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A Carrier-Independent Non-Data-Aided Real-Time SNR Estimator for M-PSK and D-MPSK Suitable for FPGAs and ASICs Yair Linn, Member, IEEE

Abstract—We present a channel signal-to-noise ratio (SNR) -ary phase shift keying (M-PSK) and differential estimator for M-PSK. The estimator is non data aided and is shown to have the following advantages: 1) It does not require prior carrier synchronization; 2) the estimator has a compact fixed-point hardware implementation suitable for field-programmable gate arrays and application-specific integrated circuits; 3) it requires only 1 sample/symbol; 4) accurate estimates can be generated in real time; and 5) the estimator is resistant to imperfections in the automatic gain control circuit. We investigate the proposed estimator theoretically and through simulations. In particular, we investigate the required quantization necessary to achieve a good estimator performance. General formulas are developed for SNR estimation in the presence of frequency-flat slow fading, and specific results are presented for Nakagami- fading. The proposed estimator is then compared with other SNR estimators, and it is shown that the proposed method requires less hardware resources while, at the same time, providing comparable or superior performance. Index Terms—Application-specific integrated circuit (ASIC), binary phase-shift keying (BPSK), carrier loop, channel estimation, demodulation, differential phase-shift keying (DPSK), digital communications, efficient, estimation, fading, field-programmable gate array (FPGA), fixed-point, hardware, implementation, M-ary phase-shift keying (MPSK), modulation, Nakagami, phase-locked loop, quaternary phase-shift keying (QPSK), real-time, receivers, signal-to-noise ratio (SNR), synchronization, wireless.

I. INTRODUCTION N MANY modern communication systems, an accurate estimate (also called a signal-to-noise-ratio (SNR) estimate in this paper) is needed both as a monitoring aid and often also as an integral part in the receiver’s operation. As an example, we mention that some error correction decoders (e.g., turbo codes [1]) can make use of SNR estimates to improve their coding gain. Systems that employ diversity reception [2, Sec. 14.4] often require SNR estimates to assign relative weights to the data obtained from the various receivers. Another

I

Manuscript received December 06, 2007; revised June 03, 2008 and August 26, 2008. First published November 11, 2008; current version published July 06, 2009. This work was supported in part by the National Sciences and Engineering Research Council of Canada and in part by Universidad Pontificia Bolivariana, Bucaramanga, Colombia. This paper was presented in part as “A Real-Time SNR Estimator for D-MPSK over Frequency-Flat Slow Fading AWGN Channels” by Y. Linn at the Proceedings of the 2006 IEEE Sarnoff Symposium, Princeton, NJ, March 27–28, 2006. This paper was recommended by Associate Editor W. Utschick. The author is with the Faculty of Electronic Engineering, Universidad Pontificia Bolivariana, 56006 Bucaramanga, Colombia (e-mail: yairlinn@gmail. com). Digital Object Identifier 10.1109/TCSI.2008.2007064

example is the adaptive data/coding rate schemes where the data and/or coding rates are altered according to the SNR (e.g., [3] and [4]). The reader is referred to [5, Sec. 1.2] and the references therein for an extensive overview of these and other applications of SNR estimates in communication systems. In this paper, we present a robust real-time SNR estimator for differential -ary phase shift keying (D-MPSK). This estimator is a modification of the Linn–Peleg estimator [6], [7] for coherent M-PSK, and as such, it retains the advantages which were observed for the latter. Specifically, the estimator is shown to have excellent performance and an exceptionally compact hardware implementation which is quite suitable for implementation within field-programmable gate arrays (FPGAs) or application-specific integrated circuits (ASICs). The resilience to automatic gain control (AGC) circuit imperfections that was observed for the Linn–Peleg estimator [6], [7] is also observed for the current structure. There have been many SNR estimation algorithms proposed by various researchers in the past. For example, the reader is referred to [5] and [8]–[26]. We shall not address all of those estimators individually, since this would take an inordinate amount of space, and moreover, as we shall show, this is unnecessary. In lieu of that, we shall conduct our quantitative comparison versus some of the most widely used SNR estimation methods, estimator [9] and 2) the signal-to-varianamely, 1) the tion-ratio (SVR) estimator [9]. We shall supplement this quantitative comparison with qualitative comparisons versus other SNR estimators, which will show that the estimator proposed here possesses several important advantages over these previously proposed estimators. II. SYSTEM MODEL The signal and receiver characteristics are assumed identical to those in [6, Sec. II] and [7, Sec. II], except that, in this paper, we do not assume carrier synchronization. The reader is strongly urged to take a thorough look at [6] and [7], since notations and results from both papers will be used extensively. The baseband , with being PSK signal is the pulse shape and , , with . The modulated signal is . For the purposes of this paper, it is simpler to treat the system as a D-MPSK system, whether the actual system is M-PSK or D-MPSK. This is because a D-MPSK signal is identical to an M-PSK system in terms of the transmitted signal waveform, with the possible exception of the D-MPSK waveform being differentially coded, whereas,

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To highlight the motivation for the approach undertaken, note that the estimator of [6] and [7] cannot be used directly on a D-MPSK signal due to the lack of carrier coherence between the local and received carrier, which is a prerequisite for the SNR estimator of [6] and [7] to function. We therefore resort to using the estimator of [6] and [7] upon the normalized pseudocoherently demodulated M-PSK signal. As we shall see, this approach yields an estimator that preserves the innate advantages of the Linn–Peleg estimator [6], [7] but which, in contrast, dispenses with the requirement of prior carrier coherence. Fig. 1. Front end of D-MPSK receiver (simplified diagram).

sometimes,1 M-PSK systems do not include such coding. In terms of demodulation, the only difference between M-PSK and D-MPSK is in how these are demodulated (the former, coherently, and the latter, differentially). However, we do not assume carrier synchronization, and the proposed estimator is non data aided (NDA); hence, the issues of carrier synchronization and differential coding/decoding are not relevant for the discussion of the proposed SNR estimator. Therefore, it is admissible to treat the system as a D-MPSK system whether the actual system is M-PSK or D-MPSK. A simplified diagram of the front end of the D-MPSK receiver under discussion is shown in Fig. 1. The matched filter response , symbol timing synchronization is assumed, and the is Nyquist criterion for zero ISI [2, Sec. 9.2.1] is assumed obeyed at the outputs of the matched filters. From [6, Sec. II], we have and , and , with where (see [2, eq. (4.1-24)]). is the equivalent (AGC-controlled) – arm gain [6, Sec. II]. As noted in [6, Sec. II], is a slow function of time and is controlled by the AGC in order to attain the desired signal level at the inputs of the and samplers so that they are not saturated yet their full dynamic range is utilized. See also [28, Sec. 1.5] for a thorough discussion of the AGC and the parameter . The phase of the complex symbol is . We then have . Here, unlike in [6] and [7], we do not assume that , but rather , which is the standard assumption that is made in D-MPSK receivers (e.g., [29, Sec. 10.19]). III. MOTIVATION AND ESTIMATOR STRUCTURE A. Motivation The detection of D-MPSK signals is often facilitated by first generating a pseudocoherently demodulated M-PSK signal and then applying M-PSK decision regions upon . The motivation here is similar, but we add a twist: The idea is to use the Linn–Peleg NDA M-PSK SNR estimator [6], [7] upon a normalized pseudocoherently demodulated M-PSK signal . As we shall see, using instead of yields a simpler hardware implementation. 1In coherent M-PSK systems, sometimes, the signal is differentially coded, and then, after coherent demodulation, the received sequence is differentially decoded in order to resolve the inherent carrier synchronization ambiguity of 360/M degrees. See [27, pp. 203–205].

B. Estimator Structure and Operation Principle We define (Note: To avoid confusion with [6] and [7], throughout this paper, we use the superscript “D” in variables pertaining to D-MPSK structures). The is defined as Linn–Peleg detector [6], [7] applied to

(1) Here, we do not use as a lock detector [since there is no carrier phase-locked loop (PLL)] but rather only as an SNR esfor all ; therefore, theoretically, timator. Note that we could have defined . However, when quantization effects are taken into account, we see that does not always hold, and then, has distinct implementation and performance advantages (the same as outlined in [6, Sec. III-B]). In this paper, we present a general method for SNR estimation in the presence of fading. We assume that the channel is underspread, i.e., that the fading is frequency flat and slow, which means that , where is the channel’s coherence bandwidth, and that , where is the channel coherence time ([2, Sec. 14.1.1 and 14.2]). The channel coherence time is defined as the inverse2 of the fading channel’s Doppler spread , i.e., , as defined in [2, Ch. 14] and [30, Ch. 2]. We use the notation to refer to the instantaneous SNR and the notation to denote the average SNR (i.e., ). The conditional probability density function (pdf) of the SNR due to fading will be denoted as . For example, from [31, Table 2], we have for Nakagami- fading3 . We differentiate between two cases: 1) and 2) . For case 1), during the averaging over symbol intervals that is done in (1), the channel SNR will not 2The definition of the Doppler spread and, consequently, the definition of the coherence time are somewhat arbitrary ([2, Ch. 14]). However, as we shall shortly see, we only need an order-of-magnitude accuracy in the determination of T , and we only need to know whether 2NT T , 2NT T , or 2NT T = T (in which case, N needs to be increased to ensure that 2NT T ). 3In this paper, due to space and focus constraints, it is not possible to delve into the causes and derivation of fading channel statistics. For an introductory treatment of this subject, the reader is referred to [2, Ch. 14], and for an in-depth treatment along with many references, the reader is referred to [32].

1 

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LINN: CARRIER-INDEPENDENT NON-DATA-AIDED REAL-TIME SNR ESTIMATOR FOR M-PSK AND D-MPSK

N T

Fig. 2. Example figure illustrating fading and the relationship between 2 1 and for Nakagami- fading. The curve represents the instantaneous SNR ). Moreof the signal (also denoted in the paper by the notations and 1 over, shown are computation times for the following example cases: 1) 2  ; 2) 2 1  ; 3) 2 1 ' ; and 4) 2 1 ' . The average SNR is  = 20 dB, and we chose = 50 1 and = 2 as the Nakagami- parameter. As can be seen in the figure, for the , we encounter only the instantaneous SNR, for the case  case 2 , the SNR distribution encountered during the computation pe2  ( j ), while, for the case 2 , the encountered SNR riod is ' distribution is unpredictable.

m

T

T T N T T  T m m NT T NT T p 



E =N N N T T N T T T

NT

T

have changed much; hence, SNR estimation from will yield an estimate of the instantaneous SNR ratio . For case is much larger than the channel coherence time, 2), since will yield an estimate of the average SNR estimation from SNR ratio . Before continuing, it is important to note that the case (where “ ” means the same order of magnitude) is undesirable since, in that case, the SNR distribution during the estimator computation interval cannot be predicted (it would be impossible to know which part of the fading pdf we experience during the computation interval). Thankfully, can always be averted by choosing a large-enough , which ensures that [case 2)]. In general, however, we would ideally like to produce SNR estimates which are instantly available and can be fed in real time to the decoder, equalizer, or other receiver components which could make good use of them. Hence, ideally, we would be served by perfect knowledge of the instantaneous SNR ratio (which, if desired, could be averaged over time in order to produce an estimate of ). However, the estimation of is not always possible, due to the fact that may be too short as compared with the estimation period which is necessary in order to achieve an acceptable accuracy in the SNR estimation (see Section VII). Nonetheless, if , timely knowledge of the average SNR is often sufficient to facilitate substantial performance gains [1], [4]. In Fig. 2, we see an illustration of the cases , , and . SNR estimation is achieved following a procedure analogous to that in [7, eq. (9)], i.e., we estimate the SNR via the following. Case 1) : Instantaneous SNR is estimated via

Fig. 3. Fixed-point hardware generation of



or 

.

: The average SNR is estimated via

Case 2)

(3)

dB where

.

C. Hardware Implementation A fixed-point (two’s complement) hardware implementation of the estimator is shown in Fig. 3. The lookup tables (LUTs) require (4)

bits. See [6, Sec. III-B and III-C] and [7, Sec. III-B] for discussions applicable to LUT #3 and LUT #4, as well as for a should be a power of two. The use of discussion of why rather than significantly reduces the hardware resources needed to compute ; this is because the normalized constellation has less dynamic range (it is [ 1,1]). Therefore, this reduces the and required to achieve an acceptable degradation due to quantization. Finally, it is noted that the integrate and dump (IAD) module is very simple to implement: It is essentially a register with an adder in the feedback path and some control logic ([33, Fig. 18]). IV. CONDITIONAL DISTRIBUTION OF In this section, we shall derive the conditional probability distribution of . These stochastic properties will then be used to develop the SNR estimation method in Section V. Without loss of generality (see [6, Sec. IV]), we assume that , , whereupon

(2)

dB where

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(5)

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Let us define (similar to [6, eq. (32)])

independent ([6, Sec. III], [38, Sec. 7.1.1]), we have , i.e., (10)

(6) , then the physical meaning Since we assumed that , is clear: It is the phase error in the received symbol that of and (to see this, substican be attributed to tute in the expressions for and , and then, ). Since (5) has the same , we deform as [6, eq. (31)] with replaced by as defined in (6) is distributed the same as as rive that defined in [6, eq. (32)], namely, at , it has a Rician phase pdf given by [6, eq. (19)]

We now have the tools to investigate the distribution of A. Expectation of

Given

.

for

For , we can assume that the ratio is constant over the estimation interval and is equal to the instantaneous SNR , i.e., this is equivalent to a case where no fading is present during the estimator’s computation process. Hence,

(7) where stitutions show that

. Now, let us investigate

. Trivial sub-

where . Observe that , although the true phase is . We could have indeed performed the modulo operation and confined the range to ; this is, in fact, the approach undertaken in of [34]–[36]. In contrast, we choose to follow the approach of [37] because, as we and to maintain the pretence shall see, it simplifies the analysis. However, note that, since , this choice has no bearing upon the results. From (6), we then have

(11) At a high SNR, we use (10) to obtain a useful approximation ([39, (using eq. (15.73)])

(8) Let us define

; note that, since . The pdf of is easily found since it is a convolution of the distributions of and [since and are independent (see [6, Sec. III] and [38, Sec. 7.1.1])]. Namely (for ) ,

(12) (see Section II), it follows that , and the degradation in (11) and (12) due to the carrier frequency error is negligible. Thus, for simplicity, we set , although (11) and (12) provide an easy way to model small frequency errors. Equation (12) is quite useful because it allows the designer to predict the value of rather accurately by computing a single exponential. It is noted that, using Fourier analysis, exact closed-form expressions for can be found. This is done in Appendix A, and the expressions are given in (30)–(34). Since

(9) which is straightforward to evaluate numerically. Furthermore, a simple expression for the distribution at high SNR is easily obtained: From [6, eq. (22)] , ; now, since and since and are

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LINN: CARRIER-INDEPENDENT NON-DATA-AIDED REAL-TIME SNR ESTIMATOR FOR M-PSK AND D-MPSK

Fig. 4. f () for the case of 2N T

T

versus instantaneous SNR .

The plots of (11) and (12) and the simulated results for are given in Fig. 4; we see that (12) is an excellent approximation. The simulations in Fig. 4 which include quantization effects are quite realistic since they model the following AGC effects: 1) sampler input signal-level backoff (samplers are assumed to be driven at a root mean square (RMS) of 80% of the samplers’ full-scale voltage range) (see [28, Sec. 1.5]) and 2) clamping by the samplers when they are saturated. Hence, the simulations presented should be a good prediction of achievable results. If we assume that, for example, (which would imply an 8-bit SNR measurement4 in decibels), then, from (4), we have for the simulated quantized systems in Fig. 4 that , 30 720, 124928, respectively, all of which are very reasonable considering the amount of dedicated memory available in FPGAs (e.g., the various Xilinx Virtex families [40]) or which can be implemented in ASICs. Fig. 4 shows that, for low ’s, only coarse quantization is needed, while (as expected) higher ’s require finer quantization to achieve good agreement with the predicted value of . The hardware implementation will be discussed further in Section IX. B. Expectation of The expectation of as follows:

Given

for

conditioned upon

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Fig. 5. f ( ) versus   for Nakagami-m fading: Theory and simulations. Quantization effects are ignored.

(13) Due to the infinite number of possible fading distributions, we obviously cannot present results of (13) for all fading types. Rather, we shall investigate its behavior under Nakagamifading, which is a fading statistic commonly found in systems which use D-MPSK ([41], [42]). We again assume and plot theoretical and simulated results for (13) in Fig. 5 for various types of Nakagami- statistics (note that the “No Fading” curves also correspond to case 1), as verified by comparing Fig. 5 with Fig. 4). The theoretical results were derived using the procedure outlined in Appendix B. Comparing Fig. 5 with Fig. 4, we see that the effect of fading upon the curve of is rather mild. To evaluate the effects of the fading pdf upon the quantization requirements, we can plot (13) for the various quantizations in Fig. 6. As we see by used in Fig. 4. This is shown for comparing Fig. 6 with Fig. 4, the quantization which was sufficient for case 1) (as shown in Fig. 4) is also sufficient for case 2). Hence, there is no appreciable impact of fading upon the hardware resources required for the implementation of the proposed structure.

is [using (11)] C. Variance of It can be shown that, for slow fading, the cross-correlation coefficients of defined as

4Note that such a measurement could include digits after the binary point. For example, if we put the binary point to the left of the least significant bit, then we have, for an 8-bit output, the following: one sign bit followed by six wholenumber bits and one fractional bit, which would allow, in two’s complement notation, the representation of the interval 64 to +63.5 dB in 0.5-dB intervals, which is usually a quite sufficient range and quantization.

0

satisfy for

if

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if

and for

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Fig. 6. Demonstration of the effects of quantization on the measured value of ) for Nakagami-m fading with m = 2. f (

Fig. 7. [for case 1); these are the “No Fading” curves; see Section IV-A] ]. or  [for case 2) versus ^l

where (see Appendix C). Moreover, we can still use [6, Sec. III-D] to surmise that , . Now

would be undefined. This is solved by setting the output5 of for (not shown in Fig. 7). LUT #4 to (which This correctly reflects the SNR estimate for should be dB, i.e., no signal) within the limits of the available quantization. Note that the analysis of quantization effects for LUT #4 is not provided because this LUT contributes only a small amount to the total size of the estimator’s implementation. This can be seen in (4): For example, even for and , we the relatively robust quantizations of have the fact that LUT #4 is bits, which is quite small. Also, see footnote 4. There is a very strong relationship between Figs. 7 and 5. To see this, recall that, to graph the inverse of any function, all that one has to do is reflect the graph over the line . Thus, if we , then we arrive reflect the curves of Fig. 5 over the line at Fig. 7.

and using

and

, we have

(14) Finally, from the central limit theorem for ables [43, Th. 7.3.1, p. 196], we derive that

-dependent variis Gaussian.

D. Summary: Conditional Distribution of Let us unite what we have learned in the previous subsections. We have the following. 1) For case 1) : where is given in (11) and . 2) For case 2) : where is given in (13) and . V. LUTS FOR SNR ESTIMATION As noted in Section III-B, for case 1), the instantaneous SNR is estimated through (2), while for case 2), the average SNR is estimated through (3). Graphs of and are shown in Fig. 7. These curves are the value of LUT #4 in Fig. 3, and the curve to use would be chosen according to the fading characteristics of the channel. There is an additional small point that needs to be addressed: Theoretically, we can encounter negative values of , in which case and

VI. RESILIENCE TO AGC IMPERFECTIONS The proposed SNR estimator is resilient to AGC imperfections, i.e., imperfect AGC of the parameter . This is a direct result of the fact that the Linn–Peleg estimator is resilient to such effects [6, Sec. III-C], [7, Sec. V]. Similarly to [6] and [7], the limitations on this are twofold. First, the AGC must make sure that the samplers and their preceding signal chains are not driven to saturation. Second, the AGC must still ensure that the signal levels at the sampler inputs span enough sampler quantization bits so that quantization noise does not cause an unacceptable degradation. These are two relatively loose constraints upon the AGC, which essentially amount to stating that the AGC is performing its basic function and no more than that (i.e., so long as these conditions are fulfilled, we do not care about the AGC’s exact operating point, its dynamic range, or its transient 5This is the value of the LUT output if the representation is of whole numbers (not necessarily the case, see footnote 4). Generally, the idea is to use the lowest SNR expressible via the LUTs quantization.

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response). This resilience vis-à-vis the AGC is a particular advantage when fading is encountered, since in such a case, the AGC often acts in a decidedly nonideal manner.

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(from [9, eqs. (43) and (44)], with adaptation to the notations used in this paper):

VII. PERFORMANCE COMPARISON VERSUS OTHER BLIND 1-SAMPLE/SYMBOL SNR ESTIMATORS Although, as stated in Section I, it would be impossible, due to space constraints, to engage in quantitative comparisons with all other SNR estimators, we shall engage in quantitative comestimator and the SVR estimator parisons versus the which are analyzed in [9]. This is because the performance of the estimator) aforementioned estimators (particularly the has been found in [9] to be very good [in fact, the estimator is judged one of the “best” SNR estimators [9, Sec. V]. and SVR estimators are blind methods Moreover, the that require a sampling rate of 1 sample/symbol, which is exactly the characteristic of the proposed SNR estimator. Hence, a comparison is appropriate. and SVR First, following [9], let us review the estimators. A.

Estimator

. The estimator Recall that utilizes the second and fourth moments of the signal, defined as

(15)

(21) Equation (20), with the moments (21), is the estimator.

and

computed as in

B. SVR Estimator To define the SVR estimator, we define the auxiliary variable [9, eq. (45)] (22) The SVR estimator is defined as [9, eq. (48)]

(23) For the M-PSK/D-MPSK signals discussed in this paper, it can be shown [9] that and , so that (23) simplifies to (24) In practice, is estimated from a finite number of symbols, as follows (from [9, eq. (53)], with adaptation to the notations used in this paper):

It can be shown in [9, eqs. (35) and (36)] that we have (16) (17) where

is the signal power,

is the noise variance, is the signal kurtosis, and is the noise kurtosis (where ). We can solve (16) and to yield [9, eqs. (37) and (38)]

(25) Equation (24), with computed as in (25), is the SVR estimator. C. Comparison Metrics Versus the

and SVR Estimators

For the M-PSK/D-MPSK signals discussed in this paper, it can be shown [9] that and , and using (18) and (19), we find the SNR estimate as [9, eq. (39)]

In order to facilitate the comparison between the proposed estimator and the and SVR estimators, we shall use the same metrics used in [9]. The first of these metrics is [9, eq. (65)] the normalized mean squared error (NMSE), i.e., we shall , which is the NMSE for the proposed compare and SVR estimators, method, with the NMSEs for the which are and , respectively. The second metric is the normalized bias [9, eq. (68)], i.e., we shall compare with and . In order to help us evaluate the results, we shall also look at the Cramér–Rao bound (CRB) for the NMSE, which is [9, eq. (65)] as follows:

(20)

(26)

Obviously, in an actual implementation, the moments and are estimated from a finite number of symbols, as follows

The CRB is the lowest theoretically attainable NMSE for unbiased estimators. As for the best theoretically attainable bias, obviously that limit is zero.

we define (17) for and

(18) (19)

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l Fig. 8. NMSE comparison of estimation via ^ versus estimation via the M M and SVR estimators. M = 2, with 2N = 1024 symbols used to compute each estimator.

Fig. 9. NMSE comparison of estimation via ^ versus estimation via the l M M and SVR estimators. M = 4, with 2N = 1024 symbols used to compute each estimator.

In order to understand the significance of the NMSE metric, . it is helpful to note that the MSE is Thus, if we have, for example, an NMSE of , this would mean that the average RMS error of the estimation would be , which, in decibels, is a posidB and a negtive RMS deviation of dB, which ative RMS deviation of is a respectably small error. Even an NMSE of , which will have an RMS error of , which is a positive RMS deviation of dB and a negative RMS deviation of dB, is still satisfactory for many receivers.

D. Quantitative NMSE and Bias Results for the Case

First, let us discuss the NMSE results. As can be seen in Fig. 8, the proposed estimator outperforms the and SVR

Fig. 10. NMSE comparison of estimation via ^ versus estimation via the l M M and SVR estimators. M = 8, with 2N = 1024 symbols used to compute each estimator.

estimators at low and medium SNRs. As can be seen in Figs. 9 and 10, estimation via performs very respectfully at medium and high SNRs for QPSK and 8-PSK. At those SNRs, it is better than the SVR estimator and only slightly worse than estimator. We can see in Figs. 8–10 that, at high the SNR, estimation via tends to a high-SNR bound that is estimator. This is 50% higher than the estimation via the not surprising, since operates on the pseudodemodulated constellation, whose phase perturbation variance can be easily seen to be higher than that of the original constellation upon and SVR estimators operate (an easy way which the to intuitively see this is to compare the Gaussian distribution approximation of (10) to the lower variance distribution of [6, eq. (21)]). However, this disadvantage may be overcome if the used to compute is simply increased by 50%, with the tradeoff being a longer estimation period. Similarly, regarding operation at low SNRs, the NMSE may be decreased by increasing , with the tradeoff again being a longer estimation period.6 Such a tradeoff may be acceptable, since an increase in causes a negligible increase in the complexity of : The only increase in complexity is the augmentation of the accumulator register and adder in the IAD structure by a few bits—see Fig. 3 (for example, if we increase to , we would need to augment these structures by bits). Thus, if the designer of the system for which SNR estimates are produced is more concerned about hardware efficiency than about estimation for D-MPSK (or M-PSK in the latency, estimation via case of a lack of carrier synchronization) may be an attractive choice over the estimator (for a coherent M-PSK when the carrier PLL is locked, estimation via the method of [7] is a much better alternative to both, as seen in [28]). As for bias results, as Fig. 11 shows, all three methods (the proposed method and the and SVR estimators) have small biases that are very near the optimal value of zero for . Results for QPSK and 8-PSK are very similar and are thus omitted.



6One must take care, when increasing N , that the condition 2N T T still holds. If not, the designer must ensure that N is large enough so that 2N T T and then use (3) in order to estimate the average SNR.



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Fig. 11. Normalized bias comparison of estimation via ^ versus estimation l via the M M and SVR estimators. M = 2, with 2N = 1024.

Fig. 13. NMSE comparison of SNR estimation via estimation via ^ l

, 2N =

Fig. 14. NMSE comparison of SNR estimation via estimation via ^ l

, 2N =

1024, for M = 4 and Nakagami-m fading.

1024, for M = 8 and Nakagami-m fading. Fig. 12. NMSE comparison of SNR estimations via estimation via ^ l 2N = 1024, for M = 2 and Nakagami-m fading.

,

E. Quantitative NMSE and Bias Results for the Case As noted in Section III-C, when , the estimation is of the average SNR, and that estimate is obtained through dB . When fading is present, the SNR estimation approach taken in recent years has been to estimate the SNR using Viterbi algorithm-based [4] or expectation-maximization-algorithm-based estimators [18], [19]. Obviously, such estimators have many orders of magnitude more complex than the proposed estimator. As for the previously studied blind 1-sample/symbol moment-based estimation, it has been documented that the performs very poorly in fading conditions [19], and the same has been seen by the author regarding the SVR estimator. In order to nonetheless evaluate the effects of fading upon the proposed estimator’s NMSE, simulation results are obtained for the case of Nakagami- fading for various values of , and the NMSE is compared with the NMSE without fading. This is shown in Figs. 12–14.

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We see from Figs. 12–14 that, over much of the SNR range of interest, an increase in the NMSE is observed (particularly at high SNR, although this is not particularly problematic since SNR estimates usually need not be very accurate at high SNRs, since only minor performance gains of coders/equalizers are achievable at high SNRs from precise knowledge of the SNR [1]). Moreover, as noted in Section VII-D, the NMSE can be rethat is duced simply by increasing the number of symbols used in order to compute the estimate, which will have a negligible effect on the estimator’s complexity. As for the biases of estimation via under fading conditions with , the bias of the proposed method is very near zero for all fadings and modulations, similar to the bias curve shown in Fig. 11. Thus, for brevity, those curves are omitted from this paper. F. Qualitative Comparisons versus Other Estimators Several additional SNR estimation methods are presented in [5] and [8]–[26]. While a quantitative and exhaustive comparison versus all of those methods cannot be undertaken in this paper due to space constraints, this is, moreover, unwarranted since the qualitative characteristics of the proposed SNR estimator make it so attractive that it renders such a comparison

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unnecessary. To this end, it shall be commented that, regarding the cited aforementioned methods, the following are true. 1) Some are unique to a specific receiver structure. 2) Most require some form of symbol decisions to be made. 3) Most require more than 1 sample/symbol. 4) Some require prior carrier synchronization. 5) Most do not possess the same resilience to AGC effects. 6) Most importantly, none of those methods appears to have a hardware implementation nearly as compact as the one suggested in this paper. Thus, while quantitative comparisons with all of the aforementioned estimators are beyond the capacity of this paper, the qualitative advantages of the proposed method, particularly the very compact hardware implementation, make it a very attractive choice as compared with those estimators. VIII. COMMENTS REGARDING SIMULATION MODELS Nakagami- and Rayleigh channels in this paper were done using Matlab [44]. Specifically, the model used the “rayleighchan” Matlab function which uses the simulation model as described in [45, Sec. 9.1.3.5.2], which simulates Rayleigh channel behavior by passing white Gaussian noise through a filter whose magnitude corresponds to the square root of the Jakes Doppler spectrum response ([2, Sec. 14.1]). Nakagami- channel modeling was then obtained through the procedure outlined in [46]. IX. CONCLUSION In this paper, we have presented a new NDA SNR estimator for M-PSK and D-MPSK operating at 1 sample/symbol. It was found, both theoretically and through simulations that included AGC and quantization effects, that the estimator has a simple fixed-point hardware implementation that can be easily and compactly implemented within contemporary FPGAs or ASICs. The sensitivity of the proposed estimator with regard to the AGC loop was discussed, and it was found that the proposed SNR estimation method is quite resilient to AGC imperfections. Quantitative comparisons were conducted and SVR estimators using the NMSE metric, versus the and NMSE results were also provided for the estimator’s performance in Nakagami- fading. In terms of speed and accuracy performance of the estimator, the proposed estimator and SVR estimators was found to be better than both the for . For and , the proposed estimator was found to be a competitive estimator as compared with the and SVR estimators, particularly at medium and high SNRs, and the disadvantage at low SNRs could be remedied by increasing the estimation interval (at the expense of additional latency but very little complexity penalty). While fading had an impact upon the NMSE performance of the proposed estimator, it was found once again that increasing the estimation interval allowed for that accuracy to be recuperated, with the price of an additional estimation latency but very little additional hardware complexity. For all of the preceding reasons, the SNR estimation method proposed in this paper has immediate applications in contemporary D-MPSK communication systems as well as in SNR estimation for coherent M-PSK systems when carrier synchronization has not yet been achieved.

Finally, it is mentioned that the techniques of this paper (i.e., the extension of the Linn–Peleg detector to D-MPSK constellations) can conceivably be generalized for SNR estimation for PSK modulations (defined in [38, Ch. 8.1]), an issue that is currently being explored by the author. However, this issue is beyond the scope of this paper. APPENDIX A CLOSED-FORM EXPRESSIONS FOR Through the use of Fourier analysis, closed-form expressions . To do so, we first redefine for may be attained for over the interval convenience the density function (this is possible because the true phase is in the interval , as noted in Section IV). We name this distribution (where ), and it is given by [38, p. 441, eq. (7.3)]

(27) With this definition of , we find from (11) that (assuming that, for simplicity, ) (28) ). Now, (Note that the limits of the integral in (28) are because the domain of is finite, the periodic extension of can be expressed as a Fourier series, i.e.,

where, using the fact that

is even (29)

(See [49, Appendix 4A]). From comparing (29) with (28), we . The coefficients were see that computed in [49, App. 4A], from which it follows that (see [49, eq. (4.A.18)])

(30) where is the th-order modified Bessel function of the first kind (see [39, Ch. 24]). An inspection of (30) and [50, eq. (6)] shows that we have . Hence, using [50, eq. (6)], we have

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(31)

LINN: CARRIER-INDEPENDENT NON-DATA-AIDED REAL-TIME SNR ESTIMATOR FOR M-PSK AND D-MPSK

Expanding and then simplifying, we have

1535

In order to compute (35), several steps are needed. First, using the definite integral ([39, eq. (15.76)]), we define, as in [50, eq. (12)]

(36)

(32)

and . A slight difficulty is encounwhere and is a nonpositive integer, tered when using (36): When tends to positive or negative infinity ([39, Fig. 16.1]). This difficulty is easily circumvented by substituting instead of in (36), where , such as . This negligibly affects the result while evading the singularity . Therefore, we define the following function: of

From (32), we arrive at the following: (33)

(34)

(37) , , and . Using (32) where , in (35), with the aid of (36) and (37), we have (38), as shown at the bottom of this page. , we have For example, for

Expressions for can also be found. However, since those expressions are very tedious and since they can be arrived at following the same procedure outlined previously, they are omitted due to space constraints. Moreover, we note that the ap(see Fig. 4). proximation (12) is extremely accurate for APPENDIX B EXACT CLOSED-FORM EXPRESSIONS FOR PRESENCE OF NAKAGAMI- FADING

IN THE

(39)

Another application of (30)–(34) is in the computation of via (13). For certain fading distributions, this can lead . A case in point is again to closed-form expressions for the very important Nakagami- distribution. From (13), exact can be obtained for Nakclosed-form expressions for and all through the computation agami- fading for all of the definite integral (35)

and . If or [In (39), we assume that , we use or , respectively, in (39), can be obtained in a similar as per (37)]. Expressions for straightforward manner, although the resulting expressions are extremely long. It is important to note that, through the method for Nakagami- fading presented here, we can compute using only elementary and gamma functions, something that can be easily done using numerical computational packages such as Matlab. These exact expressions were used in the computation

(38)

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of Fig. 5, where we see that the theoretical results agree completely with those obtained through simulations.

APPENDIX C CROSS-CORRELATION COEFFICIENTS In this appendix, we discuss the cross-correlation coefficients for used in Section IV-C, and we prove that if if , where and is the cross-correlation coefficient of defined as (40)

Let us first assume that no fading is involved. We begin by noting that . Since we assumed that , this simplifies to . We note that the variare mutually independent. From this, it ables and will be independent immediately follows that , which means that for . for all Moreover, the fact that is a fundamental result from probability theory which can also be verified by inspection . Thus, the only remaining issue is of the definition of for , namely, the the characterization of characterization of the function . As for operation with fading, we note that, since we assumed slow fading, we can assume that the SNR remains constant over two symbol intervals. Thus, ipso facto, we will have the same for as for the nonfading correlation coefficient and , the arguments case. Regarding the cases that led to the conclusion that and for remain valid in the presence of slow fading. Hence, are unaffected by slow we conclude that the coefficients fading. Pursuant to the preceding analysis, we now engage upon characterizing for , which is the only thing which remains in order to fully qualify these variables. This can be done through stochastic simulations, i.e., through the computa(the notation that we use for for ) tion of using the simulated sequences of and . This is shown in Fig. 15. As seen there, we indeed have which was the assumption used in Section IV-C. As seen in Fig. 15, we have and . We can actually justify these asymptotic values theoretically, an endeavor that we shall presently undertake. First, let us take a look at the case of . In that limit, we see that there is no signal component in the values of and . It is then easy to show (and is also clear intuitively) that for tends to zero (i.e., lack of correlation). Let us now take a look at and prove that . Assuming that , we have (this can be easily seen from the distribution of (10)). Therefore, we derive from the Taylor series expansion

Fig. 15.  for jn . SNR

(= )

0

kj

= 1 (also denoted as  ()) as a function of the

that (omitting intermediate results)

At a high SNR, that (41)

. Thus,

(41) (see Section IV) so . Thus, from

and

(42) We now turn our attention to the numerator of (40). We as(a similar derivation will give the same sume that result for ). Since and since (15.73)])

, we have (using [39, eq.

(43) Furthermore

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(44)

LINN: CARRIER-INDEPENDENT NON-DATA-AIDED REAL-TIME SNR ESTIMATOR FOR M-PSK AND D-MPSK

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(49)

(50)

Let us define . At high SNR, we have all . Hence, since easy to show that we have Thus, (using [39, eq. (15.73)])

ACKNOWLEDGMENT

and for are mutually independent, it is and .

The author would like to thank, in particular, Prof. A. A. Monclou and Prof. S. M. Cristancho from Universidad Pontificia Bolivariana and the editor, Prof. W. Utschick, and the anonymous reviewers for their time, effort, and thoughtful comments, which helped improve the manuscript considerably. REFERENCES

(45)

(46) Substituting (45) and (46) into (44), we get

(47) Substituting (47), (43), and (42) into (40) and writing we get

(48) The limit is equivalent to . Taking this limit upon (48) and using L’hôpital’s rule, we have (49), as shown at the top of the page. Both the denominator and numerator still tend to zero; therefore, we use L’hôpital’s rule again to yield (50), as shown at the top of the page, which is what we set out to prove. To summarize this appendix, we have used stochastic simulations to show that for if if where . Furthermore, we used heuristic and mathematical derivations to justify the asymptotic values of , and . namely, that

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m

m

m

Yair Linn (M’01) received the B.Sc. (with honors) degree in computer engineering from Technion Israel Institute of Technology, Haifa, Israel, in 1996 and the Ph.D. degree in electrical engineering from the University of British Columbia (UBC), Vancouver, BC, Canada, in 2007. In 1996–2001, he was an Electrical Engineer with the Israeli Ministry of Defense, where he worked with the development, implementation, and deployment of wireless communication systems. He is currently a Visiting Professor at the Faculty of Electronic Engineering, Universidad Pontificia Bolivariana, Bucaramanga, Colombia. His research interests include synchronization in wireless receivers, estimation of wireless channel parameters, and implementation of real-time digital signal processing algorithms in field-programmable gate arrays.

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