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IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 11, NOVEMBER 2005
Threshold Setting Strategies for a Quantized
Total Power Radiometer
Janne J. Lehtomäki, Student Member, IEEE, Markku Juntti, Senior Member, IEEE, Harri Saarnisaari, Member, IEEE,
and Sami Koivu, Student Member, IEEE
Abstract—We analyze the impact of a uniform quantizer on the
false-alarm probability of a total power radiometer. Different possibilities to set the detection threshold are discussed. The main emphasis is on methods that use the estimated noise level. In particular, we analyze the cell-averaging (CA) constant false-alarm rate
threshold setting strategy. The numerical results show that the CA
strategy offers the desired false-alarm probability.
native. It uses as a decision statistic the energy of the received
signal [4], i.e.,
Index Terms—Constant false-alarm rate (CFAR), detection
threshold, quantization, radiometer, signal detection.
The received signal is usually quantized in practical equipment,
and the detection decision is made based on the quantized
received signal. Proper threshold setting may be difficult,
especially when quantization is used and the noise variance is
unknown.
Expressions for the mean and variance of the quantized total
power radiometer outputs with a white Gaussian input, taking
into account the effects of the passband filter and saturation,
have been derived in [5]. Although the goal therein was to measure the signal power (as in [6]), the mean and variance can also
be used for evaluating the detection performance of a quantized
radiometer with the normal approximation [2]. Quantized radiometer performance has been found with simulations in [7].
The noise process before quantization was assumed to be white
and Gaussian, and the detection threshold of an analog total
power radiometer was used. Noise level was estimated based
on quantized reference samples in [8].
In cell-averaging constant false-alarm rate (CA-CFAR)
detection [9], the detection threshold is the sum of the squared
noise-only reference samples multiplied by a scaling factor.
The strategy used in [8] is similar to CA-CFAR but with a
different scaling factor. Previously, the effects of quantization
in CA-CFAR signal detection have been theoretically studied in
[10]. Therein, the order statistic (OS) CFAR detector was also
studied. However, the analog square-law device was assumed
so that the input follows the exponential distribution.
In this letter, the false-alarm probability of a quantized radiometer 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 probability corresponding to a fixed threshold is found in Section II.
Noise variance estimation based on quantized samples is discussed in Section III. In Section IV, we use a randomized decision rule that gives exactly the required false-alarm probability,
assuming that the noise variance is known (or an accurate estimate is available). The main focus of this letter is analyzing
false-alarm probability of a detector that uses the CA-CFAR
strategy. The analysis is carried out in Section V. Numerical results are presented in Section VI, and the conclusions are drawn
in Section VII.
I. INTRODUCTION
T
HE DETECTION of unknown signal(s) is an important
goal in electronic support (ES) [1] and radio monitoring.
A recent detection application is finding signal-free frequency
bands for cognitive radios [2], [3]. The (binary) detection
problem can be formulated as choosing between the noise-only
and the signal(s)-and-noise hypothesis
, i.e.,
hypothesis
the goal is to decide between
(1)
where
is the received real-valued signal sample at time instant ,
is the noise process sample,
is a sample of
is the total
a typically unknown signal to be detected, and
number of samples used for one decision.
The detection is usually based on some statistic that is compared to a threshold. If the threshold is exceeded, it is decided
is true. The probability of false alarm
is the probathat
bility that the decision statistic exceeds the threshold when only
noise is present. Usually, the required probability of false alarm
is specified, and the detection threshold is set so that the probability of the false alarm does not exceed the desired value. If
the threshold is too high, false-alarm probability will be smaller
than the desired one, and detection performance suffers.
The likelihood ratio is the optimal decision statistic in the
Neyman–Pearson sense. It often requires more information than
is available or is too complex to implement. The total power radiometer (or the energy detector) is a simple yet powerful alterManuscript received March 18, 2005; revised May 16, 2005. This work was
supported by the Finnish Defence Forces Technical Research Centre. The work
of J. J. Lehtomäki was also supported by the GETA Graduate School and the
Nokia Foundation. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Olivier Besson.
The authors are with the Centre for Wireless Communications, University of
Oulu, FIN-90014 Oulu, Finland (e-mail: janne.lehtomaki@ee.oulu.fi).
Digital Object Identifier 10.1109/LSP.2005.855521
1070-9908/$20.00 © 2005 IEEE
LEHTOMÄKI et al.: THRESHOLD SETTING STRATEGIES FOR A QUANTIZED TOTAL POWER RADIOMETER
797
, where
is the ceiling function. Now,
is
the probability of a false alarm corresponding to the threshold
is
(7)
Fig. 1.
Uniform midriser quantizer with B = 2 quantization bits.
II. RADIOMETER WITH QUANTIZATION
The quantized total power radiometer uses
(2)
where
, where is assumed to be a deterministic
,
midriser quantization operator. It has output levels
where the quantization levels are indexed with integer values
, is the number of
quantization bits, and is the quantization step so that the dy, as illustrated in Fig. 1. Uniform quannamic range is
tization is widely used in practice due to its simplicity. In the
context of the energy detection, half of the quantization levels
are “wasted” because the sign information is not needed.
The detection threshold is set according to the properties of
. Noise is assumed to
the detector in the noise-only case
be a discrete, zero-mean, white Gaussian random process with
variance . Thus, the probability that the th quantization level
of 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 the
, i.e,
current observation interval, where
the quantized values are
.
Instead of (2), it is more convenient to use the equivalent integer-valued decision variable
One possibility is to use the detection threshold corresponding
to an analog total power radiometer [7]. In that case,
,
,
is the chi-square cumuwhere
degrees of freedom,
lative distribution function (CDF) with
is the desired false-alarm probability [11]. It was
and
found in [7] that the actual probability of false alarm is different
from the desired value, especially when the number of quantization bits is low. It is well known that quantization noise can be
approximated to be zero-mean Gaussian with variance
[8]. In this case,
, i.e., the threshold depends
also on the step size.
III. NOISE LEVEL ESTIMATION BASED
ON QUANTIZED SAMPLES
Noise level estimation based on quantized reference samples is needed in various applications. For example, the detection threshold used in [7] requires knowledge of the noise
level. Also, the randomized decision rule in Section IV requires
knowledge of the noise level. The mean of the squared quantized noise-only samples normalized with the (known) step size
is
(8)
where the index has been dropped because the samples are
is
(the quani.i.d. The maximum value of
tization step size is so small that only the largest output values
are selected), and the minimum value is 1/4 (the quantization
step size is so large that only the smallest output values are selected). 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 substituting the sample mean in place of the statistical mean, i.e., by
, where
is the number of
using
noise-only reference samples.
(4)
IV. EXACT RANDOMIZED DECISION RULE
Let us denote the probability density function of
, i.e.,
with
(5)
which can be easily found using the probability weights
is
Now, the probability density function of
.
(6)
Let us assume that a fixed threshold is used with a quantized
total power radiometer. The equivalent integer-valued threshold
The decision variable (4) has a finite number of possible
output values. Therefore, using a randomized decision rule
is necessary for obtaining arbitrary false-alarm probabilities
and . If the observed
[13]. It is specified by the threshold
is larger than , an alarm always occurs. If
, an
alarm occurs with probability , for example, a random number
generator is used within the intercept receiver. The proper
is the maximum value satisfying
threshold
(10)
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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 falsealarm probability is obtained, assuming that is known. If it is
estimated, a large number of reference samples may be necessary so that false-alarm probability is close to the desired value.
It may be desirable to choose so that the randomization factor
is 0 or 1 (no randomization) [13].
V. PERFORMANCE ANALYSIS OF THE CA-CFAR STRATEGY
If the noise variance is unknown, and reference samples are
, where
available, it is intuitive to use as a threshold
(12)
(zero-mean) reference
is the variance estimate based on
, and is a scaling factor chosen so that, on the avsamples
erage, the false-alarm probability has the desired value. Using
the results in the CA-CFAR literature (see, for example, [15]
and [16]), the correct scaling factor for a detector without quantization is found to be
Fig. 2.
False-alarm probabilities:
N = 512 and N
where
, and
conditional false-alarm probability, assuming
(14)
When quantization is performed, the correct CA-CFAR
scaling factor depends on the unknown . Therefore, it is not
possible to always use the “correct” scaling factor. Instead,
or the scaling factor
for example, the scaling factor
(13) is used. We evaluate the exact false-alarm probability
corresponding to an arbitrary scaling factor and some . Let
denote the indices of the chosen quantization levels for the
reference signal, i.e.,
. Now
. The
, is
(15)
Now, the probability of false alarm can be found with
(13)
where
is the
(Fisher) CDF. The scaling factor (13)
gives exactly the desired false-alarm probability. When the
number of reference samples increases, the scaling factor
approaches 1. In other words, when the number of reference
samples is large, less scaling is needed.
has been used in [8]. The theoretical
The scaling factor
(assuming no
false-alarm probability corresponding to
quantization) is
= 256.
(16)
where
denotes the density function of the
found
[see (6)]. Together, (15)
similarly as the density function of
and (16) allow the calculation of the theoretical false-alarm
probability of the quantized total power radiometer using the
CA-CFAR threshold setting method.
In [10], similar methods have been used for investigating
data quantization effects for exponentially distributed input.
The exponential distribution results, for example, from squaring
the envelope of a complex Gaussian random variable or from
summing the squares of two i.i.d. real-valued Gaussian random
variables. It was found that 6–12 quantization bits are typically
necessary.
VI. NUMERICAL RESULTS
Fig. 2 shows false-alarm probabilities as a function of the dy,
, and
.
namic range when
,
The theoretical false-alarm probability corresponds to
and no quantization was calculated using (14). The CA refers
to the scaling factor (13). It can been seen that when the CA
, the system is operating propscaling factor is used and
erly when dynamic range
. When the CA scaling
, it should be that
.
factor is used and
gives significantly higher than deThe scaling factor
sired false-alarm probabilities. The results when
(not shown here) were similar to those in Fig. 2, except that the
were closer to
false-alarm probabilities obtained using
the desired value. Let us now study the situation where the CA
is used and
. Fig. 3 shows the false-alarm probabilities
as a function of the dynamic range with different reference set
LEHTOMÄKI et al.: THRESHOLD SETTING STRATEGIES FOR A QUANTIZED TOTAL POWER RADIOMETER
799
), the lower and upper bounds are
and
.
.
The allowed values of are in the range
Equivalently, for a fixed step size, the noise standard deviation
.
is allowed to have values in the range
An automatic gain control (AGC) device could be used for controlling the standard deviation so that it belongs to the allowed
range.
VII. CONCLUSION
Fig. 3.
False-alarm probabilities: P
= 10
, N = 512, and B = 3.
A quantized total power radiometer was studied in two
cases: 1) the noise power is unknown and 2) the noise power is
known. In case 1), the focus was on analyzing the CA-CFAR
threshold setting strategy with different scaling factors. In case
2), three different threshold setting strategies were studied
and compared. The threshold corresponding to the Gaussian
quantization noise assumption performed surprisingly well.
Typically, about three to four quantization bits are required for
obtaining the desired false-alarm probability (both cases).
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