5 Proposed System
The objective of designing a new system for blind equalization is to improve the convergence rate and tracking property over that of the conventional blind equalizer. The conventional blind equalizer using the constant modulus algorithm (CMA) has a very slow convergence rate. To improve its convergence rate, a recursive linear predictor was used to predict the amplitude at the output of the equalizer. The predictor uses past samples to predict the present value of the equalizer output amplitude. The predicted amplitude is then used to normalize the equalizer output. A block diagram for the designed system is shown in Figure 5.1.
Absolute Value Operator
•
yenv(n-1)
Linear Predictor ^ yenv(n)
Z-1 x(k) Fractionally Spaced Equalizer f (k) Filter Adaptive Procedure ye(n)
y(n) ÷
e(n)
Property Measurement
Figure 5.1: Proposed FSBEEP System Model.
The idea for such a system comes from the fact that, in a fading channel, all of the multi-paths have different fading envelopes. Again in Chapter 4, it was shown, that a recursive linear predictor is able to predict the envelope of a flat fading channel without having prior knowledge of the channel. Therefore, if the equalizer can remove the ISI, the output of the equalizer should have the same properties as if it was the output of a flat fading channel. In that situation the recursive predictor would be able to predict the present value of the fading envelope based on the past samples. The predicted value of the envelope can then
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be used to normalize the equalizer output. In practice, the equalizer and the predictor do not work separately. Since the predictor is used for envelope prediction, the proposed system will be named a fractionally spaced blind equalizer with envelope prediction (FSBEEP). The predictor predicts the envelope amplitude at the equalizer output based on the correlation among the amplitudes of the equalizer output samples y env (n) . So, if any white random noise is added to the input sequence to the predictor, theoretically the output of the predictor will not be affected by that noise. In practice, since the autocorrelation matrix needs to be estimated, the variance of this autocorrelation estimate is affected by the additive white noise. As a result the noise will increase the variance of the predictor output. Now let us consider the stationary channel case. At the beginning of the equalization process, when the equalizer is not perfectly adapted to the channel, the output samples of the equalizer will have some correlation with the adjacent samples. The predictor will then rely on that correlation to predict the shape of the envelope of the equalizer output signal and will also try to de-correlate the overall output samples. Therefore, the combined system is expected to provide a better convergence rate than conventional CMA based blind equalizers. The coefficients of the predictor used with FSBEEP (Figure 5.1) will be updated by updating the auto-covariance matrix, Λ, of the predictor input sequence. The auto-covariance matrix is updated by using the input sequence of the predictor as follows:
H Ë (n) = y env (n) y env (n)
(5.1)
In (5.1), y env (n) = [ y env (n − 1)
y env (n − 2) L y env (n − W − 1)] T is the predictor input
vector. If the length of the predictor is W, the length of yenv(n) will be W+1. Now the first column of Ë (n) provides an estimate of the auto-correlation function of the predictor input sequence at instant n. Therefore, the first column of Ë (n) will be used to update the predictor coefficients of FSBEEP. The method of updating the predictor coefficients is given in Table 4.1 (Step 4 only).
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Due to the highly non-linear nature of the FSBEEP approach (shown in Figure 5.1), it is difficult to analyze the convergence behavior of the designed system analytically. Therefore, simulation is employed to analyze the convergence behavior of FSBEEP. However, for simulation, the equations used to update the filter (equalizer) coefficients need to be derived. These update equations should include the effect of the predictor while updating the filter coefficients. The update equations will be derived in the next section.
5.1 Equalizer Update Equations
All the variables used in the derivation are shown in Figure 5.1. Let us assume that the length of the equalizer is L and the length of the predictor is W. Now let us define the following vectors: x(n) = [x(n) x(n − 1) L x(n − L + 1)] T f (n) = [ f (0, n) f (1, n) L f ( L − 1, n)]
T
(5.2)
Here x(n) is the equalizer input sequence, x(n) is the input vector to the equalizer at time instant n, and f(n) is the equalizer coefficient vector at instant n. The second index in the coefficients of the equalizer is used to indicate the time varying property of the equalizer coefficients. For the output of the equalizer we then have,
L −1 l =0
y e (n) =
∑ f * (l, n) x(n − l ) = f H (n)x(n)
(5.3)
f*(l,n) indicates the conjugate of f(l,n), while fH(n) indicates the transpose conjugate of f(n). The output of the predictor can be written as: ˆ y env (n) =
m =1
∑ h(m, n) yenv (n − m)
W
(5.4)
ˆ In (5.4), y env (n) , which could be negative, is the estimate of the amplitude of the equalizer output. Since the amplitude of the equalizer output is always a positive quantity, a
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hard-limiter can be used at the predictor output. The hard-limiter will not allow the output amplitude of the predictor to go below a certain positive level. As the predictor builds up its input signal statistics in a recursive way, the coefficients of the predictor are also timevarying. Therefore a second index n is used to indicate the time-varying property of the predictor coefficients. However, this time-varying property of the predictor coefficients depends on the correlation of its input signal. If the correlation does not change with time, as shown in the previous section, after 40 samples the coefficients of the predictor become almost constant in a noiseless environment. Now the overall output of the system satisfies: y ( n) = y e ( n) f H ( n ) x( n ) = ˆ ˆ y env (n) y env (n)
(5.5)
Let us assume that a constant modulus is the desired property of the signal and A is the desired amplitude level. Let us assume also that the error signal is defined as: e( n ) = y ( n ) − A 2 , with the cost function, J [e(n)] = E e 2 (n)
2
(5.6)
{
}
(5.7)
Here, E{•} denotes statistical expectation. The objective of updating the equalizer coefficients is to minimize the cost function given in (5.7). To accomplish this objective a gradient descent algorithm will be used to update the filter coefficients. According to the gradient descent algorithm the filter coefficients need to be updated in the following way: f (n + 1) = f (n) − µ∇ f J . (5.8)
In the derivation of the coefficient update equation the true gradient at time instant n will be approximated by its instantaneous value. Therefore, ˆ ∇ f J ≈ ∇ f e 2 (n) = 2e(n)
{
}
∂ 2 y ( n) − A 2 ∂ f (n)
{
}
(5.9)
However, 73
∂ ∂ f H ( n ) x( n ) 2 y ( n) = ˆ ∂ f ( n) ∂ f (n) y env (n)
2
(5.10)
ˆ Note that y env (n) , the estimated value of the envelope amplitude, is always a positive real number. Therefore this quantity can be taken outside the modulus operation and (5.10) can be written as: ∂ ∂ f H (n)x(n)x H (n)f (n) 2 y ( n) = ∂ f ( n) ∂ f ( n) ˆ2 y env (n) ∂ f H (n) A (n)f (n) = ∂ f ( n) ˆ2 y env (n) with
A ( n ) = x ( n) x H ( n )
(5.11)
(5.12)
To express (5.11) in a simplified form, the following will be assumed. AS 1) Each input vector x(n) is statistically independent of all previous input vectors x(j), where j