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Defence Research and Recherche et développement Development Canada pour la défense Canada

& DEFENCE DÉFENSE

Ambiguity and Cross-ambiguity Properties of Some Reverberation Suppressing Waveforms

Sean Pecknold

Defence R&D Canada Technical Memorandum DRDC Atlantic TM 2002-129 July 2002 This page intentionally left blank. Copy No:

Ambiguity and Cross-ambiguity Properties of Some Reverberation Suppressing Waveforms

Sean P. Pecknold

Defence R&D Canada – Atlantic Technical Memorandum DRDC Atlantic TM 2002-129 July 2002

Abstract

The theoretical performance of different active pulse types is examined through the use of ambiguity functions, cross-ambiguity functions, and Q-functions. The commonly used continuous wave (CW) and frequency modulated (FM) waveforms are reviewed, along with more complex wave forms and pulse sequences designed for superior reverberation performance or Doppler/range resolution, including sinusoidally frequency modulated (SFM) waves and frequency hopped trains of pulses, such as Costas codes and quadratic congruential codes.

Résumé

La performance théorique de différents types d'impulsions de sonar actif est examinée à l'aide de fonctions d'ambiguïté, de fonctions de contre-ambiguïté et de fonctions Q. Les formes d'ondes continues (CW) et les formes d'ondes à modulation de fréquence (FM) couramment utilisées sont examinées, de même que les formes d'ondes plus complexes et les séquences d'impulsions conçues pour donner de meilleures performances sur les plans de la réverbération ou de la résolution Doppler/résolution en portée, y compris les ondes à modulation de fréquence sinusoïdale (SFM) et les trains d'impulsions à sauts de fréquences, comme les codes Costas et les codes quadratiques congruents (QCC).

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Executive summary

Introduction The role of pulse design in an active sonar system is an important one. Different choices of transmitted pulse types will affect resolution in both range and Doppler, and will also affect the ability of the sonar to discriminate a target in a reverberation limited background. The Towed Integrated Active-Passive Sonar project (TIAPS) requires an analysis of the properties of different pulse types to determine the requirements for signal generation for its active sonar component. These properties are examined in this paper.

Results The narrowband and wideband ambiguity functions are defined and evaluated for CW pulses, LFM and HFM waveforms, a “comb”-type ambiguity function waveform, the SFM, and three “thumbtack”-like ambiguity function waveforms, the PRN pulse, the Costas waveform, and the QCC waveform. Doppler and range resolution are examined using the wideband ambiguity function, and the waveforms fall into the categories of those with good Doppler resolution (CW and SFM), those with good range resolution (LFM/HFM), and those that resolve both Doppler and range (PRN, Costas, QCC).

The Q-function is defined and used to examine the theoretical reverberation suppression capabilities of the waveforms. It is found that the SFM can be chosen to provide the best reverberation suppression at higher target velocities, and that the PRN and frequency-hopping Costas and QCC waveforms offer slightly better reverberation suppression than the LFM at velocities greater than nearly stationary.

Finally, the cross-ambiguity functions between different waveforms and waveforms of the same type with slightly different parameters are examined, and examples indicating the approximate expected levels of mutual interference given several simultaneous active sonar signals are provided.

Future Work Further work is required for a complete understanding of the pulse type requirements for the TIAPS active sonar. In particular, both modelling of and analysis of experimental data of pulse propagation of different waveforms in various environments is needed. As the TIAPS project continues, more of this work will be undertaken.

Pecknold, S. 2002. Ambiguity and Cross-ambiguity Properties of Some Reverberation Suppressing Waveforms. DRDC Atlantic TM 2002-129

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Sommaire

Les fonctions d'ambiguïté de bande étroite et de large bande sont définies et évaluées pour les impulsions en ondes entretenues, les formes d'ondes LFM et HFM, une forme d'onde à fonction d'ambiguïté de type « peigne », les formes d'ondes SFM, trois formes d'ondes à fonction d'ambiguïté en pointe (« punaise »), l'impulsion Bruit pseudo-aléatoire (BPN), la forme d'onde Costas et la forme d'onde QCC. La résolution Doppler et la résolution en portée sont examinées à l'aide de la fonction d'ambiguïté de large bande, et les formes d'ondes se classent dans les catégories suivantes : celles qui donnent une bonne résolution Doppler (CW et SFM), celles qui donnent une bonne résolution en portée (LFM/HFM), et celles qui donnent une bonne résolution Doppler et une bonne résolution en portée (BPN, Costas, QCC).

La fonction Q est définie et utilisée pour examiner les capacités théoriques de suppression de réverbération des formes d'ondes. On constate que la forme d'onde SFM permet la meilleure suppression de réverbération aux vitesses supérieures des cibles et que les impulsions BPN et les formes d'ondes à sauts de fréquences Costas et QCC permettent une suppression de la réverbération légèrement supérieure à celle que permettent les formes d'ondes LFM aux vitesses supérieures à celles de la condition quasi stationnaire.

Enfin, les fonctions de contre-ambiguïté entre des formes d'ondes différentes et entre des formes d'ondes du même type avec paramètres légèrement différents sont examinées, et des exemples montrant les niveaux approximatifs prévus de brouillage mutuel pour plusieurs signaux de sonar actif simultanés sont donnés.

Pecknold, S. 2002. Ambiguity and Cross-ambiguity Properties of Some Reverberation Suppressing Waveforms (Propriétés d'ambiguïté et de contre-ambiguïté de certaines formes d'ondes supprimant la réverbération). RDDC Atlantique, TM 2002-129

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Table of contents

Abstract...... i

Executive summary ...... iii

Sommaire...... iv

Table of contents ...... v

List of figures ...... vi

1. Introduction ...... 1

2. Narrowband and wideband ambiguity functions...... 2

3. Pulse types...... 3 3.1 Continuous waves (CW)...... 3 3.2 Frequency-modulated waves...... 4 3.3 Pseudo-random noise...... 6 3.4 Frequency-hopped pulse trains...... 7

4. Performance in reverberation – the Q-function...... 13

5. Cross-ambiguity ...... 17

6. Conclusions ...... 24

7. References ...... 25

List of symbols/abbreviations/acronyms/initialisms ...... 27

Distribution list...... 28

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List of figures

Figure 1. Ambiguity function for continuous wave...... 4

Figure 2. Ambiguity function for LFM wave...... 5

Figure 3. Ambiguity function for PRN wave ...... 7

Figure 4. Ambiguity function for SFM wave ...... 8

Figure 5. Ambiguity function for Costas 12-code...... 10

Figure 6. Ambiguity function for QC 11-code ...... 11

Figure 7. Normalized Q-functions for CW, SFM, and LFM. Centre frequencies of 1000 Hz, duration 1 second. Modulation frequencies of SFM waveforms are 10 Hz and 30 Hz..... 15

Figure 8. Q-functions for thumbtack-type ambiguity function waveforms versus LFM. Time bandwidth product is approximately 150...... 16

Figure 9. Cross-ambiguity function for two LFMs – centre frequencies 1000 Hz and 1100 Hz...... 18

Figure 10. Cross-ambiguity function for two SFMs – centre frequencies 1000 Hz and 1050 Hz...... 19

Figure 11. Cross-ambiguity function for SFM and CW – centre frequencies 1000 Hz, SFM modulation frequency 10 Hz...... 20

Figure 12. Ambiguity function for Costas (top) and QC (bottom) codes. N=31...... 22

Figure 13. Cross-ambiguity function for Costas (top) and QC (bottom) codes. N=31...... 23

List of tables

Table 1. Range and Doppler -3dB resolution for various waveforms...... 12

Table 2. Cross-ambiguity function maximum relative to auto-ambiguity maximum value..... 21

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1. Introduction

The development of an active sonar system requires knowledge of the properties of the waveforms transmitted by that system. The choice of waveform will determine the ability of the system to resolve targets in range and Doppler, and will also impact the detection capabilities of the system. The range resolution of a signal is generally proportional to its bandwidth – a larger bandwidth provides a better distinction between two signals in range (see e.g. [1]). The theoretical optimal signal-to-noise ratio of a signal, and thus theoretical detection probabilities in a passive or noise-limited situation, depends only on the noise power density and the total energy of the signal [2], and therefore its duration and power efficiency. In order to improve detection performance while maintaining range resolution, it is therefore desirable to increase the bandwidth of the signal without a corresponding decrease in the duration of the signal. This implies that the time-bandwidth product of the transmitted signal must be made larger than unity. Modulating the frequency or the amplitude of the transmitted waveform will result in an increased time-bandwidth product. Amplitude modulation, however, results in a decrease of power efficiency, so it is more common to use frequency modulated signals. It should be noted that various windowing or shading techniques used for increased resolution on either the transmitted signal or the received signal will also affect the available bandwidth or power efficiency.

Active sonar performance, on the other hand, depends also on reverberation rejection, and thus on pulse design. In fact, since echo to reverberation ratio is theoretically independent of power level, in a reverberation limited situation, the effects of signal type and windowing on power efficiency can be ignored. Here, we will be primarily concerned with the resolution capabilities of various types of signal and with their ability to reject reverberation.

The resolution capabilities for a given active waveform can be examined through the use of the ambiguity function. This function represents the envelope of the response to a target in both range and Doppler. The ability of a signal to resolve a target can be estimated based on the width of the main lobe of the ambiguity function. The sidelobes of this function will also help determine how well a signal can resolve multiple targets, or a target within a reverberant environment.

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2. Narrowband and wideband ambiguity functions

Given a transmitted signal with amplitude s a function of time t of the form s(ta)=()tcos(ωt+φ()t), or in complex form st()= at()eit(ωφ+ (t)) with complex conjugate s *(t), where a(t) is a (slowly) time-varying window function, ω = 2π f the angular frequency and φ is the possibly time-dependent phase shift, then the output of a matched filter receiver is given by the cross-correlation:

+∞ ωτ τ τ ττit[(ωη1++) (1−η)+φ(t+)−φ(η(t+))] X (,τη)=+at[ ]a[η(t−)]e 222dt ∫ 22 −∞ (1) where η is the Doppler scale factor and τ the time delay. Higher order terms are neglected, i.e. relative velocity is assumed constant. Under the assumption that the signal is narrowband with frequency ω0= 2π f0, the signal compression is neglected, and the expression resolves to (translating the delay axis for convenience to allow the maximum response at t =τ ):

+∞ X (,τη)=−esiiωη00(1−−)τ∫ (t)s*(tτ)eω(1η)tdt −∞ (2)

Neglecting the term outside the integral, the approximation η ≈1+=2v 1+f (where cf0

2vf0 f = c is the Doppler shift, c the sound speed and v the relative speed of target and receiver) simplifies the cross-correlation X to the narrowband ambiguity function of time delay and Doppler shift [3]:

+∞ χτ(,f )=∫ st()s*(t−τ)e−2π ift dt (3) −∞

For a wideband signal, the effect of target velocity must also include the compression of the signal. The wideband ambiguity function of time delay and Doppler scale factor is thus defined as [4]:

+∞ χτ(,η)=∫ s(ts)*(η(t−τ))dt (4) −∞

The quantity of interest is usually the square of the envelope of the matched filter response, and the ambiguity function is thus often seen defined as χτ(,f )2 .

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3. Pulse types

3.1 Continuous waves (CW)

The continuous wave (CW) waveform is simply a (windowed) sinusoidal wave,

st()= a()t sin(2π ft0 +φ) (5) with frequency f0 and phase shift φ (in radians). The function a(t) is a slowly varying window or envelope function, e.g. a square or Hanning window (see Harris [5] for a discussion of window types).

The ambiguity function of the continuous wave pulse of duration T, with a square window normalized so that the ambiguity function peak at zero time delay is unity, i.e.

−1 at()= T 2 0≤≤t T; (6) at()=>0 t T is given as a function of time delay and Doppler shift frequency by:

()T − τ sin (π fT( − τ )) χτ(,f )= (7) Tfπτ()T−

This behaves as a sinc function with respect to Doppler frequency, and behaves as 1− τ T in time delay. This can be noted from the ambiguity diagram (Figure 1). In this diagram and the ambiguity diagrams that follow, the time delay has been transformed to range R = cτ , and the Doppler shift frequency to velocity, v = cf , using a value for c of 1500 m/s. It is seen that the 2 f0 (long) continuous wave is very Doppler sensitive, but has a high degree of ambiguity with respect to range. A frequency of 1000 Hz and a pulse length of 1 second was used for this figure, with a square window.

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u 0.8 F y t i 0.6 gu bi

m 0.4 A d e s

i 0.2 al m r

o 0 N 10 1000 0 0 -10 -1000 Velocity [knots] Range [m]

Figure 1. Ambiguity function for continuous wave

3.2 Frequency-modulated waves

In order to maintain peak transmitted power, various forms of frequency modulation are typically used. These include linear-frequency modulation (LFM), with a transmit waveform given by:

Bt  st()= at()sin2π f0 ±t+φ  (8) 2T where B is the bandwidth and f0 the centre frequency. This can be generalized to higher orders of modulation, such as quadratic and exponentially frequency modulated waves. More commonly used are hyperbolic-frequency modulation (HFM) or equivalently LPM (linear period-modulated) waveforms, given by:

TBt st()=±a()tsin 2π f2 ln 1 +φ (9) 0 m  Bf(20 − B)T

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The HFM waveform has the property of being a Doppler tolerant signal, i.e. one with unbiased delay estimation and a very slow decrease of the maximum matched filter response with Doppler [6]. The envelope or window function a(t) can be chosen, depending on constraints, to create an optimally Doppler tolerant signal; an example is the rectangular window for constrained energy and bandwidth [7].

The narrowband ambiguity function of an LFM waveform may be expressed analytically:

Bτ T − τ sin(π ( fT−−T )( τ )) χτ(,f )= (10) Bτ Tfπτ()−−T (T)

The characteristics of the ambiguity function can be seen in Figure 2: an LFM waveform with duration 0.1 second, centre frequency 1000 Hz, bandwidth 200 Hz. The function exhibits a ridge in the time delay-Doppler (or equivalently range-velocity, as graphed) plane.

on 1 i t c n

u 0.8 F y t i 0.6 gu bi

m 0.4 ed A s

i 0.2 al m r

o 0 N 10 100 0 0 -10 -100 Velocity [knots] Range [m]

Figure 2. Ambiguity function for LFM wave

This choice of time-bandwidth product TB was to enable closer examination of the structure of the ambiguity function.

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3.3 Pseudo-random noise

A pseudo-random noise signal is one in which either the signal amplitude or phase is modulated in a pseudo-random manner. Amplitude modulated signals are in general not used as active sonar signals, due to the desirability of maintaining peak power output for maximum signal to noise ratio. These can include correlated or uncorrelated Gaussian noise based on a pseudo-random sequence. As this signal cannot be expressed analytically, a replica of the transmitted signal is saved for matched filtering.

More commonly encountered are pseudo-randomly phase modulated signals. These include signals such as binary- and quaternary-phase shift key signals (BPSK and QPSK) (see for example Rihaczek [8]). The BPSK signal is a pseudo-random sequence of ±1 that modulates the phase of a sinusoidal wave. The BPSK signal code has a bandwidth determined by the length of the code. Ternary, quaternary and other phase shift keying schemes modulate the phase by other multiples. These and other phase codes, such as Barker codes, all have thumbtack-like ambiguity functions, with a sharp central spike and a sidelobe “plateau.” Figure 3, the ambiguity function for a BPSK waveform with bandwidth 200 Hz, centre frequency 1000 Hz and duration 1 second, shows this characteristic thumbtack shape. For a large time-bandwidth product (TB 1), a pseudo-random noise signal has an ambiguity function that can be written as [9]:

2 Tf− ττsin()πτ B sin()πτfT( − ) 1   χτ(,f )=+1−1− (11) TBπτ π f()T− τ TBTB

A thumbtack ambiguity function is indicative of the ability to simultaneously resolve both range and Doppler of a target. The trade-off is the existence of the pedestal region. For a multiple target situation, or one in which there are significant scatterers or reverberation, the pedestal creates self-clutter near and around the target range and Doppler, which can result in ambiguous detections. The background clutter is sufficiently high that for a significant area in the velocity-range plane, it may be impossible to distinguish a weak target from a strong scatterer.

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on 1 i t c n

u 0.8 F y t i 0.6 gu bi

m 0.4 A ed s

i 0.2 al m r

o 0 N 10 1000 0 0 -10 -1000 Velocity [knots] Range [m]

Figure 3. Ambiguity function for PRN wave

3.4 Frequency-hopped pulse trains

Several types of ambiguity function behaviour can be observed by taking sequences of frequency-hopped continuous waves or frequency modulated waves. A simple example of frequency-hopping is given by a geometric or Cox comb [10], a pulse constructed from adding individual tonals:

N st()=a()t∑sin(2π fn (t+α)) (12) n=1 where α is an arbitrary time shift parameter and the frequency is given recursively by n−2 fnn=+fr−11∆f, with r being slightly larger than one and ∆f1 being the frequency difference between the first two tonals. The ambiguity function that this generates has several high peaks with reduced levels of self-clutter. The advantage to this is that these peaks can be moved to areas of the range-Doppler space where they will not interfere with the target. It has the disadvantage of splitting the available energy.

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There are other choices of signal that yield a similar comb-type spectrum and ambiguity shape. One of the simplest is the sinusoidal frequency modulated waveform (SFM):

s()ta=()tsin(2πβf0t+sin(2πfmt)) (13)

where β is the modulation index and fm is the modulation frequency. Frequency-hopped FMs can also be used, as in the case of the Newhall train [11]. The ambiguity function for an SFM is shown in Figure 4. The parameters used were a centre frequency of 1000 Hz, pulse duration of one second, modulation frequency of 10 Hz and modulation index of 9. The effective bandwidth is given by [12] B ≈ 2 fm (1 + β) : this example has a bandwidth of 200 Hz. These values were chosen for illustrative purposes: the ridges that appear along lines of constant Doppler can be moved outside of likely target areas by choosing an appropriate modulation frequency relative to the centre frequency. The first Doppler sidelobe ridges will be found for 2vc= fm f0 , so an increase in modulation frequency to 30 Hz will move these ridges out to ±45 knots.

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u 0.8 F y t i 0.6 gu bi

m 0.4 A d e s

i 0.2 al m r

o 0 N 10 1000 0 0 -10 -1000 Velocity [knots] Range [m]

Figure 4. Ambiguity function for SFM wave

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Frequency hops can be made in a non-uniform and non-geometric manner. If these hops are based on pseudo-random codes, the signal effectively becomes a pseudo-random noise and will exhibit thumbtack ambiguity properties. One method of generating these codes is to select a train of N sub-pulses over time and frequency, each one of which falls into a unique time and frequency bin. These are Costas codes [13]:

N −1 s()ts=∑ n (t−nT) (14) n=0 where

s ()te=≤jf2π nt, 0 t≤T n (15) stn ( ) = 0, elsewhere.

yn The frequency of each of these sub-pulses is given by fn = T , with yn being a set of ordered integers with the following property relating their differences:

yyrL++− r≠−ysL ys,r≠s (16) for every L, i.e. the differences between adjacent numbers are unique, the differences between next-adjacent numbers are unique, etc. Certain properties of the ambiguity function of this frequency-hopped pulse can be expressed analytically. Along the Doppler-shift axis, the ambiguity function is identical to that of a CW signal with duration NT:

sin(Nπ fT ) χ(0, f ) = (17) Nπ fT

The time-delay axis behaviour cannot be expressed this simply in general, and must be calculated numerically for differing frequency-hopping codes. An example of the ambiguity function of a 12-element code with total duration 5 seconds, centre frequency 1000 Hz, and bandwidth 30 Hz is found in Figure 5. Note that using CW pulses, filling the frequency bins entails the use of N pulses with duration T/N and a total time-bandwidth product of approximately N2 [13].

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u 0.8 F y t i 0.6 gu bi

m 0.4 A d e s

i 0.2 al m r

o 0 N 10 5000 0 0 -10 -5000 Velocity [knots] Range [m]

Figure 5. Ambiguity function for Costas 12-code

Frequencies can also be chosen based on the theory of either linear [14] or quadratic congruences [15]. The latter case is a frequency-hopped code defined similarly to the Costas code, above. However, in each time segment of the pulse, the frequency of the N sinusoidal waves is given by

B ff=+y (18) n0 nN where N is an odd integer, and the ordered set of integers is given by

nn(1+ ) ykn = ,0≤≤nN−1 (19) 2 mod N where k∈{1,2,K, N−1}. This therefore generates a set of N −1 codes, which can be shown to have very good cross-correlation properties. However, these codes, unlike Costas codes, are not full: not all frequency channels are used. This can result in increased sidelobes particularly along the zero-Doppler axis, and a decreased detection ability for zero-Doppler targets. An example of an 11-term Quadratic Congruential Code (QCC) with parameters as for the Costas waveform is given in Figure 6.

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u 0.8 F y t i 0.6 gu bi

m 0.4 A d e s

i 0.2 al m r

o 0 N 10 5000 0 0 -10 -5000 Velocity [knots] Range [m]

Figure 6. Ambiguity function for QC 11-code

The overall range and Doppler resolution capabilities of these waveforms can be summarized by comparing the width of the -3dB level of the magnitude-squared definition of the ambiguity function along each axis, as done in Table 1. The comparison is made using waveforms with a time-bandwidth product of about 150.

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Table 1. Range and Doppler -3dB resolution for various waveforms

WAVEFORM DOPPLER RESOLUTION RANGE RESOLUTION

Doppler resolvent

Continuous wave ±0.8 knot ±500 m Duration 1s, 1000Hz

SFM ±0.8 knot ±400 m 1 Duration 1s, 1000Hz, B=150 Hz Modulation frequency 30 Hz

Range resolvent

LFM/HFM ±20 knots ±25 m Duration 1s, 1000Hz, B=150 Hz

Doppler and range resolvent

PRN (BPSK) ±0.8 knot ±25 m Duration 1s, 1000Hz, B=150 Hz

Costas 12-code ±1 knot ±100 m Duration 3s, 1000Hz

QC 11-code ±1.2 knot ±100 m Duration 3s, 1000Hz

1. The SFM ridge has a series of peaks and nulls along the range axis that fall within these bounds

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4. Performance in reverberation – the Q-function

The detection capability of various ping types versus active signal spectrum reverberation is often modelled using the Q-function [16]. This is simply the integration of the ambiguity function along the range axis; i.e., the ability of the signal to discriminate against targets with different Doppler:

+∞ Q()η = ∫ χτ(,η)2 dτ (20) −∞

To motivate this, a simple reverberation model must be considered. Under the assumption that the reverberation field is caused by a time-invariant scattering surface composed of uniformly distributed uncorrelated scatterers, the estimated reverberation level can be factored into a function of target range and Doppler [11]. Following the derivation presented in this reference, the scattering-field response function can be written as

ht()=A()uλδ()u (t−τ)du (21) ∫ u where at range u, δ is the Dirac delta-function, λ(u) is the propagation loss, τu the time delay to the scatterer, and A(u) the complex random variable describing scattering amplitude and phase. A received signal r(t) is then the convolution of the transmitted signal s(t) with this scattering response, and its matched filter response at range u0 and Doppler scale factor η is the further convolution of this Doppler shifted response signal with the original:

d(,u ηη)=−s∗ (t τ)s(t −τ)A(u)λ(u)dudt (22) 0 ∫∫ ( uu0 )

The expected reverberation level is the integration over ranges (time-delays) of the squared amplitude of du(,0 η). Using the assumptions that A(u) is uncorrelated and that the range u0 is much greater than the scattering area (far-field approximation), this factors into a function that is dependent on range and one that is dependent on Doppler factor

2 du(,0η) = Fu(0)Q(η) (23) where F is a function of range and Q is the Q-function as given as in equation (20).The Q- function can therefore be used to compare the reverberation processing capabilities of different pulse types. The model used to motivate this derivation of the Q-function is very simple; for more detailed comparisons of pulse types and their reverberation processing capabilities in more complex environments, this model may be extended in several ways, including the use of an ocean surface motion model to define a generalized Q-function [11].

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The effects of spectral spreading have also not been considered; a discussion of reverberation and frequency-spread modelling may be found in Theriault and Buckley [17].

The pulse types described in Section 2 may now be compared. Figure 7 shows the normalized Q-functions for three types of waveforms – continuous wave (CW), linear frequency modulation (LFM), and sinusoidal frequency modulation (SFM). The Q-function has been normalized by the total energy of the transmitted signal. The centre frequencies are 1000 Hz, with the bandwidth for the LFM and SFM being 150 Hz. Two SFMs are shown, with modulation frequencies of 10 Hz and 30 Hz. It is easy to see that the LFM chirp provides the best reverberation suppression (i.e. lowest Q-function) for lower Doppler targets. It is superior to the CW for speeds in the range of ±6 knots. The SFM exhibits several peaks: adjustment of the modulating frequency can move these peaks out to larger Doppler positions, at the expense of slightly reduced reverberation suppression. The SFM is superior to the CW until these secondary peaks are reached, and is bettered by the LFM only at these peaks and for very low Doppler targets.

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Figure 7. Normalized Q-functions for CW, SFM, and LFM. Centre frequencies of 1000 Hz, duration 1 second. Modulation frequencies of SFM waveforms are 10 Hz and 30 Hz.

A comparison of the capabilities of the various waveforms with thumbtack-type ambiguity functions, i.e. Costas, QCC, and PRN shows that they have Q-functions fairly similar to those of the LFM, i.e. a much narrower range of variation over Doppler. Figure 8 shows the Q- functions for these signals using the parameters used for Table 1. In each case, these waveforms provide significant extra reverberation suppression except for very low Doppler targets: the QCC waveform has 4dB to 6dB better reverberation suppression at anything over 4 knots, for example. Comparing this to Figure 7 shows that these waveforms do fare more poorly than the CW and SFM waveforms at higher Dopplers (6-12 knots in these examples).

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Figure 8. Q-functions for thumbtack-type ambiguity function waveforms versus LFM. Time bandwidth product is approximately 150.

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5. Cross-ambiguity

It is quite possible to have a scenario with multiple active sonar systems in the same area. It is desirable in this case to have an active transmit waveform that is not subject to a high degree of interference. The lower the interference, the less the possibility of either jamming or target detection error. In a similar vein, a single active system could utilize non-interfering signals to transmit in several directions simultaneously. The ambiguity function is essentially the correlation of a signal with its Doppler-shifted and compressed replicas, or equivalently the matched-filter response to a Doppler-shifted signal. The generalization of this to a cross- correlation or cross-ambiguity between an incoming signal and replicas of a different output signal, or equivalently the matched-filter response to a different signal, gives information about the mutual interference of different transmit waveforms.

The first and simplest situation to consider is one in which two waveforms of the same type are being transmitted, but in different frequency bands. The behaviour of the cross-ambiguity function, and thus of the mutual interference, can be observed by extending the ambiguity function to the Doppler shift corresponding to the frequency difference. Thus, a transmitted signal at 1100 Hz operating simultaneously with a 1000 Hz signal will produce apparent targets shifted by about 150 knots; however, the peaks caused by a 1020 Hz signal will appear shifted by 30 knots, which is much more restricting.

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o 1 i t c n

u 0.8 F y t i 0.6 gu i b

m 0.4 A ed s

i 0.2 al m r

o 0 N 10 100 0 0 -10 -100 Velocity [knots] Range [m]

Figure 9. Cross-ambiguity function for two LFMs – centre frequencies 1000 Hz and 1100 Hz.

For wideband signals, the situation is similar, but some degree of signal compression also takes place. Figure 9 shows the cross-ambiguity function of two 0.1 second LFMs with bandwidth 100 Hz, at centre frequencies 1000 Hz and 1100 Hz (i.e. just non-overlapping). There is obviously a significant degree of interference throughout the indicated Doppler shift (velocity range) of ±15 knots with an apparent shift here of 100 m. Figure 10 shows the cross-ambiguity function for two SFMs with modulation frequency 10 Hz, duration 1 second and centre frequencies 1000 Hz and 1100 Hz. Again, there is some interference, but in this case primarily restricted to the higher Doppler targets. For the cases of PRN signals and frequency-hopped codes, there are many different potential codes: the situation of an exact duplicate is not as interesting as that of a similar signal using a different code.

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o 1 i t c n

u 0.8 F y t i 0.6 gu bi

m 0.4 ed A s

i 0.2 al m r

o 0 N 20 1000 0 0 -20 -1000 Velocity [knots] Range [m]

Figure 10. Cross-ambiguity function for two SFMs – centre frequencies 1000 Hz and 1050 Hz.

The case of simultaneous signals with different waveforms in the same frequency range may also be considered. In some cases, such as for a CW and an LFM or HFM, there is a very low degree of interference. Likewise, the interference between an LFM/HFM and thumbtack type waveform or SFM is also very low. The case of an SFM and a CW shows significantly more interference (Figure 11). The cross-ambiguity function for a CW and Costas or QCC waveform will depend on the length (number of time/frequency bins) of code, as there will be few overlapping frequency bins between the two. Examples of the maximum expected values for cross-ambiguity functions normalized by the maximum of the ambiguity function for several waveforms are given in Table 2. The waveforms used have the same bandwidth, duration, and/or code length, but differ in centre frequency for the CW, LFM, and SFM. The Costas, PRN, and QCC signals differ in the code used.

DRDC Atlantic TM 2002-129 19

on 1 i t c n

u 0.8 F y t i 0.6 gu bi

m 0.4 A ed s

i 0.2 al m r

o 0 N 20 1000 0 0 -20 -1000 Velocity [knots] Range [m]

Figure 11. Cross-ambiguity function for SFM and CW – centre frequencies 1000 Hz, SFM modulation frequency 10 Hz.

20 DRDC Atlantic TM 2002-129

Table 2. Cross-ambiguity function maximum relative to auto-ambiguity maximum value.

WAVEFORM CW LFM PRN1 SFM Costas1 QCC1 (centre frequency) 1050 Hz 1100 Hz 1000 Hz 1100 Hz 1000 Hz 1000 Hz

Continuous wave -20dB -31dB -7.1dB -12dB -7.1dB -5.8dB T= 1s, 1000Hz LFM -31dB -1.7dB -19dB -32dB -26dB -26dB T= 1s, 1000Hz, B=150 Hz PRN (BPSK) -8.0dB -17dB -6.9dB -9.0dB -8.3dB -9.0dB T= 1s, 1000Hz, B=150 Hz SFM -2.8dB -31dB -7.1dB -5.7dB -8.2dB -9.0dB T= 1s, 1000Hz, B=150 Hz Modulation frequency 30 Hz Costas 12-code -12dB -18dB -7.0dB -13dB -4.7dB -1.0dB T= 3s, 1000Hz QC 11-code -13dB -19dB -6.4dB -13dB -3.8dB -4.2dB T= 3s, 1000Hz

Maxima are over the range ±30 knots Doppler and ±5000 metres.

1. PRN, Costas and QCC signals using different code of the same temporal and code length.

Given a situation where a large number of independent signals must be generated within the same frequency range, it is interesting in particular to look at the cross-ambiguity properties of sets of codes, including Costas and QCC. It can be shown [18] that an optimal choice for the time-bandwidth product TB for these codes gives an asymptotic upper bound to the sidelobes of the auto-ambiguity function of 2/N with increasing code length N. The cross- ambiguity function, on the other hand, has an asymptotic result of 4/N for the quadratic congruential codes across the whole set of codes, while the Costas code cross-ambiguity function is at best 5/N, and at worst tends to a fixed number (with no N dependence), depending on code choice. Thus, the QCC codes for large time-bandwidth and code length will have in general significantly better cross-ambiguity properties and thus reduced mutual interference as compared to Costas codes. Examples of the auto- and cross-ambiguity functions for Costas and QCC codes of length 29 and time-bandwidth product 1000 at centre frequency 1000 Hz are shown in Figures 12 and 13. The maximum value attained by the cross-ambiguity functions, -9.5dB for the Costas waveform and -8.0dB for the QCC, seems to contradict this conclusion. This might indicate that for code lengths and TB products that are practical in an underwater acoustic setting, the differences in cross-ambiguity properties are negligible.

DRDC Atlantic TM 2002-129 21

1 n o ti

c 0.8 n u

F 0.6 ty i u g i 0.4 mb A - 0.2 to u A 0

20 1000 0 500 0 -20 -500 1Velocity [knots] -1000 Range [m] n o ti

c 0.8 n u

F 0.6 ty i u g i 0.4 mb A - 0.2 to u A 0

20 1000 0 500 0 -20 -500 Velocity [knots] -1000 Range [m]

Figure 12. Ambiguity function for Costas (top) and QC (bottom) codes. N=31.

22 DRDC Atlantic TM 2002-129

1 on i t

c 0.8 n u F y

t 0.6 i gu

bi 0.4 m A -

s 0.2 os r

C 0 20 1000 0 500 0 -20 -500 Velocity [knots] -1000 Range [m]

1 on i t

c 0.8 n u F y

t 0.6 i gu

bi 0.4 m A 0.2 oss- r

C 0

20 1000 0 500 0 -20 -500 Velocity [knots] -1000 Range [m]

Figure 13. Cross-ambiguity function for Costas (top) and QC (bottom) codes. N=31.

DRDC Atlantic TM 2002-129 23

6. Conclusions

The theoretical performance of several types of active sonar waveforms has been investigated using the ambiguity function and Q-function. These waveforms included the continuous wave pulse, linear frequency modulated pulses, the SFM, an example of a “comb”-type waveform, and several Doppler- and range-resolvent waveforms, namely the pseudo-random noise pulse and two examples of frequency-stepped wave trains, the Costas code and the QCC. As expected, there are tradeoffs in selecting a waveform for the best performance. The SFM is seen to be a good alternative to the CW, giving the best Doppler rejection at higher target velocities and having a 5dB to 10dB advantage over the CW at lower target velocities. Nevertheless, it and the CW both suffer from poor range resolution. The LFM and HFM have poor Doppler resolution, but better reverberation rejection at low target velocities. The waveforms with “thumbtack”-like ambiguity functions, however, (PRN, Costas, and QCC), have up to 6dB advantage over the FM waveforms at anything above a nearly stationary relative Doppler, and both the QCC and Costas waveforms have slightly better performance even at zero Doppler. They also provide simultaneous resolution of range and Doppler. The PRN pulse suffers from the possibility that, having a long effective code word length, it may suffer more from channel spreading than the other waveforms.

The cross-ambiguity function was used in determining the degree of interference between different waveforms for a multiple signal environment. The situation of most interest in these cases is for several coexisting signals of the same type in the same frequency bandwidth. Both Costas waveforms and QCC waveforms of frequency-hopped CWs provide a large set of signals that can be used with interference maxima up to 10dB lower than the matched response while still being feasible to produce. Given these cross-ambiguity properties and the favourable reverberation rejection and resolution properties of these waveforms, they may prove worthwhile in multiple signal active sonar scenarios.

24 DRDC Atlantic TM 2002-129

7. References

1. Burdic, W.S. ‘Underwater acoustic system analysis,’ Prentice-Hall, Eaglewood Cliffs, NJ, 1984.

2. Turin, G.L. ‘An introduction to matched filters’ IRE Trans.Inform.Theory, 1960, 6, pp. 311-329.

3. Woodward, P.M. ‘Probability and information theory with application to ,’ Pergamon, London,1953.

4. Kelly, E.J., and Wishner, R.P. ‘Matched-filter theory for high-velocity accelerating targets,’ IEEE Trans. Mil. Electron., 1965, MIL-9, pp. 59-69.

5. Harris, F.J. ‘On the use of windows for harmonic analysis with the discrete ,’ Proc. IEEE, 1978, 66 (1), pp. 51-83.

6. Altes, R.A. ‘Some invariance properties of the wideband ambiguity function,’ J. Acoust. Soc. Am., 1973, 53, pp. 1154-1160.

7. Altes, R.A. ‘Optimum waveforms for sonar velocity discrimination,’ Proc. IEEE, 1971, pp. 1615-1617.

8. Rihaczek, A.W. ‘Principles of high-resolution radar,’ McGraw-Hill, New York, 1969.

9. Costas, J.P. ‘Medium constraints on sonar design and performance,’ EASCON 75, Washington, D.C., 1975, pp. 68A-68L.

10. Cox, H. and Lai, H. ‘Geometric comb waveforms for reverberation suppression,’ Proc. of 28th Asilomar Conference on Signals, Systems and Computers, 1994, 2, pp. 1185-1189.

11. Brill, M.H., Zabal, X., Harman, M.E., Eller, A.I. ‘Doppler-based detection in reverberation-limited channels: Effects of surface motion and signal spectrum,’ Proc. of IEEE Conf. Oceans ’93, 1993, pp. 1220-1224.

12. Cooper, G.R. and McGillem, C.D. ‘Modern communications and spread spectrum,’ McGraw-Hill, 1986.

13. Costas, J.P. ‘A study of a class of detection waveforms having nearly ideal range- Doppler ambiguity properties,’ Proc. IEEE, 1984, 72 (8), pp. 996-1009.

DRDC Atlantic TM 2002-129 25

14. Titlebaum, E.L. ‘Time-frequency hop signals Part I: Coding based upon the theory of linear congruences,’ IEEE Trans. Aero. Elec. Systems, 1981, AES-17 (4), pp. 490-493.

15. Bellegarda, J.R. and Titlebaum, E.L. ‘Time-frequency hop codes based upon extended quadratic congruences,’ IEEE Trans. Aero. Elec. Systems, 1988, AES-24 (6), pp. 726-742.

16. Deley, G.W. ‘Waveform Design,’ in Radar Handbook, ed. Skolnik, M.I., McGraw-Hill, 1970.

17. Theriault, J.A. and Buckley, K.B.W. ‘Doppler induced frequency spreading of reverberation’, 2001, DREA Technical Memorandum TM 2001-129.

18. Titlebaum, E.L., Marić, S.V., Bellegarda, J.R. ‘Ambiguity properties of quadratic congruential coding,’ IEEE Trans. Aero. Elec. Systems, 1991, AES-27 (1), pp. 18- 29.

26 DRDC Atlantic TM 2002-129

List of symbols/abbreviations/acronyms/initialisms

BPSK Binary phase-shift keying CW Continuous wave DND Department of National Defence HFM Hyperbolic frequency-modulated LFM Linear frequency-modulated LPM Linear period-modulated PRN Pseudo-random noise QCC Quadratic congruential coded SFM Sinusoidal frequency-modulated

DRDC Atlantic TM 2002-129 27

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