Design of Zigzag FM Signals

Since no practical method is available for synthesizing radar waveforms, a sizable effort has been directed into studies of the matched-filter response, or ambiguity function, of many waveforms. In this paper, we investigate the class of FM signals whose instantaneous frequency varies in a zigzag pattern. The waveforms thus consist of linear FM segments and are relatively easy to generate and process. The paper discusses the relation between the characteristics of the waveform and the features of the associated ambiguity function. The effects studied include those of signal repetition, changes in the FM slope, phase-shift and frequency-shift coding, and staggering of frequency step and segment duration. Ambiguity functions of interesting waveforms illustrate the general results. These ambiguity functions are computer-plotted projections of the three-dimensional surface above the delay-Doppler plane.     

Removing autocorrelation sidelobes by overlaying orthogonal coding on any train of identical pulses

A coherent train of identical linear FM (LFM) pulses is used extensively in radar because of its good range and Doppler resolution. Its relatively high autocorrelation function (ACF) sidelobes are sometimes reduced through spectrum shaping (e.g., nonlinear FM, or intrapulse weighting on receive). We show how to completely remove most of the ACF sidelobes about the mainlobe peak, without any increase to the mainlobe width, by diversifying the pulses through overlaying them with orthonormal coding. A helpful byproduct of this design is reduced ACF recurrent lobes. The overlaid signal also results in reduced Doppler tolerance, which can be considered as a drawback for some applications. The method is applied to several trains of identical pulses (LFM and others) using several orthonormal codes. The effect on the three important properties of the radar signal: ACF, ambiguity function (AY), and frequency spectrum is presented. The effect on Doppler tolerance is studied, and implementation issues are discussed. The new design is also compared with complementary and sub-complementary pulse trains and is shown to be superior in many aspects.     

Generalized Response of a Linear FM Pulse Compression Matched Filter

A pulse compression matched filter is analyzed so that the response may be computed when the pulse width, FM rate, and center frequency simultaneously differ from design conditions. Unilateral and bilateral time domain amplitude weighting for sidelobe reduction is included. A general cross-ambiguity function is defined to include these effects and some basic computed results are presented for the peak envelope response with various degrees of Hamming weighting. Computer evaluation of this cross-ambiguity function allows one to choose a combination of mismatches for signal design trade-off between resolution and detection performance. Since no restrictions are placed upon the mismatch parameters, this analysis may also be used to evaluate the filter discrimination against various interfering signals.     

Self-Clutter Cancellation and Ambiguity Properties of Subcomplementary Sequences

In radar signal design it is well known that a fixed volume under the ambiguity surface representing signal energy can only be shifted but not eliminated in the delay-Doppler plane because of the constraint imposed by Woodward's total volume invariance. Rihaczek has shown that periodic signal repetition, though appealing to increased energy, increases the time-bandwidth product at the expense of introducing pronounced ambiguities in the delay-Doppler plane, and thus self-clutter is generated when signals are repeated in the time domain to increase energy. The undesirable self-clutter has a masking effect on targets in different resolution cells thereby limiting performance. An analysis is presented to show that a class of waveforms described in an earlier paper as the subcomplementary set of sequences which are basically repetitive and Hadamard coded, exhibit the property of cancelling self-clutter completely in the delay-Doppler plane if their ambiguity functions are combined. By this technique it is possible to repeat contiguously a basic waveform N times in a prescribed manner to increase signal energy and to cancel totally the resulting self-clutter by combining the ambiguity functions of N different repetitive waveforms which are Hadamard coded. A convenient matrix method to combine the ambiguity functions of subcomplementary sequences, which is an extension of known methods to derive the ambiguity function of repetitive waveforms, is presented. Radar implementation considerations and comparison of performance with various forms of linear frequency modulation (FM) are also discussed.     

Time-Frequency HOP Signals Part II: Coding Based upon Quadratic Congruences

High-efficiency multicomponent signals for maximization of signalto-noise ratio are investigated. Maximization of signal-to-noise ratio in colored noise requires control of volume distribution of the signal ambiguity function and transmission of unity efficiency signals. Signal efficiency is defined as the ratio of average power to the peak power. It is concluded that the signals must be frequency hop pulse trains. Quadratic congruences are chosen to place the components in time-frequency space. The number-theoretic properties of these signals provide bounds on the position and amplitude of the various peaks of the signal ambiguity function. The tradeoffs are shown between volume removal, number of component signals, and the time-bandwidth product.     

Fourier Transform and Ambiguity Function of Piecewise Polynomial Functions

A general systematic procedure is described for computing the Fourier transform and the ambiguity function of waveforms that are piecewise polynomial. The procedure can be implemented by hand or programmed for execution by a digital computer. The main advantage of the technique is that integration is replaced by a finite summation. Examples include the computation of characteristic function, moments, and the ambiguity function of an amplitude modulated linear FM signal.     

Ambiguity function of quadriphase coded radar pulse

Taylor's quadriphase coding (J.W. Taylor, Jr. and H.J. Blinchikoff, ibid., vol.23, no.2, p.156-70, Mar. 1988) is investigated for nonzero Doppler shifts. While the zero-Doppler cut of the ambiguity function (i.e. the autocorrelation) strongly resembles that of the corresponding biphase code, the remaining ambiguity function differs considerably. The ambiguity function of quadriphase code 13 is typified by a diagonal ridge as found in linear FM signals. The ambiguity function of quadriphase code 28A resembles the three parallel ridges of Frank code 16     

Resolution in Doppler and Acceleration with Coherent Pulse Trains

The resolution properties and clutter performance of a simultaneous Doppler and acceleration measurement are investigated in detail with particular emphasis given to coherent pulse trains. The analysis is based on the concept of a matched-filter receiver, although receiver weighting of the type that reduces Doppler sidelobes is also analyzed in detail. Near the main lobe of the acceleration response is a pedestal-ike sidelobe region, the height of which is about 1/N of the main response lobe power where N is the number of pulses in the train. The extent of this pedestal along the acceleration axis is proportional to N. The acceleration measurement in a clutter environment is best performed when both targets and clutter are confined to this pedestal region, since some response sidelobes outside of this region are extremely large.     

Radar Waveforms for Suppression of Extended Clutter

A common problem in waveform design is to adapt the transmitted signal to the target environment in order that the interference from extended fields of scatterers is reduced. This problem is investigated here for the special case of detection of a single target in the ``vicinity' of an extended clutter space. The paper considers the possibility of confining the matched-filter response in delay and Doppler, or ambiguity function, to a narrow strip with arbitrary orientation in the delay-Doppler plane. It is shown that strict confinement of the response is achievable only with waveforms that are unlimited in both time and frequency domain. With practical waveforms, which are necessarily of finite extent, one merely can trade close-target separability against detectability in the background clutter. Thus, one form of the resolution problem is exchanged against the other. The paper examines these effects quantitatively.     

Linear FM Signal Formats for Beacon and Communication Systems

This paper examines the capabilities of the class of linear FM spread-spectrum signals within the context of potential communications systems usage in order to establish some performance criteria and bounds that permit comparison with other spread-spectrum formats. A systematic basis is provided for parameter selection for this class of signals by examining the interaction a mong the frequency-modulation indices, time-bandwidth product, and cross-talk criteria that determine the number of effective linear FM signals (or channels) that can be used within the constraints of a bounded time-frequency region. A general expression is derived relating N, the number of useful signals, R2, a cross-talk parameter, ToWo, the mean time-bandwidth product, and ?max and ?min, the maximum and minimum FM rates of the signal set. Canonic signal processor structures are described for ensembles of linear FM signals that have either constant duration or constant bandwidth. It is then shown that the signal modulation format can be modified in accordance with classical paired-echo theory to expand the utility of this class of signals in both synchronous and nonsynchronous operations to yield the equivalent of time-division and code multiplexing. Possible applications for this signal format are discussed.     

Slope Coding of Coherent Pulse-Burst Waveforms

Useful new properties are obtained with a coherent pulse-burst waveform if the FM slopes are allowed to change from pulse to pulse. A design feature using such a "slope-coded" waveform is the ability to transfer, in a controllable manner, ambiguity volume from the range ambiguity peaks to regions located between the peaks (in the frequency direction). Slope coding thus provides a useful mechanism for matching the waveform to the environment; i. e., it permits the signal designer to make some compromise between range ambiguity-peak heights and low-velocity clutter rejection.     

Phase-Coded Interrupted CW Signals and Processors

High resolution radars require signals with large time-bandwidth product such as CW signal and coherent pulse train (CPT). We discuss a phase-coded interrupted CW (ICW) signal which is the combination of CW signal and CPT. Phase codes used here are with perfect periodic autocorrelation. The periodic ambiguity function of ICW signals is studied including single-carrier signal and multi-carrier signal. It is interesting that the gate function has different effects on two signals and contributes to a multi-carrier ICW signal which yields nearly perfect autocorrelation. Meanwhile we also suggest an efficient receiver approach to ICW signals, which can reduce the computational burden of the processor and utilize the good properties of P3 and P4 codes.     

Optimum Signals for Radar

The research reported herein deals with the general problem of the selection of radar waveforms. The investigation is specifically concerned with the synthesis of radar signals which are optimum in the sense that they are characterized by ambiguity surfaces minimized over certain predetermined regions of the ambiguity plane. The weighted ambiguity surface is utilized as the weighted error criterion. This error criterion is mathematically tractable and pertinent to radar system performance but is not unduly restrictive as some orientation parameters are left unspecified for subsequent cost or penalty function analysis. The signal optimization is approached by variational techniques augmented by equality and inequality constraints, for example, limiting the amount of bandwidth or frequency modulation to be less than some system requirement. Several examples are presented demonstrating the optimization techniques and providing a minimum error for the stated problem. It is shown that for any given type of amplitude modulation of the radar signal, the variance or dispersion of the ambiguity surface is not decreased for any type of phase modulation added. The optimum signal for an elliptical weighting function is derived for several cases. The minimum error is shown to depend upon the constraints and the unspecified orientation parameters and, for one case, on the second moment of the signal.     

Resolution Performance of Pulse Trains with Large Time-Bandwidth Products

The coherent pulse train has good clutter suppression performance because the energy in its matched-filter response is essentially concentrated within sharp ambiguous spikes. However, this is so only when the Doppler distortions are neglected, so that the Doppler effect is taken as a simple translation of the carrier frequency. This paper analyzes the consequences of Doppler distortions on the resolution performance of pulse trains. It is found that Doppler distortions widen the Doppler ambiguities of the pulse train response, with the widening factor proportional to the order of the Doppler ambiguity. This reduces the interval between Doppler ambiguities, and hence the Doppler width of a clutter space that can be accommodated without severe clutter interference. For an operation in a Doppler-ambiguous mode, it also degrades nominal Doppler resolution performance. A detailed analysis of the effects is presented, and numerical results on the widening of the Doppler ambiguities are obtained.     

Weight effects on the periodic ambiguity function

CW radar signals and processors are discussed. The use of the periodic ambiguity function (PAF) to analyze the delay-Doppler performance of CW signals and their corresponding correlation receivers, is extended to include weight function effects. This work provides tools which can predict the delay-Doppler response of almost any phase-coded CW radar. Examples demonstrate that a combination of CW signals having perfect periodic autocorrelation, a matched reference signal with a large number of modulation periods and a smooth weight function, can create a delay-Doppler response with extremely low sidelobes, strongly resembling the response of a coherent pulse train     

The Calculation of Frequency-Modulation Distortion in Linear Networks

A numerical method is developed for the calculation of steady-state FM distortion in a linear passive network from the character of the FM signal input and the steady-state transfer characteristics of the network. The method of calculating FM distortion requires the solution of a finite set of linear equations which is accomplished readily by a digital computer. To illustrate the use of the method, the FM distortion introduced by a Chebyshev-response bandpass filter is calculated for a range of network and input signal parameters.     

Variable Format Radar Receiver Using a SAW Convolver

A technique for receiving radar pulse trains is presented [which can be of a variable format in the sense that they vary from pulse to pulse]. The heart of the receiver is a sufrace ascoustic wave (SAW) convolver. In addition to prsenting experimental results for variable format waveform reception, it is shown that the convolver can easily generate ambiguity functions for virtually any waveform, and specific results for signals such as Barker codes and linear FM (chirp) waveforms are presented.     

Nullifying ACF grating lobes in stepped-frequency train of LFM pulses

An effective way to increase the bandwidth of a coherent pulse-train is to add a frequency step /spl Delta/f between consecutive pulses. A large /spl Delta/f implies a large total bandwidth, hence improved range resolution. However, when the product of the frequency step times the pulse-duration t/sub p/, is larger than one (t/sub p/ /spl Delta/f > 1), the autocorrelation function (ACF) of the stepped-frequency pulse-train suffers from ambiguous peaks, known as "grating lobes." It is well known that replacing the fixed-frequency pulses with linear FM (LFM) pulses of bandwidth B can reduce those grating lobes. We present a simple analytic expression for the ambiguity function (AF) and ACF of such a signal and derive from it very simple relationships between /spl Delta/f, B, and t/sub p/ that will place s exactly where the grating lobes are located, and thus remove them completely.     

Matched-Filter Response to a Linear FM Signal Transmitted Through a Phased Array

The response of a linear phased array with a matched filter connected at its output is investigated when a linear FM signal is incident on the array at an arbitrary angle. The filter is assumed to be matched to the linear FM signal. The dispersion produced by the array results in a mismatch at the receiver which depends on the scan angle and on the type of feed system used with the array. The distortion of the compressed pulse is studied for the series end-fed arrays, the series     

Dispersion Effects in a Circular Phased Array

The dispersion effects that are produced when wideband linear frequency modulation (FM) signals are transmitted through a circular phased array are considered for both one-way and two-way transmissions through the array. The distortions resulting at the output of a matched fi'lter which is matched to the undistorted linear FM signal are studied. These distortions are 1) loss in peak pulse amplitude, 2) widening of the pulse, and 3) reduction in sidelobe level.