Post Discrete Fourier Transform Integration

In the usual implementation of the DFT, one electronic mod­ule is assigned to each center frequency, and that module pro­cesses all range cells within the repetition interval. Current technology allows these modules to be designed as integrated circuits using VLSI. Although each module is very small, the sheer number required can place a severe strain on available signal processor real estate. Also, there is, a cost impact. Thus, there will be a limitation on the number of filters available.

Given that the number of filters is limited to Nf, then the spacing between filters is PRF/Nf. However, a particular tar­get Doppler may fall midway between centers. As the integra­tion length increases, the bandwidth of each filter decreases and a loss is imparted to the midway signal. Thus, there is an optimum integration length that maximizes the SNR gain for that signal. It may be shown that the optimum length is 0.74 Nf for an unweighted DFT.

In many applications, the available pulses during a beam dwell will far exceed the optimum integration length. For ex­ample, with a beamwidth of 5°, a scan rate of 60 RPM and a PRI of 30 да, the number of pulses over the beam is approxi­mately 463. If only 50 filters were implemented, the optimum DFT length would be 37. Thus, there would be 12 or 13 DFT outputs per beam position. The careful designer should try to combine these outputs into a single decision. This combining requires post-DFT integration.

The post-DFT integrator might be a Markov chain, which is a simple and effective approach. However, the magnitude integrator has superior performance. In that approach, indi­vidual DFT output magnitudes are summed and the result applied to a threshold test. The only disadvantage to this technique is that probability of false alarm is difficult to pre­dict. Apparently, the only viable means for predicting false alarm characteristics is by Monte Carlo simulation.

Another aspect of magnitude integration is block versus sliding integration. In the block integrator, several DFT out­puts are allowed to accumulate before thresholding. Then, an­other block is collected. In this approach, it may turn out that both blocks lie on the beam skirts and full benefit of main bean gain is not achieved. A preferred approach is to allow the DFT summation to slide across the beam where a new sum is produced after each DFT on a first-in, first-out basis. This requires more computer memory but ensures that at least one block will encompass the maximum antenna gain.

[1] Elevators. Elevators are attached to the pitch stabilizer on the tail to control pitch motion. They are raised and lowered together.

[2] Monowing Configuration. Ramjet missiles have two in­lets external to the main body and there is room for only one pair of wings (i. e., monowing).

[3] G4.I

Clutter Tracking

In many applications, shipboard and airborne, the radar plat­form is in motion. In these cases, the received clutter Doppler may be nonzero and its spectrum may not be canceled suffi­ciently in the relatively narrow null of the MTD response. Improved clutter rejection may be obtained by incorporating a clutter tracker. This subsystem measures the clutter Dopp­ler and, through various tuning schemes, positions the clutter spectrum exactly at the MTD null for maximum suppression. The clutter tracker is a special purpose, automatic frequency control (AFC) loop.

Archaic designs might use analog frequency discriminators for clutter frequency sensing. However, a preferred approach is to use a pair of DFT filters. One filter is centered at some frequency below clutter and the other is centered above. An almost linear error function is obtained by forming the follow­ing ratio:

F(f) + Fh(f)

where Fl and Fh are the frequency responses of the lower and higher filters, respectively. The spacing of the filters and the required integration length are dictated by the clutter sce­nario and geometry. The AFC loop is used to drive this error to 0 where maximum cancellation is achieved.

Tuning of the clutter spectrum may be achieved using a variable frequency oscillator as the final local oscillator. That approach is discouraged, because free-running oscillators tend to be noisy and, invariably, produce unwanted intermod­ulation products within the video bandwidth. A more attrac­tive option is to tune the MTD itself using digital control. As an example, consider the first-order canceler. Without tuning, its output is given by

F0 = g(n) — g(n — 1)

where g(n) = cos(2n/Tr) + j sin(2n/Tr), f is the Doppler fre­quency, and Tr is the repetition interval. This function may be shown to have a null at zero frequency. Tuning is accom­plished by multiplying the delayed pulse by a complex con­stant. Thus,

F0 = g(n) — [cos Ф1 + j sin Ф1] g(n — 1)

where ф1 = 2nf1Tr. It may be shown that the null of this func­tion has been shifted to f1.

When using cancelers having multiple delays, two or more separate nulls may be obtained. This would be useful, for ex­ample, in placing sea clutter and rain clutter, which are sepa­rated in Doppler, into their own nulls. Of course, the design of a dual-frequency AFC loop does present a challenge. Be­cause the clutter Doppler oscillates as the antenna bearing angle rotates relative to the platform velocity vector, tracking may be aided by introducing antenna angle into the AFC loop. With this aid, the tracking loop can be a simple, first-order implementation which is not required to compensate for Doppler accelerations introduced by rotation. Wilson (12) sug­gests an adaptive clutter tracking algorithm.

Discrete Fourier Transform

The discrete Fourier transform (DFT) is a digital signal pro­cessing technique that provides perfect, coherent integration of incoming pulse trains and filter discrimination among tar­get and clutter Doppler shifts. Usually, it is implemented as a bank of equally spaced digital filters that spans a frequency band equal to the PRF. The response of a given filter repeats at integral multiples of the PRF.

These filters are generated by summing the product of the incoming samples with a time-varying sequence of coeffi­cients. The latter is designed to produce a narrowband filter at some prescribed frequency. The output of a given filter is

N— 1

F = ^2 A[cos 2nfnTr + j sin 2nfnTr]

n =0

x [cos2nfcnTr — j sin2nfcnTr]

where A is the peak amplitude of the input signal, f is the frequency of the input signal, f c is the center frequency of the filter and Tr is the pulse repetition interval. N is the number of pulses integrated. Note that the input and coefficient se­quences are expressed as complex quantities. The real part of the input is taken from the in-phase video channel, whereas the imaginary part is taken from the quadrature. The output is also complex valued.

The real, or in-phase, output may be shown to be

N —1

Fi =^2 A cos2n( f — fc )nTr

n =0

and the imaginary, or quadrature, output is


Fq =^2 A sin2n( f — fc )nTr


When the input signal frequency coincides with the filter cen­ter frequency, the outputs are

Fi = NA


Fq = 0

The square of the output magnitude, which is output power, is, then,

M2 = Fi2 + Fq2 = N2A2

The filter response to noise is given by

Discrete Fourier Transform

Figure 8. Frequency response of a discrete Fourier transform filter with uniform window. The DFT response is relatively narrow but ex­hibits sidelobes, which may cause extraneous detections.


F =^2[xn + jyn][cos2nfcnTr – jsin2nfcnTr]


where xn and yn are noise samples taken from the in-phase and quadrature channels, respectively. These samples are as­sumed to be independent from channel to channel and sample to sample. They have a Gaussian probability distribution with zero mean and variance a2. The in-phase output is


Fi = ^2 xn cos 2пfcnTr + yn sin 2пfcnTr


The variance of that output is


V(Fi) = ^2 V(xn )cos2 2п fcnTr + V(yn )sin2 2пfcnTr

n=0 N – 1

= a2[cos2 2пfcnTr + sin2 2пfcnTr] = Na2


norm —

It may be shown also that

V(Fq) = Na2 Then, the output SNR is

N"2 A 2 a 2

8Ш° = Ж^=М2^=М-8Ш>

where SNRi is the input, per-pulse SNR. Thus, the DFT pro­vides an SNR gain equal to the length of integration. For ex­ample, if that length was 100, the gain would be a very im­pressive 20 dB. No other known technique yields a higher SNR gain than the DFT.

The normalized frequency response of the DFT filter may be shown to be

sin Nф N sin ф


Ф = П f/fr

F =^2 AW (n)[cos2nfnTr + j sin2nfnTr]


[cos2n fcnTr – j sin2nfcnTr] where W ( ) is the window function. A typical function is W(n) = sin2

which reduces the first sidelobe to a level approximately 32 dB below the main lobe response. There is almost an infinite number of windows that might be applied in a given applica­tion. Harris (10) presents an entire catalog of windows and lists advantages and disadvantages.

The use of windows is not without penalty. Invariably, the main lobe gain will be reduced and the bandwidth will be increased. Thus, a trade-off exists between sidelobe response and main lobe performance.

The DFT dwell length is

Td = NTr

and the filter bandwidth is, roughly,

N Тл

f is frequency relative to filter center frequency and fr is the PRF. This response is plotted in Fig. 8 for the case N = 10. In this plot, a frequency normalization factor, k, is used. When k = 0, f = 0 and when k = 1, f = fr.

It will be noted that the response exhibits sidelobes. The first sidelobe is 13 dB below the main lobe response. Also, note that the response repeats at integral multiples of the PRF.

If the relatively high sidelobes are not sufficient to provide desired clutter attenuation, then window functions may be applied to suppress these sidelobes. A window function is a real-valued, time-varying sequence applied to the input data for all filters. With a window, the DFT response is given by

Normally, the nominal number of filters implemented is equal to the integration length, N. Then, the filter spacing is almost equal to the bandwidth. This is not an inviolable rule. When the number of filters exceeds N, the loss to frequencies be­tween filter centers is reduced but the filter outputs are corre­lated somewhat. Fewer filters result in more loss.

A very interesting signal processing scheme is the one in which a nonrecursive MTD is used preceding the DFT. This type of MTD, which does not employ feedback, has the very attractive feature that no transients are produced when radar frequency is changed, beams are transitioned, or PRF is changed. Thus, wasteful receiver blanking is avoided. This feature is achieved because the combined MTD-DFT re­
sponse is constant over most of the PRF bandwidth. Although the MTD attenuates low-frequency signals, it also attenuates noise in their vicinity within the narrow DFT bandwidth. The result is an almost constant SNR over most of the band cov­ered. Thus, the combined response can be made to approxi­mate that of a well-designed MTD with feedback, but tran­sients will be absent. Wilson (11) presents a derivation of SNR loss using this combination.

Active Arrays

The next generation of search radar may well use the concept of the active array. This is a natural extension of the phased array. In that design, each array element is provided with its own transmitter and receiver module. This solid-state module contains a low-power transmission amplifier, receiver protec­tion, a low-noise RF preamplifier, filtering, and frequency con­version. It also contains a digitally controlled phase shifter for beam control. On transmission, all modules are driven by the same coherent signal from a frequency synthesizer. That element also provides one or more local oscillator signals for frequency conversion to IF. The IF signals are combined to form one channel for signal processing.

A chief advantage of the modular array is the elimination of a large, expensive, hard-tube transmitter. Its disadvantage is in the huge number of modules required. If 4356 elements are required, then that number of modules is also required. If the cost of each module were only $100, then the total array cost would approach $500,000. For this reason, the modular array has not found a large application. However, as technol­ogy improves and costs decrease, this approach may become very popular in future designs.

Phased Array Antennas

Phased array antennas are planar arrays of waveguide slots. Variable phase shifters are used to drive a group of slots and, thus, to effect electronic steering of the antenna beam. This technique can eliminate the necessity for bulky mechanical devices such as motors and gimbaled platforms.

The basic theory of phased arrays is described best by con­sidering the simplest case, which is a two-element array. The radiated fields from two adjacent sources combine in space to form a radiation pattern. When a phase shift is applied to one element, the directivity of that pattern may be altered. In this simple case, it may be shown that the relative gain of the array is

G =

2 + 2cos(n sin a – ф) 4

where a is the space angle relative to perpendicular and ф is the introduced phase shift. Maximum gain occurs when

a = sin-1 (ф/п)

Thus, for small angles, the ratio of phase shift to steered angle is approximately a factor of three. In no way does this example present the design equations for a complex array. It simply shows the effect of phase shift on boresight shift.

The concept is extended easily to linear arrays of any length, N, and to two-dimensional arrays having N X M ele­ments (9). In practice, the pattern produced by an array will
be the product of the pattern from each element and the array factor, which is determined by the element spacing. The opti­mum spacing is one-half wavelength. At wider spacing, the pattern begins to develop unwanted sidelobes called grating lobes. These can be as large as the main lobe and may cause confusion or interference. In addition, coupling between ele­ments can alter the actual antenna pattern. Phased array de­sign is a complex process.

When the array beam is steered off-axis, the beam will broaden and the gain will decrease. In general, this effect is in proportion to the cosine of the steered angle. For example, a steering angle of 45° may result in a loss of 1.5 dB relative to on-axis gain.

When the element spacing is one-half wavelength, the number of elements and required phase shifters can become quite large. For example, consider a design at X band where one wavelength is 0.03 m. An antenna 1 m on a side would require 4356 phase shifters to enable steering in both planes. The sheer cost and weight of this system might be prohibitive.

A good compromise design is one in which the antenna array is rotated mechanically in azimuth while being steered electronically in elevation. The example given previously would then require only approximately 66 phase shifters to provide elevation-only steering. This approach also allows for raster scanning. Rather than holding the elevation position constant over a full 360° rotation, the beam could be directed to visit several elevation positions during one scan. This not only reduces the time required to illuminate a given volume but, since the beam traverses the target in both dimensions, beam splitting in elevation and azimuth could be imple­mented.

An ultimate phased array design is the conformal array. Here, the array is designed as an integral part of an existing geometry. That geometry might be the fuselage or wing of an aircraft or the hull of a ship. The ideal conformal array would be a sphere or hemisphere. This design could eliminate off – axis steering loss, because the beam would always be perpen­dicular to the array surface.

Another advantage of phased arrays is their capability for instant target verification. An initial detection could be fol­lowed by freezing the beam in the direction of the target de­tection. Then, a longer dwell could be chosen to both reduce measurement error and increase confidence level. The time savings relative to scan-to-scan verification could be sig­nificant.

A final application of phased arrays is in platform motion compensation. When the radar is carried by an aircraft or ship, it is desirable that the beam position be maintained rel­ative to earth coordinates independent of platform motion. This is implemented easily using beam steering and its use eliminates the necessity for complex motor-gimbal apparatus.