Digital communications systems


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Digital communications systems

  1. 1. Digital Communications Systems<br />
  2. 2. Chapter 3: Baseband Demodulation/Detection<br /><ul><li> Detection of Binary Signal in Gaussian Noise
  3. 3. Matched Filters and Correlators
  4. 4. Baye’s Decision Criterion
  5. 5. Maximum Likelihood Detector
  6. 6. Error Performance</li></li></ul><li>Baseband Demodulation/Detection<br />In case of baseband signaling, the received signal is already in pulse-like form. Why is then is demodulator required?<br />Arriving baseband pulses are not in the form of ideal pulse shapes, each one occupying its own symbol interval.<br />The channel (as well as any filtering at the transmitter) causes intersymbol interference (ISI).<br />Channel noise is another reason that may cause bit error.<br />
  7. 7. Effect of Noise<br />
  8. 8. 3.1.2 Demodulation and Detection<br />Figure 3.1: Two basic steps in the demodulation/detection of digital signals<br /><ul><li>The digital receiver performs two basic functions:
  9. 9. Demodulation, to recover a waveform to be sampled at t = nT.
  10. 10. Detection, decision-making process of selecting possible digital symbol </li></li></ul><li>3.2 Detection of Binary Signal in Gaussian Noise<br />For any binary channel, the transmitted signal over a symbol interval (0,T) is: <br />The received signal r(t) degraded by noise n(t) and possibly degraded by the impulse response of the channel hc(t), is<br /> (3.1)<br />Where n(t) is assumed to be zero mean AWGN process<br />For ideal distortionless channel where hc(t) is an impulse function and convolution with hc(t) produces no degradation, r(t) can be represented as:<br /> (3.2)<br />
  11. 11. 3.2 Detection of Binary Signal in Gaussian Noise<br />The recovery of signal at the receiver consist of two parts<br />Filter<br />Reduces the effect of noise (as well as Tx induced ISI)<br />The output of the filter is sampled at t=T.This reduces the received signal to a single variable z(T) called the test statistics<br />Detector (or decision circuit)<br /> Compares the z(T) to some threshold level 0 , i.e.,<br /> where H1and H2are the two possible binary hypothesis<br />
  12. 12. Receiver Functionality <br />The recovery of signal at the receiver consist of two parts:<br />Waveform-to-sample transformation (Blue dotted block)<br />Demodulator followed by a sampler<br />At the end of each symbol duration T, predetection point yields a sample z(T), called test statistic<br />(3.3)<br /> Where ai(T) is the desired signal component, <br /> and no(T) is the noise component<br />Detection of symbol<br />Assume that input noise is a Gaussian random process and receiving filter is linear<br /> (3.4)<br />
  13. 13. Then output is another Gaussian random process<br />Where 02 is the noise variance<br />The ratio of instantaneous signal power to average noise power , (S/N)T, at a time t=T, out of the sampler is: <br /> (3.45)<br />Need to achieve maximum (S/N)T<br />
  14. 14. 3.2.2 The Matched Filter <br />Objective: To maximizes (S/N)T<br />Expressing signal ai(t) at filter output in terms of filter transfer function H(f) (Inverse Fourier transform of the product H(f)S(f)).<br /> (3.46)<br /> where S(f) is the Fourier transform of input signal s(t)<br />Output noise power can be expressed as:<br /> (3.47)<br />Expressing (S/N)T as:<br /> (3.48)<br />
  15. 15. Now according to Schwarz’s Inequality:<br /> (3.49)<br />Equality holds if f1(x) = k f*2(x) where k is arbitrary constant and * indicates complex conjugate<br />Associate H(f) with f1(x) and S(f) ej2 fT with f2(x) to get:<br /> (3.50)<br />Substitute in eq-3.48 to yield:<br /> (3.51)<br />
  16. 16. Or and energy E of the input signal s(t):<br />Thus (S/N)T depends on input signal energy<br /> and power spectral density of noise and<br /> NOT on the particular shape of the waveform<br />Equality for holds for optimum filter transfer function H0(f) <br /> such that: (3.54) <br /> (3.55)<br />For real valued s(t):<br /> (3.56)<br />
  17. 17. The impulse response of a filter producing maximum output signal-to-noise ratio is the mirror image of message signal s(t), delayed by symbol time duration T.<br />The filter designed is called a MATCHED FILTER <br />Defined as:<br /> a linear filter designed to provide the maximum <br /> signal-to-noise power ratio at its output for a given <br /> transmitted symbol waveform <br />
  18. 18. 3.2.3 Correlation realization of Matched filter<br />A filter that is matched to the waveform s(t), has an impulse response<br />h(t) is a delayed version of the mirror image of the original signal waveform<br />Signal Waveform<br />Mirror image of signal waveform<br />Impulse response of matched filter<br />Figure 3.7<br />
  19. 19. This is a causal system<br />Recall that a system is causal if before an excitation is applied at time t = T, the response is zero for - < t < T<br />The signal waveform at the output of the matched filter is<br /> (3.57)<br />Substituting h(t) to yield:<br /> (3.58)<br />When t=T,<br /> (3.59)<br />
  20. 20. The function of the correlator and matched filter are the same<br />Compare (a) and (b)<br />From (a) <br />
  21. 21. From (b)<br /> But<br />At the sampling instant t = T, we have<br />This is the same result obtained in (a)<br />Hence<br />
  22. 22. Detection <br />Matched filter reduces the received signal to a single variable z(T), after which the detection of symbol is carried out<br />The concept of maximum likelihood detectoris based on Statistical Decision Theory<br />It allows us to <br />formulate the decision rule that operates on the data<br />optimize the detection criterion<br />
  23. 23. Probabilities Review<br />P[s1], P[s2]  a priori probabilities<br /> These probabilities are known before transmission<br />P[z]<br /> probability of the received sample<br />p(z|s1), p(z|s2)<br /> conditional pdf of received signal z, conditioned on the class si<br />P[s1|z], P[s2|z]  a posteriori probabilities<br /> After examining the sample, we make a refinement of our previous knowledge<br />P[s1|s2], P[s2|s1]<br /> wrong decision (error)<br />P[s1|s1], P[s2|s2]<br /> correct decision<br />
  24. 24. How to Choose the threshold?<br />Maximum Likelihood Ratio test and Maximum a posteriori (MAP) criterion:<br />If <br />else<br />Problem is that a posteriori probabilities are not known.<br />Solution: Use Bay’s theorem:<br />
  25. 25. <ul><li>MAP criterion:
  26. 26. When the two signals, s1(t) and s2(t), are equally likely, i.e., P(s2) = P(s1) = 0.5, then the decision rule becomes
  27. 27. This is known as maximum likelihood ratio test because we are selecting the hypothesis that corresponds to the signal with the maximum likelihood.
  28. 28. In terms of the Bayes criterion, it implies that the cost of both types of error is the same</li></li></ul><li>Substituting the pdfs<br />
  29. 29. <ul><li>Hence:</li></ul>Taking the log of both sides will give<br />
  30. 30. Hence<br />where z is the minimum error criterion and 0 is optimum threshold<br />For antipodal signal, s1(t) = - s2 (t) a1 = - a2<br />
  31. 31. <ul><li>This means that if received signal was positive, s1 (t) was sent, else s2(t) was sent</li></li></ul><li>Probability of Error<br />Error will occur if<br />s1 is sent  s2 is received<br />s2 is sent  s1 is received<br /><ul><li>The total probability of error is the sum of the errors</li></li></ul><li>If signals are equally probable<br />Numerically, PBis the area under the tail of either of the conditional distributions p(z|s1) or p(z|s2) and is given by:<br />
  32. 32. The above equation cannot be evaluated in closed form (Q-function)<br /> Hence,<br />
  33. 33. Table for computing of Q-Functions<br />
  34. 34. A vector View of Signals and Noise<br /><ul><li> N-dimensional orthonormal space characterized by N linearly independent basis function {ψj(t)}, where:
  35. 35. From a geometric point of view, each ψj(t) is mutually perpendicular to each of the other{ψj(t)} for j not equal to k. </li></li></ul><li><ul><li>Representation of any set of M energy signals { si(t) } as a linear combinations of N orthogonal basis functions where NM.</li></ul>where:<br />
  36. 36. <ul><li>Therefore we can represent set of M energy signals {si(t) } as:
  37. 37. Waveform energy: </li></ul>Representing (M=3) signals, with (N=2) orthonormal basis functions<br />
  38. 38. Question 1: Why use orthormal functions?<br /><ul><li> In many situations N is much smaller than M. Requiring few matched filters at the receiver.
  39. 39. Easy to calculate Euclidean distances
  40. 40. Compact representation for both baseband and passband systems.
  41. 41. Gram-Schmidt orthogonalization procedure.</li></ul>Question 2: How to calculate orthormal functions?<br />
  42. 42. <ul><li>Examples</li></li></ul><li><ul><li>Examples (continued)</li></li></ul><li>Generalized One Dimensional Signals<br />One Dimensional Signal Constellation<br />
  43. 43. Binary Baseband Orthogonal Signals<br />Binary Antipodal Signals<br />Binary orthogonal Signals<br />
  44. 44. Constellation Diagram<br />Is a method of representing the symbol states of modulated bandpass signals in terms of their amplitude and phase<br />In other words, it is a geometric representation of signals<br />There are three types of binary signals:<br />Antipodal<br /> Two signals are said to be antipodal if one signal is the negative of the other<br />The signal have equal energy with signal point on the real line<br />ON-OFF<br /> Are one dimensional signals either ON or OFF with signaling points falling<br />on the real line<br />
  45. 45. With OOK, there are just 2 symbol states to map onto the constellation space<br />a(t) = 0 (no carrier amplitude, giving a point at the origin)<br />a(t) = A cos wct (giving a point on the positive horizontal axis at a distance A from the origin)<br />Orthogonal<br /> Requires a two dimensional geometric representation since there are two linearly independent functions s1(t) and s0(t)<br />
  46. 46. Typically, the horizontal axis is taken as a reference for symbols that are Inphase with the carrier cos wct, and the vertical axis represents the Quadrature carrier component, sin wct<br />Error Probability of Binary Signals<br />Recall:<br />Where we have replaced a2 by a0.<br />
  47. 47. To minimize PB, we need to maximize:<br /> or<br />We have<br />Therefore,<br />
  48. 48. <ul><li>The probability of bit error is given by:</li></li></ul><li><ul><li>The probability of bit error for antipodal signals:
  49. 49. The probability of bit error for orthogonal signals:
  50. 50. The probability of bit error for unipolar signals:</li></li></ul><li>Bipolar signals require a factor of 2 increase in energy compared to orthogonal signals<br />Since 10log102 = 3 dB, we say that bipolar signaling offers a 3 dB better performance than orthogonal<br />
  51. 51. Comparing BER Performance<br />For the same received signal to noise ratio, antipodal provides lower bit error rate than orthogonal<br />
  52. 52. Baseband Communication System<br />We have been considering the following baseband system<br /><ul><li>The transmitted signal is created by the line coderaccording to</li></ul>where anis the symbol mappingand g(t) is the pulse shape<br />Problems with Line Codes<br /><ul><li>One big problem with the line codes is that they are not bandlimited
  53. 53. The absolute bandwidth is infinite
  54. 54. The power outside the 1st null bandwidth is not negligible. That is, the power in the sidelobes can be quite high</li></li></ul><li>If the transmission channel is bandlimited, then high frequency components will be cut off<br />High frequency components correspond to sharp transition in pulses<br />Hence, the pulse will spread out<br />If the pulse spreads out into the adjacent symbol period, then intersymbol interference (ISI) has occurred<br />Intersymbol Interference (ISI)<br />Intersymbol interference (ISI) occurs when a pulse spreads out in such a way that it interferes with adjacent pulses at the sample instant<br />Causes<br />Channel induced distortion which spreads or disperses the pulses<br />Multipath effects (echo)<br />
  55. 55. Due to improper filtering (@ Tx and/or Rx), the received pulses overlap one another thus making detection difficult<br />Example of ISI<br />Assume polar NRZ line code<br />
  56. 56. Inter Sybol Interference<br />Input data stream and bit superposition<br />The channel output is the sum of the contributions from each bit<br />
  57. 57. Note:<br />ISI can occur whenever a non-bandlimitedline code is used over a bandlimited channel<br />ISI can occur only at the sampling instants<br />Overlapping pulses will not cause ISI if they have zero amplitude at the time the signal is sampled<br />
  58. 58. ISI Baseband Communication System Model<br />
  59. 59. AWGN term<br />Desired symbol scaled by gain parameters h0<br />Effect of other symbols at the sampling instants t=kT<br />Note that he(t) is the equivalent impulse response of the receiving filter<br />To recover the information sequence {an}, the output y(t) is sampled at t = kT, k = 0, 1, 2, …<br />Input data stream and bit superposition<br />The sampled sequence is<br /> or equivalently <br />h0 is an arbitrary constant<br />
  60. 60. Signal Design for Bandlimited Channel<br />Zero ISI<br />To remove ISI, it is necessary and sufficient to make the term<br />This means that the overall communication system has to be designed such that<br />A pulse will produce zero ISI if it satisfies the following condition:<br />Nyquist studied this problem many years ago<br />
  61. 61. Nyquist condition for zero ISI<br />A pulse will produce zero ISI at sampling instants if it satisfies<br /> provided that its Fourier Transform satisfy (necessary and sufficient condition)<br />Proof: (Page 557 to 560, Digital Comm., Fourth Edition by J. G. Proakis)<br /><ul><li> In general, he(t) is the inverse Fourier of He(f)</li></li></ul><li>At sampling instants t = nT, <br />Mathematically breaking up the integral to cover the finite range 1/T,<br />where:<br />
  62. 62. Note that B(f) is a periodic function with period 1/T, therefore it can be expanded in terms of its Fourier series coefficients {bn} as:<br /> where<br />Comparing (6) and (3)<br />
  63. 63. Therefore necessary and sufficient conditions are,<br />Substituting (8) in (5) yields<br />
  64. 64. <ul><li> Suppose channel bandwidth is B, then HC(f) = 0, |f| > B </li></ul> and He(f) = 0 for |f| > B<br /><ul><li> Case I :Pulse shape such that (fs is used to denote symbol rate and is equal to 1/T):
  65. 65. He(f) consist of non-overlapping replicas separated by fs = 1/T
  66. 66. In this case, elimination of ISI is not possible. Why?
  67. 67. We cannot design He(f) to ensure that H(f) = T</li></li></ul><li>Case II: Pulse shape is such that:<br />In this case, the pulses touch and almost begin to overlap<br />There exist one He(f) for which H(f) = T<br />
  68. 68. Pulse shape that satisfy this criteria is Sinc(.) function, e.g.,<br />The smallest value of T for which transmission with zero ISI is possible is<br />Problems with Sinc(.) function<br />It is not possible to create Sinc pulses due to<br />Infinite time duration<br />Sharp transition band in the frequency domain<br />Sinc(.) pulse shape can cause ISI in the presence of timing errors<br />If the received signal is not sampled at exactly the bit instant, then ISI will occur<br />
  69. 69. We seek a pulse shape that<br />Has a more gradual transition in the frequency domain<br />Is more robust to timing errors<br />Yet still satisfies Nyquist’s condition for zero ISI<br />Case III:Consider a pulse shape that satisfies:<br />In this case, pulse touch and almost begin to overlap<br />There are many He(f) for which H(f) = T<br />
  70. 70. Raised Cosine (RC) Pulse<br />The following pulse shape satisfies Nyquist’s criterion for zero ISI<br />The Fourier Transform of this pulse shape is<br /> where a is the roll-off factor that determines the bandwidth (0=a= 1)<br />
  71. 71. Bandwidth occupied beyond 1/2T is called the excess bandwidth (EB)<br />EB is usually expressed as a %tage of the Nyquist frequency, e.g.,<br />a = 1/2 ===> excess bandwidth is 50 %<br />a = 1 ===> excess bandwidth is 100 %<br />The sharpness of the filter is controlled by the parameter a<br />When a = 0 this corresponds to an ideal rectangular pulse<br />Bandwidth B occupied by the filtered pulse is increased from its minimum value:<br /> to actual modulation bandwidth<br />
  72. 72. The Nyquist pulse shape can also be written as<br /> such that the Fourier Transform becomes<br />This is equivalent to equation 3.78 in your text<br />
  73. 73. If we denote<br />A RC rolloff pulse shape is defined in this case by the rolloff factor<br />f1and f▲ are related to the pulse bandwidth B (or W) as shown in the figure:<br />
  74. 74. <ul><li> Solving for bandwidth in terms of the roll off factor and symbol rate, we have:
  75. 75. The DSB bandwidth can be written as:</li></ul>Example 3.3: Find the minimum bandwidth for the baseband transmission of a 4 level PAM having a R=2400bits/sec and r=1.<br />
  76. 76. Practical Issues with Pulse Shaping<br />Can be digitally implemented with an FIR filter<br />Analog filters such as Butterworth filters may approximate the tight shape of this spectrum. <br />Like the Sa(.) pulse, Raised Cosine rolloff pulses extend infinitely in time<br /> However, a very good approximation can be obtained by truncating the pulse. Truncation leads to sidelobes - even in RC pulses. (E.g. h(t) extend from -3Tbto +3Tb)<br />RC rolloff pulses are less sensitive to timing errors than Sa(.) pulses<br /> Larger values of a are more robust against timing errors<br />US Digital Cellular (IS-54/136) uses root RC rolloff pulse shaping with α= 0.35<br />IS-95 uses pulse shape that is slightly different from RC rolloff shape<br />European GSM uses Gaussian shaped pulses<br />
  77. 77. Root RC rolloff Pulse Shaping<br />We saw earlier that the noise is minimized at the receiver by using a matched filter<br />If the transmit filter is H(f), then the receive filter should be H*(f) <br />The combination of transmit and receive filters must satisfy Nyquist’s first method for zero ISI<br />Transmit filter with the above response is called the root raised cosine-rolloff filter<br />Root RC rolloff pulse shapes are used in many applications such as IS- 54 and IS-136<br />
  78. 78.
  79. 79. Eye Patterns<br />An eye pattern is obtained by superimposing the actual waveforms for large numbers of transmitted or received symbols<br />Perfect eye pattern for noise-free, bandwidth-limited transmission of an alphabet of two digital waveforms encoding a binary signal (1’s and 0’s)<br />Actual eye patterns are used to estimate the bit error rate and the signal to- noise ratio<br />
  80. 80. Concept of the eye pattern <br />
  81. 81. Concept of Eye diagram Mask. Waveform must not intrude into the shaded regions.<br />
  82. 82. Eye pattern for 5-level PAM (PAM-5), as used to operate gigabit Ethernet over 4 unshielded twisted pairs:<br />
  83. 83. Eye Diagrams for Raised Cosine Filtered Data<br />As a is reduced, the eye opening dramatically narrows, requiring the accuracy of symbol timing to be even more exact<br />The ‘overshoot’ of the pulse caused by filtering is greater for small a, increasing the peak to mean ratio of the energy in the data signal<br />Hence the peak signal handling requirement of the modulator and demodulator circuits is increased<br />Benefits of small a<br /> Maximum bandwidth efficiency achieved<br />
  84. 84. Eye Diagrams<br />Eye Diagram after Root raised cosine filter <br />Eye Diagram after Raised cosine filter <br />
  85. 85.
  86. 86.
  87. 87.
  88. 88. close all<br />clear all<br />T = 1; % Bit period<br />tau = 1; % Time constant of channel<br />dt = 0.01; % Sampling time in simulation<br />N = 100; % Number of bits to generate<br />% Create output pulse: rectangular pulse convolved with first-order<br />% low-pass filter impulse response.<br />t1 = (dt:dt:T)';<br />t2 = (T+dt:dt:T+5*tau)';<br />c=[sin(2*pi*(0.5)*[1/100:1/100:1])]';<br />c=[c;zeros(500,1)];<br />figure(1)<br />plot([t1;t2], c)<br />xlabel('Time (sec)')<br />ylabel('c(t)')<br />title('Pulse c(t)')<br />
  89. 89. % Generate bit stream<br />b = rand(N,1);<br />z0 = find(b < 0.5);<br />z1 = find(b >= 0.5);<br />b(z0) = -1*ones(size(z0));<br />b(z1) = +1*ones(size(z1));<br />% Create received signal <br />nT = T/dt;<br />nc = length(c);<br />nx = N*nT;<br />x = zeros(nx, 1);<br />for n=1:N<br /> i1 = (n-1)*nT;<br /> y = [zeros(i1,1); b(n)*c; zeros(N*nT-i1-nc,1)];<br /> x = x + y(1:nx);<br />end<br />% Plot eye diagram<br />figure(2);t3 = dt:dt:2;plot(t3, x(1:200));<br />hold on<br />for n=3:2:N<br /> plot(t3, x((n-1)*nT+1:(n+1)*nT));<br />end<br />hold off;xlabel('Time (sec)');title('EYE DIAGRAM');<br />
  90. 90. 3.4 Equalization<br />Nyquist filtering and pulse shaping schemes assumes that the channel is precisely known and its characteristics do not change with time<br />However, in practice we encounter channels whose frequency response are either unknown or change with time<br />For example, each time we dial a telephone number, the communication channel will be different because the communication route will be different<br />However, when we make a connection, the channel becomes time-invariant<br />The characteristics of such channels are not known a priori<br />Examples of time-varying channels are radio channels<br />These channels are characterized by time-varying frequency response characteristics<br />
  91. 91. To compensate for channel induced ISI we use a process known as Equalization: a technique of correcting the frequency response of the channel<br />The filter used to perform such a process is called an equalizer<br />Since HR(f) is matched to HT(f), we usually worry about HC(f)<br />The goal is to pick the frequency response HEQ(f) of the equalizersuch that<br />where<br /> and the phase characteristics<br />
  92. 92. Problems with Equalization<br />It can be difficult to determine the inverse of the channel response<br />If the channel response is zero at any frequency, then the inverse is not defined at that frequency<br />The receiver generally does not know what the channel response is. <br />Channel changes in real time so equalization must be adaptive<br />The equalizer can have an infinite impulse response even if the channel has a finite impulse response<br />The impulse response of the equalizer is usually truncated<br />
  93. 93. Equalization Techniques or Structures<br />Three Basic Equalization Structures<br />Linear Transversal Filter<br />Simple implementation using Tap Delay Line or FIR filters<br />FIR filter has guaranteed stability (although adaptive algorithm which determines coefficients may still be unstable)<br />Decision Feedback Equalizer<br />Extra step in subtracting estimated residual error from signal<br />Maximal Likelihood Sequence Estimator (Viterbi)<br />“Optimal” performance<br />High complexity and implementation problem.<br />
  94. 94. Linear Transversal Equalizer<br />This is simply a linear filter with adjustable parameters<br />The parameters are adjusted on the basis of the measurement of the channel characteristics<br />A common choice for implementation is the transversal filter(Tap Delay Line) or the FIR filter with adjustable tap coefficient<br />Total number of taps = 2N+1<br />Total delay = 2Nt; In Figure 3.26,  is chosen as high as T<br />Fig. 3.26<br />
  95. 95. N is chosen sufficiently large so that equalizer spans length of the ISI.<br />Normally the ISI is assumed to be limited to a finite number of samples<br />The output ykof the Tap Delay Line equalizer in response to the input sequence {xk} (which is the output of the matched filter) is<br />where cnis the weight of the nth tap<br />At the equalizer output we have:<br />
  96. 96. Since there are 2N+1 equalizer coefficients, we may express in matrix form as:<br />y=Xc<br />where: <br /> X = (4N+1) x(4N+1) matrix(because K=-2N to 2N)with elements x(kT - n), We consider (2N+1)x(2N+1) values of the received data<br /> c = (2N+1) column coefficient vector<br /> y = (2N+1) column vector<br />In Figure 3.26,  is chosen as high as T<br /> = T  Symbol-spaced equalizer;  < T Fractional-spaced equalizer<br />
  97. 97. Zero-Forcing Solution<br />We obtain a set of (2N+1) linear equations for the ZFE and force y[n] other than at the sampling instant to be zero.<br />For N=1<br />
  98. 98. For N=2<br />Generalizing results:<br />
  99. 99. Zero Forcing Equalizer: Noise Enhancement<br />The Fourier transform of the the received signal can be written as: <br />Where S(ω) is the Fourier transform of the transmitted signal and N(ω). Now defining the equalizer response Heq(ω) as the inverse of the channel:<br /><ul><li>Using (A) in (B), we get
  100. 100. This shows that at the output of the ZFE the noise will be enhanced if the channel frequency response has spectral nulls </li></li></ul><li>Minimum MSE Solution<br />A more robust equalizer can be obtained if {cn} tap weights are chosen to minimize the mean square error(MSE) of all ISI terms plus noise power at the output of equalizer<br />MSE is defined as:<br /> the expected value of the squared difference between<br /> the desired data symbol and estimated data symbol <br />Whereas <br />
  101. 101.
  102. 102. Deterministic Case:<br />Example 3.6: A Minimum 7-Tap Equalizer<br /> Consider that the tap weights of an equalizing transversal filter are to be determined by transmitting a single impulse as a training signal. Let the equalizer circuit be made up of 7 taps. Given a received distorted set of pulse samples{x(k)}, with values 0.0110, 0.0227, -0.1749, 1.000, 0.1617, -0.0558,0.0108, use a minimum MSE solution to find the weights {cn} that will minimize the ISI. With these weights, calculate the resulting values of the equalized pulse samples at the following times:<br />What is the largest magnitude sample contributing to ISI, and what is the sum of all the ISI magnitudes?<br />
  103. 103. Solution: For a 7-tap filter (N=3)<br /> Dimensions for matrix ‘x’ will be 4N+1 by 2N+1 = 13x7 <br />
  104. 104. %% Example 3.6<br />x=[0.0108 -0.0558 0.1617 1 -0.1749 .0227 0.011]<br />%%%%%%%%%%%%%%<br />N=length(x);<br />z=[0 0 0 0 0 0 1 0 0 0 0 0 0]’;<br />X=zeros(2*N-1,N);<br />for n=1:N<br /> X(n+N-1:-1:n,n)=x';<br />end<br />R_xx=X'*X<br />R_xz=X'*z<br />c=inv(R_xx)*R_xz<br />z1=conv(x(end:-1:1),c)<br />z_ISI=[z1(1:(N-1));z1((N+1:2*N-1))];<br />Max_ISI=max(abs(z_ISI))<br />Sum_ISI=sum(abs(z_ISI))<br />
  105. 105. c =<br /> -0.0116<br /> 0.0108<br /> 0.1659<br /> 0.9495<br /> -0.1318<br /> 0.0670<br /> -0.0269<br />Max_ISI = 0.0095<br />Sum_ISI = 0.0197<br />
  106. 106. Using matrix ‘x’, form autocorrelation matrix Rxx and cross correlation matrix Rzx. Solution for tap weights is:<br />Using these weights, the 13 equalized samples {y(k)} at times <br /> :<br />The largest magnitude sample contributing to ISI : 0.0095 <br />The sum of all the ISI magnitudes : 0.0197<br />
  107. 107. Steepest Descent Algorithm <br /><ul><li>Difficult to find the inverse of a large matrix.
  108. 108. Use gradient based iterative techniques
  109. 109. Cost function
  110. 110. Start with an initial estimate of c0 and update it by moving in the opposite direction of the gradient of J.
  111. 111. Keep on updating the old estimate till convergence is reached.</li></li></ul><li>Steepest Descent Algorithm <br />
  112. 112. Steepest Descent Algorithm <br />Consider the coefficients:<br />The steepest descent algorithm is given by:<br />Where<br />If we use instantaneous estimate of at each sample we have:<br />Which is called the LMS algorithm. As in the previous case z is the desired signal and x is the received signal.<br />
  113. 113. Matlab Example :<br />clear all;close all<br />L=3; % Signal Duration in seconds<br />fs=8000; % Sampling frequency<br />N=21; % Number of filter taps<br />%Training Signal<br />z=randn(1,L*fs); <br />% Impulse response of the channel<br />h=[1,0.7,0.2,-0.5,-0.8,-0.4,0,0.25,0.1,0.05,0,0];<br />x_r=filter(h,1,z);<br />% Intialization<br />% Delay line<br />x=zeros(1,N);<br />% Filter Coefficients<br />c=zeros(1,N);<br />% Step Size<br />mu=0.001; <br />
  114. 114. % LMS Algorithm<br />for n=1:L*fs<br /> x=[x(2:N) x_r(n)];<br /> e(n)=z(n)-c*x’;<br /> c=c+mu*e(n)*x;<br />end<br />figure(1)<br />H=fftshift(fft(h,fs));H=abs(H);H=H/max(H);<br />plot(0:fs/2-1,H(fs/2+1:fs));hold on<br />Cf=fftshift(fft(c,fs));Cf=abs(Cf);Cf=Cf/max(Cf);<br />plot(0:fs/2-1,Cf(fs/2+1:fs),'r');grid;xlabel('Frequency (Hz)');<br />gtext('|H(f)|')<br />gtext('|C(f)|')<br />
  115. 115. Results:<br />
  116. 116. Fractionally Spaced Equalizer<br /><ul><li> The spectrum property of the baud-rate (or symbol rate) and fractionally spaced equalizer.</li></li></ul><li>Decision Feedback Equalizer<br />A decision-feedback equalizer(DFE) is a nonlinear equalizer that employs previous decisions to eliminate the ISI caused by previously detected symbol<br />It consists of a feedforward section a feedback section and a detector connected together as shown<br />The feed forward filter is usually fractionally spaced FIR with adjustable tap coefficients. Feed-back section is symbol spaced.<br />The detector is a symbol-by-symbol detector<br />
  117. 117. DFE is based on the principle that once you have determined the value of the current transmitted symbol, you can exactly remove the ISI contribution of that symbol to future received symbols<br />The nonlinear feature is due to the feed back portion. The detector attempts to determine which symbol of a set of discrete levels was actually transmitted.<br />Once the current symbol has been decided, the filter structure can calculate the ISI effect it would tend to have on subsequent received symbols and compensate the input to the decision device for the next samples.<br />Thus postcursor ISI removal is accomplished by the use of a feedback filter structure.<br />
  118. 118. Training Mode vs. Decision Directed mode<br />
  119. 119. Adaptive Equalization for Digital Cellular Telephony<br />The direct sequence spreading employed by CDMA (IS-95) obviates the need for a traditional equalizer.<br />The TDMA systems (for example, GSM and IS-54), on the other hand, make great use of equalization to contend with the effects of:<br /> multipath-induced fading, <br />additive received noise,<br />channel-induced spectral distortion, etc<br />Of the nonlinear equalizers, the DFE is currently the most practical system to implement in a consumer system.<br />Other designs that outperform the DFE in terms of convergence or noise performance, but these generally come at the expense of greatly increased system complexity.<br />