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Cost Efficient Frequency Hopping Radar Waveform for Range And Cost Efficient Frequency Hopping Radar Waveform for Range and Doppler Estimation Benjamin Nuss1, Johannes Fink2, Friedrich Jondral2 1Institut fur¨ Hochfrequenztechnik und Elektronik (IHE) 2Communications Engineering Lab (CEL) Karlsruhe Institute of Technology (KIT), Germany E-mail: [email protected], johannes.fi[email protected], [email protected] Abstract—A simple and cost efficient frequency hopping radar chose exactly the same sequence π is given by waveform which has been developed to be robust against mutual 1 interference is presented. The waveform is based on frequency P (Π = π) = . (1) hopping on the one hand and Linear Frequency Modulation Shift N! Keying (LFMSK) on the other hand. The signal model as well as The height of one step is ∆f = B , where B is the total the radar signal processing are described in detail, focusing on N the efficient estimation of range and Doppler of multiple targets occupied bandwidth of the frequency hopping signal. Each in parallel. Finally, the probability of interference is estimated possible frequency is transmitted exactly once due to the and techniques for interference mitigation are presented showing random permutation. Fig. 1 shows an exemplary hopping a good suppression of mutual interference of the same type of sequence for N = 1024 and B = 150 MHz. radar. 150 I. INTRODUCTION In recent years more and more radar systems in unlicensed 100 bands have come to market resulting in an increasing interference level. Thus, new waveforms have to be developed being robust against mutual interference. Furthermore, in order f in MHz 50 to successfully compete in the market, the radars have to be cheap and thus, the hard- and software have to be as simple and cost efficient as possible. A well known and widely used radar 0 waveform in automotive and automation markets is Linear 0 2 4 6 8 10 12 14 Frequency Modulation Shift Keying (LFMSK) [1] which t in ms forms the basis of the developed frequency hopping radar Fig. 1. Exemplary frequency hopping sequence of the proposed waveform waveform. It is based on a proposal in [2] where the author suggests to scramble the linear increasing frequency steps of The analytical transmit (Tx) signal can be described as a LFMSK correspondent to a random permutation. However, T sum of complex oscillations of the particular length N and this method leads to some problems in a dynamic scenario the frequency f(n) = B π(n), with multiple moving targets. To overcome these difficulties, N N−1 T ! especially the signal processing part of the waveform has been X j2π f + B π(n) t t − N n s(t) = A e ( c N ) · rect , (2) extended. Tx T N This paper is organized as follows: In Section II the n=0 frequency hopping signal is presented followed by the radar where T is the duration of the whole radar signal, fc is the signal processing techniques in Section III. Section IV presents carrier frequency, π(n) is the n-th number of the permutation some analytical aspects as well as simulation results of the and rect(t) the rectangular function defined in [3]. proposed radar system. The receive (Rx) signal is an attenuated, time and frequency shifted version of the Tx signal. In the case of multiple targets, the reflected signals of all N objects are superimposed at the II. SIGNAL MODEL t receiver resulting in The radar signal is a modified version of a LFMSK signal Nt N−1 B 2·(Ri−vi·t) X X j2π(fc+ N π(n)) t− c where the linear increasing frequency steps are replaced by a r(t) = ARx;i e · random permutation of these steps. This results in a frequency i=1 n=0 (3) hopping like signal where the hopping sequence is a random Ri−vi·t T ! t − c − N n permutation π of length N, where N is the number of different rect T . steps. Thus, the probability that two independent radar systems N The down converted and low pass filtered Rx signal is is described by a vector x = [x(0) x(1) : : : x(N − 1)] N sampled at a sampling rate of fs = T correspondent to of length N which allows all following signal processing LFMSK. Thus, from each frequency step only one sample is steps to be written in matrix vector notation. To eliminate taken resulting in a total of N complex values. The sampling a frequency offset, this vector has to be multiplied element- point is chosen at the end of each step due to the settling time wise with a complex cosine signal of the same length and of the oscillator after each frequency hop in order to avoid the negative Doppler shift. Because multiple Doppler shifts transients. Neglecting the rectangular functions, the sampled shall be removed in parallel, a matrix X 2 CN×N is formed baseband signal is given by consisting of N-times the vector x: 0 1 Nt x X −j2π 2fcRi j2π 2vi f + B π(n) T n x(n) = A · e c · e c ( c N ) N · . i X = B . C . (8) i=1 (4) @ A 2R x −j2π B π(n) i e N c . An efficient way to perform the elimination in parallel is to Assuming that the bandwidth is much smaller than the carrier multiply X element-wise with a DFT-matrix WN , frequency, B fc, (4) can be simplified to Xe = X ◦ WN . (9) Nt X −j2π 2fcRi j2π 2fcviT n −j2π 2RiB π(n) Here, x(n) ≈ Ai ·e c ·e c N ·e c N . (5) 0 (− N )·1 (− N )·(N−1) 1 i=1 −j2π 2 −j2π 2 1 e N2 ::: e N2 B C III. RADAR PROCESSING − N +1 ·1 − N +1 ·(N−1) B ( 2 ) ( 2 ) C B −j2π 2 −j2π 2 C For simplification the signal processing is outlined initially B1 e N ::: e N C WN = B C . for a single target scenario. Further on also the case of multiple B. .. C B. C targets is treated. If only one target with no Doppler (v = 0) @ ( N −1)·1 ( N −1)·(N−1) A −j2π 2 −j2π 2 is considered, (5) can be rewritten to 1 e N2 ::: e N2 −j2π 2fcR −j2π 2RB π(n) (10) x(n) = A · e c ·e c N . (6) | {z } is a vertically shifted version of the original DFT-matrix =A0 [4]. Thus, the frequency offset of zero appears at row index The first exponential term is just a constant phase offset m = N , m 2 f0;:::;N − 1g. Furthermore, the vertical axis 0 2 and can be combined with A to the complex amplitude A directly corresponds to the velocity axis. whereas the second term can be used to extract the range Afterwards, the columns of Xe have to be reordered according information. Therefore, the samples are reordered according to to the inverse permutation: the permutation sequence used on Tx side. Using the relation ord −1 π−1 ◦ π = π ◦ π−1 ≡ id, where id is the identity function, the Xe (m; n) = Xe (m; π (n)) . (11) reordered signal is given by To get the range axis, for each row of Xe ord(m; n) an IFFT ord −1 0 −j2π 2RB n has to be calculated, resulting in x (n) = x π (n) = A · e c N , (7) n o Xord(m; k) = IFFT Xord(m; n) . (12) where the second exponential term shows a linear increasing e e phase. Thus, an IFFT can be used to estimate the range of the To detect the targets, a threshold γ can be applied, target by determining the maximum of the IFFT spectrum. identifying all values with Xe ord(m; ^ k^) ≥ γ to be a target, However, if the target is moving, the Rx signal is frequency whose range and velocity can be directly estimated from the shifted by a Doppler offset. This results in a linear increasing two-dimensional (2D) radar image. Fig. 2 shows an example phase term in x(n) as it can be seen in (5). After reordering with to targets at 20 m and 25 m and velocities of −48 km=h the samples, on the one hand the second term shows a linear and 16 km=h. phase, but on the other hand the linearity of the first term is The separation of multiple targets is possible due to the sum destroyed resulting in a noise like spectrum. To avoid this, orthogonality within the IFFT. Only the reflected signal that the Doppler shift has to be eliminated before reordering the corresponds to the actual frequency offset that is corrected samples. One possibility would be to estimate the frequency shows a linear increasing phase after the correction resulting offset by an additional constant CW tone emitted before in a peak after the IFFT. This can be seen if we consider the 2fcv1 l each hopping sequence. However, this would increase the case fD;1 = c = T , where fD;1 is the Doppler shift of duration of one Tx cycle and possibly be susceptible for target 1: interference. Additionally, multiple targets can have different ord Xe (m = l; n) = Doppler shifts and thus the frequency offsets have to be −1 2fcv1T π (n) 2RiB n 0 j2π ( c −l) N −j2π c N assigned unambiguously to the different distances of these A1 · e · e + | {z } (13) N =0 targets. To overcome these disadvantages multiple frequency t −1 X 2fcviT π (n) 2RiB n 0 j2π c −l N −j2π c N offsets are eliminated in parallel regardless of the real Doppler Ai · e · e .
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