Waveform analysis for high-frequency FMICW radar

Waveform analysis for high-frequency FMICW radar R.H. Khan D.K. Mitchell Indexing term: High-frequency radar, FMICW waveforms, Ambiguityfunction, Ocean surveillance Abstract: The paper describes the waveform analysis for an experimental long-range high- frequency radar system with a design capability for over-the-horizon detection of ocean surface targets up to 400 km from shore with a range resolution of 400m. Motivated by the need to maximise the transmitter duty cycle in order to detect long-range targets, the radar is based on a frequency modulated interrupted continuous wave (FMICW) type waveform. The interrupt sequence, which is necessary for a high power monostatic operation, complicates the analysis of the radar performance in terms of the ambiguity function (AF). This paper analyses the range dependent AF of FMICW waveforms and proposes practical interrupt sequences for the synthesis of desired AFs. 1 Introduction The authors are involved in the construction of a shore- based long-range high-frequency (HF) radar facility for iceberg detection, ship tracking and ocean environmental measurements. At these frequencies, electromagnetic waves are capable of over-the-horizon (OTH) propaga- tion in either the ground wave (GW) or sky wave mode. This prototype system relies on the GW mode of propa- gation and has a design capability for OTH detection and tracking of targets up to 400 km from shore with a range resolution of 400 m [l]. The main objective is to develop techniques for the radar waveform analysis and synthesis to achieve the desired performance specifi- cations for the prototype radar. In a conventional single-frequency pulsed radar system, the desired range resolution determines the pulse length. Range is computed as the time delay of returned echoes. The maximum range is determined by the repeti- tion rate of the pulses and the hardware limit on the maximum achievable pulse peak power. It is well known (see for example Reference 2) that pulse compression techniques can be used to realise high energy pulses with low peak powers. These waveforms can be compressed, after reception, using simple hardware and/or software operations. For example, using a linear frequency modu- lated (FM) sweep during the radar pulse, large pulse compression ratios can be achieved. The range measure- Paper 8091F (ElS), first received 1st June and in revised form 29th November 1990 Mr. Khan is with the Centre for Cold Ocean Resources Engineering, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1B 3x5 Mr. Mitchell is with Northern Radar Systems Ltd., PO Box 8323, Stn. A, Bartlett Building, St. John’s, Newfoundland, Canada A l B 3N7 IEE PROCEEDINGS-F, Vol. 138, NO. 5 , OCTOBER 1991 ment of FM radars is based on the difference in instanta- neous frequency between the transmitted and received waveforms. The received waveform from a stationary target, with a two-way travel time of to seconds, differs in instantaneous frequency from the received waveform by (fo = at,), where a is the frequency sweep rate. However, the linear FM pulse waveform has a range/Doppler ambiguity. This ambiguity may be resolved by using a frequency modulated continuous wave (FMCW) sweep; effectively many realisations of linear FM waveforms are present and target velocity may be unambiguously esti- mated as the rate of change in range [3]. FMCW radars have a range resolution competitive with pulse radars when the same signal bandwidth is used. A survey of numerous papers addressing different aspects of FMCW radars has been presented by Skolnik [4]. The FMCW waveform, with a 100% duty cycle, is ideal for bistatic or low-power monostatic operation. For the high-power monostatic application of our prototype radar, there are two problems with the FMCW approach. First, the necessary isolation between the transmitter and receiver for continuous operation at H F frequencies is very difficult to achieve. Secondly, due to the propagation losses, the signal levels decrease rapidly with distance and the dynamic range of the signals exceeds that of any available receiver hardware. To overcome both the above mentioned problems the authors have chosen a frequency modulated interrupted continuous wave (FMICW) type waveform [SI for the prototype radar. This is simply an FMCW waveform which is gated on and off with a well defined sequence, typically a pseudorandom sequence. The concept is to transmit for a period of time and, after the transmitter has been turned off, to receive during the quiet period. Thus, compared to an FMCW waveform, both the trans- mit and receive waveforms are subjected to different interrupt sequences. The introduction of these interrupt sequences complicates the analysis of the radar per- formance in terms of the well known ambiguity function (AF). For example, it can be easily shown that the effec- tive gating on the received signal varies with the target range and introduces a range dependence on the AF. The form of this AF has been discussed in the literature [SI, however, in Section 3, we will use some properties of the AF to simplify the AF analysis of FMICW radar wave- forms with arbitrary transmit and receive interrupt sequences. The inverse problem of determining the parameters of an FMICW waveform to realise a desired AF is more difficult. Some simplification is possible for a GW-HF surveillance radar, where many targets of interest are slow moving and the zero-Doppler ambiguity function is suffcient to evaluate the performance of the waveform. This zero-Doppler ambiguity function can be easily obtained from the general form of the AF and offers us 41 1 an insight into the effect of different interrupt sequences on the FMICW waveform. Both random and deterministic interrupt sequences are considered in Section 4. It is shown that random interrupt sequences result in an AF with a high noise floor, which can obscure weaker targets, whereas deter- ministic sequences cause ambiguity pop-ups. Section 5 details how these ambiguity pop-ups may be successfully controlled, resulting in an AF with a very low noise floor, a signal dynamic range with the capability of available receivers, and a reasonable signal-to-noise ratio (SNR) with respect to FMCW. 2 Background Modern radar technolgy relies on the AF to evaluate the performance characteristics, such as range and Doppler resolution, of waveforms for different applications [2]. For a radar waveform p(t), the AF is generally defined in terms of the response function ~ ( z , 4): where The response function has two parameters, time delay z and Doppler frequency 4, and can be interpreted as a measure of the correlation of the waveform p(t) with a delayed and/or Doppler shifted version of itself. This is used to evaluate the capability of a radar waveform to resolve targets close to each other in range and/or Doppler. For example, the well known ‘chirp’ waveform is expressed as g(t) = ~ e j ( ~ ‘ ~ ) / * all t (2) where K is a scaling constant and a is the frequency sweep rate. This ideal ‘chirp’ waveform is an infinite linear frequency modulated sweep and its response func- tion can be easily evaluated using eqn. I : x,(79 4) = I K I22.d(4 - (3) The response function in eqn. 3, clearly shows the well known range/Doppler ambiguity of the ‘chirp’ waveform. The characteristics of the response function for more complicated radar waveforms are not so obvious. In par- ticular, the evaluation of the response function for FMICW waveforms is complicated by the finite extent of the linear frequency sweep as well as the effect of the interrupt sequence used. Poole [2] has studied the ‘absol- ute delay-dependent cross-ambiguity function’ for FMICW waveforms with pseudorandom interrupt sequences; an important result reported is that the AF is not independent of target range. In the next Section, we present an analysis of the AF of FMICW waveforms in a manner which will allow the subsequent synthesis of desired performance characteristics by using different waveform. Thus some form of interruption is required on transmission (FMICW waveform) so that the receiver may be activated during the intervals between transmis- sions. Our objective here is to evaluate the AF for an FMICW waveform. The different gating sequences imposed on the transmitted and received signals compli- cates the analysis and also introduces a range depen- dence of the AF. We will use the properties of the AF to simplify the analysis of some physically realisable FMICW waveforms. It is important to note that the receiver must always be off whenever the transmitter is on, so that the trans- mitter and receiver interrupt sequences are not identical. Further differences between the two sequences are intro- duced by the ‘soft gating’ used to suppress transients which would be caused if the interrupt sequences were switched rapidly between the on and off states. Depend- ing on the target range, the transmitted waveform is not received in its entirety at the receiver and the radar per- formance needs to be evaluated by the ‘cross response function’ between the transmitted and received signals. The effect of target range on the received signal is demonstrated in Fig. 1 where, for the same transmitted signal, the received signals are shown for two different target ranges. Although the returned signals are similar, the relative time delays in the receiver gating sequences significantly change the actual signals gated into the receiver. The AF will obviously be different for these two ranges and this effect must be carefully considered when designing waveforms for long-range applications. 3.1 Evaluation of the response function components We will now evaluate the AF for a general FMICW waveform. The soft gating mentioned above is not con- sidered; the low pass filtering effect due to this gating can be easily added at any stage of the analysis. It is possible to write an expression for the AF based on the strict defi- nition [2] but that approach gives little indication of the effect of changing the FMICW parameters on the AF. The derivation presented here evaluates the AF in terms of a number of component response functions which are combined using linear operations. Specifically, only the ‘cross response’ function for the interrupt sequence needs to be computed. The rest of the derivation involves linear operations and the well known response function of an FM sweep. For example, a single period of an FMCW waveform can be described as the time-domain product of an ideal FM sweep or ‘chirp’ waveform g(t) and a gate function with width corresponding to the FMCW sweep period TR. The full FMCW waveform can then be rea- lised by convolving this single period with an impulse sequence having a constant period TR. The transmitted and received FMICW waveforms are obtained by multi- plying the FMCW waveform with the transmit and receive interrupt sequence, respectively. The interrupt sequence corresponding to the transmit- ted waveform is defined by - interrupt sequences. 0 ~ A d t ) ~ l - m < r < m (4) Similarly, the interrupt sequence corresponding to the received waveform is defined by - 00 < t < 00 ( 5 ) 3 Ambiguity function for FMICW waveforms The practical implementation of the ‘chirp’ waveform sets 0 Q AR(t) < 1 a finite limit on the frequency sweep; a periodic repeti- Of this finite sweep is known as the FMCW wave- form. In the H F band, for a high-power monostatic operation, the transmitter and receiver cannot be effec- A fundamental aspect of FMICW operation constrains the receiver to be disconnected whenever the transmitter is operating i.e. tively isolated for the 100% duty cycle of the FMCW AR(t ) = 0 if AT(t) # 0 (6) 412 I E E PROCEEDINGS-F, Vol. 138, No . 5 , OCTOBER I991 D fl, 0 d 01 I I I I I; 1 I I 11 e 01 t2 f Fig. 1 (I Generated FMCW signal b Transmitter gating sequence c Receiver gating sequence (complement of transmit sequence) d FMICW signal using the gating sequence of b e FMICW radar echo from target with two-way travel time off, seconds .f FMICW radar echo from target with two-way travel time oft, seconds Effect of target range on FMICW radar signal It is well known that the response function of the product of two time-domain signals can be evaluated by convolv- ing their individual response functions along the Doppler variable. Thus the ideal response function of the ‘chirp’ waveform in eqn. 3 is broadened by convolution with the response function of an ideal gate to achieve the response function of a finite FM sweep pulse. The response func- tion of an ideal gate is shown in Fig. 2. From the finite gate G(t) of Fig. 2a, the kernel of eqn. 1 is formed as a two dimensional function in t and T. The response func- tion is obtained by computing the Fourier transform in (t) as per eqn. 1. Note that this linear broadening of the ideal response of the chirp signal may be implemented later to retain simplicity in our derivation. The same comment applies to the operation of convolution with an impulse sequence to form the repetitive sweeps of the FMCW waveform. I ‘ VI\ / I \ c response function x ~ ( r . 4 ) delav Fig. 2 gate function using eqn. I Evaluation of the response function corresponding to an ideal Thus, expressing the transmitted waveform as a gated version of an ideal ‘chirp’ waveform fAt) = A A W ) (7) The radar returns from targets are delayed by the two-way travel time to and are further modified by the receiver gating sequence f a = A J W A r - to)g(t - to ) fR(t + to ) = AR(t + to )AAMt) (8) (9) Eqn. 8 may be simply rewritten as Examination of eqns. 7 and 9 shows that the response function for the FMICW waveform can be evaluated by (a) Evaluating the response function for g(t) . (b) Evaluating the ‘cross response’ function for AAt) (c) Convolving the above two response functions along and AR(t + to)AAt). the Doppler variable. 413 IEE PROCEEDINGS-F, Vol. 138, NO. 5 , OCTOBER 1991 (d) Convolving the response function of the infinite sequences, with the impulse response of the finite band- width. (e) The response function is periodically repeated, in the delay variable, due to the periodic nature of the FMICW sweep. The response function of the ‘chirp’ waveform is given by eqn. 3 and contains the effects of any Doppler shift due to target velocity. The ‘cross response’ function for the transmitter and receiver gating sequences can be easily evaluated using: Note that this two-dimensional response function depends on the delay time, i.e. the ambiguity function will vary with target range. The response function of eqn. 10 can be convolved, along the Doppler variable, with the response function of the ideal ‘chirp’ waveform to give the response function of the FMICW waveform. This ideal response function can now be broadened to account for the soft gating as well as the finite bandwidth of the FM sweep. 3.2 Zero Doppler ambiguity function In many applications of H F radars, the target velocities are very small and the zero-Doppler ambiguity function is sufficient to evaluate the performance of the waveform. This zero-Doppler ambiguity function can be easily evaluated. It can be seen that the convolution of the response function of eqn. 10 with the response function of the ideal chirp of eqn. 3, at each value of the delay parameter, will shift the response function (eqn. 10) in fre- quency. This frequency shift is a linear function of the delay parameter z. A little reflection will show that the response function needs to be evaluated for a set of delay- Doppler parameters such that the values will be mapped to the zero Doppler axis after convolution with the simple response function (eqn. 6) of the ‘chirp’ waveform. For this set, the Doppler frequency corresponding to any delay z is simply -az, where a is the frequency sweep rate of the ‘chirp’ waveform. Thus the zero-Doppler response function for an FMICW waveform can be com- puted using x/(z? a, = t + i ) R ( + - i) x A,( , - ;)dzna7‘ dt (1 1) As the mapping to the zero doppler axis depends on the frequency sweep rate, a now appears as a parameter in the evaluation of the response function. From a compu- tational point of view, eqn. 11 may be simplified by repla- cing t with ( t + ~ / 2 ) xf(‘9 ‘7 = j:mAdt + + t O ) A T ( t ) (12) e j z n d t + r / 2 ) dt Eqn. 12 may be interpreted as the correlation of two sequences P(t , to ) and Q(t, z) given by P(t , to ) = AR(t + to)Az(t) Q(t, Z) = Adt + z ) & ~ ~ ~ ~ ( * + ~ ’ ~ ) (13) The following Section presents some examples of FMICW waveform ambiguity functions evaluated using the above technique. 4 Numerical evaluation of ambiguity function The two examples presented here are evaluated using the following specifications of the prototype H F long-range radar: sweep bandwidth, B 375 kHz sweep repetition period, TR 0.524288 s range resolution 400m 409.6 km unambiguous range For the computer simulations the 0.524288 s FMICW sweep repetition period was quantised to 8192 samples corresponding to a sample interval of 6 4 p . Note also that the AF is plotted as a function of range in contrast to the usual delay parameter. The main reason for this is that the AF of FMICW waveforms varies with range; the sample delay of 64 p s corresponds to a range of 9.6 km. 4.1 FMCW FMCW may be thought of as a special case of FMICW, where AT(t) = AR(t) = 1. However, it varies from FMICW in that AR(t) # 0, when A d t ) # 0. This case is useful as a benchmark with which to compare FMICW waveforms. In Fig. 3, it is shown that the design resolution of 400 m has been obtained. 4.2 FMICW with random interrupt sequence Random gating sequences are conventionally chosen as a means to distribute the sidelobe energy of the ambiguity function. Fig. 4a shows the interrupt sequence for a simple random gating sequence. The durations T of the periods within the sequence have a normal distribution, as shown in Fig. 4b. This distribution was chosen as it can be shown that, for this case, the ratio of the receive power to the transmitted power is constant over all target ranges. This minimises periodicities and produces a near white noise floor of the ambiguity function as shown in Fig. 4c. Note that the ambiguity noise floor is only about 30 dB below the target level. This will seri- ously limit the radar performance for weak targets in the presence of stronger targets. For ground wave H F radar, the radar range equation [4) is defined as: P G G oA2W; ( 4 7 ~ ) ~ R ~ p,= where P , = transmit power P , = receive power G , = transmit antenna gain G R = receive antenna gain o = radar cross-section I = wave length W, = attenuation function R = range Thus, the ratio of receive power from two ranges is and for a transmit frequency of 6.75 MHz, the variation in single strength from 50 km to 400 km is 0.7882 400 4 = dB ( T X i i K 3 ) 414 1EE PROCEEDINGS-F, Vol. 138, NO. 5, OCTOBER 1991 -100 ' 1 0 50 100 150 I I 200 2 50 300 350 400 range, km Fig. 3 Computer simulation of AF of FMCW to = T/2 = 196 km t a 1 - I 90 t b Fig. 4 interrupt sequence AF of FMICW with random J a Transmit interrupt sequence b Distribution of interrupt periods T with variance U' 200 300 400 500 600 700 -100; 100 range, km C IEE PROCEEDINGS-F, Vol. 138, NO. 5, OCTOBER 1991 c AF corresponding to interrupt sequence in a 415 After acceptable processing gains the ambiguity noise floor mus