Radar-Verfahren und -Signalverarbeitung

Transcription

1 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, Wachtberg joachim.ender@fhr.fraunhofer.de

2 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

3 Coherent radar - complex envelope Reference frequency (RF) Real valued RF signal Complex envelope, base band signal Figure: Bypass of quadrature modulator and demodulator 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

4 Coherent radar - generic radar system r * Point target Antenna T/R Switch r c 0 t N(t) s(t;r) a Distance to a point scatterer Velocity of light Traveling time White noise Received waveform complex amplitude QM QDM f 0 Traveling time s(t) as(t;r) N(t) Traveling distance Received signal Z(t) Figure: Radar system with baseband signals - 4 -

5 - 5 - RADAR FUNDAMENTALS I Coherent radar - received waveform Complex envelope Received waveform Wave length and wave number: R k R c R f f t k f c

6 Coherent radar - Fourier transform of the received waveform Fourier transform: Reference frequency Baseband frequency RF frequency Wave number in range direction - 6 -

7 Coherent radar - optimum receive filter f 0 as(t;r) QDM N(t) Y(t) h(t) Z(t) i.e

8 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 - 8 -

9 Coherent radar - correlation with the transmit signal - 9 -

10 Coherent radar - point spread function

11 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

12 Definitions of resolution c r t 2 r Rayleigh c 2b Figure: Definitions of resolution

13 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

14 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 s RF (t) effect the same point spread function for matched filtering, if s RF (t) is generated by s(t) by passing it through a filter with a transfer function of magnitude

15 Generation of high bandwidth signals Re Im Analogue Phase modulation with phase shifter Frequency modulation by a VCO SAW filter Frequency multiples (non-linear devices, extraction of higher harmonics) Digital Arbitrary wave form generator (AWG) Direct digital synthesizer (DDS) VCO coupled to DDS D/A D/A deglitch deglitch AWG principle Memory (writeable, fast read out) Clock (e.g. 1 GHz) Filter

16 Memory slow read out (e.g. 50 MHz) RADAR FUNDAMENTALS I Generation of high bandwidth signals cos sin fast accumulator (mod 2) read pointer Look-up table (fast read out) Fast logic (e.g. GaAs) D/A D/A deglitch deglitch Clock (e.g. 1 GHz) Filter DDS principle

17 Digital pulse compression in the frequency domain Receive signal z e A/D The reference signal can be the designed (wanted) signal the measured signal in a calibration mode Z e FFT Z a S * S reference complex conj. FFT signal s FFT z a This part is performed only during calibration

18 Pulse compression in the time and in the frequency domain t t Raw data T Compression filter Range compressed data T For each range line h(t)=s*(-t) t f Raw data T For each range line Range FFT T H(f)=S*(f) Range compressed data t T Range IFFT f T Figure: Range compression with the matched filter. Left: direct convolution, right: processing in the frequency domain

19 Pulse compression - Anatomy of a chirp I Rectangular chirp t s( t) rect exp jt t s f ( t) t Instantaneous frequency: R{s(t)} 2 Frequency span (= bandwidth for large ): Time-bandwidth product: f b t s bt 2 s t s t f=t t -t s /2 t s /2 -t s /2 t s /2-19 -

20 Pulse compression - Anatomy of a chirp I Fourier transform of a rectangular chirp

21 Pulse compression - Anatomy of a chirp I Figure: Magnitude of the Fourier transforms of chirps with growing time basis

22 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 t s. For infinite duration the Fourier transform is again a chirp with rate -1/

23 Pulse compression - Compression of a chirp Chirp Compression result t t ac t F F -1 Spectrum f t. 2 Power Spectrum f

24 De-ramping

25 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 = c 0 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

26 Matched filter / inverse filter / robustified filter We regard a receive filter with transfer function

27 Matched filter / inverse filter / robustified filter Figure: Robustified inverse filter

28 The k-set

29 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)

30 Pre-processing to the normal form (inverse filter) We regard a noise-free signal of a point scatterer at R=R 0 :

31 Coherent radar Pulse repetition frequency: PRF (~ 100 Hz khz) Intrapulse sampling frequency: f s (~ 10 MHz... 1 GHz) F f s s PRF 1 t 1 T T= 1ms: Covered range =150 km t= 1ns: Range sampling = 15 cm Figure: Two time scales for pulse radar

32 Pre-processing to the normal form (inverse filter) Figure: Pre-processing in the k-domain

33 Doppler effect Basic component of the Doppler frequency: Resolution of the waveform in spectral components:

34 Doppler effect Object motion negligible during the wave's travelling time (stop and go approximation):

35 Doppler effect Exact expression The Doppler frequency of a moving target is given by For the stop-and-go approximation this is simplified to

36 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 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

37 Doppler effect - modulation and time expansion

38 Doppler effect - in the two-times domain slow time fast time Figure: Doppler-effect for a pulse train

39 Range-Doppler processing Pre-processed k r data T Slow-time FFT k r Double frequency data F Range-Doppler processing with subsequent pulse compression and Doppler filtering Range IFFT Range- R Doppler data F Range-Doppler processing via the double frequency domain

40 Ambiguity function Ambiguity function = response of a matched filter to a signal shifted in time and Doppler frequency Figure: Ambiguity function of a rectangular pulse

41 Ambiguity function Figure: Ambiguity function of a chirp with Gaussian envelope

42 Doppler tolerance of a chirp f Doppler shifted signal f 0 + f 0 time of best fit t Matched filter for f 0 Area of phase match time shift t 0 -t t 0 t

43 Doppler tolerance of a chirp Obviously, for the chirp waveform there is an ambiguity between Doppler and range. If one of the two is known, the other variable can be measured with high accuracy. The Doppler frequency may be measured over a sequence of pulses and used for a correction of range. The chirp wave form is Doppler tolerant, i.e. a Doppler shift of the echo with respect to the reference chirp leads only to a moderate SNR loss corresponding to the nonoverlapping part of the signal spectra. A time shift is effected which is proportional to the Doppler shift

44 Ambiguity function Figure: Ambiguity function of a train of 5 rectangular pulses

45 Ambiguity of range and Doppler Doppler ambiguity: F Fr 1 PRF modes: T c 0 2 Range ambiguity: r c0 2 T PRF Area of ambiguity rectangle: Low PRF: unambiguous range, ambiguous Doppler High PRF: ambiguous range, unambiguous Doppler Medium PRF: in between For a pulse train repeated with T, the product of temporal and frequency ambiguity is equal to 1. The product of range ambiguity and Doppler ambiguity is equal to c 0 /2. The product of range ambiguity and radial velocity ambiguity is c 0 / = f

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