2

Optimum Reception in White Noise

2.1 Summary
An active radar or sonar can be used to answer two questions that are, a
priori, very different:

• Is there a target?
• What are the range, the velocity, and other parameters of the target?

The first question corresponds to the problem of detection. There are

two hypotheses to choose from—target present or absent. The second ques-
tion raises a problem of estimation, or measurement, of the parameters of
a target assumed to be present. As the questions are different, it might be
expected that the structures of the appropriate radars or sonars would also
be different and that, in particular, the processing of the received signals
would be different. We shall see that this is not the case and that the structure
of the optimum receiver is the same—except for the final decision criterion—
whether the objective is detection or parameter estimation.
To determine the structure of the optimum receiver, an optimality
criterion must first be chosen. For the estimation problem (measurement
of, for example, range and velocity) many criteria can be considered, such
as the least maximum error (minimax criterion) and the least mean square
error. A natural choice consists of taking as the measurement result the value
␪ˆ of the parameter that would, with the highest probability, result in the
signal actually being received. In other words, given the received signal r ,

13
14 Principles of Radar and Sonar Signal Processing

the probability of receiving a signal such as r is calculated for each possible

value ␪ of the parameters, and the value for which the probability of receiving
r is the highest, is taken as the estimated value. This is a commonsense
approach, proceeded by a query: if ␪ was the true value of the parameter,
what would be the chances of the signal r actually being received? The best
choice for ␪ , that is, ␪ˆ , is then obviously that for which these chances are
the greatest.
This method for determining ␪ is called the maximum likelihood
method (the likelihood being defined as the probability of receiving r for a
value ␪ of the parameters). Apart from its intuitive character, this method
has an important advantage, related to the existence of the Cramer-Rao
bounds. For a given estimation problem, that is, for a law describing the
received signal as a function of the parameters ␪ to be measured, these
bounds define an absolute limit to the accuracy of any estimator, whatever its
principle, the only condition being that it gives on average the correct value
of ␪ , which seems the least one could ask. These bounds are calculated from
the likelihood mentioned previously. We then show that if a receiver achieves
this limiting accuracy, then the receiver built according to the maximum
likelihood criterion also achieves it. This very general result thus confirms
the value of this maximum likelihood criterion. In addition, we show that
in the case in which the noise is Gaussian, the structure of the receiver is
the same for a wide variety of criteria, including the maximum likelihood
and the least mean square error. This set of arguments—intuitive approach,
Cramer-Rao bounds, equivalence with other criteria—is a strong incentive
to use the maximum likelihood as a basic tool throughout the analysis of
radar processing, even though in some cases the least squares criterion gives
the same result using simpler calculations.
For the detection problem, a natural criterion appears rapidly if we
put ourselves in the situation of the radar operator, who from time to time
sees the equipment give an alert: detection.
This operator hopes:

• On the one hand, that the alert has not been triggered by mistake,
that is, that the level of false alarms is low;
• On the other hand, that the targets are detected as reliably as possible,
that is, that the detection probability is high.

These two hopes are obviously antithetical, since at the limit a zero
false alarm rate can be achieved by a receiver that has failed, at the price of
a detection probability that is also zero; whereas, a detection probability of
 Optimum Reception in White Noise 15

1 is obtained when the alert is triggered systematically at any time, at the

price of a false alarm probability of 1. The operator seeks a compromise
between these two extreme situations, and the natural approach consists of
specifying that the false alarm probability P fa does not exceed a given value—
dependent on the extent of the area to be covered but also on the cost of
the basic alert, which differs widely according to whether it is a radar triggering
the automatic opening of a door or a ballistic missile detection radar. Under
this condition of fixed P fa , the operator wants to obtain the highest detection
probability. The receiver optimization criterion is thus to maximize the
detection probability at a fixed false alarm probability (Neyman-Pearson
criterion).
This idea of a trade-off between the false alarm probability and the
detection probability is fundamental in radar and sonar, and it is therefore
important to know their orders of magnitude. For a radar to operate effi-
ciently, the false alarm probability must generally be on the order of 10−5
to 10−8 (10−3 to 10−6 in sonar), and the detection probability (for a single
scan of the target) must be on the order of 0.5 to 0.9 at maximum range.
Although these figures concern only average situations, they clearly reveal the
dissymmetry between the two requirements and the high stability necessary to
obtain such false alarm probabilities.
Under these conditions (maximum likelihood for estimation and Ney-
man-Pearson criterion for detection), we show that the structure of the
optimum receiver is the same for both problems, apart from the final decision
logic. This receiver comprises consecutively: correlation with a replica of
the expected signal (for each possible parameter ␪ ), quadrature detection,
normalization with respect to the replica power, and determination of the
maximum on all the obtained values corresponding to all the possible parame-
ters ␪ (in the case of estimation) (Figure 2.1), or comparison of these values
with a threshold, defined according to the required false alarm probability
(in the case of detection) (Figure 2.2).
Although not obvious, since the questions raised and the quality criteria
chosen were not the same, this structural identity between the two receivers—
detector or estimator—can be explained intuitively; in both cases, the receiver
attempts to measure the similarity between the expected situation (target of
parameter ␪ ) and the observed situation (signal r received). It measures this
similarity by correlating the received signal with the expected signal, performs
square-law envelope detection to eliminate the phase—unknown—of the
received signal, then normalizes so as not to distort this comparison by any
multiplication factor of the replica (to avoid being ‘‘attracted’’ by the high-
power replicas). Finally, it compares the results obtained for the different
16 Principles of Radar and Sonar Signal Processing

values of the parameters—estimation—or compares these results with a

threshold above which it is considered that the similarity to the transmitted
signal is not fortuitous but results from the presence of a target.
With respect to this intuitive argument, theory can define:

• The way to measure the similarity, by correlation;

• The way to eliminate the phase, by square-law envelope detection.

These two definitions are direct consequences of the Gaussian additive

noise assumption. For other noise assumptions, these operations must be
carried out in another way (correlation with the sign of the replica for
Laplacian noise, for example). It thus appears that the structure of the
receiver—square-law envelope detection, correlation, normalization, and
decision—does not depend on the form of the modulation introduced by
the parameters ␪ . Since it is assumed that the form is known, and it is thus
possible to build replicas of the expected signal, the structure of the receiver
is the same. This result was not obvious, since it might be expected that,
depending on the type of modulation introduced by the parameters (e.g.,
phase, frequency, linear, quadratic), the structure of the receiver would also
be different.
This argument thus gives the structure of the optimum receiver and
demonstrates a characteristic approach of radar or sonar, consisting of scan-
ning all possible hypotheses to make the decision (the approach will be
different for tracking radars or sonars, which evaluate the value of the required
parameter ␪ directly, in an explicit form, and no longer by scanning). This
approach thus generates an image of the observed scene in the range-velocity
plane, or in the elevation-bearing-range-velocity space if these are the required
parameters. This image is built up from elementary points whose brightness
represents the probability, in the widest sense, that the corresponding parame-
ters are those of the target.
Like any imaging equipment, the radar or sonar receiver has an equip-
ment function, which can be defined as the response to a point target of
unit amplitude in the absence of noise. Ideally, this response should be equal
to unity for the set of parameters corresponding to the position of the target,
and zero for all other sets of parameters. In practice, the inherent limitations
of the equipment—such as duration and form of the transmitted signal and
layouts of the transmitters and receivers—result in an equipment function
that, like the equipment function of an optical system, has two types of
faults:
 Optimum Reception in White Noise 17

• The peak located at the position of the target is not infinitely narrow,
but has the shape of a more-or-less pronounced maximum.
• Apart from this maximum, more-or-less bright secondary increases
may exist for some values of the parameters.

In radar and sonar, this equipment function is called the ambiguity

function (a strong secondary increase induces ambiguity in the position of
the target) and it provides an indicator of the quality of the radar or sonar
(Figure 2.3).
In particular, it is easy to comprehend that the accuracy of the parameter
measurements is directly related to the more-or-less pronounced character
of the main maximum. The theoretical analysis confirms this intuition, since
it demonstrates (once again by means of the Cramer-Rao bounds) that
when the additive noise is relatively low, the accuracy of the parameter
measurements is proportional to:

• The inverse of the curvature of the ambiguity function at its maxi-

mum, that is, to the ‘‘sharpness’’ of this maximum;
• The ratio of the energy of the transmitted signal to the spectral density
of the noise.

And this accuracy depends on the transmitted signal only by these

two quantities. The ambiguity function thus completely characterizes the
estimation problem, and this provides an initial explanation for the prime
role that ambiguity plays in radar. Other explanations will appear in the
next chapter, when the measured parameters (range and velocity) will be
defined.

2.2 Principle
Let us consider the general problem of the reception of a signal s (t , ␪ ), a
known (possibly vectorial) function of time and of an unknown parameter
(or parameter vector) ␪ , with additive Gaussian noise n (t ), whose characteris-
tics will be defined later. The ‘‘reception’’ of this signal may involve two
distinct problems [1, 2].
The first problem is detection, that is, the decision between two hypoth-
eses: one, H0 , corresponding to the absence of a signal [the signal received
then consists solely of noise n (t )], the other, H1 , for which the received
18 Principles of Radar and Sonar Signal Processing

signal is the sum of the desired signal s (t , ␪ ) for a given value of ␪ and the
noise n (t ):

再 H0 : r (t ) = n (t )
H1 : r (t ) = s (t , ␪ ) + n (t )
(2.1)

The second problem is estimation, that is, the determination of the

value ␪ of the parameter from the received signal r (t ):

␪ ? r (t ) = s (t , ␪ ) + n (t ) (2.2)

The optimum receiver is obviously defined with respect to an ‘‘opti-

mality criterion.’’ The purpose of this chapter is to build the receiver for
each of these two problems in terms of the maximum likelihood criterion, the
use of which will be justified in Section 2.2.1, and to define the performance
evaluation criteria of these receivers.

2.2.1 Estimation of a Parameter

Consider a signal s (t , ␪ ), a known function of time t and of an unknown
parameter ␪ , with additive noise n (t ).
The signal is described by

r (t ) = s (t , ␪ ) + n (t )

The estimator ␪ˆ of the value ␪ of the observation parameter of the

received signal r (t ) will be a function of the received signal and of the known
assumed dependence law s (t , ␪ ) of the signal as a function of the parameter
␪ . A natural optimality criterion consists of choosing the value ␪ˆ of the
parameter that results with the highest probability in the signal actually
received r (t ).
This involves calculating the probability p r /␪ of receiving the signal
r (t ) for each possible value of the parameter ␪ , and determining ␪ˆ as the
value of ␪ that maximizes p r /␪ for the signal r (t ) actually received:

␪ˆ : p r /␪ maximum/␪

If this probability p r /␪ can be calculated, the value ␪ˆ will be a function

of the received signal r (t ) and of the form of s (t , ␪ ) which will contribute
to p r /␪ .
 Optimum Reception in White Noise 19

The efficiency of an estimator can be measured by its variance

Var(␪ˆ − ␪ ) (i.e., variance of the deviation from the actual value ␪ ). In the
case in which the mean value of the estimator is correct on average—
unbiased—that is, where:

E(␪ˆ ) = ␪ , E: mathematical expectation

the lower limit of this variance is set by the Cramer-Rao bounds (demon-
strated in Appendix 2A):

冋冉 冊册
2
␦ log p r /␪
Var(␪ˆ − ␪ ) ≥ 1/E (2.3)
␦␪

or

Var(␪ˆ − ␪ ) ≥ 1/E 冋 ␦ 2 log p r /␪

␦␪ 2
册 (2.4)

These bounds thus constitute a lower limit to the variance of any

unbiased estimator. When they are reached, the estimator ␪ˆ is said to be
efficient.
It is then shown [(2A.1), described in Appendix 2A] that if an efficient
estimator exists, then the maximum likelihood estimator is efficient. This
very strong property is an additional reason for adopting this estimator
hereafter.

2.2.2 Simultaneous Estimation of Several Parameters

The previous properties can easily be extended to the case of several parameters
by defining ␪ as a vector of components ␪ i [see (2.1)]. The Fisher matrix
J, of elements J ij , is then defined as

J = [ J ij ] with J ij = E 冋 ␦ log p r /␪ ␦ log p r /␪

␦␪ i ␦␪ j 册 (2.5)

or

J ij = −E 冋 ␦ 2 log p r /␪
␦␪ i ␦␪ j 册 (2.6)
20 Principles of Radar and Sonar Signal Processing

If J ij is the current element of the inverse matrix J −1:

J −1 = [ J ij ] (2.7)

then

Var(␪ˆ i − ␪ i ) ≥ J ii (2.8)

2.2.3 Optimum Detection

The problem of detecting a signal s (t ) in noise n (t ) consists of choosing
between two hypotheses, depending on the received signal r (t ):

H1 : r (t ) = s (t ) + n (t )
H0 : r (t ) = n (t )

Various criteria can be used; the most natural, corresponding to the

approach actually used in practice, consists of considering that two different
types of error can be made:

• Deciding that there is a target (H1 ) when only noise is present (H0 ):
this is a false alarm, with probability P fa for a given decision rule;
• Deciding that there is no target (H0 ) when hypothesis H1 is in fact
true: this is a nondetection, of probability Pnd .

It is clear that in practice, the two errors are antithetical. Minimizing

the probability of one of these errors will automatically increase the probability
that the other will occur. The practical criterion used, that of Neyman-
Pearson, consists of setting the false alarm probability P fa to a specified value
␣ and minimizing the nondetection probability:

再 P fa = ␣
Min P nd
(2.9)

Let us consider the case in which the received signal is a vector r of

measurement samples. Defining a decision rule means defining an area Z 0
in the space of the vectors r such that if r lies within this area, hypothesis
H0 will be chosen, and if not, hypothesis H1 will be chosen. The false alarm
and nondetection probabilities are then given by:
 Optimum Reception in White Noise 21

P fa = 1 − 冕 p r /H0 (r )dr (2.10)

Z0

Pnd = 1 − 冕 p r /H1 (r )dr (2.11)

Z0

where p r /H0 , and p r /H1 , are the probability densities for r in the absence
and presence of a target, respectively.
The minimization problem (2.9) that the radar receiver must solve can
be expressed as follows, using the Lagrange multiplier:

再 Min F = P nd + ␭ ( P fa − ␣ )
with Pfa = ␣

by adjustment of the boundary of the area Z 0 . This is written:

Min F = ␭ (1 − ␣ ) + 冕 [ p r /H1 (r ) − ␭ p r /H0 (r )]dr

Z0

F will be minimal if Z 0 is chosen so that the term into brackets is

negative at all points r of Z 0 . The decision rule is therefore to decide H0
if:

p r /H1 (r )
<␭ (2.12)
p r /H0 (r )

the threshold value being set by the condition: P fa = ␣ .

The receiver formally compares p r /H1 and p r /H0 and decides that a
target is present if the former is more than ␭ times greater than the latter.
The structure of the receiver for detection is thus identical to that of the
receiver for estimation (calculation of p r /H1 , and calculation of p r /␪ ), the
determination of the maximum being replaced here by a comparison with
a threshold. This identity of form between the optimum receiver for detection
(according to the Neyman-Pearson criterion) and the optimum receiver for
estimation according to the maximum likelihood is another reason for adopt-
ing this maximum likelihood criterion.
22 Principles of Radar and Sonar Signal Processing

2.3 Optimum Receiver

2.3.1 Optimum Estimator

The optimum receiver is thus the one that maximizes the probability
p r /␪ (r ) with respect to ␪ for the received signal r ; the definition of the
optimum receiver therefore requires the calculation of this probability
p r /␪ (R ).
The received signal is given by:

r (t ) = s (t , ␪ ) + n (t ) (2.13)

Let us express the signals in the form obtained in Chapter 1:

s (t , ␪ ) = ␣ a (t , ␪ )e j␻ 0 t (2.14)

r (t ) = z (t )e j␻ 0 t (2.15)

n (t ) = b (t )e j␻ 0 t (2.16)

where ␻ 0 is the angular frequency of the carrier (the choice of this carrier
in the band occupied by the signal will be explained later), ␣ is an unknown
complex coefficient accounting for the attenuation and the phase-shift during
the two-way path, and a (t , ␪ ) is the complex envelope of the received signal
including any Doppler effect.
Let us assume that b (t ) is ‘‘band-limited complex white noise,’’ that
is, limited to a frequency band [−F max , +F max ] (bandwidth of the receivers),
with a correlation function B (t ):

B (t ) = N 0 sinc (2F max t )

where

sin ␲ t

冦
sinc t =
␲t (2.17)
N 0 = E | b (t ) |
2

where N 0 is the power spectral density of the noise.

 Optimum Reception in White Noise 23

Let us sample this signal at the rate given by Shannon’s criterion

⌬t = 1/(2F max ). The noise samples b (t i ) obtained are Gaussian and uncorre-
lated since B (k /(2F max )) = 0 for all nonzero integer values of k ; they are
therefore independent, and the probability density of the set of samples is
therefore the product of the probability densities of the individual samples:

⌳= 写 i
1
␲ N0 冋
exp − (z*i − ␣ *a i* (␪ ))
1
(z − ␣ a i (␪ ))
N0 i 册 (2.18)

Strictly speaking, as the noise spectrum is limited in bandwidth, the

duration of z (t ) is infinite. However, we shall assume that this duration is
finite (2K samples), and then make the number K tend to infinity. It can
be shown that, under these conditions of band-limited white noise, the
required convergence conditions are satisfied, because the expansion of z (t )
into a set of samples z (t i ) is a Karhunen-Loève expansion (Appendix 4A).
The maximum of ⌳ will be obtained by taking the logarithm and
retaining only terms that are functions of ␪ , that is, maximizing the following
expression:

+K +K
Max /␪ ∑ (z*i ␣ a i (␪ ) + z i ␣ *a i* (␪ )) − ∑ | ␣ a i (␪ ) | 2 (2.19)
−K −K

This contains scalar product expressions, which can be rewritten by

letting K tend to infinity and returning to continuous signals. The optimum
receiver must therefore determine the value of ␪ and ␣ , which maximize
this expression for a received signal r (t ) (or z (t ) after demodulation):

冕
Max /␪ , ␣ 2 Re[␣ z (t )a *(t , ␪ )]dt − | ␣ |
2
冕| a (t , ␪ ) | dt
2
(2.20)

冤| 冕
|| 冕
|冥
2 2

z (t )a *(t , ␪ )dt z (t )a *(t , ␪ )dt

Max /␪ , ␣ − 冕| a (t , ␪ ) | dt
2
␣−
冕| a (t , ␪ ) | dt
2
−
冕| a (t , ␪ ) | dt
2

Consequently
24 Principles of Radar and Sonar Signal Processing

|冕 |
2
z (t )a *(t , ␪ )dt
␪ˆ : Maximum/␪
冕|
(2.21)
a (t , ␪ ) | dt
2

␣ˆ =
冕 z (t )a *(t , ␪ˆ )dt

冕|
(2.22)
a (t , ␪ˆ ) | dt
2

In general, the value of ␣ is not determined by the radar, which just

determines ␪ , satisfying condition (2.21).
The expressions (2.21) and (2.22) thus constitute the solution to the
very general problem described by (2.2) of estimating the parameters of a
signal of known form and unknown amplitude and phase in the presence
of white Gaussian noise (narrowband). The most direct approach to this
estimation procedure is to give a set of discrete neighboring values and
determine the maximum with respect to ␪ by a systematic exploration of
all these values. The procedure, illustrated by the diagram in Figure 2.1,
consists of correlating the received signal z (t ) after demodulation of the
carrier e j␻ 0 t, with a set of replicas of the expected signal a (t , ␪ i ), detecting
and normalizing each channel, and choosing ␪ˆ corresponding to the highest
result.

Figure 2.1 Optimum estimator in white noise.

 Optimum Reception in White Noise 25

2.3.2 Optimum Detector

It is then easy to derive the optimum detection procedure from the estimation
procedure; given the analysis performed in Section 2.2.3, the structure of
the receiver is in fact identical. In other words, if one wants to detect a
target characterized by its given parameters, ␪ , p r /H1 (for this value ␪ of the
parameters) and p r /H0 must be calculated. In fact, p r /H1 is the quantity called
p r /␪ in the context of the estimation problem (probability of r if a signal of
parameter ␪ is present).
After demodulation, the signals can be expressed as follows:

s (t , ␪ ) = ␣ a (t , ␪ )e j␻ 0 t, ␣ unknown

r (t ) = z (t )e j␻ 0 t

n (t ) = b (t )e j␻ 0 t

In the sampled situation described in the previous paragraph:

p r /H1 = 写i
1
␲ N0 冋
exp − (z*i − ␣ *a*i (␪ ))
1
(z − ␣ a i (␪ ))
N0 i 册
In this same situation, p r /H0 is given by:

p r /H0 = 写
i
1
␲ N0
1
冋
exp − | z i |
N0
2
册
and the decision rule consists, in accordance with (2.12), of comparing the
quantity:

p r /H1
p r /H0

or its logarithm:

p r /H 1 +K +K
log = ∑ (z*i ␣ a i (␪ ) + z i ␣ *a*i (␪ )) − ∑ | ␣ a i (␪ ) | 2 (2.23)
p r /H 0 −K −K

with a threshold (function of the desired false alarm rate).

26 Principles of Radar and Sonar Signal Processing

Equation (2.23) can be recognized as the quantity whose maximum

had to be determined in the estimation problem (2.19).
Letting K tend to infinity, as previously, and returning to continuous
signals, we obtain as decision rule the comparison of the following quantity
with a threshold:

冕
2 Re[␣ z (t )a *(t , ␪ )]dt − | ␣ |
2
冕| a (t , ␪ ) | dt
2
(2.24)

This quantity cannot be used as such because it includes the unknown

quantity ␣ (it is assumed that ␪ is known; we are trying to determine, by
choosing between hypotheses H0 and H1 , whether a target of parameter ␪
is present). The problem is solved by using the maximum likelihood criterion
to estimate this parameter ␣ . If a target is present (hypothesis H1 ), then its
parameter ␣ is that given by the maximum likelihood criterion, that is,
maximizing, in accordance with the argument of the preceding paragraph,
the quantity (2.24) expressed as:

冤| 冕
|| 冕
|冥
2 2

z (t )a *(t , ␪ )dt z (t )a *(t , ␪ )dt

⇒ 冕| a (t , ␪ ) | dt
2
␣−
冕| a (t , ␪ ) | dt
2
−
冕| a (t , ␪ ) | dt
2

(2.25)

⇒ ␣ˆ =
冕 z (t )a *(t , ␪ )dt

冕|
(2.26)
a (t , ␪ ) | dt
2

The optimum detection rule then compares (2.25) with a threshold,

with ␣ = ␣ˆ

|冕 |
2
z (t )a *(t , ␪ )dt H1
⭵T
冕|
(2.27)
a (t , ␪ ) | dt
2 H0
 Optimum Reception in White Noise 27

and, if the hypothesis H1 is confirmed (target detected), the estimated

coefficient ␣ˆ of the received signal is:

␣ˆ =
冕z (t )a *(t , ␪ )dt

冕|
(2.28)
a (t , ␪ ) | dt
2

In practice, the target presence or absence question must be raised for

a set of possible values of the parameters ␪ ; consequently, the detection test
will be carried out for all these possible values of the parameters. This results
in a detector structure (shown in Figure 2.2) that is similar to that of the
estimator: correlation of the received signal z (t ) with a set of replicas
a (t , ␪ i ), square-law envelope detection and normalization of each channel,
and comparison with a threshold (the normalization of each channel being
equivalent to modifying the threshold value).
It is interesting to note that in the optimum receiver (for estimation
or for detection) with the structure described above:

• Square-law envelope detection is a consequence of the need to

estimate ␣ (or at least to eliminate the unknown ␣ ). Without this
need, the optimum receiver would simply calculate the real part of
the correlation between received signal and replica, in accordance
with (2.20). This same need for quadratic detection would result if

Figure 2.2 Optimum detector in white noise.

28 Principles of Radar and Sonar Signal Processing

we assumed a random signal s (t , ␪ ), which is another way of

modeling the complete lack of knowledge of the phase of the received
signal.
• The correlation, or matched filtering, is a consequence of the Gaussian
character of the noise. It is the Gaussian character that generates
the double product terms containing z i a i* (␪ ) in the function to be
maximized, (2.19).

Furthermore, the optimum estimator in Figure 2.1 determines the

maximum of the function (2.21) by systematic exploration of the parameter
␪ . This procedure may appear unsophisticated—and we shall see in our
analysis of tracking radars, that it is sometimes possible to calculate the
estimated value ␪ˆ in the form of an analytical expression—but it is natural
in radar and sonar, in which the question asked is ‘‘Where are the targets?’’
that is, in a situation where the following operations must be performed
simultaneously:

• Detection of any target in any possible position;

• Estimation of its position (parameter ␪ ).

2.3.3 Estimation Performance: The Ambiguity Function

Let us apply (2.21) to be maximized to obtain ␪ˆ in the case in which the
received signal corresponds to a value ␪ 0 of the parameters:

z (t ) = ␣ 0 a (t , ␪ 0 ) + b (t ) with | ␣ 0 | = A 0
2
(2.29)

The expression to be maximized with respect to ␪ becomes:

| 冕 冕 |
2
1
Q (␪ , ␪ 0 ) = ␣ 0 a (t , ␪ 0 )a *(t , ␪ )dt + b (t )a *(t , ␪ )dt
冕 | a (t , ␪ ) | 2dt

|冕 冕 |
2
A0 1
Q (␪ , ␪ 0 ) = a (t , ␪ 0 )a *(t , ␪ )dt + b (t )a *(t , ␪ )dt
冕 | a (t , ␪ ) | 2dt
␣0
 Optimum Reception in White Noise 29

Let

␹ (␪ , ␪ 0 ) = 冕 a (t , ␪ 0 )a *(t , ␪ )dt (2.30)

and

X (␪ ) =
1
␣0 冕
b (t )a *(t , ␪ )dt (2.31)

We shall now examine the case in which 冕| a (t , ␪ ) | dt = 1, for all

2

values of ␪ (in this case, A 0 is the energy of the received signal). This situation
is the one generally encountered in radar, and the argument can easily be
generalized to the case in which the signal energy depends on the param-
eter ␪ .

Q (␪ , ␪ 0 ) = A 0 | ␹ (␪ , ␪ 0 ) + X (␪ ) |
2
(2.32)

X (␪ ) is a zero-mean Gaussian random variable resulting from the

passage of the white Gaussian noise b (t ) through the filter whose impulse
1
response is a * (−t , ␪ ). The power of X (␪ ) is N 0 /A 0 since:
␣0

E [X (␪ )X *(␪ )] =
1
A0 冕a *(t , ␪ )E [b (t )b *(u )]a (u , ␪ )du dt =
N0
A0

Consequently, the average value of receiver output Q (␪ , ␪ 0 ) is:

E [Q (␪ , ␪ 0 )] = A 0 | ␹ (␪ , ␪ 0 ) | + N 0
2
(2.33)

It is useful to interpret these quantities: Q (␪ , ␪ 0 ) is the receiver output

‘‘at ␪ ’’ when a target of parameters ␪ 0 is present; | ␹ (␪ , ␪ 0 ) | is called the
2

ambiguity function. It is the receiver output in the absence of noise. This

function has a maximum for ␪ = ␪ 0 .

| ␹ ( ␪ , ␪ 0 ) | 2 ≤ | ␹ (␪ 0 , ␪ 0 ) | 2 = 1 (2.34)
30 Principles of Radar and Sonar Signal Processing

given the normalization to 1 of the energy of a (t , ␪ ). Consequently, in the

absence of noise, the estimated value ␪ˆ is unbiased: ␪ˆ = ␪ 0 .
In general, this ambiguity function has multiple local maxima, which,
in the presence of noise, may be taken for the absolute maximum. This
confusion corresponds to a genuine ambiguity—confusion between two
completely distinct values of the parameters ␪ . An example of an ambiguity
function is shown in Figure 2.3, for a situation in which ␹ has two parameters
t and f .
If the noise is not too high, it can be assumed that the value ␪ˆ is
located ‘‘on the main peak’’ of this ambiguity function; it is clear that in
this case the more pronounced the peak, that is, the higher the curvature
near the maximum, the better the estimation.
This intuitive assessment of the quality of the estimation as a function
of the shape of the ambiguity function can be expressed rigorously. In [2],
it is shown that asymptotically (for high signal-to-noise ratios A 0 /N 0 ):

• The estimator is unbiased;

• The estimator is efficient; that is, the Cramer-Rao bounds are
reached;
• These bounds are expressed directly as a function of the second
derivatives of the ambiguity function at ␪ = ␪ 0 : If J is the Fisher
matrix (2.5) with general element J ij :

Figure 2.3 Ambiguity function.

 Optimum Reception in White Noise 31

J = | Jij |

A0 ␦2
Jij = −
N 0 ␦␪ i ␦␪ j
| ␹ (␪ , ␪ 0 ) | 2 for ␪ = ␪ 0 (2.35)

E | (␪ˆ k − ␪ 0k )(␪ˆ 1 − ␪ 01 ) | = J kl (2.36)

J kl being the general element of matrix J −1, inverse of J. The demonstration

is performed neglecting | X (␪ ) | in the expansion of Q (␪ , ␪ 0 )—in (2.32)—
2

and expanding Q in the neighborhood of ␪ 0 .

This important general result—not specific to the case of radar or sonar
signals—shows that the ambiguity function | ␹ (␪ , ␪ 0 ) | , which is the output
2

of the optimum receiver (see Figure 2.1) in the absence of noise when a
target is present at ␪ = ␪ 0 , characterizes the entire estimation problem—
true ambiguities (when the presence of noise causes a secondary maximum
of the ambiguity function to be chosen for ␪ˆ ) and variance of the estimation
about the actual value ␪ 0 [by (2.35) and (2.36)].
More intuitively, this ambiguity function can also be interpreted as
an ‘‘equipment function,’’ which transforms the target landscape into a
‘‘degraded’’ landscape. This intuitive representation gives an idea of the
importance of this notion of ambiguity function, which, moreover, depends
only on the form of the signals s (t , ␪ ).
Finally, the form of this function depends on the number of channels
in the optimum receiver (Figures 2.1 or 2.2). This number is determined
by the sampling interval of ␪ (which was not specified until now); this
interval must obviously be chosen so that the ambiguity function is correctly
sampled, that is, so that there are at least a few reception channels on the
width of a peak of this function. Any reduction in the number of channels
will lead to a degradation in performance, and the performance calculated
previously is, strictly speaking, obtained only for continuous exploration
of ␪ .
In addition, it is important to note that the estimation accuracy given
by (2.35) and (2.36) is a function of the ratio of the received signal energy,
A, to the noise spectral density, N 0 .

2.3.4 Detection Performance

To calculate the performance of the optimum detector defined by (2.27),
consider the situation in which the energy of a (t , ␪ ) is normalized:
32 Principles of Radar and Sonar Signal Processing

冕| a (t , ␪ ) | dt = 1
2

The optimum detector then performs the following operation for a

given value ␪ of the parameter.

|冕 |
2 H0
x= z (t )a *(t , ␪ )dt ⭴T (2.37)
H1

Just as the natural detector optimization criterion consisted of maximiz-

ing the detection probability for a given false alarm probability, the detector
performance will be measured by the change in the detection probability Pd
as a function of the false alarm probability P fa .
These two quantities are easily calculated from the two probability
densities:

• p x /H1 : Probability density of the quantity x defined by (2.37) under

the hypothesis in which the target is present:

H 1 : z (t ) = ␣ a (t , ␪ ) + b (t ), with | ␣ | = A : energy of the received signal

2

(2.38)

• p x /H0 : Probability density of the same quantity x under the hypothesis

in which no target is present:

H 0 : z (t ) = b (t ) (2.39)

The detection probability is then the probability of exceeding the

threshold T, under hypothesis H1 :

+∞

Pd = 冕 p x /H1 (x )dx (2.40)

T

Similarly, the false alarm probability is the probability of exceeding

the threshold under hypothesis H0 :
 Optimum Reception in White Noise 33

+∞

P fa = 冕 p x /H0 (x )dx (2.41)

T

These two probabilities are represented in Figure 2.4 by the hatched

areas. This figure clearly shows that a change in the threshold T can improve
P d only at the price of an increase in P fa .
The false alarm probability can be calculated in the Gaussian case
under hypothesis H0 . The variable x is given by:

|冕 |
2
= |b1 |
2
x= z (t )a *(t , ␪ )dt (2.42)

where

b1 = 冕 b (t )a *(t , ␪ )dt (2.43)

is a complex zero-mean Gaussian noise

E | b 1 | = E | b (t ) | = N 0
2 2
(2.44)

Thus, x is the square of the modulus of a complex zero-mean Gaussian

variable. Its probability density is therefore, according to a classic result, an
exponential law [3]:

Figure 2.4 Detection probability P d and false alarm probability P fa .

34 Principles of Radar and Sonar Signal Processing

x
1 −N
p x /H0 (x ) = e 0 (2.45)
N0

when the false alarm probability:

+∞

冕
x T
1 − −
P fa = e N 0 dx =e N0 (2.46)
N0
T

This simple result enables the value of the threshold T to be determined

for a required false alarm probability P fa when N 0 is known.
Similarly, under the target-present hypothesis, H1 , the variable x is
given, from (2.37) and (2.38), by:

|冕 |
2
= |␣ + b1 |
2
x= z (t )a *(t , ␪ )dt (2.47)

where b 1 is defined by (2.43).

Thus, x is the square of the modulus of the sum of a constant (nonfluctu-
ating target) and a Gaussian complex variable. It is shown in [2] and [3]
that the probability density of such a quantity is given by:

冉 冊
2√xA
x+A
1 − N0 I
p x /H1 (x ) = e 0 (2.48)
N0 N0

where I 0 is the zero-order Bessel function.

The calculation of the detection probability P d by integration of
p x /H1 (x ) according to (2.40) must be done numerically (various analytical
approximations of this probability are given in [4]).
The characteristics of the receiver can be summarized by the graphs
plotted in Figure 2.5, known as the receiver operating curves (ROC). This
figure illustrates the trade-off between detection probability and false alarm
probability. The practical application to calculation of radar range is examined
in Appendix 3B, since it requires prior analysis of radar resolution in angle,
range, and Doppler (Chapter 3).

2.3.5 Conclusion
Summarizing, in the presence of white noise, the optimum estimation proce-
dure consists of correlating the received signal with a set of replicas corre-
 Optimum Reception in White Noise 35

Figure 2.5 Detection performance, detection on a single pulse, and nonfluctuating target.

sponding to the different possible values of the parameters, detecting and

normalizing these correlation results with respect to the energy of the replica,
and choosing the parameter corresponding to the channel that gives the
maximum result. The optimum detector has the same structure, the determi-
nation of the maximum being replaced by comparison with a threshold
determined by the desired false alarm probability.
This optimum maximum likelihood estimator is also asymptotically
efficient, which means that asymptotically, when the signal-to-noise ratio
tends to infinity, the Cramer-Rao bounds are reached—and thus, at the
limit, the probability estimator also minimizes the variance of the estimation
error. This property is an a posteriori justification of our choice of the
maximum likelihood criterion. Furthermore, examination of the performance
of the estimator reveals the importance of the role of the ambiguity function
| ␹ (␪ , ␪ 0 ) | 2 :
• The curvature of this surface near the maximum ␪ = ␪ 0 limits the
variance of the estimator.
• The ‘‘spurious’’ increases on this surface (local maxima) define the
possible ambiguities in the presence of noise.

Finally, remember that the ambiguity function is the output of the

optimum receiver shown in Figure 2.1 for all values of ␪ when the received
signal is noise free (2.33). It is the equipment function of the receiver.
36 Principles of Radar and Sonar Signal Processing

Throughout this chapter, the form of the received signal, which deter-
mines the form of the ambiguity function, has not been specified. The results
obtained are thus applicable to any problem involving the estimation of the
parameters of a signal of unknown amplitude and phase—or the detection
of such a signal whatever its bandwidth and, above all, whatever the form
of its dependence on ␪ . The only assumption is that this form is known. It
thus appears that this single assumption, together with the additive Gaussian
character of the noise, is sufficient to define the structure of the optimum
estimator and detector by correlation and square-law envelope detection.

References

[1] Van Trees, H., Detection, Modulation, and Estimation Theory, New York: Wiley, 1971.
[2] Levine, M., Fondements Théoriques de la Radiotechnique Statistique, Moscow: Mir (Trans-
lation from Russian), 1973.
[3] Rice, S. O., ‘‘Mathematical Analysis of Random Noise,’’ Bell System Technical Journal,
Vol. 23, No. 3, July 1944, pp. 282–332, and Vol. 24, No. 1, Jan. 1945, pp. 46–156.
[4] Darricau, M., Physique et Théorie du Radar, Paris, France: Sodipe, 1994.

Appendix 2A: Cramer-Rao Equality

Consider the case in which the received signal consists of a measurement
sample vector r. Let p r /␪ (r ) be the probability density of r , conditional to
the value ␪ of the parameters. Let ␪ˆ (r ) be an estimator of ␪ based on
the observation r . If ␪ˆ (r ) is unbiased, then:

E [ ␪ˆ (r ) − ␪ ] = 0
⇔ 冕 p r /␪ (r ) [␪ˆ (r ) − ␪ ] dr = 0

Differentiating this equation with respect to 0:

冕 ␦ p r /␪ (r ) ˆ
␦␪
[␪ (r ) − ␪ ]dr = 1

Moreover, the following identity can be written:

 Optimum Reception in White Noise 37

␦ p r /␪ (r ) ␦ log p r /␪ (r )
⇒ = ⭈ p r /␪ (r )
␦␪ ␦␪

⇒ 冕冋 ␦ log p r /␪ (r )
␦␪ 册
√ p r /␪ (r ) {√ p r /␪ (r ) [␪ˆ (r ) − ␪ ]}dr = 1

Applying Schwarz’s inequality, we obtain:

冋 册
2
␦ log p r /␪ (r )
E [ ␪ˆ (r ) − ␪ ]2 ≥ 1/E
␦␪

Another expression of this inequality can be obtained by differentiating

the equality:

冕 p r /␪ (r )dr = 1

⇒ 冕 ␦ p r /␪ (r )
␦␪
dr = 冕 ␦ log p r /␪ (r )
␦␪
p r /␪ (r )dr = 0

Differentiating a second time, we obtain:

冕 冕冉 冊
2
␦ 2 log p r /␪ (r ) ␦ log p r /␪ (r )
p r /␪ (r )dr + p r /␪ (r )dr = 0
␦␪ 2 ␦␪

That is

冋 册 冋 册
2
␦ 2 log p r /␪ (r ) ␦ log p r /␪ (r )
E = −E
␦␪ 2 ␦␪

Consequently

E [ ␪ˆ (r ) − ␪ ]2 ≥ −1/E 冋
␦ 2 log p r /␪ (r )
␦␪ 2
册
Furthermore, Schwarz’s inequality becomes an equality when the two
functions under the integral are proportional:
38 Principles of Radar and Sonar Signal Processing

␦ log p r /␪ (r ) ˆ
= [␪ (r − ␪ )]k (␪ ) (2A.1)
␦␪

However, if ␪ˆ mv (r ) is the estimator in terms of maximum likelihood,

then by definition:

␦ log p r /␪ (r )
= 0 for ␪ = ␪ˆ mv (r )
␦␪

Substituting in the inequality (2A.1), we obtain:

␪ˆ (r ) − ␪ˆ mv (r ) = 0

Thus if ␪ˆ (r ) is an efficient estimator, then it is also the maximum

likelihood estimator.