Channel Estimation with Binary Sequences for MIMO Wireless Systems

Transcription

1 University of Padova Department of Information Engineering (DEI) Channel Estimation with Binary Sequences for MIMO Wireless Systems Marco Rotoloni Lorenzo Vangelista Phone: Fax: /18

2 Presentation abstract Theoretical Introduction: Problem statement Brief introduction to Multiple Input Multiple Output (MIMO) systems Brief introduction to Complementary Sets of Sequences (CSS) Paper framework: Implementation of a channel estimator using CSS for a MIMO singlecarrier transmission system Implementation of a LS channel estimator for a MIMO-OFDM system Performance comparison and conclusions October

3 Problem Statement Why, nowadays, channel estimation is a feature systems have to perform? Receivers need to acquire the knowledge about the channel impulse response to perform equalization, data detection and to implement new transmission schemes (MIMO) Even the transmitters needs to have information about the channel impulse response (OFDM, MIMO) October

4 Multiple Input Multiple Output (MIMO) (1) Exploit channel DIVERSITY (multipath) to fight FADING MIMO should be able to discern different paths The transmission scheme need to acquire the channel impulse response of the channel on each path October

5 Multiple Input Multiple Output (MIMO) (2) The MIMO scheme is able to discern different data paths (coming from each transmission antenna) if it knows the channel impulse response for each path. We define subchannel the channel impulse response which affect each signal on a different path: that is the relation between the signal transmitted (i) and the signal received (j), where 1 < i < N T and 1 < j < N R. NOTE: we use path and subchannel as synonyms October

6 Theoretical introduction to CSS (1) Defining the autocorrelation function for a finite length sequence x(n) as r xx ( k) = N-k-n= 0 x( n) x( n + k) 0 k, n N -1 Two binary sequences {a(n),b(n)}, with elements in the alphabet {-1,+1}, are a pair of complementary sequences when: r aa (k) + r bb (k) = 0 r aa (0) + r bb (0) = 2N k 0 moreover we define a second CSS {c(n),d(n)} mate of {a(n),b(n)} when: r ac (k) + r bd (k) = 0 k October

7 Auto&Cross-correlation for CSS Autocorrelation of a(n) Autocorrelation of b(n) Sum of autocorrelations Crosscorrelation of a(n) and b(n) October

8 Theoretical introduction to CSS (2) Budisin Algorithm: a (m) (n) = a (m-1) (n) + w(m-1) b (m-1) (n-d(m-1)) b (m) (n) = a (m-1) (n) - w(m-1) b (m-1) (n-d(m-1)) a (0) (n) = b (0) (n) = δ(n) Implementation of filters with complementary sequences as impulse responses (FGC e EGC): FGC Those schemes lead to efficient correlator implementations to be used at the receiver side of a transmission system. October

9 Signal representation (time-domain & freq-domain) We use matrix notation to represent the overall MIMO signal, that is the collection of the signals affected by each subchannel. Defining the overall transmitted (X) and received (Y) signal the relation between the signals for both time-domain and frequency-domain is given by: Where H is the matrix collecting each subchannel impulse (or frequency) response called MIMO channel matrix, and E is the matrix collecting the receiver noise components. The difference between time and frequency domain is the following: For time-domain HX perform the convolution between each signal of X and each subchannel impulse response of H; For frequency-domain HX represents the scalar product between each signal collected by X with each subchannel frequency response of H. October

10 Channel Estimation using CSS (1) Data-aided channel estimation for MIMO-SC (single-carrier) Signal MIMO-SC LS estimation of the channel matrix Expression of TMSE Minimum TMSE condition CSS: Complementary Sets of sequences With a pair of complementary sequences the minimum TMSE condition is reached. October

11 Channel Estimation using CSS (2) Frame structure of modern transmission systems using CSS: Sequences of a CSS, or of its mate, fill the Preamble and Postamble fields of the frame structure. CP fields prevent intersymbol interference (ISI) Estimation procedure: Each couple of transmitting antennas is associated with a CSS ~ ~ and its mate {b,-a} Exploiting a particular arrangement of the sequences of preamble and postamble each receiver is able to estimate independently the opportune subchannels The subchannels estimated by n r -th receiver are those between the n r -th receiving antenna and the transmitting antenna associated to the same couple CSS-mate. October

12 Transmission system MIMO-SC G 1 yellow subchannels G 2 white subchannels Rows of H October

13 Efficient estimation scheme (receiver side) ¾ Periodic Budisin Correlator (all receivers have the same structure): Section where autocorrelations are combined to obtain: October X1X1T + X2X2T = I 13

14 Example of estimation with CSS Let consider a MIMO-SC transmission system with N T = 2 transmitting antennas and N R = 2 receiving antennas. Moreover, let consider a MIMO channel composed by four ideal subchannels: that is h nr,n t (n) = δ(n) for each n r,n t. At the n r -th receiving antenna we have: At the first output of the Budisin correlator ( n rh o ) we obtain: It can be demonstrated ~ ~ that r bb (k) = r bb (k) ~ ~ and r ba (k) = r -ab (k), hence: October

15 MIMO-OFDM Channel Estimation Overall MIMO-OFDM signal in the frequency domain (only for pilots of k-th OFDM simbol) Using Q symbols in transmission for each antenna (Q = N T ) we have to ensure that for each instant k the matrices that collect the Q symbols are an orthonormal basis. ~ LS (Least Square) estimation of the matrix H(i), the estimation is independent for each pilot. Matrices S(i) collect the transmission symbols X k (i) from each antenna, for each instant k, on pilot i: since S is a matrix which column are an orthonormal basis (because OFDM symbols are a basis), LS estimation is optimal. Overall MIMO-OFDM signal of all the Q symbol LS estimation October

16 Results Performance is measured as Mean Square Error (MSE) (E[ X-µ 2 ]). TMSE for MIMO-OFDM estimator TMSE for estimator using CSS October

17 Performance comparison Estimator using CSS TMSE reaches CRLB, the lower bound for the variance of estimation Insensibility of estimate to temporal delay between different transmitted signals Normalizing factor of the transmitted power (ρ/n T ) The number of transmitting antennas must be even The number of known symbols to transmitt at each antenna to perform channe estimation is always 2 Estimator for MIMO-OFDM TMSE che raggiunge il limite assoluto per la varianza fissato dal CRLB Insensibilità della stima a sfasamenti tra i frame trasmessi su antenne diverse Normalizing factor of the transmitted power (ρi /N T ), that is I more than the estimator using CSS The obstacle to discover a matrices family S(i) which is a basis of the MIMO frequency channel matrices space, limits the number of transmitting antennas (in practice it is not grater than 8) The number of known symbols to transmit at each antenna increase linearly with the number of transmitting antennas October

18 Conclusions Analyzing the features of the estimation systems is we highlight the differences that itemize each method, even if both of them reach the same performance. However, in practice, we recall we have to consider real-time systems, where a critical parameter is the computational complexity: from this viewpoint a filter bank is more efficient than invert large matrices. The aim of the paper was to evaluate the features of the estimator which use CSS by comparing it with another estimator: our conclusion is that the estimator using CSS is more efficient. Another merit of the method chosen is that it can be integrated in a OFDM transmission systems, however it could show some problems in the synchronization mechanism. October

19 Matrices H, X e Y of SC signal single-carrier received signal Transmitting signal matrix CIR matrix Received signal matrix October

20 Complementary Sequences & Periodic Shift Sequenze complementari (CSS {A(n),B(n)} con N = 10) A = [ ] B = [ ] r AA (0) + r BB (0) = = 20 = 2N r AA (1) + r BB (1) = A(0) A(1) + A(1) A(2) + + A(9) A(10) + B(0) B(1) + B(1) B(2) + + B(9) B(10) = (-3) + 3 = 0 ecc. Periodic Shift (traslazione/shift periodico di una sequenza) Sia A(n) = [ ] una sequenza qualsiasi Per shift periodico q si intende una traslazione a destra di q posizioni degli elementi della sequenza A. Ovvero l elemento di posizione n viene posto nella posizione n + q. I q elementi per cui n + q > N-1 vengono posti all inizio, cosicché per esempio: Â(n) = [ ] è la sequenza ottenuta da A(n) con uno shift periodico 2 October

21 Autocorrelazione & Crosscorr. periodiche Sequenze di CSS aperiodiche, sono allo stesso tempo anche periodiche Le stesse sequenze soddisfano la proprietà di ottimalita sia con la funzione di correlazione aperiodica che con quella periodica October

22 Costruzione e struttura della matrice S Simbolo seed Prodotto di kronecker tra il seed e delle matrici ortogonali particolari October

23 Definizione del CRLB Il Cramer-Rao Lower Bound (CRLB) è un limite inferiore assoluto per la varianza di uno stimatore (non polarizzato) di un parametro deterministico sconosciuto Se p H (x) è la densità di probabilità della matrice di canale H, allora il CRLB è il reciproco della funzione di informazione di Fisher I(H): 1 2 CRLB = Ι( H ) = -E[ log ( )] Ι( H) 2 ph x θ La varianza dello stimatore soddisfa la relazione: var( Hˆ ) Ι 1 ( H) Con gli stimatori usati il CRLB è il seguente: October