EEE358S Fundamentals of Communications Engineering

1. EEE358SFundamentals of Communications Engineering Emmanuel O Bejide ebejide@ebe.uct.ac.za http://www.uct.ac.za/depts/staff/rebejide/ Department of Electrical Engineering University of Cape Town

2. Channel Communication channels may be characterized by Hc(?) = |Hc(?)| exp{j?c(?)} Amplitude distortion results if |Hc(?)| is not constant within the bandwidth Phase distortion results if ?c(?) is not linear function of ?, i.e., the delay is not constant. The result is signal dispersion (smearing). The overlap of symbols owing to smearing is called ISI.

3. Equalization H(?) = Ht(?) Hc(?) Hr(?) He(?) To avoid ISI, we need: H(?) = HRC(?)

4. Equalization H(?) = Ht(?) Hc(?) Hr(?) He(?) Channel characteristics may change so that it may not be possible for the receiving filter to compensate for it. Therefore, pulse shaping with the receiving filter is usually such that Ht(?) Hr(?) = HRC(?) Then equalization is needed such that: He(?) = 1/Hc(?)

5. Equalizer types Transversal equalizers versus decision feedback equalizers Transversal equalizers contain feedforward elements only and is linear Decision feedback equalizers contain both feedforward and feedback elements Preset equalizers versus adaptive equalizers Symbol spaced versus fractionally spaced

6. Transversal equalizer A transversal equalizer consists of a delay line with T-second taps. The current and past values of the received signal are linearly weighted with equalizer coefficients or tap weights, an , and are then summed to produce the output. The tap coefficients an are set to subtract the effects of interference from symbols that are adjacent in time to the current symbol.

7. Transversal equalizer Consider the case of 2N + 1 taps with coefficients (tap weights) a-N, � , a0, � , aN . Let yk = y(kT) be the filter input at t=kT and zk = z(kT+NT) be the filter output at t=(k+N)T (Note the delay of NT makes the filter casual.) The equalizer output at the sample points is the convolution of the inputs yk and the 2N+1 tap weights an: zk = Sn=-NN yk-n an , where k = -2N, �, 2N, n = -N, �, N

8. Transversal equalizer

9. Transversal equalizer The system of equations can be represented in matrix form as

10. Transversal equalizer Confining the inputs to y-N, �, yN, the system of equations becomes

11. Transversal equalizer The matrix equation can also be represented as Z=YA, where in A (a-N to aN), there are 2N+1 tap weight values of the filter to be determined. Z (z-2N to z2N), there are 4N+1 values of zk which can be measured if the transmitted signals are known. Y is 4N+1 by 2N+1 matrix The equations are an overdetermined set, i.e., there are more equations than unknowns.

12. Transversal equalizer The criterion for selecting the filter coefficients is typically based on minimizing the peak distortion, leading to the zero forcing solution or minimizing the mean-square distortion leading to the Minimum mean square error solution.

13. Transversal equalizerZero forcing solution The zero-forcing solution minimizes the peak ISI distortion by selecting the tap weights so that the equalizer output is forced to zero at N sample points before and after the desired pulse. That means the equalizer outputs satisfy zk = 1 for k = 0 zk = 0 for k = �1, � , �N . Note that ISI can be present beyond �N

14. Transversal equalizerZero forcing solution The conditions on the zero-forcing equalizer can be written as S n=-N N an yn-k = 1 for k = 0 S n=-N N an yn-k = 0 for k = �1, � , �N . This represents 2N + 1 equations in the 2N + 1 unknowns a-N, � , a0, � , aN .

15. Transversal equalizerZero forcing solution The 2N+1 equations in matrix form are The solution is A=Y-1Z

16. Transversal equalizerMinimum Mean Square Error Solution In the matrix equation Z=YA, Y is 4N+1 by 2N+1 non-square matrix. Multiply with the transpose of Y: YTZ=YTYA YTY has now become a 2N+1 by 2N+1 square autocorrelation matrix of the input noisy signal YTY is cross-correlation vector with 2N+1 rows The correlations are estimated by transmitting a test signal over the channel and using time average estimates to solve for the tap weights.

17. Transversal equalizerMinimum Mean Square Error Solution The solution is then A = (YTY)-1 YTZ The minimum MSE solution does not drop the 2N equations to reduce from 4N+1 equations to 2N+1 equations as in the case of zero forcing solution. It is therefore more robust. Most telephone-line modems use MSE weight criteria.

18. Preset Equalization Preset equalization may set the tap weights according to some average knowledge of the channel. The early low speed (say 2.4kbps) data transmission over voice-grade telephone lines had used this method. In another method of preset equalization, the transmitter emits a training sequence that is compared by the receiver to a locally generated sequence. The differences between the two are used to set the filter coefficients. The preset method is usually done at the start of transmission, and then repeated usually after any break in transmission. Also, if the channel is time-varying then performance suffers.

19. Preset Equalization The taps can be adjusted to minimize the mean-square error (MSE) between the received signal zk and the known training signal ck . For K samples the MSE is At the optimal tap weight setting the error is minimized, so for all the tap weights an .

20. Preset Equalization Differentiating gives the necessary condition Note that this is equivalent to the condition that where Rex(n) is the cross correlation between the error sequence and the input sequence. We therefore require that these sequences be uncorrelated. Weight adjustment procedures can be developed to enforce this condition.

21. Adaptive Equalization Adaptive equalization adjusts tap-weights periodically or continually to track slow time variations in the channel response. Periodic adjustments use preambler or short training sequence of digital data known in advance by the receiver. Continual adjustments use the transmitted data symbols estimated from the equalizer output and treat them as known data.

22. Adaptive Equalization Automatic equalizers use iterative techniques to estimate the optimum coefficients. Decision-directed adaptive equalization refers to adaptive equalization performed continually and automatically. The initial probability of error due to ISI should not exceed one percent for decision-directed adaptive equalization to converge. A common solution is to use preset equalization initially, and then switch to adaptive equalization once normal transmission begins.