Phasor products require special care. This can be seen by considering the product:
q(t)r(t) =
=
a cos(t + a )b cos(t + b )
1
ab cfw_cos(2t + a + b ) + cos(a b )
2
The product of two sinusoidally-varying signals yields a constant term plus another term var
where the integrals can be evaluated over any complete period of f (t). Fourier decomposition applies equally well to
real and complex functions. With the aid of the Euler theorem, the decomposition above can be shown to be equivalent
to:
f (t)
cn e jnt
=
As stated above, the reciprocity theorem assumes that the antennas are matched electrically to the transmission
lines, which we take to be matched to the transmitters and receivers. If not, then reections will occur at the antenna
terminals, and power wil
we are left with (analogous to (1.1)
f (t)
1
2
=
F()e jt d
(1.5)
where the continuous function F() replaces the discrete set of amplitudes cn in a Fourier series. Together, these
two formulas dene the Fourier transform and its inverse. Note that the place
The quantity in the square brackets is clearly the convolution of f (t) and g(t). Evidently then, the Fourier transform
of the convolution of two functions is the product of the Fourier transforms of the functions. This suggests an easy
and efcient means
where the Dmax notation is nonstandard but used here to avoid confusion. The beam solid angle of the ideal dipole is
8/3 Str.
2.2 Reciprocity theorem
The proceeding discussion emphasized the performance of antennas used for transmitting radiation. Antenna
Returning to the AM car radio example, despite the poor efciency and impedance matching problems, an electrically short antenna provides acceptable performance. What ultimately matters in a communications (or radar) link is
the signal-to-noise ratio rathe
Y(
B
2 T
Figure 1.5: Illustration of the sampling theorem. The spectrum of the sampled signal is an endless succession of
replicas of the original spectrum. To avoid frequency aliasing, the sampling theorem requires that B 2/T .
fn
=
1 N1
Fm e j2nm/N ,
N
Whereas antenna directivity and gain are more naturally discussed in terms of transmission, effective area is more of
a reception concept. However, directivity and effective area are intimately related, and high gain antennas have large
effective areas.
E
regularity of the experiment, the distribution will converge on a limit as the number of trials increases. We can assign
a probability P(A) to an outcome A as the relative frequency of occurrence of that outcome in the limit of an innite
number of trials.
Central limit theorem
The central role of MVN random variables in a great many important processes in nature is borne out by the central
limit theorem, which governs random variables that hinge on a large number of individual, independent events. Underlyi
T incorporating Parsevals theorem above:
1
T T
T /2
lim
T /2
1
|F()|2 d
T 2T
1
S()d
=
2
1
S() lim |F()|2
T T
| f (t)|2 dt
=
lim
(1.13)
This is the conventional denition of the PSD, the spectrum of a signal. In practice, an approximation is made where
T
information about how the random variables representing different times are correlated. That information is contained
in the autocorrelation function, itself derivable from the JDF. In the case of a random process with zero mean and unity
variance, we hav
where is the Gamma function, (x) = 0 x1 exi d, and where the parameter is referred to as the number of
degrees of freedom. It can be shown that if the random variable xi is N(0, 1), then the random variable
n
z = x2
i
i=1
is 2 with n degrees of freedom. I
1. Var(x) 0
2. Var(x + a) = Var(x)
3. Var(ax) = a2 Var(x)
4. Var(x + y) = Var(x)+2Cov(x, y)+Var(y)
5. (x, y) 1
These ideas apply not only to different random variables but also to realizations of the same random variable at different
times. In this case,
Power spectral density of a random process
Communication and radar engineers are often concerned with the frequency content of a signal. If the signal in
question is a stationary random variable, at least in the wide sense, then it must exist over an inni
er
z
e
e
r
A
y
dl
J
x
Figure 2.1: Diagram of an elemental current element antenna.
Working with potentials rather than electric and magnetic elds allows Maxwells equations in a vacuum to be recast
into the following second-order partial differential equat
z=L/2
(r, )
R(z)
z
I(z)dz
E ,H
r
z=0
z=-L/2
Figure 2.5: Long wire antenna geometry. The observation point is (r, , ).
The differential contribution from each elemental dipole to the observed electric eld then becomes:
=
dE
jZ k
I(z)dz sin e jkr e jkz cos
then given by
|I1 Z21 |2
8R2
=
P12
where R2 = Z22 = Zl . This gives the ratio
P12
Pt
|Z21 |2
4R1 R2
=
(2.6)
which must be consistent with the result from (2.5).
Now, reverse the roles of the sites so that transmission occurs from site 2 and reception at s
The average power density propagating away from an antenna centered at the origin is given by the dot product of the
Poynting vector with the unit radial direction, r. For the elemental dipole, this is
P r =
Pr = |I|2 (dl)2 Z k2 sin2
1
(W /m2 )
322 r2
(2
Chapter 1
Introduction
This text investigates how radars can be used to explore the environment and interrogate natural and manufactured
objects in it remotely. Everyone is familiar with popular portrayals of radar engineers straining to see blips on old-
common antenna. The resulting system, which operated in the VHF band, was perhaps the rst "true" radar, capable
of making direct measurements of the range and bearing to a target. Practical radar sets were in production by 1938,
and a portable version was
The experiment in question followed earlier, successful ones involving shorter propagation paths as well as numerous
mishaps and unsuccessful transatlantic attempts involving different North American receiving sites. (The CornwallSt. Johns route is about
surface ships.
Magnetrons continued to see service during the cold war in the lines of radars spanning North America and intended as a deterrent against bomber raids from the Soviet Union. Radars along the northern-most line, the "distant
early warning" o
Tx
Rx
R
Pr
Pt
Figure 1.1: RF communication link.
= Ptx
Prx
Gtx
Aeff
4R2
Pinc
with the received power Prx being proportional to the transmitted power Ptx assuming that the media between the
stations is linear. The incident power density falling on the rece
of ways in which antennas could be characterized and evaluated. In practice, a system has developed over time for
describing and comparing antennas on the basis of a few parameters. The system is highly intuitive and requires a
minimum of computation. It