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# Matched filter

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### Matched filter

1. 1. Receiver Structure Matched filter: match source impulse and maximize SNR – grx to maximize the SNR at the sampling time/output Equalizer: remove ISI Timing – When to sample. Eye diagram Decision – d(i) is 0 or 1 Figure 7.20 Noise na(t) i ⋅Td(i) gTx(t) gRx(t) r (iT ) = r0 (iT ) + n(iT ) S → max ? N EE 541/451 Fall 2006
2. 2. Matched Filter Input signal s(t)+n(t) Maximize the sampled SNR=s(T0)/n(T0) at time T0 EE 541/451 Fall 2006
3. 3. Matched filter example  Received SNR is maximized at time T0 S Matched Filter: optimal receive filter for maximized Nexample: gTx (t ) gTx (−t ) gTx (T0 − t ) = g Rx (t ) T0 t T0 t T0 t transmit filter receive filter (matched) EE 541/451 Fall 2006
4. 4. Equalizer When the channel is not ideal, or when signaling is not Nyquist, There is ISI at the receiver side. In time domain, equalizer removes ISR. In frequency domain, equalizer flat the overall responses. In practice, we equalize the channel response using an equalizer EE 541/451 Fall 2006
5. 5. Zero-Forcing Equalizer The overall response at the detector input must satisfy Nyquist’s criterion for no ISI: The noise variance at the output of the equalizer is: – If the channel has spectral nulls, there may be significant noise enhancement. EE 541/451 Fall 2006
6. 6. Transversal Transversal Zero-Forcing Equalizer If Ts<T, we have a fractionally-spaced equalizer For no ISI, let: EE 541/451 Fall 2006
7. 7. Zero-Forcing Equalizer continue Zero-forcing equalizer, figure 7.21 and example 7.3 Example: Consider a baud-rate sampled equalizer for a system for which Design a zero-forcing equalizer having 5 taps. EE 541/451 Fall 2006
8. 8. MMSE Equalizer In the ISI channel model, we need to estimate data input sequence xk from the output sequence yk Minimize the mean square error. EE 541/451 Fall 2006
9. 9. Adaptive Equalizer Adapt to channel changes; training sequence EE 541/451 Fall 2006
10. 10. Decision Feedback Equalizer To use data decisions made on the basis of precursors to take care of postcursors Consists of feedforward, feedback, and decision sections (nonlinear) DFE outperforms the linear equalizer when the channel has severe amplitude distortion or shape out off. EE 541/451 Fall 2006
11. 11. Different types of equalizers Zero-forcing equalizers ignore the additive noise and may significantly amplify noise for channels with spectral nulls Minimum-mean-square error (MMSE) equalizers minimize the mean- square error between the output of the equalizer and the transmitted symbol. They require knowledge of some auto and cross-correlation functions, which in practice can be estimated by transmitting a known signal over the channel Adaptive equalizers are needed for channels that are time-varying Blind equalizers are needed when no preamble/training sequence is allowed, nonlinear Decision-feedback equalizers (DFE’s) use tentative symbol decisions to eliminate ISI, nonlinear Ultimately, the optimum equalizer is a maximum-likelihood sequence estimator, nonlinear EE 541/451 Fall 2006
12. 12. Timing Extraction Received digital signal needs to be sampled at precise instants. Otherwise, the SNR reduced. The reason, eye diagram Three general methods – Derivation from a primary or a secondary standard. GPS, atomic closk x Tower of base station x Backbone of Internet – Transmitting a separate synchronizing signal, (pilot clock, beacon) x Satellite – Self-synchronization, where the timing information is extracted from the received signal itself x Wireless x Cable, Fiber EE 541/451 Fall 2006
13. 13. Example Self Clocking, RZ Contain some clocking information. PLL EE 541/451 Fall 2006
14. 14. Timing/Synchronization Block Diagram Figure 2.3 After equalizer, rectifier and clipper Timing extractor to get the edge and then amplifier Train the phase shifter which is usually PLL Limiter gets the square wave of the signal Pulse generator gets the impulse responses EE 541/451 Fall 2006
15. 15. Timing Jitter Random forms of jitter: noise, interferences, and mistuning of the clock circuits. Pattern-dependent jitter results from clock mistuning and, amplitude-to-phase conversion in the clock circuit, and ISI, which alters the position of the peaks of the input signal according to the pattern. Pattern-dependent jitter propagates Jitter reduction – Anti-jitter circuits – Jitter buffers – Dejitterizer EE 541/451 Fall 2006
16. 16. Bit Error Probability Noise na(t) i ⋅T d(i) gTx(t) gRx(t) r0 (i T ) + n(iT )We assume: • binary transmission with d (i ) ∈ {d 0 , d1} • transmission system fulfills 1st Nyquist criterion • noise n(iT), independent of data source p N (n ) Probability density function (pdf) of n(iT) Mean and variance n EE 541/451 Fall 2006
17. 17. Conditional pdfsThe transmission system induces two conditional pdfs depending on d (i ) • if d (i ) = d 0 • if d (i ) = d1 p0 ( x ) = p N ( x − d 0 ) p1 ( x) = p N ( x − d1 ) p0 ( x ) p1 ( x) x d0 d1 x EE 541/451 Fall 2006
18. 18. Probability of wrong decisions Placing a threshold S p0 ( x ) p1 ( x) Probability of wrong decision x x d0 S S d1 ∞ S Q0 = ∫ p0 ( x) dx Q1 = ∫ p ( x)dx 1 S −∞When we define P0 and P1 as equal a-priori probabilities of d 0 and d1 (P0 = P = 1 )we will get the bit error probability 1 2 ∞ S SPb = P0Q0 + P Q1 = 1 1 2 ∫s p ( x)dx + ∫ p ( x)dx = S 0 1 2 −∞ 1 1 2 + ∫[ −∞ 1 2 p1 ( x) − 1 p0 ( x ) ] dx 2 1 24 4 3 S 1− ∫ p0 ( x ) dx −∞ EE 541/451 Fall 2006
19. 19. Conditions for illustrative solution 1 d 0 + d1 With  P1 = P0 = and  pN (− x) = pN ( x) ⇒ S= 2 2 S 1  S Pb = 1 + ∫ p1 ( x) dx − ∫ p0 ( x ) dx  2  −∞ −∞  d 0 − d1 d +d S ′= S S= 0 1 S 2 2 ∫ p ( x) dx = ∫ p 1 N ( x − d1 )dx ∫ p ( x) dx 1 = ∫p N ( x ′ )d x ′ equivalently−∞ −∞ −∞ −∞ S with substituting x −d1 = x ′ d −d d −d ∫ p0 ( x ) dx = d +d 0 1 1 0 2 −∞ for x =S = 0 1 1 2 1 2 = + ∫ p N ( x ′ )d x ′ = − ∫ p N ( x ′ )d x ′ d1 − d 0 d 0 + d1 d 0− d 1 2 0 2 0 1 2⇒S ′ = − d1= + ∫ p N ( x ) dx 2 2 d −d 1 0 2 0 1 2  Pb = 1 − 2 ∫ p N ( x )dx  2 0  EE 541/451 Fall 2006
20. 20. Special Case: Gaussian distributed noise Motivation: • many independent interferers • central limit theorem • Gaussian distribution d1− d 0 n 2  x  2 − 2 − 1 1 2  e 2σ ∫ 2 2σ pN ( n ) = 2 N  Pb = 1 − e dx  N 2π σ N 2 2π σ N 0 0   1 24  4 3 no closed solution Definition of Error Function and Error Function Complement x 2 − x′ 2 erf( x) = ∫ e d x′ π 0 erfc( x) = 1 − erf( x ) EE 541/451 Fall 2006
21. 21. Error function and its complement function y = Q(x) y = 0.5*erfc(x/sqrt(2)); 2.5 erf(x) erfc(x) 2 1.5 erf(x), erfc(x) 1 0.5 0 -0.5 -1 -1.5 -3 -2 -1 0 1 2 3 x EE 541/451 Fall 2006
22. 22. Bit error rate with error function complement  d1 − d 0 x2  1 2 1 2 −  1  d − d0  ∫ 2 2σ N Pb = 1 − e d x  Pb = erfc 1  2  π 2σ N 0  2  2 2σ N    Expressions with E S and N 0antipodal: d1 = + d ; d 0 = − d unipolar d1= + d ; d 0 = 0 1  d −d  1  d  1  d  1  d2 Pb = erfc  1 0  = erfc   Pb = erfc   2 2σ  = erfc   2  2  2 2σ  2  2σ  2  8σ N 2   N   N   N    1  d2  1  SNR  1  d2 / 2  1  SNR  = erfc   = erfc    = erfc   = erfc   2  2σ N  2  2   2  2  4σ N  2 2   4    d2 ES d2 / 2 ESSNR = 2 = SNR = 2 = σN matched N / 2 0 σ N matched N 0 / 2 1  ES  1  ES  Pb = erfc  N  Q function Pb = erfc   2  0  2  2 N0    EE 541/451 Fall 2006
23. 23. Bit error rate for unipolar and antipodal transmission BER vs. SNR theoretical -1 10 simulation unipolar -2 10 BER antipodal -3 10 -4 10 -2 0 2 4 6 8 10 ES in dB N0 EE 541/451 Fall 2006