Page 19 - Radar Technology Encyclopedia

algorithm, Burg algorithm, Volder 9

mated, the spectrum is computed in a regular way, by insert-
ing them into the filter transfer function. SAL RW = mS
Ref.: Nitzberg (1992), pp. 296–298; Galati (1993), p. 391.
where R is the matrix of the cross-covariances of the signals
The Cooley-Tukey algorithm is a time-saving computational in the N channels; W is a vector of the weights; ms an arbi-
i
technique in spectral analysis. The exponential Fourier trans- trary nonzero constant, and S is vector representation of the
form of periodic function F(t), desired signal in each channel.
The main algorithms to determine the optimum weights
¥
1 W are Kalman methods, the Howells-Applebaum control
=
G w() ------ ò Ft () exp(jwt) dt
2p loop, the Widrow algorithm, the Gram-Schmidt algorithm,
– ¥ direct matrix inverse, and inverse matrix updating. The possi-
ble implementations of SLC algorithms are all-analog, all-
can be written in discrete matrix form as
digital, and hybrid. The principal advantage of analog imple-
nk
=
[
] F ] (1)
G ] [ W [ mentation is simplicity of RF channel matching. The major
n k
disadvantages are lack of flexibility and slow convergence.
n = 0, 1, 2,... (N - 1)
On the contrary, the all-digital SLC is flexible and possesses
k = 0, 1, 2,... (K - 1)
high-speed convergence but meets difficulties in matching
where
RF-to-digital receiver chains. The most common hybrid SLC
T is the cascaded A/D canceler, which offers the advantages of
F = -----------Ft () ,
k 2pK k both digital and analog systems. SAL
,
W = e –2p j/N , w = nDw, t = kDt, and DwDt are increments of Ref.: Cantafio (1989), pp. 465–467; Monzingo (1980).
n k
sampling in frequency and time domain correspondingly. The step transform algorithm is a special algorithm in
The Cooley-Tukey method of computing the [G ] matrix matched filter processing using subaperture processing to
n
nk
is based on expressing the [W ] matrix in terms of products reduce the size of fast Fourier transform (FFT). A block dia-
of Y square matrices, where Y is an integer in the increment gram of the step transform matched filter is shown in
Y
equation K = 2 . It offers the possibility of reducing the num- Fig. A19. A short duration signal is mixed with the incoming
ber of multiplications to YN (where N is the order of these signal and the output is fed to the FFT processor. The time
2
square matrices) as compared to N for the direct evaluation aperture of the processor is equal to the duration of the refer-
of the matrix of (1). The Cooley-Tukey algorithm is exten- ence waveform but is less than the total waveform length. The
sively applied in spectral analysis and digital use of a series of short-aperture FFTs gives the possibility of
signal processing. SAL reducing hardware requirements by up 50% from conven-
tional FFT approaches. The step transform algorithm is espe-
Ref.: Cooley (1965), pp. 297–301; Hovanessian (1984), pp. 251–264.
cially efficient for synthetic aperture radar (see also
The dynamic programming algorithm (DPA) is used to
TRANSFORM, Fourier). SAL
define the optimum paths over which a system can make tran-
Ref.: Brookner (1977), pp. 163–169.
sitions from one state to another. The possible paths the sys-
tem can take are assigned numerical values through a merit
FFT output Matched
function. The most common of these functions are efficiency coefficient Coherent filter
Input Baseband Short- storage combination output
and cost. The DPA was originally developed by Bellman for conversion aperture over a of coefficients
and A/D FFT number of using FFT
control problems. Later it found uses in target detection, sig- time
apertures
nal processing, and radar system analysis and tradeoff studies. Reference
ramp
For these uses it is more commonly known as the Viterbi
Figure A19 Step transform matched filter (after Brookner, 1977,
algorithm, which is a forward DPA, recursively updating the
Fig. 4, p. 165).
merit function through consecutive stages of the system. SAL
Ref.: Bellman (1957); Bar-Shalom, (1990), pp. 85–153. Viterbi algorithm (see dynamic programming algorithm).
Gram-Schmidt algorithm (see CANCELER, Gram- The Volder algorithm is a recursive procedure for determin-
Schmidt). ing the coordinates of a vector when it is rotated through a
given angle. It was developed by J. E. Volder in 1956 for the
Howells-Applebaum algorithm (see CANCELER, How-
calculation of trigonometric and hyperbolic functions. The
ells-Applebaum). - i
algorithm uses a sequence of rotation angles a = arctan(2 ),
i
Sidelobe cancelation (SLC) algorithms are the algorithms which makes it possible to reduce the rotation of the vector to
used in the sidelobe cancellation technique to place a null in a series of additions, subtractions, and shifts (division by 2),
the required direction. For an adaptive processor with N which are readily implemented in digital hardware. The algo-
degrees of freedom, the N input channels are each multiplied rithm is also known as the coordinate rotation digital com-
by a complex weight and summed to form the output: puter (CORDIC) algorithm.