SNR Estimation Techniques for Low SNR Signals

SNR Estimation Techniques for Low SNR Signals fred harris San Diego State University San Diego, CA [email protected]

Chris Dick Xilinx Corp San Jose, CA 95124 USA [email protected]

Abstract—Radio receivers contain a set of adaptive algorithms that estimate the received signal’s unknown parameters required by the receiver to demodulate the signal. Often missing from the standard parameter list is Signal to Noise ratio (SNR) or equivalently Eb/No. The SNR estimate is the ubiquitous scale factor associated with all maximum likelihood estimators. The SNR qualifies the signal quality letting the estimator algorithms know whether the observables are reliable, hence should make significant contribution to the estimate or are unreliable and should make limited contribution to the estimate. Error correcting algorithms also use SNR to set soft decision probabilities and likelihood ratios. Many SNR estimates are accurate at high SNR when we really don’t need the estimates and are inaccurate at low SNR when we have most need for them. This paper discusses two SNR estimator techniques which maintain estimation accuracy down to very low SNR values.

estimate includes carrier frequency and phase, symbol timing frequency and phase, channel impulse response inverse, automatic gain control level, and Signal to Noise ratio (SNR) or equivalently Eb/N0. The SNR estimate is the ubiquitous scale factor associated with all maximum likelihood (ML) estimators. We examine the ML timing recovery process. Let A y( ) be the amplitude of the matched filter output for a two level input signal of amplitude A with offset . The log likelihood function for this signal is shown in (1). We normalize the matched filter output to unity by dividing by its peak amplitude yPK=AT as shown in (2) to obtain the expression (3). Note that this normalization emphasizes the SNR scale factor embedded in the ML estimator. We now differentiate the log likelihood function with respect to and set it to zero to determine the requirement to obtain the peak of the matched filter output.

ln

Keywords: SNR Estimators, Low SNR Estimators, Unbiased SNR Estimators, software defined radio.

LNA

Carrier Waveform

VGA RF Carrier

Carrier

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A2T yk ( ) (2) N 0 / 2 yPK

ln cosh[vk ( )]

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IF Waveform Analog Bandpass Filter

2 A yk ( )] N0

AT 1 Ayk ( ) yPK N 0 / 2

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Radio receivers contain a set of adaptive algorithms that estimate the received signal’s unknown parameters required by the receiver to demodulate the signal. Figure 1 is a block diagram of a DSP based or software defined radio that shows the various adaptive algorithms performing the required signal processing tasks. The list of parameters that the algorithms Analog

ln cosh[ k

I. INTRODUCTION

Gain Control

( )

SNR SNR Estimate

Baseband Waveform Matched Filter SNR Timing Control

SNR Equalize Quantized Control Amplitude Amplitude Equalizer

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AGC Control SNR

Error Detect & Correct SNR

DDS

Phase Control

Figure 1. Receiver Block Diagram Containing SNR Estimator and Various Algorithms that Require SNR Estimate

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vk ( ) tanh[vk ( )] 0 k

The block diagram that reflects the relationship of (4) is shown in Figure 2. Here the output of the matched filter is scaled by SNR and the tanh of the scaled term is multiplied by the time derivate matched filter output. For large and small argument, the tanh(v) function can be replaced by v and by sign(v) respectively. These two operating conditions represent the small SNR and the large SNR approximations to ML phase locked loop (PLL). Of course we must first obtain the SNR estimate in order to move the tanh(v) to its correct operating range. y(nT+ T, )

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Figure 2. Maximum Likelihood Timing Recovery Loop We note that the SNR is not an average over all signal levels but rather the SNR per symbol. The noise per symbol may be constant but the signal level varies over the constellation space. Constellation points near the origin have a smaller SNR than those near the corners. We also note that the matched filter output should be scaled by the SNR but it is not necessary to scale the derivate by SNR since the product of (4) will go to zero without regard to scale factors on the derivative term.

SNR1

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At high SNR this is perfectly good SNR estimator. Figure 3. shows the conditional density functions for two operating regimes; high SNR and low SNR. By high SNR we mean the conditional densities do not cross the mid-point threshold and by low SNR we mean that the conditional densities do cross the zero threshold We see in Figure 3 that for high SNR the density after the magnitude operation is the same density as before. On the other hand, as low SNR, the density due to the magnitude operation contains aliases or folds area from the negative axis to the positive axis and no longer matches the original conditional density. Due to the folding, the sample mean will be too high and sample variance will be to low and the estimated SNR will greater than the true SNR. Figure 4 shows the sample histogram of the signal y(n) and of |y(n)| for a Gaussian noise sequence with mean m=1 and variance 2 =0.75. We see the mean of the |y(n)| is 6% above and p(y|x=-A)

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Figure 3. Conditional Densities of y(t) and of |y(t)| for High and Low SNR

II CONVENTIONAL SNR ESTIMATOR The standard approach to estimating the SNR of a signal is to estimate its first two moments, the sample mean and the sample variance of the signal’s conditional density function. The problem is we don’t know the condition and we may not know the sample time to observe the signal. Let us assume we know the sample time but not the condition. We suppress the condition by finding the sample mean and variance of the absolute value of the signal. The estimated SNR will be as shown in (5) and (6) with the expectation being replaced by the average sums as shown in (7).

SNR1

E 2{| y (n) |} E{[| y (n) | E{| y (n) |}]2 }

Figure 4. Sample Mean and Variance for y(n) and |y(n)| (5)

the variance is 12% below the same statistics for y(n). Figure 4 presents the bias in mean and the bias in variance as a function of true SNR. Figure 5 presents the estimated SNR formed with the biased mean and variance and the estimation error as a function of the true input SNR. We note the estimate starts significant deviation from the true SNR for values of input SNR below 8 dB.

standard deviation. The sampled skewness is defined in 9 where we have replaced the expectation with sample means formed by average sums.

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E{[ y (n) E{ y (n)}]3} E{[ y (n) E{ y (n)}]2 1 N

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Figure4. Bias in Sample Mean and Sample Variance as Functions of True SNR.

Figure 5. Comparison of Estimated SNR to True SNR and Error in SNR Estimate.

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The upper subplot of Figure 6 shows how skewness varies with input SNR. We see that the skewness is zero for large SNR when the input conditional density essential does not cross the threshold and is thus symmetric. Skewness is seen to rise for low values of SNR corresponding to more severe folding of the conditional density function tail and hence more pronounced asymmetry. The lower subplot of Figure 6 shows a simple approximation to SNR estimation error by five times skewness squared plus skewness. There are other approximations to the SNR estimate error as a weighted sum of powers of the skewness function. The one we show here is fairly accurate but does require a square root to form the denominator of (9). If we desire to avoid the square root another useful approximation to the estimation error is simply seven times skewness squared.

III CORRECTED SNR ESTIMATOR We now introduce an estimator which starts with the standard estimator and corrects the offset between the biased estimate of SNR and the true value of SNR. One method of correcting the offset is the use of a look-up table or a polynomial approximation to the offset error as a function of the biased estimate. Another option is to find a function which approximates the offset error as a function of true SNR. Such a method must be sensitive to the degree of asymmetry of the |(y(n)| density function that occurs as a result of the folded tails of the conditional density function. In seeking a statistic sensitive to asymmetry we quickly find the statistic skewness. Skewness is defined in (8) as the third central moment normalized by the cube of the

Figure 6. Sampled Skewness as function of input SNR and Simple Approximation of SNR Error with 5 2+ . Figure 7 shows the input output relationship of the original biased SNR estimator and the same

relationship for the corrected SNR estimator obtained by subtracting the correction of Figure 6 from the estimate from the biased estimate of Figure 5.

Figure 7.Biased SNR Estimator and Corrected SNR estimator as Functions of input SNR IV FFT BASED SNR ESTIMATOR A particularly accurate SNR estimator of SNR can be formed by an FFT based spectrum analyzer coupled to an ensemble averager. What the FFT based SNR estimator offers is an SNR estimate prior to any signal processing; we won’t require the matched filter to reduce the noise bandwidth to the signal bandwidth. The transform accomplishes the separation for us. Figure 8 shows the block diagram and signal processing flow of a standard spectrum analyzer that performs a sequence of overlapped sliding windowed Discrete Fourier Transforms as part of a spectrum analysis process. The bars above the variables remind us that these variables are N-tuples. d(n) d w (n)

Dw (k)

FFT Algorithm

P(k) 2

Post Detection Average Buffers

PAVG (k)

SNR

Feature Detection

multiple independent data collection intervals and in doing so significantly decrease the estimated standard deviation. Figure 9 shows a time signal at the output of a matched filter, the constellation spread due to additive noise, and the sample density functions of the signal. The SNR of this simulation is 9 dB. Figure 10 shows the spectrum of this same 9 dB SNR QAM signal. Shown are the raw spectrum with its obvious large variance spread and the averaged spectrum with its significantly reduced variance spread. Here we see the signal plus noise level is indeed 9 dB above the noise level. Here the noise is so far below the signal, the signal plus noise term is barely moved from the signal only level. In noisy environments the signal level would be estimated from the expression in (10) where the parameters are represented in linear levels as opposed to dB levels.

Figure 9. Time Series, Constellation, and Histogram of 9 dB SNR QAM Signal at Output of Matched Filter.

Window

Figure 8. Spectrum Analyzer Based SNR Estimator By the central limit theorem we know that Random signals, be they random modulation signals plus noise or noise only, contribute Gaussian distributed spectral components to the output of a transform. The raw power obtained from a single windowed transform has terrible statistics. This is because the raw power, the sum of the squares of I(k) and Q(k) at each frequency k, is Chi-Square with 2-degrees of freedom. ChiSquare, 2-degrees of freedom variable has a standard deviation equal to its mean. That can be interpreted as a zero dB SNR measurement. In a sense a Chi-Square, 2degrees of freedom measurement says “here is your data with an uncertainty spread of 100%” .This is not very useful information! We improve the quality of the estimate by performing an ensemble average over

Figure 10. Raw and Averaged Spectrum of 9 dB SNR QAM Modulated Signal Prior to Matched Filter.

Signallevel-est (Signal+Noise)level-est

(Noise)level-est

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Figure 11 presents the time profiles of three SNR estimators for the two test conditions of 6 dB SNR and for 3 dB SNR. The three estimators are tagged as SNR1

Figure 13. Time profiles for Three SNR Estimators, SNR_1 = Mean/Std, SNR_2 = SNR_1+Skew Correct, and SNR_3 = Spectral Ratio, for 0 dB SNR V CONCLUSIONS AND CLOSING COMMENTS Figure 11. Time profiles for Three SNR Estimators, SNR_1 = Mean/Std, SNR_2 = SNR_1+Skew Correct, and SNR_3 = Spectral Ratio, for 6 dB SNR and for 3 dB SNR. for our first mean to variance estimator, as SNR2 for the skew corrected SNR1 and SNR3 for the spectral ratio estimate. We first notice there are transients of 5-to-10 symbol time intervals till the estimators reach their steady state behavior. After the initial transient the estimates exhibit a time varying trajectory due to the variance of the finite averaging interval of the running average or leaky integrators. We the note that, as expected, the SNR1 estimate offers the biased estimate with the larger offset for the smaller SNR. We then note that the skewness corrected SNR and the spectral ratio SNR have smaller estimation errors with the spectral ratio performing slightly better than the skew corrected estimate. The SNR estimators are random processes with variance levels that vary with input SNR. We believe the estimators to be unbiased. For thoroughness, Figures 12 and 13 shows the spectrum and the SNR time profiles for the 0 dB SNR QAM signals.

We have presented a short discussion of why a modern radio receiver has need for a reliable estimate of SNR. We then presented a review of the most obvious SNR estimator, one that works very well at high SNR and then demonstrated why it becomes biased and unreliable at low SNR. The reason of course is the folding of the conditional density tail when we form the absolute value of the observed signal. This prompted us to examine a statistic that is sensitive to density function asymmetry. Skewness, the 3-rd central moment proved to be an appropriate statistic which could easily supply the correction to the first estimate. We also reviewed the benefit and the performance of FFT based spectral measurement based SNR estimators. We demonstrated that the latter two estimators performed well down to 0 dB SNR. As an aside we have demonstrated QAM receivers that can maintain phase and timing lock to below 0 dB SNR using the skew corrected mean/std SNR estimator. REFERENCES [1] David Pauluzzi and Norman Beaulieu, “A Comparison of SNR Estimation Techniques for the AWGN Channel”, IEEE Trans on Com, Vol. 48, No. 10, Oct. 2000,pp 1681-1691. [2] Richard Blahut, Chapter 8.5, “Modem Theory: An Introduction to Telecommunications”, Cambridge University Press, 2010. [3] fred harris, Course Notes: “Modem Design and Synchronization Techniques for Digital Receivers” [4] fred harris and Wade Lowdermilk, “Software Defined radio (part 22 in a Series of Tutorials), IEEE Instrumentation & Measurement Mag. Feb. 2010, pp. 23-32.

Figure 12. Raw and Averaged Spectrum of 0 dB SNR QAM Modulated Signal Prior to Matched Filter.

[5] C. Gong; B. N. Zhang; A. J. Liu, and D. X. Guo, “A High Accurate and Low Bias SNR Estimator: Algorithm and Implementation”, Radioengineering, Vol. 20, Issue 4, pp 976-981, 2011.