Decision Weighted Adaptive Algorithms with Applications to Wireless Channel Estimation

Transcription

1 Decision Weighted Adaptive Algorithms with Applications to Wireless Channel Estimation Shane Martin Haas April 12, 1999 Thesis Defense for the Degree of Master of Science in Electrical Engineering Department of Electrical Engineering and Computer Science University of Kansas

2 2 ) What is an Adaptive Algorithm? ) What is System Identification? Introduction x(n) Reference Signal Channel Desired Signal d(n) Adjustable Filter ^ d(n) w + - Error Signal e(n) Adaptive Algorithm ^ (Accepts x(n), d(n), d(n), and e(n) as input) ) Benefits of Channel Estimation ) Training Sequence Versus Blind Estimation

3 3 Theoretical Development ) * Problem Formulation * Multiple Phase Shift Keying Presentation Overview * Characterizing Wireless Communication Channels * Bandpass to Low-Pass Conversion of Signals and Systems * Adaptive Algorithms Linear and LMS Estimation Algorithms Properties of Decision Weighted Algorithms ) Simulation Methodology ) Simulation Results ) Conclusions

4 4 Problem Formulation Training Sequence Estimator Channel Estimate Tx Symbols Rx Symbols (Hard Decisions) Symbol Source Modulator Channel Demod Decision Symbol Sink Tx Signal Rx Signal Modulator Soft Decisions Reconstructed Tx Signal Decision Directed Estimator Channel Estimate

5 s i (t) = 1(t) = a i1 = a i2 = i = 2i M T cos (2f ct) sin (2f ct) T p cos (i ) E p E sin (i ) 5 ) Modulation Multiple Phase Shift Keying (MPSK) 2E cos 2f c t + 2i T for 0 t < T r M with = a i1 1 (t) + a i2 2 (t) r 2 2(t) =, r 2

6 6 ) Demodulation ϕ (t) 2 s i(t) a i1 φ i a i2 ϕ (t) 1 ϕ 1(t) ~ X Τ 0 X r (t) i arctan(y/x) θi Compute φ θ i i Choose Smallest Detected Symbol X T 0 Y ~ ϕ (t) 2

7 7 Characterizing Wireless Communication Channels ) Multipath X y(t) = n (t)s(t, n (t)) n path n... path 3 path 2 path 1 path 0 (line-of-sight) Transmitter Receiver Channel Models ) * Radio Relay Three-Path (Rummler) Model * Mobile Radio Channel Model

8 es i (t) = 2E cos 2i M T r 2E j + sin 2i M T 8 Bandpass to Low-Pass Conversion ) The Complex Envelope of the MPSK Signalling Waveform r Main Idea: Convolution of real bandpass signals is the same as the ) convolution of their complex envelope low-pass equivalents

9 9 ) System Identification Problems Adaptive Algorithms w(n) Unknown System u(n) b + y(n) β ^ y(n) - + e(n) Model w(n) Unknown System Decision Model u(n) b + y(n) x(n) β + - e(n) ^ y(n)

10 10 Linear Estimators ) Define for the training sequence estimation problem: y(n) = b 1 u 1 (n) + +b M u M (n)+w(n) by(n) = 1 u 1 (n)++ M u M (n) e(n) = y(n),by(n) =y(n),( 1 u 1 (n)++ M u M (n)) Define for the decision directed estimation problem: y(n) = b 1 u 1 (n) + +b M u M (n)+w(n) by(n) = 1 x 1 (n)++ M x M (n) e(n) = y(n),by(n) =y(n),( 1 x 1 (n)++ M x M (n))

11 w = e = b = = U = X = u 1 (N ) u M (N) x 1 (N ) x M (N) i T i T i T 11 Observe the system for N sample periods and write y = y(1) y(2) y(n) h w(1) w(2) w(n) h e(1) e(2) e(n) h h b 1 b 2 b M i T h 1 2 M i T 1 (1) u M (1) u (1) x M (1) x

12 12 The channel output is y = Ub + w The error for the training sequence estimation problem is e = y, U While that for the decision directed estimation is e = y, X The error or loss function is J() = e T Re where R is a N x N matrix of weighting coefficients.

13 b =, U T RU,1 U T Ry n = b n,1 +a n H,1 n u(n)e(n) b e(n) = y(n),u(n) T b n,1 i T 13 ) Linear Estimator ) Recursive Weighted Least Squares Estimator u(n) = u 1 (n) u 2 (n) u M (n) h R =diag( n,1 a 1 ;::: ;a n,1;a N ) 0< 1 H n = H n,1 +a n u(n)u T (n)

14 b n = b n,1 + n u(n)e(n) e(n) = y(n), u(n) T b n,1 14 ) Least Mean Squares (LMS) Estimator Decision weighted estimators are decision directed estimators whose ) weights depend on the quality of the decisions. * Ideal decision weighted estimators use knowledge of decision errors to calculate their weights. Specifically, X T RX = X T RU * Soft decision weighted estimators use receiver soft decisions to calculate their weights.

15 a n = p n p n,1 p n,m+1 15 More on Ideal Decision Weighted Linear ) Estimators * Q: How does R one choose X such RX = X that RU? T T * A: IfX and U differ in the jth row, choose the jth column of R orthogonal to each column in X. * Q: AreX T RX = X T RU and X T RX non-singular conflicting conditions? * A: No, R let be an identity matrix with its jth column set to zero X if U and differ in the jth row. Under slightly more restrictive assumptions placed X on than in ordinary training sequence X estimators, RX is non-singular. T More on Soft Decision Weighted Estimators ) For MPSK modulation define a soft decision as i = 1, j i, i j p =S A possible choice for the soft decision weight is

16 n b o = cov fsubg + E SVS T 16 Biasness of Decision Directed Linear Estimators ) E fwjx; Ug = 0 If then = E n b o n, X T RX,1 X T RU o E b For ideal decision weighted estimators X T RX = X T RU, and therefore the estimator is unbiased. Covariance of Decision Directed Linear Estimators ) E fwjx; Ug = 0 If then cov S = where decision weighted SU = estimators I., X T RX,1 X T R and V = E ww T j X; U. Notice for ideal

17 17 ) Algorithm Summary Simulation Methodology * Training Sequence LMS (TLMS): Uses training sequence LMS with n = 1 * Blind LMS (BLMS): Uses decision directed LMS with n = 1 * Soft Decision Weighted LMS (SDWLMS): Uses decision directed LMS with soft decision weights * Ideal Decision Weighted LMS (IDWLMS): Uses decision directed LMS with n = 1 if x(n) = u(n) and zero otherwise * Training Sequence RLS (TRLS): Uses training sequence WRLS with a n = 1 * Blind RLS (BRLS): Uses decision directed WRLS with a n = 1 * Soft Decision Weighted RLS (SDWRLS): Uses decision directed WRLS with soft decision weights

18 18 * Ideal Decision Weighted RLS (IDWRLS): Uses decision directed WRLS with a n = 1 if x(n) = u(n) and zero otherwise * Modified Soft Decision Weighted RLS (MSDWRLS): Uses decision directed WRLS with soft decision weights; however, we modify the matrix update H n by removing the weight a n, resulting in H n = H n,1 + x(n)x T (n). * Modified Ideal Decision Weighted RLS (MSDWRLS): Uses decision directed WRLS with a n = 1 if x(n) = u(n) and zero otherwise; however, we modify the matrix update H n by removing the weight a n, resulting in H n = H n,1 + x(n)x T (n).

19 19 General Methods for Delay Spread, SNR, and Doppler Frequency Tests ) * LMS = 0:3 Gain: * RLS Forgetting Factor: = 0:99 * Sampling Rate: 1 sample per second * Symbol Interval: 4 samples per symbol * Modulation: QPSK * Number of Symbols per Individual Simulation: 300 symbols * Number of Individual Simulations to Perform per Test Point Iteration: 20 simulations * Maximum Symbol Error Rate (SER): 0.2 * Number of Symbols to Skip Before Calculating Estimation Error (N 0 ): 100 symbols * Initial Estimate: the true response * Performance Criteria: median average estimation error

20 20 Simulation Results 10 0 Estimation Error as a Function of Delay Spread Median Average Squared Error from True Impulse Response tlms blms sdwlms idwlms Delay Spread [s] Figure 1: Median of the average squared error of LMS algorithms as a function of delay spread (SNR = 10 db)

21 21 Median Average Squared Error from True Impulse Response 10 0 Estimation Error as a Function of Delay Spread trls brls sdwrls idwrls msdwrls midwrls Delay Spread [s] Figure 2: Median of the average squared error of RLS algorithms as a function of delay spread (SNR = 10 db)

22 22 Median Average Squared Error from True Impulse Response 10 1 Estimation Error as a Function of SNR tlms blms sdwlms idwlms SNR [db] Figure 3: Median of the average squared error of LMS algorithms as a function of SNR (Delay Spread = 1 symbol interval)

23 23 Median Average Squared Error from True Impulse Response 10 1 Estimation Error as a Function of SNR trls brls sdwrls idwrls msdwrls midwrls SNR [db] Figure 4: Median of the average squared error of RLS algorithms as a function of SNR (Delay Spread = 1 symbol interval)

24 Estimation Error as a Function of Doppler Frequency (SNR = 10 db) Median Average Squared Error from True Impulse Response tlms blms sdwlms idwlms Doppler Frequency [Hz] Figure 5: Median of the average squared error of LMS algorithms as a function of Doppler frequency (SNR = 10 db)

25 25 Median Average Squared Error from True Impulse Response Estimation Error as a Function of Doppler Frequency (SNR = 10 db) trls brls sdwrls idwrls msdwrls midwrls Doppler Frequency [Hz] Figure 6: Median of the average squared error of RLS algorithms as a function of Doppler frequency (SNR = 10 db)

26 26 Conclusions Summary of Performance Test Results ) * Soft decision weighted LMS (SDWLMS) performed better than the other LMS algorithms in delay spread (by a factor of 2 to 100, Figure 1) and SNR (by a factor of 2, Figure 3) tests * Soft decision weighted RLS (SDWRLS) performed worse than the other RLS algorithms in delay spread (by a factor of 2, Figure 2) and SNR (by a factor of 3, Figure 4) tests * Modified soft decision weighted RLS (MSDWRLS) performed better than the other RLS algorithms in delay spread (by a factor of 2 to 20, Figure 2) and SNR (by a factor of 2, Figure 2) tests * SDWLMS performed better at normalized Doppler frequencies less than 10,5 and worse at higher Doppler frequencies than the other LMS algorithms (Figure 5) * SDWRLS and MSDWRLS performed worse over all Doppler frequencies than the other RLS algorithms (Figure 6)

27 27 * Ideal decision weighted LMS and RLS (IDWLMS and IDWRLS) performed similar to their training sequence versions in all tests, and generally better than their ordinary decision-directed counterparts (Figures 1 through 6). * Decision weighted estimators defined, analyzed, and simulated ) General Conclusions * SDWLMS shows most promise for implementation * SDWRLS performed poorly, but MSDWRLS performed well * Ideal decision weighted algorithms performed similar to training sequence algorithms