RADIO SYSTEMS – ETI Contents Lecture no: 8 051 • Inter-symbol interference • Linear equalizers • Decision-feedback equalizers • Maximum-likelihood sequence estimation Equalization Ove Edfors, Department of Electrical and Information Technology Ove.Edfors@eit.lth.se 2010-04-29 Ove Edfors - ETI 051 2 Inter-symbol interference Background Even if we have designed the basis pulses of our modulation to be interference free in time, i.e. no leakage of energy between consecutive symbols, multi-path propagation in our channel will cause a delay-spread and inter-symbol interference (ISI) . Transmitted symbols Received symbols INTER-SYMBOL INTERFERENCE Channel with delay spread ISI will degrade performance of our receiver, unless mitigated by some mechanism. This mechanism is called an equalizer . 2010-04-29 Ove Edfors - ETI 051 3 2010-04-29 Ove Edfors - ETI 051 4
Inter-symbol interference Inter-symbol interference Including a channel impulse response Including a channel impulse response We can create a discrete time equivalent of the “new” system: What we have used so far (PAM and optimal receiver): n ( ) k n t kT c ϕ ( ) ϕ δ − ( ) c t kT k ( ) k z − 1 F z F * k k ( ) ( ) * − g t g T t ISI-free and where we can say that F (z) represent the basis pulse and channel, while PAM Matched filter white noise F* ( z -1 ) represent the matched filter. (This is an abuse of signal theory!) with proper pulses g( t ) Including a channel impulse response h ( t ): We can now achieve white noise quite easily, if (the not unique) F ( z ) is ( ) chosen wisely ( F *( z - 1 ) has a stable inverse) : n t kT ( ) δ − ϕ c t kT n k k ( ) ( ) k ? ∗ T − t g t h t g ∗ h c ϕ u ( ) ( ) k ( ) k k z − 1 z − F z F * 1 1/ F * Matched filter This one is no PAM NOTE: longer ISI-free and Noise F *( z -1 )/ F *( z -1 )=1 Can be seen as a “new” whitening noise is not white basis pulse filter 2010-04-29 Ove Edfors - ETI 051 5 2010-04-29 Ove Edfors - ETI 051 6 Inter-symbol interference The discrete-time channel model With the application of a noise-whitening filter, we arrive at a discrete-time model n k c u k ( ) k F z This is the model we are where we have ISI and white additive noise, in the form going to use LINEAR EQUALIZER when L u k = ∑ f j c k − j n k designing equalizers. j = 0 The coefficients f j represent the causal impulse response of the discrete-time equivalent of the channel F ( z ), with an ISI that extends over L symbols. 2010-04-29 Ove Edfors - ETI 051 7 2010-04-29 Ove Edfors - ETI 051 8
Linear equalizer Linear equalizer Principle Zero-forcing equalizer The principle of a linear equalizer is very simple: Apply a filter E ( z ) at the The zero-forcing equalizer is designed to remove the ISI completely receiver, mitigating the effect of ISI: n k c $ u c n k ( ) ( ) k k F z 1/ F z k $ c u c k ( ) ( ) k k F z E z ZF equalizer Linear FREQUENCY DOMAIN equalizer Information Information Channel Noise Equalizer and noise Now we have two different strategies: 1) Design E ( z ) so that the ISI is totally removed Zero-forcing 2) Design E ( z ) so that we minimize the mean MSE − $ ε = f f f f f squared-error of c c k k k Noise enhancement! 2010-04-29 Ove Edfors - ETI 051 9 2010-04-29 Ove Edfors - ETI 051 10 Linear equalizer Linear equalizer Zero-forcing equalizer, cont. MSE equalizer The MSE equalizer is designed to minimize the error variance A serious problem with the zero-forcing equalizer is the noise n enhancement , which can result in infinite noise power spectral densities k $ c ( ) u c σ 2 * − 1 F z after the equalizer. k ( ) k k s F z ( ) 2 σ 2 + F z N s 0 The noise is enhanced (amplified) at frequencies where the channel has a high attenuation. MSE equalizer Another, related, problem is that the resulting noise is colored, which makes FREQUENCY DOMAIN an optimal detector quite complicated. Information Information Channel Noise Equalizer and noise By applying the minimum mean squared-error criterion instead, we can at least remove some of these unwanted effects. f f f f f Less noise enhancement than Z-F! 2010-04-29 Ove Edfors - ETI 051 11 2010-04-29 Ove Edfors - ETI 051 12
Linear equalizer MSE equalizer, cont. The MSE equalizer removes the most problematic noise enhancements as compared to the ZF equalizer. The noise power spectral density cannot go to infinity any more. This improvement from a noise perspective comes at the cost of not totally removing the ISI. DECISION-FEEDBACK The noise is still colored after the MSE equalizer which, in combination EQUALIZER with the residual ISI, makes an optimal detector quite complicated. 2010-04-29 Ove Edfors - ETI 051 13 2010-04-29 Ove Edfors - ETI 051 14 Decision-feedback equalizer Decision-feedback equalizer Principle Principle, cont. We have seen that taking care of the ISI using only a linear filter will cause Decision (sometimes severe) noise coloring. device n A slightly more sophisticated approach is to subtract the interference k $ c c caused by already detected data (symbols). + k ( ) k ( ) F z E z - This principle of detecting symbols and using feedback to remove the If we make ISI they cause (before detecting the next symbol), is called decision- Forward a wrong feedback equalization (DFE). ( ) filter D z decision here, we may Feedback increase the filter ISI instead of remove This part removes ISI on This part shapes “future” symbols from it. the signal to the currently detected work well with symbol. the decision feedback. 2010-04-29 Ove Edfors - ETI 051 15 2010-04-29 Ove Edfors - ETI 051 16
Decision-feedback equalizer Decision-feedback equalizer Zero-forcing DFE Zero-forcing DFE, cont. In the design of a ZF-DFE, we want to completely remove all ISI before Like in the linear ZF equalizer, forcing the ISI to zero before the decision the detection. device of the DFE will cause noise enhancement. ISI-free n k $ c Noise enhancement can lead to high probabilities for making the wrong c + k k ( ) ( ) F z E z decisions ... which in turn can cause error propagation, since we may - add ISI instead of removing it in the decision-feedback loop. Due to the noise color, an optimal decision device is quite complex and ( ) D z causes a delay that we cannot afford, since we need them immediately in the feedback loop. This enforces a relation between the E ( z ) and D ( z ), which is (we assume that we make correct decisions!) ( ) ( ) ( ) − = F z E z D z 1 As soon as we have chosen E ( z ), we can determine D ( z ). (See textbook for details!) 2010-04-29 Ove Edfors - ETI 051 17 2010-04-29 Ove Edfors - ETI 051 18 Decision-feedback equalizer Decision-feedback equalizer MSE-DFE MSE-DFE, cont. To limit noise enhancement problems, we can concentrate on minimizing By concentrating on minimal MSE before the detector, we can reduce mean squared-error (MSE) before the decision device instead of totally the noise enhancements in the MSE-DFE, as compared to the ZF-DFE. removing the ISI. minimal MSE By concentrating on minimal MSE before the detector, we can reduce n the noise enhancements in the MSE-DFE, as compared to the ZF-DFE. k $ c c + k ( ) k ( ) E z F z - The performance of the MSE-DFE equalizer is (in most cases) higher than the previous equalizers ... but we still have the error propagation problem that can occur if we make an incorrect decision. ( ) D z The overall strategy for minimizing the MSE is the same as for the linear MSE equalizer (again assuming that we make correct decisions). (See textbook for details!) 2010-04-29 Ove Edfors - ETI 051 19 2010-04-29 Ove Edfors - ETI 051 20
Recommend
More recommend
