ADAPTIVE EQUALIZER EXPERIMENTS

by

Hulya Seferoglu
Faculty of Engineering and Natural Science, Sabanci University, Istanbul, Turkey

Spring 2004

1. Introduction

The goal of equalizers is to eliminate intersymbol interference (ISI) and the additive noise as much as possible. Intersymbol interference arises because of the spreading of a transmitted pulse due to the dispersive nature of the channel, which results in overlap of adjacent pulses. In Fig. 1, there is a four-level pulse amplitude modulated signal (PAM), x(t). This signal is transmitted through the channel with impulse response h(t). Then noise n(t) is added. The received signal r(t) is a distorted signal.

Figure 1: Block diagram of transmission channel

Equalizers are used to overcome the negative effects of the channel. In general, equalization is partitioned into two broad categories;

• Maximum likelihood sequence estimation (MLSE) which entails making measurement of channel impulse response and then providing a means for adjusting the receiver to the transmission environment. (Example: Viterbi equalization)
• Equalization with filters, uses filters to compansate the distorted pulses. The general channel and equalizer pair is shown in Figure.2.

Figure 2: General channel and equalizer pair

These type of equalizers can be grouped as preset or adaptive equalizers. Preset equalizers assume that the channel is time invariant and try to find H(f) and design equalizer depending on H(f). The examples of these equalizers are zero forcing equalizer, minimum mean square error equalizer, and desicion feedback equalizer. Adaptive equalizers assume channel is time varying channel and try to design equalizer filter whose filter coefficients are varying in time according to the change of channel, and try to eliminate ISI and additive noise at each time. The implicit assumption of adaptive equalizers is that the channel is varying slowly.

In the following, there will be brief explanation on how adaptive equalizers work, then linear minimum mean square error equalizer (stochastic gradient algorithm) with simulations and adaptive desicion feedback equalizer with simulations will be shown.

2. Brief overview of adaptive equalizers

The general block diagram of an adaptive equalizer is shown in Fig. 3.

Figure 3: General block diagram of an adaptive equalizer

The working principles of adaptive equalizers are in the following:

• The received signal is applied to receive filter. In here, receive filter is not matched filter. Because we do not know the channel impulse response. The receive filter in here is just a low-pass filter that rejects all out of band noise.
• The output of the recive filter is sampled at the symbol rate or twice the symbol rate.
• Sampled signal is applied to adaptive transversal filter equalizer. Transversal filters are actually FIR discrete time filters.
• The object is to adapt the coefficients to minimize the noise and intersymbol interference (depending on the type of equalizer) at the output.
• The adaptation of the equalizer is driven by an error signal.

There are two modes that adaptive equalizers work;

• Decision Directed Mode: This means that the receiver decisions are used to generate the error signal. Decision directed equalizer adjustment is effective in tracking slow variations in the channel response. However, this approach is not effective during initial acqusition .
• Training Mode: To make equalizer suitable in the initial acqusition duration, a training signal is needed. In this mode of operation, the transmitter generates a data symbol sequence known to the receiver. The receiver therefore, substitutes this known training signal in place of the slicer output. Once an agreed time has elapsed, the slicer output is substituted and the actual data transmission begins.

3. Stochastic Gradient Algorithm

The main idea behind stochastic gradient algorithm (linear adaptive equalizer) is to minimize the mean square error between the signals that one is the output of the equalizer, and another is the transmitted signal. However, since the number of samples that receiver gets is finite, mean square is calculated by using time averages instead of ensemble averages. Now that, there is an error definition, the aim is to find equalizer coefficients that minimize the mean square error. The resultant adaptation algorithm (it is also named LMS algorithm) becomes;

(3.1)

Where is the coefficient vector at time k+1 , is the coefficient vector at time k, is the step size parameter, is the error signal at time k between the equalized signal and transmitted signal in training mode and between the equalized signal and the equalized and quantized signal in desicion directed mode. For further information about derivation of adaptation algorithm (3.1) and the working principle of general LMS algorithm please refer to [1] , [2], and [3].

In the following, there will be simulations of stochastic gradient algorithm for two different channels. The impulse response of the first channel is raised cosine function and it results spectral expansion, and the second channel is a channel having nulls on the spectrum. In addition, the first parts of the simulations will work on the training mode and the second parts will work on the desicion directed mode. The simulations will be performed for 3rd degree adaptive filter and 11th degree adaptive filter.

Case I:

• The degree of the adaptive filter is 11.
• The impulse response of the channel is a raised cosine function;

• The signal applied to the channel consists of a Bernoulli sequence taking values with probability 0.5.
• The additive noise is Gaussian noise with mean zero, and variance .
• The step size parameter of the adaptation algorithm (the filter) is for the training mode and for the desicion directed mode.
• The overall signal-to-noise ratio; SNR = 26.9897 dB
• The adaptive filter is working on training mode for the first 1000 samples
• The adaptive filter is working on desicion directed mode for the samples between 1000 samples and 5000 samples
• The initial filter coefficients are equated zero. At each iteration these coefficients are modified and at the beginning of desicion directed mode the filter coefficients of the last iteration of the training mode are taken as initial coefficients.
• The Matlab code of this type of equalizer is adaptive_equalizer1.m
• The equalized signal is passed through the slicer, the slicer is actually a quantizer. The rule of the quantizer in this simulation is that it quantizes the signal to 1 when the signal is greater than 0.5 and quantizes the signal to -1 when the signal is less than 0.5

Figure 4: Channel magnitude response

Figure 5: Original signal, observed signal, equalized signal and quantized signal in the training mode.

Figure 6: Error signals before/after the slicer while equalizer is on training mode.

Figure 7: Magnitude spectrum and impulse response of the equalizer working on training

mode and at sample 1000

Figure 8: Original signal, observed signal, equalized signal and quantized signal in the

desicion directed mode.

Figure 9: Error signals before the slicer and after the slicer while equalizer working on its

desicion directed mode.

Figure 10: Magnitude spectrum and impulse response of the equalizer working on desicion

directed mode and at sample 5000

Comments:

Fig.4 shows that the channel frequency response of the transmission system behaves as a low-pass filter. Since equalizers try to eliminate the negative effects of the channel, we can say that the adaptive equalizer should be a high-pass filter. Fig. 7 and 10 shows that our guess is true. If we examine the error signals, we can see that at the beginning of the training mode there are errors. The reason of this is that the equalizer tries to adapt itself to the channel and achives this after some period of time. Another point in here is that after training mode is completed, desicion directed mode begins. In desicion directed mode actual data transmission begins. The reference signal is output of the equalizer instead of the training signal. In spite of this, the error level is still low in the desicion directed mode. The reason of this is that the equalizer adapted itself to the channel at the end of training mode. After this time the channel is not changed. The only change comes from the randomness of the noise. However, if the channel is slowly varying channel instead of our channel, the equalizer continues to track channel for some time duration. Even in our case, after sime time interval, error level becomes to increase, in Fig. 9. To overcome this problem, training signal is sent after some time duration. If the number of equalizer filter coefficients are smaller, for example, the number of coefficients is 3, the adaptation time in the training mode reduces, however, the amount of error in the desicion directed mode increases. The example code can be found in adaptive_equalizer2.m

In the following, there are simulation results for a different channel. This channel has nulls on the spectrum.

Case II:

• The degree of the adaptive filter is 11.
• The impulse response of the channel is [0.2, -0.15, 1.0, 0.21, 0.03]. This channel has a spectral null in the middle frequency region.

• The signal applied to the channel consists of a Bernoulli sequence taking values with probability 0.5
• The additive noise is Gaussian noise with mean zero, and variance .
• The step size parameter of the adaptation algorithm (the filter) is for the training mode and for the desicion directed mode.
• The overall signal-to-noise ratio; SNR = 26.9897 dB
• The adaptive filter is working on training mode for the first 1000 samples.
• The adaptive filter is working on desicion directed mode for the samples between 1000 samples and 5000 samples.
• The initial filter coefficients are equated zero. At each iteration these coefficients are modified and at the beginning of desicion directed mode the filter coefficients of the last iteration of the training mode are taken as initial coefficients.
• The Matlab code of this type of equalizer is adaptive_equalizer3.m
• The equalized signal is passed through the slicer, the slicer is actually a quantizer. The rule of the quantizer in this simulation is that it quantizes the signal to 1 when the signal is greater than 0.5 and quantizes the signal to -1 when the signal is less than 0.5

In the following, there are simulation results of this case.

Figure 11: Channel magnitude response

Figure 12: Original signal, observed signal, equalized signal and quantized signal in the

training mode.

Figure 13: Error signals before the slicer and after the slicer while equalizer working on

its training mode.

Figure 14: Magnitude spectrum and impulse response of the equalizer working on training

mode and at sample 1000

Figure 15: Original signal, observed signal, equalized signal and quantized signal in the

desicion directed mode.

Figure 16: Error signals before the slicer and after the slicer while equalizer working on its

desicion directed mode.

Figure 17: Magnitude spectrum and impulse response of the equalizer working on desicion

directed mode and at sample 5000.

Comments:

In many ways Case II is similar to Case I. However, channel magnitude spectrum has a null in the middle frequency region. Due to this property of the channel, amount of error that equalizer results increase rapidly after training mode is ended. (The average error in the desicion directed mode is E = -31.7064 dB). This drawback of linear adaptive equalizers can be overcomed by using desicion feedback equalizer (DFE). In the following there is a brief explanation of adaptive DFE and then simulations will be shown.

4. Adaptive Desicion Feedback Equalizer

• A decision feedback equalizer (DFE) is a nonlinear equalizer that uses previous detector decisions to eliminate the ISI on pulses that are currently being demodulated.
• The basic idea of a DFE is that if the values of the symbols previously detected are known (past decisions are assumed to be correct), then the ISI contributed by these symbols can be canceled out eactly the output of the forward filter by subtracting past symbols values with appropriate weighting. In Fig. 18, there is a general structure of the adaptive desicion feedback equalizer.

Figure 18: Block diagram of the adaptive desicion feedbak equalizer

• If we look at Fig. 18, we see that the estimated signal sequence becomes,

(4.1)

• ’s are coefficients of the precursor equalizer, ’s are coefficients of the postcursor equalizer. N is the number of precursor equalizer coefficients and M is the number of postcursor equalizer coefficients.
• Adaptive DFE algorithm is similar to stochastic gradient algorithm, with the important difference that the input to the causal portion of the filter is the decisions rather than the output of the precursor equalizer filter.
• This difference will obviously change the desired tap coefficients as well as reduce the noise enhancement due to equalization.
• The derivation of a stochastic gradient algorithm for the DFE is a simple extension of the stochastic gradient algorithm for linear case. First, define an augmented vector of N+M coefficients,

(4.2)

• and an augmented input signal vector

(4.3)

• DFE slicer error can be expressed as,

(4.4)

• The adaptation algorithm becomes,

(4.5)

In the following, there are simulations of adaptive DFE for the channel having spectral null on the middle frequency range. The simulations for the raised cosine channel is not included here, because there is not any performance enhancement for this case. Therefore, we can say that adaptive DFE performs better for the channels having spectral nulls. The Matlab code of adaptive DFE for raised cosine channel can be found in adaptive_equalizer4.m .

Case:

• The degree of the precursor equalizer is 5, and the degree of postcursor equalizer is 3.
• The impulse response of the channel is [0.2, -0.15, 1.0, 0.21, 0.03]. This channel has a spectral null in the middle frequency region.

• The signal applied to the channel consists of a Bernoulli sequence taking values with probability 0.5
• The additive noise is Gaussian noise with mean zero, and variance .
• The step size parameter of the precursor adaptation algorithm is for the training mode and for the desicion directed mode, and the step size parameter of the postcursor adaptation algortihm is for the training mode and for the desicion directed mode.
• The overall signal-to-noise ratio; SNR = 26.9897 dB
• The adaptive filter is working on training mode for the first 1000 samples
• The adaptive filter is working on desicion directed mode for the samples between 1000 and 5000.
• The initial filter coefficients are equated zero. At each iteration these coefficients are modified and at the beginning of desicion directed mode the filter coefficients of the last iteration of the training mode are taken as initial coefficients.
• The Matlab code of this type of equalizer is adaptive_equalizer5.m
• The equalized signal is passed through the slicer, the slicer is actually a quantizer. The rule of the quantizer in this simulation is that it quantizes the signal to 1 when the signal is greater than 0.5 and quantizes the signal to -1 when the signal is less than 0.5

In the following, there are simulation results of this case.

Figure 19: Magnitude spectrum of the channel

Figure 20: Original signal, observed signal, equalized signal and quantized signal in the

training mode.

Figure 21: Error signals before the slicer and after the slicer while equalizer working on

its training mode.

Figure 22: Magnitude response and impulse response of precursor equalizer working on its

training mode and at sample 997.

Figure 23: Magnitude response and impulse response of postcursor equalizer working on its

training mode and at sample 997.

Figure 24: Original signal, observed signal, equalized signal and quantized signal in the

desicion directed mode.

Figure 25: Error signals before the slicer and after the slicer while equalizer working on its
desicion directed mode.

Figure 26: Magnitude response and impulse response of precursor equalizer working on its

desicion directed mode and at sample 4997.

Figure 27: Magnitude response and impulse response of postcursor equalizer working on its

desicion directed mode and at sample 4997.

Comments:

In this experiment, the channel has spectral nulls on the middle frequency region. Adaptive desicion feedback equalizer tries to minimize the negative effects of the channel with its precursor and postcursor equalizers. Fig. 21 and 25, it is shown that the error level is less than that of the linear adaptive equalizer. In addition, there is not a significant increase in the desicion directed mode while time is passing. The average error in the desicion directed mode is E = -46.6790 dB. Adaptive desicion feedback algorithm works well for the channels with spectral nulls and low noise power condition. If the noise level is high, adaptive DFE does not perform well.

5. Conclusion

In this work, adaptive equalizers are examined. Stochastic gradient algorithm and adaptive desicion feedback equalizers are simulated for different channel responses and the results are compared. This work can be expanded by using time varying channels and by using different signal-to-noise ratio’s.

6. References:

[1] E. A. Lee, D. G. Messerschmitt, “Digital Communication”, Second Edition, Kluwer Academic Publishers, The Netherlands , 1994

[2] S. Haykin, “Adaptive Filter Theory”, Third Edition, Prentice Hall, New Jersey, 1996

[3] S. D. Stearns, R. A. David, “Signal Processing Algorithms in Matlab”, Prentice Hall, New Jersey, 1996