796 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 11, NOVEMBER 2005

Threshold Setting Strategies for a QuantizedTotal Power Radiometer

Janne J. Lehtomki, Student Member, IEEE, Markku Juntti, Senior Member, IEEE, Harri Saarnisaari, Member, IEEE,and Sami Koivu, Student Member, IEEE

AbstractWe analyze the impact of a uniform quantizer on thefalse-alarm probability of a total power radiometer. Different pos-sibilities to set the detection threshold are discussed. The main em-phasis is on methods that use the estimated noise level. In partic-ular, we analyze the cell-averaging (CA) constant false-alarm ratethreshold setting strategy. The numerical results show that the CAstrategy offers the desired false-alarm probability.

Index TermsConstant false-alarm rate (CFAR), detectionthreshold, quantization, radiometer, signal detection.

I. INTRODUCTION

THE DETECTION of unknown signal(s) is an importantgoal in electronic support (ES) [1] and radio monitoring.A recent detection application is finding signal-free frequencybands for cognitive radios [2], [3]. The (binary) detectionproblem can be formulated as choosing between the noise-onlyhypothesis and the signal(s)-and-noise hypothesis , i.e.,the goal is to decide between

(1)

where is the received real-valued signal sample at time in-stant , is the noise process sample, is a sample ofa typically unknown signal to be detected, and is the totalnumber of samples used for one decision.

The detection is usually based on some statistic that is com-pared to a threshold. If the threshold is exceeded, it is decidedthat is true. The probability of false alarm is the proba-bility that the decision statistic exceeds the threshold when onlynoise is present. Usually, the required probability of false alarmis specified, and the detection threshold is set so that the prob-ability of the false alarm does not exceed the desired value. Ifthe threshold is too high, false-alarm probability will be smallerthan the desired one, and detection performance suffers.

The likelihood ratio is the optimal decision statistic in theNeymanPearson sense. It often requires more information thanis available or is too complex to implement. The total power ra-diometer (or the energy detector) is a simple yet powerful alter-

Manuscript received March 18, 2005; revised May 16, 2005. This work wassupported by the Finnish Defence Forces Technical Research Centre. The workof J. J. Lehtomki was also supported by the GETA Graduate School and theNokia Foundation. The associate editor coordinating the review of this manu-script and approving it for publication was Dr. Olivier Besson.

The authors are with the Centre for Wireless Communications, University ofOulu, FIN-90014 Oulu, Finland (e-mail: janne.lehtomaki@ee.oulu.fi).

Digital Object Identifier 10.1109/LSP.2005.855521

native. It uses as a decision statistic the energy of the receivedsignal [4], i.e.,

The received signal is usually quantized in practical equipment,and the detection decision is made based on the quantizedreceived signal. Proper threshold setting may be difficult,especially when quantization is used and the noise variance isunknown.

Expressions for the mean and variance of the quantized totalpower radiometer outputs with a white Gaussian input, takinginto account the effects of the passband filter and saturation,have been derived in [5]. Although the goal therein was to mea-sure the signal power (as in [6]), the mean and variance can alsobe used for evaluating the detection performance of a quantizedradiometer with the normal approximation [2]. Quantized ra-diometer performance has been found with simulations in [7].The noise process before quantization was assumed to be whiteand Gaussian, and the detection threshold of an analog totalpower radiometer was used. Noise level was estimated basedon quantized reference samples in [8].

In cell-averaging constant false-alarm rate (CA-CFAR)detection [9], the detection threshold is the sum of the squarednoise-only reference samples multiplied by a scaling factor.The strategy used in [8] is similar to CA-CFAR but with adifferent scaling factor. Previously, the effects of quantizationin CA-CFAR signal detection have been theoretically studied in[10]. Therein, the order statistic (OS) CFAR detector was alsostudied. However, the analog square-law device was assumedso that the input follows the exponential distribution.

In this letter, the false-alarm probability of a quantized ra-diometer using different threshold setting strategies is analyzed.The input is assumed to follow the Gaussian distribution.

This letter is organized as follows. First, false-alarm proba-bility corresponding to a fixed threshold is found in Section II.Noise variance estimation based on quantized samples is dis-cussed in Section III. In Section IV, we use a randomized deci-sion rule that gives exactly the required false-alarm probability,assuming that the noise variance is known (or an accurate es-timate is available). The main focus of this letter is analyzingfalse-alarm probability of a detector that uses the CA-CFARstrategy. The analysis is carried out in Section V. Numerical re-sults are presented in Section VI, and the conclusions are drawnin Section VII.

1070-9908/$20.00 2005 IEEE

LEHTOMKI et al.: THRESHOLD SETTING STRATEGIES FOR A QUANTIZED TOTAL POWER RADIOMETER 797

Fig. 1. Uniform midriser quantizer with B = 2 quantization bits.

II. RADIOMETER WITH QUANTIZATIONThe quantized total power radiometer uses

(2)

where , where is assumed to be a deterministicmidriser quantization operator. It has output levels ,where the quantization levels are indexed with integer values

, is the number ofquantization bits, and is the quantization step so that the dy-namic range is , as illustrated in Fig. 1. Uniform quan-tization is widely used in practice due to its simplicity. In thecontext of the energy detection, half of the quantization levelsare wasted because the sign information is not needed.

The detection threshold is set according to the properties ofthe detector in the noise-only case . Noise is assumed tobe a discrete, zero-mean, white Gaussian random process withvariance . Thus, the probability that the th quantization levelof the midriser quantizer is chosen is [5], [6]

erf erf (3)

where erf is the error function, , and. If ,

erf . If ,erf . Let denote the indices

of the chosen quantization levels for the received signal in thecurrent observation interval, where , i.e,the quantized values are .

Instead of (2), it is more convenient to use the equivalent in-teger-valued decision variable

(4)

Let us denote the probability density function of with, i.e.,

(5)which can be easily found using the probability weights .Now, the probability density function of is

(6)

Let us assume that a fixed threshold is used with a quantizedtotal power radiometer. The equivalent integer-valued threshold

is , where is the ceiling function. Now,the probability of a false alarm corresponding to the threshold

is

(7)

One possibility is to use the detection threshold correspondingto an analog total power radiometer [7]. In that case, ,where , is the chi-square cumu-lative distribution function (CDF) with degrees of freedom,and is the desired false-alarm probability [11]. It wasfound in [7] that the actual probability of false alarm is differentfrom the desired value, especially when the number of quanti-zation bits is low. It is well known that quantization noise can beapproximated to be zero-mean Gaussian with variance[8]. In this case, , i.e., the threshold dependsalso on the step size.

III. NOISE LEVEL ESTIMATION BASEDON QUANTIZED SAMPLES

Noise level estimation based on quantized reference sam-ples is needed in various applications. For example, the de-tection threshold used in [7] requires knowledge of the noiselevel. Also, the randomized decision rule in Section IV requiresknowledge of the noise level. The mean of the squared quan-tized noise-only samples normalized with the (known) step size

is

(8)

where the index has been dropped because the samples arei.i.d. The maximum value of is (the quan-tization step size is so small that only the largest output valuesare selected), and the minimum value is 1/4 (the quantizationstep size is so large that only the smallest output values are se-lected). The variance of the input signal can be found with (see[12] for explicit results for a three-level quantizer)

(9)

The variance and, equivalently, can be estimated by substi-tuting the sample mean in place of the statistical mean, i.e., byusing , where is the number ofnoise-only reference samples.

IV. EXACT RANDOMIZED DECISION RULE

The decision variable (4) has a finite number of possibleoutput values. Therefore, using a randomized decision ruleis necessary for obtaining arbitrary false-alarm probabilities[13]. It is specified by the threshold and . If the observed

is larger than , an alarm always occurs. If , analarm occurs with probability , for example, a random numbergenerator is used within the intercept receiver. The properthreshold is the maximum value satisfying

(10)

798 IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 11, NOVEMBER 2005

and the corresponding randomization factor is [14]

(11)

By using the randomization factor (11), the exact required false-alarm probability is obtained, assuming that is known. If it isestimated, a large number of reference samples may be neces-sary so that false-alarm probability is close to the desired value.It may be desirable to choose so that the randomization factoris 0 or 1 (no randomization) [13].

V. PERFORMANCE ANALYSIS OF THE CA-CFAR STRATEGYIf the noise variance is unknown, and reference samples are

available, it is intuitive to use as a threshold , where

(12)

is the variance estimate based on (zero-mean) referencesamples , and is a scaling factor chosen so that, on the av-erage, the false-alarm probability has the desired value. Usingthe results in the CA-CFAR literature (see, for example, [15]and [16]), the correct scaling factor for a detector without quan-tization is found to be

(13)

where is the (Fisher) CDF. The scaling factor (13)gives exactly the desired false-alarm probability. When thenumber of reference samples increases, the scaling factorapproaches 1. In other words, when the number of referencesamples is large, less scaling is needed.

The scaling factor has been used in [8]. The theoreticalfalse-alarm probability corresponding to (assuming noquantization) is

(14)

When quantization is performed, the correct CA-CFARscaling factor depends on the unknown . Therefore, it is notpossible to always use the correct scaling factor. Instead,for example, the scaling factor or the scaling factor(13) is used. We evaluate the exact false-alarm probabilitycorresponding to an arbitrary scaling factor and some . Let

denote the indices of the chosen quantization levels for thereference signal, i.e., . Now

Fig. 2. False-alarm probabilities: N = 512 and N = 256.

where , and . Theconditional false-alarm probability, assuming , is

(15)

Now, the probability of false alarm can be found with

(16)

where denotes the density function of the foundsimilarly as the density function of [see (6)]. Together, (15)and (16) allow the calculation of the theoretical false-alarmprobability of the quantized total power radiometer using theCA-CFAR threshold setting method.

In [10], similar methods have been used for investigatingdata quantization effects for exponentially distributed input.The exponential distribution results, for example, from squaringthe envelope of a complex Gaussian random variable or fromsumming the squares of two i.i.d. real-valued Gaussian randomvariables. It was found that 612 quantization bits are typicallynecessary.

VI. NUMERICAL RESULTS

Fig. 2 shows false-alarm probabilities as a function of the dy-namic range when , , and .The theoretical false-alarm probability corresponds to ,and no quantization was calculated using (14). The CA refersto the scaling factor (13). It can been seen that when the CAscaling factor is used and , the system is operating prop-erly when dynamic range . When the CA scalingfactor is used and , it should be that .The scaling factor gives significantly higher than de-sired false-alarm probabilities. The results when(not shown here) were similar to those in Fig. 2, except that thefalse-alarm probabilities obtained using were closer tothe desired value. Let us now study the situation where the CA

is used and . Fig. 3 shows the false-alarm probabilitiesas a function of the dynamic range with different reference set

LEHTOMKI et al.: THRESHOLD SETTING STRATEGIES FOR A QUANTIZED TOTAL POWER RADIOMETER 799

Fig. 3. False-alarm probabilities: P = 10 , N = 512, and B = 3.

Fig. 4. False-alarm probabilities P = 10 , N = 512, threequantization bits (B = 3), and is assumed to be known.

sizes. It is observed that is not enough. With ,the results are quite close to the situation with , whichwas discussed above.

Fig. 4 shows the false-alarm probabilities when the noise vari-ance before quantization is assumed to be known. Thethreshold setting strategies analyzed were 1) the threshold corre-sponding to an analog radiometer [7], 2) the thresholdcorresponding to the Gaussian quantization noise assumption[8] with , and 3) the randomized decisionrule. In addition to the theoretical results found with the methodpresented in Section II, simulation results are shown. It is seenthat the threshold in 1) is not a good choice. The threshold in 2)yields surprisingly good performance, when the dynamic range

. The strategy in 3) gives exactly the de-sired false-alarm probability, no matter what the dynamic rangeis. With four quantization bits, the threshold in 2) gives goodresults when .

Let us assume that it is required that .For example, in the situation shown in Fig. 2 (with CA and

), the lower and upper bounds are and .The allowed values of are in the range .Equivalently, for a fixed step size, the noise standard deviationis allowed to have values in the range .An automatic gain control (AGC) device could be used for con-trolling the standard deviation so that it belongs to the allowedrange.

VII. CONCLUSION

A quantized total power radiometer was studied in twocases: 1) the noise power is unknown and 2) the noise power isknown. In case 1), the focus was on analyzing the CA-CFARthreshold setting strategy with different scaling factors. In case2), three different threshold setting strategies were studiedand compared. The threshold corresponding to the Gaussianquantization noise assumption performed surprisingly well.Typically, about three to four quantization bits are required forobtaining the desired false-alarm probability (both cases).

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tocThreshold Setting Strategies for a Quantized Total Power RadiomeJanne J. Lehtomki, Student Member, IEEE, Markku Juntti, Senior I. I NTRODUCTION

Fig.1. Uniform midriser quantizer with $B=2$ quantization bits.II. R ADIOMETER W ITH Q UANTIZATIONIII. N OISE L EVEL E STIMATION B ASED ON Q UANTIZED S AMPLESIV. E XACT R ANDOMIZED D ECISION R ULEV. P ERFORMANCE A NALYSIS OF THE CA-CFAR S TRATEGY

Fig.2. False-alarm probabilities: $N=512$ and $N_{R}=256$ .VI. N UMERICAL R ESULTS

Fig.3. False-alarm probabilities: $P_{{\rm FA},{\rm DES}}=10^{-Fig.4. False-alarm probabilities $P_{{\rm FA},{\rm DES}}=10^{-3VII. C ONCLUSIOND. C. Schleher, Introduction to Electronic Warfare . Norwood, MAA. Sahai, N. Hoven, and R. Tandra, Some fundamental limits on coS. Haykin, Cognitive radio: Brain-empowered wireless communicatiH. Urkowitz, Energy detection of unknown deterministic signals, J. E. Ohlson and J. E. Swett, Digital radiometer performance, IEM. A. Fischman and A. W. England, Sensitivity of a 1.4 GHz direcS. Koivu, H. Saarnisaari, and M. Juntti, Quantization and dynamiP. P. Gandhi and S. A. Kassam, Analysis of CFAR processors in noP. P. Gandhi, Data quantization effects in CFAR signal detectionR. A. Dillard and G. M. Dillard, Detectability of Spread-SpectruJ. R. Piepmeier and A. J. Gasiewski, Digital correlation and micP. Willett and D. Warren, The suboptimality of randomized tests Y. I. Han and T. Kim, Randomized fusion rules can be optimal in D. A. Shnidman, Radar detection probabilities and their calculatJ. J. Lehtomki, M. Juntti, and H. Saarnisaari, CFAR strategies