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Aerospace Radar Radar-Verfahren und -Signalverarbeitung - Lesson 2: RADAR FUNDAMENTALS I Hon.-Prof. Dr.-Ing. Joachim Ender Head of Fraunhoferinstitut für Hochfrequenzphysik and Radartechnik FHR Neuenahrer Str. 20, 53343 Wachtberg [email protected] RADAR FUNDAMENTALS I Coherent radar - quadrature modulator and demodulator Figure: Quadrature modulator Figure: Quadrature demodulator The QM transfers the complex baseband signal to a real valued RF signal. Ender: Radarverfahren - 2 - RADAR FUNDAMENTALS I Reference Coherent radar - complex envelope frequency (RF) Real valued Complex envelope, RF signal base band signal The QDM performs the inverse operation to that of the QM. The real valued RF-signal may be regarded as a carrier of the base band-signal s(t), able to be transmitted as RF waves over long ranges. Figure: Bypass of quadrature modulator and demodulator Ender: Radarverfahren - 3 - RADAR FUNDAMENTALS I Coherent radar - generic radar system * Point target r r Distance to a point scatterer c0 Velocity of light Antenna t Traveling time N(t) White noise T/R Switch s(t;r) Received waveform a complex amplitude QM QDM f0 Traveling time as(t;r) Traveling distance s(t) N(t) Received signal Z(t) Figure: Radar system with baseband signals Ender: Radarverfahren - 4 - RADAR FUNDAMENTALS I Coherent radar - received waveform Wave length and wave number: c0 0 f0 Complex envelope 2 k 0 Received waveform 0 R 2f0t 2f0 c0 2R 0 k0 R Ender: Radarverfahren - 5 - RADAR FUNDAMENTALS I Coherent radar - Fourier transform of the received waveform Reference Baseband RF frequency frequency frequency Fourier transform: Wave number in range direction Ender: Radarverfahren - 6 - RADAR FUNDAMENTALS I QDM Coherent radar - optimum receive filter f0 as(t;r) N(t) Y(t) h(t) Z(t) i.e. Ender: Radarverfahren - 7 - RADAR FUNDAMENTALS I Coherent radar - matched filter The pulse response of the optimum filter is equal to the time-inverted, complex conjugated signal The maximum SNR is . This filter is called matched filter. Received signal Replica Response of matched filter Ender: Radarverfahren - 8 - RADAR FUNDAMENTALS I Coherent radar - correlation with the transmit signal Ender: Radarverfahren - 9 - RADAR FUNDAMENTALS I Coherent radar - point spread function Ender: Radarverfahren - 10 - RADAR FUNDAMENTALS I Coherent radar - matched filter, point spread function Matched filtering means correlation with the transmit signal. The point spread function is the reaction of the receive filter to the transmit signal. The point spread function is equal to the autocorrelation of the transmit signal, if a matched filter is used. In this case it is the Fourier back transform of the magnitude-squared of the signal spectrum. Point spread function Reflectivity of three point targets Output of the matched filter Ender: Radarverfahren - 11 - RADAR FUNDAMENTALS I Definitions of resolution c r t 2 c r Figure: Definitions of resolution Rayleigh 2b Ender: Radarverfahren - 12 - RADAR FUNDAMENTALS I Pulse compression The solution is to expand the bandwidth by modulation of the pulse. The Rayleigh range resolution of a waveform with a rectangular spectrum S(f) of bandwidth b is given by without direct dependence on the pulse length. Ender: Radarverfahren - 13 - RADAR FUNDAMENTALS I Pulse compression For the range resolution the bandwidth of the transmitted waveform is decisive: . The gain in range resolution with respect to a rectangular pulse of same duration is called compression rate, which is equal to the time-bandwidth product. Two different waveforms s(t) and sRF (t) effect the same point spread function for matched filtering, if sRF (t) is generated by s(t) by passing it through a filter with a transfer function of magnitude 1. Ender: Radarverfahren - 14 - RADAR FUNDAMENTALS I Generation of high bandwidth signals Re Im Analogue Phase modulation with phase shifter Frequency modulation by a VCO Memory SAW filter (writeable, Frequency multiples (non-linear devices, fast read out) extraction of higher harmonics) Clock (e.g. 1 GHz) D/A D/A Digital Arbitrary wave form generator (AWG) de- de- Filter glitch glitch Direct digital synthesizer (DDS) VCO coupled to DDS AWG principle Ender: Radarverfahren - 15 - RADAR FUNDAMENTALS I Generation of high bandwidth signals Memory slow read out (e.g. 50 MHz) cos sin fast accu- Look-up table mulator (fast read out) (mod 2) read pointer Clock (e.g. 1 GHz) D/A D/A Fast logic de- de- Filter DDS principle (e.g. GaAs) glitch glitch Ender: Radarverfahren - 16 - RADAR FUNDAMENTALS I Digital pulse compression in the frequency domain Receive signal The reference signal can be the designed (wanted) signal A/D the measured signal in a calibration mode ze FFT Ze * S S reference complex conj. FFT signal s Za FFT za This part is performed only during calibration Ender: Radarverfahren - 17 - RADAR FUNDAMENTALS I Pulse compression in the time and in the frequency domain Range com- t Raw data t Raw data t pressed data T T T For each range line For each range line Compression h(t)=s*(-t) filter Range FFT Range IFFT H(f)=S*(f) Range com- t pressed data f f T T T Figure: Range compression with the matched filter. Left: direct convolution, right: processing in the frequency domain Ender: Radarverfahren - 18 - RADAR FUNDAMENTALS I Pulse compression - Anatomy of a chirp I Rectangular chirp Frequency span b ts (= bandwidth for large ): t 2 s(t) rect expjt ts 2 Instantaneous frequency: Time-bandwidth product: bts ts f (t) t R{s(t)} f f=t t t -ts/2 ts/2 -ts/2 ts/2 Ender: Radarverfahren - 19 - RADAR FUNDAMENTALS I Pulse compression - Anatomy of a chirp I Fourier transform of a rectangular chirp Ender: Radarverfahren - 20 - RADAR FUNDAMENTALS I Pulse compression - Anatomy of a chirp I Figure: Magnitude of the Fourier transforms of chirps with growing time basis Ender: Radarverfahren - 21 - RADAR FUNDAMENTALS I Pulse compression - Anatomy of a chirp I Fourier transform of a rectangular chirp The magnitude of the Fourier transform of a rectangular chirp with rate has the approximate shape of a rectangular function with bandwidth close to the frequency span ts. For infinite duration the Fourier transform is again a chirp with rate -1/ . Ender: Radarverfahren - 22 - RADAR FUNDAMENTALS I Pulse compression - Compression of a chirp Compression Chirp result tac t t F F-1 t|.|2 f f Power Spectrum Spectrum Ender: Radarverfahren - 23 - RADAR FUNDAMENTALS I De-ramping Ender: Radarverfahren - 24 - RADAR FUNDAMENTALS I Spatial interpretation of the radar signal and the receive filter From the viewpoint of focusing to images (SAR), the spatial domain is the primary one. We transform the temporal signals into spatial signals, dependent on the spatial variable R = 2r via t -> R = c0 t. We will use the symbols s, h, p as functions of R. Signal spectrum (wave number domain) Transfer function (wave number domain) Point spread function (wave number domain) For the matched filter we get The point spread function for matched filtering in the range domain is given by the inverse Fourier transform of the power of the signal spectrum in the wave number domain. Ender: Radarverfahren - 25 - RADAR FUNDAMENTALS I Matched filter / inverse filter / robustified filter We regard a receive filter with transfer function Ender: Radarverfahren - 26 - RADAR FUNDAMENTALS I Matched filter / inverse filter / robustified filter Figure: Robustified inverse filter Ender: Radarverfahren - 27 - RADAR FUNDAMENTALS I The k-set Ender: Radarverfahren - 28 - RADAR FUNDAMENTALS I The k-set For the application of the inverse filter, the point spread function is equal to the Fourier back transform of the indicator function of the carrier of the signal spectrum (k-set) Ender: Radarverfahren - 29 - RADAR FUNDAMENTALS I Pre-processing to the normal form (inverse filter) We regard a noise-free signal of a point scatterer at R=R0: Ender: Radarverfahren - 30 - RADAR FUNDAMENTALS I Coherent radar Pulse repetition frequency: PRF (~ 100 Hz ... 10 kHz) Intrapulse sampling frequency: fs (~ 10 MHz ... 1 GHz) 1 F PRF s T 1 f s t T= 1ms: Covered range =150 km t= 1ns: Range sampling = 15 cm Figure: Two time scales for pulse radar Ender: Radarverfahren - 31 - RADAR FUNDAMENTALS I Pre-processing to the normal form (inverse filter) Figure: Pre-processing in the k-domain Ender: Radarverfahren - 32 - RADAR FUNDAMENTALS I Doppler effect Basic component of the Doppler frequency: Resolution of the waveform in spectral components: Ender: Radarverfahren - 33 - RADAR FUNDAMENTALS I Doppler effect Object motion negligible during the wave's travelling time (stop and go approximation): Ender: Radarverfahren - 34 - RADAR FUNDAMENTALS I Doppler effect Exact expression The Doppler frequency of a moving target is given by For the stop-and-go approximation this is simplified to Ender: Radarverfahren - 35 - RADAR FUNDAMENTALS I Doppler effect Effects caused by target motion Phase rotation from pulse to pulse Range migration Phase modulation during one pulse Intra-pulse time stretch / compression Christian Andreas Doppler (29 November 1803 – 17 March 1853) was an Austrian mathematician and physicist. He is most famous for describing what is now called the Doppler effect, which is the apparent change in frequency and wavelength of a wave as perceived by an observer moving relative to the wave's source. Ender: Radarverfahren - 36 - RADAR FUNDAMENTALS I Doppler effect - modulation and time expansion Ender: Radarverfahren - 37 - RADAR FUNDAMENTALS I Doppler effect - in the two-times domain slow time fast time
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