_{1}

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In this paper a closed-form approximated expression is proposed for the Intersymbol Interference (ISI) as a function of time valid during the entire stages of the non-blind adaptive deconvolution process and is suitable for the noisy, real and two independent quadrature carrier input case. The obtained expression is applicable for type of channels where the resulting ISI as a function of time can be described with an exponential model having a single time constant. Based on this new expression for the ISI as a function of time, the convergence time (or number of iteration number required for convergence) of the non-blind adaptive equalizer can be calculated. Up to now, the equalizer’s performance (convergence time and ISI as a function of time) could be obtained only via simulation when the channel coefficients were known. The new proposed expression for the ISI as a function of time is based on the knowledge of the initial ISI and channel power (which is measurable) and eliminates the need to carry out any more the above mentioned simulation. Simulation results indicate a high correlation between the simulated and calculated ISI (based on our proposed expression for the ISI as a function of time) during the whole deconvolution process for the high as well as for the low signal to noise ratio (SNR) condition.

We consider a non-blind deconvolution problem in which we observe the output of an unknown, possibly nonminimum phase, linear system (single-input-single-output (SISO) finite impulse response (FIR) system) from which we want to recover its input (source) using an adjustable linear filter (equalizer) and training symbols [

In this paper, we propose for the real and two independent quadrature carrier input case, a closed-form approximated expression for the ISI as a function of time (or number of iteration number) valid during the entire stages of the iterative deconvolution process. This new expression depends on the SNR, on the step-size parameter used in the adaptation process, on the equalizer’s tap length, on the input signal statistics and on the channel’s power (which is measurable). The obtained expression is applicable for type of channels where the resulting ISI as a function of time can be described with an exponential model having a single time constant. Based on this new expression for the ISI as a function of time (or number of iteration number), the convergence time (or number of iteration number required for convergence) of the non-blind adaptive equalizer is obtained.

The paper is organized as follows: After having described the system under consideration in Section 2, the closed-form approximated expression for the ISI as a function of time (or number of iteration number) is introduced in Section 3. In Section 4 simulation results are presented and the conclusion is given in Section 5.

The system under consideration is illustrated in

1) The input sequence

2) The unknown channel

3) The equalizer

4) The noise

The sequence

where “

where

where

the function

In this section we derive a closed-form approximated expression for the ISI a function of time or as a function of number of iteration number. Based on this expression, a closed-form approximated expression is obtained for the convergence time (or number of iteration number required for convergence) of the non-blind adaptive equalizer. Since we deal with the real and two independent quadrature carrier case, we start our derivations first with the real valued case and then turn to the two independent quadrature carrier one.

Theorem 1. For the following (additional) assumptions:

1) The convolutional noise

2) The variance of the source signal

3) The convolutional noise

The ISI as a function of the discrete time is approximately expressed as:

where

Comments:

Assumptions 1 and 3 were also made in [

Proof. Recently [

which was based on [

used where

Now, we divide both sides of (5) with

with

where

can be approximately seen as a first order differential equation with the following solution for

where

Thus, by dividing both sides of (9) with

Next, we find a closed-form expression for

In order to complete our proof, we use the relation of

mind that we have to wait until the whole convolution process is finished in (6) to create

in (7). Thus, we substitute

discrete time which completes our proof.

Based on (4), the number of iteration number required for convergence can be obtained:

where

In this section we test our new proposed expression for the ISI as a function of time (4) via simulation. For this purpose we use two different constellation inputs:

A 16QAM input case (a modulation using ±{1,3} levels for in-phase and quadrature components) and the QPSK input case (a modulation using ±{1} levels for in-phase and quadrature components). The following six channels were considered:

Channel1 (initial

Channel2 (initial

Channel3 (initial

Channel4 (initial

Channel5 (initial

Channel6 (initial

The equalizer was initialized by setting the center tap equal to one and all others to zero.

Please note that according to [

enables the recovery of the transmitted sequence. A fast convergence time is obtained by choosing a higher value for the step-size parameter.

Next we turn to test the expression for the convergence time (13). For that purpose we use the following cases:

Case 1 (

The simulated convergence time according to

Case 2 (

The simulated convergence time according to

Case 3 (

The simulated convergence time according to

Case 4 (

The simulated convergence time according to

Case 5 (

The simulated convergence time according to

Case 6 (

The simulated convergence time according to

Case 7 (

The simulated convergence time according to

Case 8 (

The simulated convergence time according to

Based on the above mentioned cases (Case 1 - Case 8), there is a high correlation between the calculated (13) and simulated convergence time.

In this paper, we proposed for the real and two independent quadrature carrier input case, a closed-form approximated expression for the ISI as a function of time (or number of iteration number) valid during the entire stages of the iterative deconvolution process that depends on the SNR, on the step-size parameter used in the adaptation process, on the equalizer’s tap length, on the input signal statistics and on the channel’s power (which is measurable). The obtained expression is applicable for type of channels where the resulting ISI as a function of time can be described with an exponential model having a single time constant. Based on this new expression for the ISI as a function of time (or number of iteration number), the convergence time (or number of iteration number required for convergence) of the non-blind adaptive equalizer was obtained. Simulation results have shown a high correlation between the calculated (based on our new expression for the ISI as a function of time) and simulated ISI as a function of time. In addition, simulation results have shown that our new proposed expression for the convergence time produces approximately the same results as those obtained from the simu- lation.

We thank the Editor and the referee for their comments.

MonikaPinchas, (2016) Convergence Curve for Non-Blind Adaptive Equalizers. Journal of Signal and Information Processing,07,7-17. doi: 10.4236/jsip.2016.71002

ISI: Intersymbol Interference

SNR: Signal to Noise Ratio

SISO: Single Input Single Output

FIR: Finite Impulse Response

QAM: Quadrature Amplitude Modulation

QPSK: Quadrature Phase Shift Keying