbols X(n,k) were transmitted in various positions (tones (n) or blocks (k) ). Estimate H(n,k) and then interpolate to n,k.
Channel Estimation Types Parametric vs non-parametric Frequency and time correlation Training vs blind Adaptive vs non-adaptive Parametric based on a channel model Non-parametric based on measurements. Correlation estimation is based on previous estimates Training well known symbols Blind based on the statistical properties o the signal Adaptive estimation algorithm modiied with the channel variations
OFDM System Model X (k,0) H (k,0) W (k,0) Y (k,0) X (k,) X ( k, N ) H (k,) H ( k, N ) W (k,) Y (k,) W ( k, N ) Y ( k, N ) k N N k k k N N k Y = X H + W = H X + W k k
System Architecture-2 Input to Time Domain x ( n ) = IDFT { X ( k )} n = 0,,2,..., N 2 x Guard Interval ( n) x = x ( N + n) ( n),, n n = N g, N g = 0,,..., N +,..., 3 y Channel ( n ) h( n ) w( n ) = x + 4 y Guard Removal ( n) = y ( n) n = 0,,..., N 5 Output to Frequency Domain Y ( k ) = DFT { y ( n )} k = 0,,2,..., N 6 Output Channel ICI AWGN 7 Y ( k ) = X ( k ) H ( k ) + I ( k ) + W ( k ) X ( k) k = 0,,..., N Channel Estimation Estimated Channel Y e = k = 0,,..., N H ( k) ( k) e
For the k th carrier: Frequency Domain Equalization x k = H k s k + v k where H k = n h k (nt s ) exp(j2π k n / N) where n = 0,,. N- Frequency domain equalizer x k s k H k - Noise enhancement actor H k 2 H k - 2 bad k good k
Channel Estimation OFDM uses variations o Quadrature Amplitude Modulation (QAM) schemes or symbol mapping which require a coherent detection method in the receiver. Naturally, data detection in coherent OFDM receivers require an accurate (or near accurate) estimate o Channel State Inormation (CSI). There are two major kinds o channel estimators that are ound in literature: Pilot assisted. blind estimation. A mixture o these two, where a blind method with limited training symbols is used, is called semi-blind technique.
Types o Channel Estimation Traditional one-dimensional channel estimation techniques or the OFDM systems can be summarized as ollows: Least Squares (LS) Minimum Mean Squared Error (MMSE) Linear MMSE (LMMSE). LS estimators are very simple to constitute, but they suer rom MSE in low SNR conditions. MMSE, based on time domain estimations, are high complexity estimators that provide good perormance in sampled-spaced channels, but limited perormance in nonsample spaced channels and high SNR conditions. LMMSE provides good perormance in both sampled and nonsampled channels
Channel State Inormation In OFDM systems, the Doppler eects are kept smaller by making sure that the symbol duration is much smaller compared to the channel coherence time. In this case, the channel attenuations at successive symbol durations experience suiciently higher time correlation. Similarly, i subcarrier spacing is chosen in a way that the spacing is much smaller than the coherence bandwidth o the channel The channel attenuations at the adjacent subcarriers will be highly requency correlated. So, the estimator can exploit both o these two correlation properties
Channel State Inormation Channel estimation o a SISO-OFDM system can be done by using complete training symbols ater certain OFDM data symbols, or by inserting some training pilot tones in every OFDM symbol. In the irst case, the CSI is estimated with the training symbol and interpolated or the consecutive symbol beore the next training symbol appears. This technique renders unacceptable results when the channel variation time is comparable to OFDM symbol duration. The second method is suitable in these kinds o ast varying channels. The CSI is estimated or all the pilot tones using the pilot subcarriers rom that particular symbol and later CSI or all other subcarriers are obtained by interpolation. In that way, perect or near perect estimates are achievable. But the cost is paid in signiicant throughput reduction.
Ideal Channel Estimation Wireless channels change requently ~ 0 ms Require requent channel estimation The attenuations o the pilot symbols are measured and the attenuations o the data symbols between these pilot symbols are typically estimated/interpolated using time correlation property o ading channel Many systems use pilot tones known symbols Given s k, or k = k, k 2, k 3, solve x k = l=0l h l e -j2π k l/n s k or h l Find H k = l=0l h l e -j2π k l / N (signiicant computation) More pilot tones Better noise resilience Lower throughput (pilots are not inormative) magnitude Pilot tones requency
Channel Estimation Via Interpolation More eicient approach is interpolation Algorithm For each pilot k i ind H ki = x ki / s ki Interpolate unknown values using interpolation ilter H m = α m, H k + α m,2 H k2 + Comments Longer interpolation ilter: more computation, timing sensitivity Typical db loss in perormance in practical implementation magnitude requency
Channel Estimation algorithm LS In a matrix orm, the observed symbols ater the DFT operation in the receiver can be written as where the diagonal matrix X contains the transmitted symbols on its diagonal (either known pilot symbols or receiver decisions o inormation symbols which are assumed to be correct), the channel attenuations o one OFDM symbol (i.e. Fourier transorm o h(t) evaluated at the requency k ) is collected in vector h and the vector r contains the observed outputs o the DFT. I we maximize the channel estimates in the Least-Square (LS) sense: maximize or all possible ĥ This is a straight orward estimation technique where the received symbol on each subcarrier is divided by the transmitted symbol to obtain the estimate.
Least Squares Estimator H LS ( m, k) Y ( m, k ) X ( m, k )
Channel Estimation algorithm LMMSE Minimize the mean square error between the actual channel response and the estimated one by linear transormation to H LS The optimal Linear Minimum Mean-Square Error (LMMSE) estimate o ĥ by (minimizing or all possible linear estimators ĥ )
Design o Pilot Based Channel Estimator There are mainly two problems in designing channel estimators or wireless OFDM systems. The irst problem concerns the choice o how pilots should be inserted. The second problem is the design o the estimator as a low complexity with good channel tracking ability. The pilot symbols should be inserted properly, so that it successully estimates the requency response o the channel. The dierence between two consecutive pilot symbols in time and requency domain, S t and S respectively, can be represented as
Pilot Symbol Assisted Modulation N p pilot symbols P i are transmitted in the subcarriers within the total OFDM symbol bandwidth o N subcarriers. At the receiver, the channel transer unction at the pilot subcarriers is estimated rom the received samples The second step, the values o the channel transer unction are estimated or the unknown data symbols by interpolation using the abovementioned equation. The placement o the pilots and the interpolation technique will inluence the quality o the channel estimation
Pilot Arrangement Block Type All sub-carriers reserved or pilots with a speciic period Comb Type Some sub-carriers are reserved or pilots or each symbol
Linear Interpolation H() is the FT o the h(t). In order to sample H() according to sampling theorem, the maximum pilot spacing in OFDM symbol is p
Time domain Channel Estimation using Training Sequence Conventional estimation schemes send a stream o transmitted symbols with a modulation scheme known to the receiver, and the receiver analyzes the eect o the channel on the known symbols by observing the deviations on the received known symbols. The transmission o training symbols reduces the spectral eiciency o the system
802.a System Speciication t t2 t3 t4 t5 t6 t7 t8 t9 t0 GI2 T T2 GI OFDM Symbol GI OFDM Symbol Short training sequence: AGC and requency oset Long training sequence: Channel estimation Sampling (chip) rate: 20MHz Chip duration: 50ns Number o FFT points: 64 FFT symbol period: 3.2µs Cyclic preix period: 6 chips or 0.8µs Typical maximum indoor delay spread < 400ns OFDM rame length: 80 chips or 4µs FFT symbol length / OFDM rame length = 4/5 Modulation scheme QPSK: 2bits/sample 6QAM: 4bits/sample 64QAM: 6bits/sample Coding: rate ½ convolutional code with constraint length 7
Channel Estimation Algorithms Linear Interpolation Second Order Interpolation Maximum Likelihood (Least Square in time domain) Linear Minimum Square Error
Linear Interpolator (I) Linear Interpolator (I) Use two piloted grid closest to the grid needed to be estimated in LI; M pilot grids during the same symbol or MMSE and ML P P D d D d a where a m k H a m k H m k H < = + = 2, / ) )(, ( ˆ ), ( ˆ ), ˆ ( D
Weighted Linear Interpolator (II) Weighted Linear Interpolator (II) Extending linear to weighted linear < > > = = + + K n K m K K n m K K n K n K m a K n m a K n m a n m g 2, ) 2 ( 2 ) (, ) 2 ( 2 2, ) 2 3( 2 ), ( > + = m D m D a m K n D m n m g = = ), ( 2 Where
Second Order Interpolation (I) Second Order Interpolation (I) Use two piloted grid closest to the grid needed to be estimated in LI; M pilot grids during the same symbol or MMSE and ML ( ) ( ) ( ) ( ) ( )( ) ( ) = + + + = = + + = + = = N l m k Hp c m k H p c m k p H c l D m k H c c c where /, ˆ, ˆ 0, ˆ ) ),( (, 2, 0, 2 α α α α α α α D
Perormance o (Simpliied) Matrix Inversion 30 Output SINR 25 20 5 MMSE k = 4 Conv OFDM 0 5 0 Conventional OFDM MMSE equalization simpliied MMSE 5 0 5 20 25 30 N = 64, v = 200 km/h, c = 7 GHz, T RMS = µs, sampling at T = µs. Doppler = 3.5 khz, Subc. spacing sr = 3.25 khz: Input SNR Compare to DVB-T: v = 40 km/h, c = 800MHz: doppler = 00 Hz while sr =.7 khz
MSE vs SNR or dierent grid
Received and Recovered Signals