new receivers in digital communications, performance...

Journal of Scientific & Industrial Research Vol. 74, November 2015, pp. 618-629

New Receivers in Digital Communications, Performance Evaluation and Comparisons

R Chakka1* and A K Ahuja2 *1Rajeev Gandhi Memorial College of Engineering and Technology (RGMCET), Nandyal-518501, India

2Meerut Institute of Engineering and Technology (MIET), Meerut-250005, India

Received 27 October 2014; revised 15 May 2015; accepted 10 August 2015

Recently developed STTS-MF receiver by the authors in1,2 (referred as TTS-MF receiver in1,2) performs better than the traditional Matched Filter based receiver (referred as STS-MF receiver in1,2), but only when the SNR (expressed in dB) is positive. In this paper, we introduce a new receiver, the CTTS-MF receiver, and demonstrate that it performs better than both STTS-MF and STS-MF receivers, for both positive and negative SNR (expressed in dB) values, for the correlated digital signals known as, the p-q signals. Comparisons and relative advantages of the CTTS-MF and the STTS-MF receivers are illustrated by extensive simulation study with baseband communications as example domain.

Keywords: Statistical signal processing, Performance, QoS, Communications signal processing, De-noising, Bit error rate, Matched filter, Signal correlations, Green Technology, p-q signals.

Introduction

Faster, safer and more reliable transmission of digital data is crucial for the success of the tremendous advancements that are taking place in wired and wireless communication systems. Efforts to satisfy the user-needs have given rise to new technologies and standards such as 3G and 4G which follow IMT-2000 and IMT-advanced standards with high mobility peak data rates of 384 Kbps and 100 Mbps, respectively3,4. Demand for reliable, efficient communication has been increasing day-by-day.

Performability5 in communication systems adopts different types of performance measures at different layers of the TCP/IP model. For example, at the network layer, higher performance can be attained by reducing system failure probabilities and turn-around times6 at nodes and servers, while at the physical layer, by reducing BER (Bit Error Rate) at the receiver output1,2. The subject of this paper pertains to the latter.

Channel noise is a crucial efficacy-limiting factor in digital communication systems1,2,7. Generally, for a given power and bandwidth of the transmitted signal, higher channel noise leads to higher BER at the receiver output. Reducing BER is crucially important for the success of existing as well as futuristic telecommunication systems1,2.

Achieving BER-reduction by increasing transmitted signal power has minimal further scope, because increased EMF exposure to the living beings1,2 can indeed be quite unsafe to life and living beings8,9. On the contrary, there is a great need to reduce the transmission EMF due to many important recent studies regarding safety of present-day wireless digital communications8,9. In another method, BER-reduction is achieved by channel coding of the data bits before transmission10,11. Due to check bits or redundant bits that are added to the original data, this method results in the reduced-BER at the receiver output upon decoding at the receiver. The main disadvantage of this method is that, for a given signal power and channel B.W., this method leads to less efficient channel B.W. utilization or reduced data rate (in bps). Due to cost considerations, B.W. needs to be preserved to the possible extent. Thus, for a given signal power and channel, both B.W. utilization and BER-reduction cannot be achieved simultaneously using this method.

Recent contributions of the authors towards BER-reduction, by designing a new receiver, are explained in1,2. In their method, higher reliability (characterized by lower BER) is shown to be achieved without affecting the efficiency, alternatively it can be higher efficiency without reducing reliability, or transmission at lower EMF without affecting efficiency and reliability. Also, by this method, for a

—————— Author for correspondence E-mail: [email protected]

CHAKKA & AHUJA: NEW RECEIVERS IN DIGITAL COMMUNICATIONS

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given channel B.W. utilization and fixed BER at the receiver output, data transmission can be achieved by reduced transmitted signal power and hence at reduced EMF1,2. Thus, this method falls in the area of ‘Greening of Communications Technology’ and has been used for the BER-reduction as well in1,2. Another advantage is the amicability of this method to be augmented with the above mentioned methods in10,11

. Due to several practical phenomena such as pre-

processing and channel-fading, signals most often do become correlated before reaching receivers.The receiver introduced and analysed in1,2 is more efficient for the case when transmitted digital signals are correlated in nature with first order correlation among adjacent bits. Such signals were referred as p-q

signals and the new receiver was termed as Two-Threshold-Scheme-based-MF receiver (TTS-MF receiver) in1,2. Using extensive simulation study and analysis, it was established1,2 that the TTS-MF receiver substantially outperforms the conventionally used Single-Threshold-Scheme-based-MF-receiver (referred as STS-MF-receiver) when evaluated over considered range of parameters, p,q and SNR1,2 (for positive SNR when measured in dB). BERRF (Bit-Error-Rate-Reduction-Factor) which is defined as BERTTS/BERSTS was chosen as performance indicator for the study and analysis carried out in1,2. Here, BERSTS is the BER at the output of traditional STS-MF receiver, and, BERTTS is the BER at the output of TTS-MF receiver. Clearly, any performance improvement by TTS-MF receiver over STS-MF receiver is indicated by BERRF < 1.0. The TTS-MF receiver has limitation that it performs well only in the selected SNR range of 0 dB ≤ SNR. In other words, for the lower values of SNR in the range SNR < -1 dB, the TTS-MF receiver performs worse than the STS-MF receiver with BERRF ≥ 1.0.

Overcoming this limitation of the TTS-MF receiver is accomplished in this paper by proposing a new receiver that is applicable when the transmitted digital signals are correlated p-q signals, for all ranges of SNR. The proposed new receiver also has two thresholds which need to be computed using an iterative procedure, hence is termed as the Computed-Two-Threshold-Scheme-based-MF receiver (CTTS-MF receiver). Further, obtaining the two thresholds of the TTS-MF receiver being rather simple, the TTS-MF receiver is termed in this paper as the Simple-Two-Threshold-Scheme-based-MF receiver (STTS-MF receiver), for convenience and elegance of terminology. In this paper, the performance of the

CTTS-MF receiver is analysed extensively and compared with that of STTS-MF receiver (referred as TTS-MF receiver in1,2) and the STS-MF receiver. PRBER (Percentage-Reduction-in-BER), defined in3

as PRBER = (1 – BERRF)*100, has been used as performance-indicator for the study and analysis that is carried out in this paper. Clearly, any performance improvement is indicated by PRBER > 0. After extensive simulation study, it is found that the CTTS-MF receiver substantially dominates the other two receivers (STTS-MF and STS-MF receivers) in performance, when evaluated over a wide SNR range -10 dB ≤ SNR ≤ +10 dB. This contribution has very important significance and implications in High Speed Communications & Green Communication Technologies. This is so because, BER-reduction, as obtained from CTTS-MF receiver makes it possible to achieve existing BER and B.W. utilization efficacy at reduced SNR levels or equivalently to achieve higher data rate for given BER and SNR level. Some additional computation is involved in the design of CTTS-MF receiver. Considering the advantages that it offers over STTS-MF and STS-MF receivers, especially at low SNR values, this disadvantage of CTTS-MF receiver is indeed negligible due to the vast advancements that are taking place in the development of fast, dedicated processors and suitable parallel processing technologies.

In the second section, STS-MF receiver and STTS-MF receiver are briefly reviewed. The third section deals with the introduction of the CTTS-MF receiver. The performance evaluation of the CTTS-MF receiver and the comparison-study with the STTS-MF and the STS-MF receivers are done in the fourth section. The paper concludes in the fifth section.

Brief Review of the STS-MF and the STTS-MF

Receivers

A general block diagram of a digital baseband transmitter can be found in10,11 and conventional baseband receiver using Matched filter (MF receiver) can be seen in10,11 (refer to pages 253 and 513, respectively). In1, it was demonstrated by examples that when uncorrelated digital signals, termed as r0-r1 signals, are transmitted through AWGN channel, the STS-MF receiver does optimal detection with the single optimal threshold λopt-uc, given by the Eq. (1) of 1. In that, the subscript opt-uc of λ indicates that this scheme is optimal for the uncorrelated digital signals. Further, N0 is the power spectral density of white noise, Tb is the duration of transmitted noiseless pulse,

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A is the pulse peak amplitude, and, r0,r1 are the independent probabilities of a transmitted bit to be 0 or 1 respectively. However, in practice, there are numerous applications, reasons and situations in which correlations get induced in the adjacent binary bits1,2,12. Reasons include channel fading in wireless environment, pre-processing at the transmitter10,11. It is thus not an exaggeration to say that the digital signals reaching the receiver do have substantial correlations most often, in most practical systems.

In1,2, for the case when transmitted digital signals are correlated p-q signals, the STTS-MF receiver uses two thresholds denoted as λSTTS-c0 and λSTTS-c1 (referred as λopt-c0 and λopt-c1 in1, respectively) for the detection of the transmitted binary signals at the receiver. These two thresholds are expressed in Eqs. (7) & (8) in1, as simple, closed-form formulas. In the subscript of λ, STTS indicates that these thresholds are for STTS-MF receiver, while, ‘c0’ and ‘c1’ in the subscripts are used for indicating ‘case-0’ and ‘case-1’ respectively1,2. λSTTS-c0 is used as threshold when the previously recovered bit is 0, and λSTTS-c1 is used as threshold when the previously recovered bit is 11,2. With extensive simulation study and analysis, it was shown that the STTS-MF receiver performs substantially better than the traditional STS-MF receiver, when evaluated over considered parameter values of p,q and with SNR > -1.0 dB, for the case when correlated p-q signals are transmitted1,2. Better performance of STTS-MF receiver over STS-MF receiver was observed for the values of SNR in the range -1 dB ≤ SNR ≤ +8 dB. For the smaller values of SNR in the range SNR < -1 dB, it was observed that STTS-MF receiver performs worse than the STS-MF receiver, producing BERRF > 1.0. As per the objective of this paper, using extensive simulations which are carried out in the later part of this paper, it is shown that CTTS-MF receiver, proposed in this paper, performs substantially better than both the STTS-MF receiver and the STS-MF receiver, for all considered values of parameters p,q and SNR, including very small SNR levels. The CTTS-MF receiver has been introduced in the following section.

Proposing the CTTS-MF receiver from the

Concept of the STTS-MF receiver

The STTS-MF receiver performs better than the STS-MF receiver for SNR > -1 dB, and worse for SNR < -1 dB. This warrants to re-examine the formulas obtained for the two thresholds and their reasonableness. We know p is the probability that the

next incoming bit is 0 if the current bit is 0. However the threshold in Eq. (7) of1 was obtained from Eq. (1) of1 by “approximating” the probability of the next bit being 0 as p, if the current bit is detected-as-0 at the receiver. Please note that this becomes only an approximation, but not completely accurate. That is because, if the current bit is detected-as-0 at the receiver after comparing with appropriate threshold, it does not mean that the current bit is actually 0, indeed there exit two different possibilities – (i) the current bit may actually be 0 and detected-as-0, (ii) the current bit may actually be 1 but detected-as-0. Similar inaccuracies exist in Eq. (8) of1, also. Achieving accurate formulas for the two thresholds, in this framework, should then certainly improve the performance of the receiver. That is precisely the crux in the development of the CTTS-MF receiver. To illustrate further, let us define the following. Let x (0 or 1) be a transmitted bit which is detected as y (0 or 1) at the receiver. If y = x, there is no bit-error. Let BER01 = Pr(y = 1/x = 0) = Probability that a transmitted 0 bit is (incorrectly) detected-as-1, BER10 = Pr(y = 0/x = 1) = Probability that a transmitted 1 bit is (incorrectly) detected-as-0. This gives rise to, BER = r0BER01 + r1BER10 where BER is the Bit-Error-Rate. Also, 1 – BER01 = 1 – Pr(y = 1/x = 0) = Pr(y = 0/x = 0) and Pr(y = 1/x = 1) = 1 – BER10. We know in1, for a p-q signal, the unconditional probability of a randomly chosen bit to be 0 and 1, denoted as r0 and r1 respectively, are given in Eq. (4) of1 as functions of the parameters p,q. We have used the necessary Bayesian Analysis of the situation & succeeded to derive thresholds for the CTTS-MF receiver as,

Y

YATN

bcCTTS 1

ln4

= 00 , where

01 10

01 10

(1 )[(1 ) (1 )][(1 )(1 ) (1 )]

q BER p BER pYq BER BER p

... (1)

Z

ZATN

bcCTTS

1ln4

= 01 , where

10 01

10 01

(1 )[(1 ) (1 )][(1 )(1 ) (1 )]

p BER q BER qZp BER BER q

... (2)

When STS-MF receiver is used, required λopt-uc can be obtained from Eqns. (5) and (6) of1. The unknowns, λCTTS-c0 and λCTTS-c1 still cannot be found explicitly from Eq. (1) and (2) since explicit expressions for BER01 and BER10 in terms of the system parameters are not available. We present an algorithm to determine λCTTS-c0 and λCTTS-c1 iteratively and accurately in order to calibrate the CTTS-MF


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receiver. This algorithm involves computation of BER01 and BER10 (or, λCTTS-c0 and λCTTS-c1) iteratively. Once the receiver is calibrated by obtaining accurate values for the two thresholds, it can be used to detect the incoming digital data contaminated by noise, with the same parameters. The following procedural steps are used in order to calibrate the CTTS-MF receiver by finding λCTTS-c0 and λCTTS-c1 iteratively. Iterative procedures with fixed number of iterations, alternatively, those with appropriate termination conditions are possible. We have used the former one since we have been successful with it, in our vast amount of experimentation. The iterative procedure that we have used is given below, STEP-wise. STEP-1: Signal Parameters N0, A, Tb, p, q, SNR are

given. Notes: It is intended to calibrate the CTTS-MF

receiver by computing λCTTS-c0 and λCTTS-c1 iteratively and accurately. Once calibrated, the new receiver can be used for detecting noise-corrupted binary signals with the same given parameters. STEP-2: Initialization of the values of the two

thresholds of the CTTS-MF receiver is done in this step, as λCTTS-c0 = λCTTS-c1 = λopt-uc. These are only initial values of the two thresholds. Thus, the CTTS-MF receiver that is to be designed, is initialized.

STEP-3: Initialize the Integer ITR (Number of the Iteration) as ITR = 0.

STEP-4: Binary p-q signal is generated with the given signal parameters, A, Tb, p, q. Please note this is for the purpose of calibration of the CTTS-MF receiver.

STEP-5: The signal is passed through AWGN of the given parameters, N0, SNR.

STEP-6: The noise-contaminated digital signals are fed to the CTTS-MF receiver, which is being designed and calibrated.

STEP-7: Let the CTTS-MF receiver process the noise-contaminated digital signals that are received.

STEP-8: Comparing the detected digital signals with the generated digital signals, the errors BER01 and BER10 are computed.

STEP-9: In this step, we need to compute the new values of λCTTS-c0 and λCTTS-c1 from signal parameters and the new values of BER01, BER10, using Eqns. (1) and (2). When we used the BER01, BER10 values obtained in STEP-8, we have faced the problem of highly zig-zag convergence.

Since smooth convergence is desirable, we have used a “weighted average” of the BER01 obtained in STEP-8 of the current iteration and the BER01 obtained in the previous iteration. With this we achieved smoothing of the procedure. Thus, the new BER01 is taken as,

where is the BER01 computed in this STEP, is the BER01 computed in the previous STEP, is the BER01 computed in STEP-9 of the previous iteration, and 0 < ω < 1.0 is an arbitrary weight. Similar procedure is used for the other error as,

. With this, λCTTS-c0 and λCTTS-c1 are now computed using Eqn. (1) and (2). The newly computed thresholds will be used for the next iteration, if any. Set ITR = ITR+1, and go to STEP-4 (alternatively, to STEP-7 in which case the original noise-contaminated signal is unchanged) if ITR < FIXITR where FIXITR is the fixed number of iterations used. Notes: We have calibrated the receiver using ω =

0.5 and with fixed number of iterations, FIXITR = 10. If ω is reduced further, then the convergence was smoother but the number of iterations required has increased. Thus, a trade-off study based on various values of ω may be useful for perfecting the iterative procedure towards faster and smoother convergence. STEP-10: The receiver is now calibrated. Use it

for detection of signals with the same parameters N0, A, Tb, p, q, SNR as those used for calibration of the receiver. Numerical studies concerning performance

evaluation of the calibrated CTTS-MF receiver and its performance-comparison with other two receivers, the STTS-MF and the STS-MF receivers, has been carried out in the succeeding sections of this paper, extensively.

Performance Evaluation of the Three Receivers

and Comparisons

Performance evaluation and comparisons of the three receivers, CTTS-MF, STTS-MF, STS-MF, are carried out in this section. PRBER (Percentage-Reduction-in-BER), as defined earlier in2, has been used as the performance-indicator for this study and analysis. Three different PRBERs have been defined here as:


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100*1100*1

STS

CTTSCTTSSTS

CTTSSTS BER

BERBERRFPRBER

100*1100*1

STS

STTSSTTSSTS

STTSSTS BER

BERBERRFPRBER

1 *100 1 *100CTTS CTTS CTTS

STTS STTSSTTS

BERPRBER BERRFBER

… (3) Here BERCTTS, BERSTTS and BERSTS are the Bit-

Error-Rates at the output of CTTS-MF receiver, STTS-MF receiver and STS-MF receiver respectively. Thus, is the percentage improvement gained (through reduction in BER) by CTTS-MF receiver over STS-MF receiver, while,

is such a gain by STTS-MF receiver over STS-MF receiver and is the gain of the CTTS-MF receiver over the STTS-MF receiver. For the simulation analysis study, the following considerations have been taken into account: MATLAB® 7.6.0.324 (R2008a) is used on Work

Stations-Z400 (HP make) for the simulation and analysis study.

Before transmission, the binary data is baseband modulated using polar NRZ (non-return-to-zero) line coding at the transmitter with peak pulse amplitude (A) of ±1 volt.

The channel is characterized by Additive-White-Gaussian-Noise (AWGN).

100,000 bits are used in each simulation run and the result is averaged over 30 such simulation runs using batch-means-method (BMM)13. This is done in both the cases, that is, (i) for calibration of the CTTS-MF receiver, and (ii) while using the calibrated CTTS-MF receiver on signals, for performance testing.

The performance evaluation study is carried out for the values of SNR in the range of, -10 dB ≤ SNR ≤ +10 dB. λCTTS-c0, λCTTS-c1 are the two thresholds in the CTTS-

MF receiver, and λSTTS-c0, λSTTS-c1 are those of the STTS-MF receiver, that is the only difference between these two receivers. For the given parameters p,q and SNR, how these thresholds of CTTS-MF receiver converge with iterations is an important study. For this, Threshold vs. Iterations curve is plotted in Fig. 1(a), for p = 0.8, q = 0.9, at SNR = -8 dB. Further, for given correlation parameter p,q, how these two thresholds of each of these two receivers change with SNR is an interesting investigation. For this, a Threshold vs. SNR (in dB) curve is plotted in

Fig. 1(b), for selected values of p = 0.1, q = 0.3 (representing low values of p and q) and p = 0.8, q = 0.9 (representing high values of p and q), for both the receivers. From Figs. 1(a) and 1(b), following conclusions can be drawn: From Fig. 1(a), it is clear that, for given values of

p,q and SNR, both the thresholds of CTTS-MF receiver change with every iteration till Number of Iterations (ITR) < 8 approximately. For ITR > 8, λCTTS-c0, λCTTS-c1 converge or become almost unchanging. Similar was the observation when this experiment was conducted, a large number of times, for a wide range of parameters p,q and SNR. Results for the same are not shown here because of page limitations. Our using fixed number of iterations with FIXITR = 10 for the calibration of the CTTS-MF receiver, can thus be justified, and this has been used in the later part of this paper for performance evaluation of the CTTS-MF receiver.

From Fig. 1(b), it is clear that, for the given p,q, thresholds of the CTTS-MF are different from those of the STTS-MF receiver in the range SNR ≤ 15 dB, that is λCTTS-c0 ≠ λSTTS-c0 and λCTTS-c1 ≠ λSTTS-c1. Further, for the case when p + q < 1.0, it can be seen that λCTTS-c0 < λSTTS-c0 and λCTTS-c1 > λSTTS-c1. For the case of p + q > 1.0, λCTTS-c0 > λSTTS-c0 and λCTTS-c1 < λSTTS-c1. This dependence (on p + q) is an interesting observation. It is worth reading about the “p+q statistic” in2, and its relation to PRBER.

For a given p,q pair, as SNR is increasing (see Fig. 1(b)), the two thresholds of both CTTS-MF and STTS-MF receivers are converging to a stable value. For SNR > 15 dB (approximately), the two thresholds of both the receivers become almost constant and equal to each other, that is λCTTS-c0 = λSTTS-c0 = λCTTS-c1 = λSTTS-c1, making the performance differences among the three receivers minimal or zero. Obviously, this is due to very low noise compared to the signal.

Fig. 1(c) is an elaboration of a plot in Fig. 1(b). This is for the case p = 0.8, q = 0.9 (p + q = 1.7, rather on the larger side). Notice that after about 6 dB onwards, λCTTS-c0 = λSTTS-c0, and λCTTS-c1 = λSTTS-c1. After around 24 dB onwards, it becomes λCTTS-c0 = λSTTS-c0 = λCTTS-c1 = λSTTS-c1 = λopt-uc.

Using those thresholds of Fig. 1(c), in the respective receivers, BERs and PRBERs are evaluated for the case p = 0.8, q = 0.9, and plotted with respect to SNR, as shown in Fig. 1(d). In Fig. 1(d), notice that, after 6 dB, BERCTTS = BERSTTS


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making This result is in agreement with the result of Fig. 1(c), that is λCTTS-c0 = λSTTS-c0 for SNR ≥ 6 dB. After calibration of the CTTS-MF receiver,

performance evaluation and comparisons need to be done for the three receivers. Towards that, binary signals are generated with various meaningful sets of input parameters p, q and SNR. STS-MF, STTS-MF receivers can be used straightaway for the signal recovery, however CTTS-MF receiver needs to

be calibrated first. Among several experiments that are possible, the first one we attempted is finding PRBER vs. SNR over a larger SNR range, that is, -10 dB ≤ SNR ≤ +10 dB. vs. SNR, vs. SNR and vs. SNR curves are plotted. Fig. 2(a) shows vs. SNR and vs. SNR plots for several cases in which p = q. Fig. 2(b) shows these plots for several cases in which p ≠ q.

Fig. 1—(a) Threshold-Convergence vs. Iteration plots for CTTS-MF receiver corresponding to p = 0.8, q = 0.9, SNR = -8 dB, (b)Threshold vs. SNR plots for CTTS-MF and STTS-MF receivers corresponding to different p,q values (c) Scaled version of Fig. 1(b), corresponding to p = 0.8, q = 0.9. (d) BER vs. SNR and PRBER vs. SNR plots for three receivers, corresponding to p = 0.8, q = 0.9, SNR ≥ 0 dB.


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For p = q (refer Fig. 2(a)) and p ≠ q (refer Fig. 2(b)), performance of the CTTS-MF receiver is substantially better than that of the traditional STS-MF receiver for SNR in the range -10 dB ≤ SNR ≤ +10 dB. This is evident since > 0. It is to be highlighted here that even for negative values of SNR in the range -10 dB ≤ SNR ≤ -1 dB,

the CTTS-MF receiver performs far better than the STS-MF receiver, while in the same range of SNR, the STTS-MF receiver performs far worse than the STS-MF receiver with < 0. This clearly demonstrates the advantage of the CTTS-MF receiver over the STTS-MF and the STS-MF receivers. Also,

Fig. 2— vs. SNR and vs. SNR plots for different p,q values with: (a) p = q, (b) p ≠ q, vs. SNR plots corresponding to high SNR values, for (c) p = q, (d) p ≠ q.


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is observed throughout the entire SNR range, for both p = q (refer Fig. 2(a)) and p ≠ q (refer Fig. 2(b)). Indeed, appears to be reducing with SNR.

As SNR increases, and increase in almost all the cases. The vs. SNR curve is more linear as compared to

vs. SNR curve. Higher linearity of the curve can be more helpful in the development of analytical model, for the fast prediction of the system performance, as in2.

When p + q = 1.0, the signal would correspond to uncorrelated r0-r1 signal as explained in1,2. In such a case, all three receivers would perform identically. For p + q < 1.0, with p = q or p ≠ q, it is found that increases with decrease in p + q. For example, compare the case of p = q = 0.15 (p + q = 0.3 < 1.0) with the case of p = q = 0.25 (p + q = 0.5 < 1.0) in Fig. 2(a), and also, compare the case of p = 0.1, q = 0.2 (p + q = 0.3 < 1.0) with the case of p = 0.2, q = 0.3 (p + q = 0.5 < 1.0) in Fig. 2(b). And, for p + q > 1.0 with p = q or p ≠ q, increases with increase in p + q. This phenomenon is further confirmed from our vast results that are not presented here, due to page-limitations. Thus, signal transmitted to the receiver with larger has an inherent advantage with the CTTS-MF receiver for all values of SNR, or with STTS-MF receiver for positive values of SNR measured in dB.

Referring to Figs. 2(a) and 2(b), the improvement gained by CTTS-MF receiver over STS-MF receiver appears to depend strongly on the sum p + q and not a lot on whether p = q or p ≠ q. For example, the curve for the two cases of p = q = 0.15 and p = 0.1, q = 0.2, appear almost the same.

In the displayed results of Fig. 2(a) and 2(b), for both the cases when p = q and p ≠ q, the maximum improvement gained by CTTS-MF receiver over STS-MF receiver is seen as 32% (approx.). However, as we see in the later Figures, for certain other values of the selected parameters, p,q and SNR, the improvement goes as high as 75% (approx.). Improvement corresponding to the cases of p = q is an important observation since they correspond to r0 = r1 = 0.5 which is desired in

many practical communication systems for the purpose of efficient clock recovery at the receiver.

Further, vs. SNR plots for several values of p = q and p ≠ q are shown in Figs. 2(c) and 2(d) respectively, this is done only for SNR in the range -2 dB < SNR < 10 dB, for obvious reasons. Further conclusions are:

At higher values of SNR, say, SNR > 8 dB (approx.), and, for all considered values of p,q, the

vs. SNR curves coincide with vs. SNR curves. This observation is valid for both the cases, p = q and p ≠ q. That means, for SNR > 8 dB, improvement gained by the CTTS-MF receiver over the STS-MF receiver becomes almost equal to the improvement gained by the STTS-MF receiver over the STS-MF receiver. This is also evident from the Figs. 2(c) and 2(d) in which (approximately) for SNR > 8 dB. In other words, for SNR > 8 dB, the thresholds of the CTTS-MF receiver tend to coincide with those of the STTS-MF receiver.

Even at very low SNR value of -10 dB, maximum improvement of around 8% reduction in BER by the CTTS-MF receiver over the STS-MF receiver is observed, among the displayed results, for the case when p = q = 0.15 (refer Fig. 2(a)) and p = 0.1, q = 0.2 (refer Fig. 2(b)). This is an important result, since, for a specified BER, this would lead to reduced transmitted power requirement as explained in1, making the work highly suitable for the domain Greening of Communication Technology. Having studied the performance-improvements and

comparisons over a large range of SNR, it would be useful to study the dependence of the performance-improvements on the signal correlation parameters p and q. As discussed in1, the unconditional bit probabilities r0, r1 of a p-q signal can be derived from the correlation parameters p,q, as in Eq. (4) of1, rather uniquely. But, the converse is not true. For example, the four {p,q} value sets namely, {0.15,0.85}, {0.05,0.8342}, {0.5,0.9188} and {0.95,0.9912}, give rise to the same (r0 = 0.15, r1 = 0.85), this can be verified through Eq. (7) of1. That is, several (infinite, indeed) pairs of p,q values can give rise to the same r0 and r1

1. This was illustrated in1 using the Eq. (9) of1. Thus, reasonably, for given r0 and r1, one could vary


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p and q while satisfying Eq. (9) of1. This way can be done in order to study the effect of variations in p and q on performance-improvements, without varying r0 and r1. PRBER vs. p curves are plotted in Figs. 3(a), 3(b), 3(c), 3(d), 4(a) and 4(b), for different values of r0 (0.1, 0.3 and 0.5) and SNR (-8 dB to +6 dB), for both CTTS-MF and STTS-MF receivers. In these plots, the parameter q varies as per Eq. (9) of1, for

varying p on X-axis. Figs. 4(c) and 4(d) show PRBER vs. q plot (q varying from 0 to 1 on X-axis, p determined from q by Eq. (9) of1) for both the CTTS-MF and STTS-MF receivers keeping r1 fixed at 0.7, for different SNR values. From Figs. (3) and (4), the following conclusions can be drawn: Performance-improvement of the CTTS-MF receiver

over the STS-MF receiver is substantially better than

Fig. 3—PRBER vs. p plots for CTTS-MF and STTS-MF receivers corresponding to: (a) low SNR values, r0 = 0.1, r1 = 0.9, (b) high SNR values, r0 = 0.1, r1 = 0.9, (c) low SNR values, r0 = 0.3, r1 = 0.7, (d) high SNR values, r0 = 0.3, r1 = 0.7. In these plots, parameter q is selected as per Eq. (9) of1.


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that of the STTS-MF receiver, for all examples considered in which SNR < 6 dB. There is not even a single example in which the STTS-MF receiver performs better than the CTTS-MF receiver.

In all the cases, for values of SNR in the range SNR < -2 dB, and . That means, in the range SNR < -2 dB, the STTS-MF receiver does not perform better than the STS-

MF receiver while, in the same range, the CTTS-MF receiver performs substantially better than the STS-MF receiver. This is the most important advantage of the CTTS-MF receiver. Thus, for SNR < -2 dB, the STTS-MF receiver is the most inferior among the three and can be discarded altogether for that range of SNR.

When p + q = 1.0, from Eqs. (4) and (5) of1, we get p = r0 and q = r1 as shown in1. In such

Fig. 4—PRBER vs. p plots for CTTS-MF and STTS-MF receivers corresponding to: (a) low SNR values, r0 = r1 = 0.5, (b) high SNR values, r0 = r1 = 0.5. In both (a) and (b) plots, parameter q is selected as per Eq. (9) of1. PRBER vs. q plots for CTTS-MF and STTS-MF receivers corresponding to: (c) low SNR values, r0 = 0.7, r1 = 0.3, (d) high SNR values, r0 = 0.7, r1 = 0.3. In both (c) and (d) plots, parameter p is selected as per Eq. (9) of1.


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situations as seen in Figs. 3 and 4, both and are equal to 0. This is because, p + q = 1.0 corresponds to the case when p-q signals become uncorrelated r0-r1 signals1,2, and in such a case the performances of both CTTS-MF receiver and STTS-MF receiver become equal to that of STS-MF receiver making .

Maximum improvement of 75% (approx.) reduction in BER is observed, among the presented results, in both CTTS-MF and STTS-MF receivers at SNR = 6 dB, r0 = 0.1 (refer to Fig. 3(b)). Of course, improvement is parameter-dependent, and that dependence can be an item for further investigation.

In Figs. 4(a) and 4(b), for r0 = r1 = 0.5, both the PRBERs are symmetrical around p = 0.5. From Eq. (5) of1, at r0 = r1 = 0.5 implies p = q. This is an important observation. This shows, for a given SNR, the PRBER of the case p = q = 0.5 + ε (ε < 0.5), and the PRBER of the case p = q = 0.5 – ε, would be the same. This is true for each of the two new receivers.

As observed from Figs. 3(c), 3(d), 4(c) and 4(d), at a given SNR, the vs. p curves for r0 = 0.3 are similar to the vs. q curves for r0 = 0.7. Same is true with vs. p and vs. q curves. That is to say, for a given SNR, the PRBER vs. p plot, with r0 = c (0 < c < 1.0), r1 = 1 – c, is similar to the PRBER vs. q plot with r0 = 1 – c, r1 = c. This observation is valid for both, CTTS-MF and STTS-MF receivers.

Even at low value of SNR (-8 dB), the maximum improvement of about 35% reduction in BER by CTTS-MF receiver over STS-MF receiver is observed, among displayed results, when r0 = 0.3, 0.5 (refer to Figs. 3(c) and 4(a)). As explained in1,2, this advantage can be translated into an equivalent reduction of transmission EMF intensity without reducing BER, hence is an important result in view of Greening of Communications Technology.

Conclusion

Different ways and means exist to improve the system performance and reliability at different layers of TCP/IP model. The method of BER-reduction by the development of new receivers that are more

efficient than the traditional MF-receiver, and their comparative performance evaluation is considered in this paper.

It was shown in1,2 that the STTS-MF receiver works substantially better than the conventionally used STS-MF (Single-Threshold-based-MF) receiver, when evaluated over considered ranges of parameters p, q and, for SNR > -1 dB1,2. BERRF (Bit-Error-Rate-Reduction-Factor)1, and PRBER (Percentage-reduction-in-BER)2 were used as performance-improvement indicator for that study and analysis. For smaller values of SNR (SNR < -1 dB), it was found that the performance of traditional STS-MF receiver was better than that of the STTS-MF receiver. The objective of this paper is to develop a new receiver that can work better than the STS-MF receiver even for SNR < -1 dB.

In this paper, a new receiver (CTTS-MF receiver) is proposed for efficient detection of the transmitted p-q signals. Performances of CTTS-MF, STTS-MF, STS-MF receivers are evaluated for SNR values in the wide range of -10 dB ≤ SNR ≤ +10 dB, and compared. Using extensive simulation results, it was found that for all the considered values of parameters p,q and SNR, the CTTS-MF receiver performs substantially better than both the STTS-MF receiver and the STS-MF receiver. Unlike the STTS-MF receiver, the CTTS-MF receiver performs better than the STS-MF receiver when SNR < -1 dB. STTS-MF receiver performs worst of three for SNR < -1 dB. The only advantage of STTS-MF over CTTS-MF is the computation of thresholds in former is simple, it can be used for SNR > 0 dB, if processing time is not available for the calibration of the CTTS-MF receiver. Availability of high speed processors, advancing parallel processing paradigm justify the use, implementation and operation of the CTTS-MF receiver.

This is an important achievement since, for given BER and channel B.W. utilization, the achieved results can in principle lead, alternatively, to reduced transmitted power and hence reduced EMF exposure to living beings especially in wireless communication systems. Thus, the work falls also in the domain of Greening of Communications Technology. The adopted methodology is applicable, with appropriate modifications, for the pass-band communications also.

Acknowledgement

This work forms a part of the ongoing Doctoral research work of the second author.


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