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 fast time
Figure: Doppler-effect for a pulse train
Ender: Radarverfahren - 38 - RADAR FUNDAMENTALS I Range-Doppler processing
Pre-processed
k r data T
Slow-time FFT
Double fre-
k r quency data F
Range-Doppler Range IFFT Range-Doppler processing with processing via the subsequent pulse double frequency Range- compression and domain R Doppler data Doppler filtering F
Ender: Radarverfahren - 39 - RADAR FUNDAMENTALS I 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
Ender: Radarverfahren - 40 - RADAR FUNDAMENTALS I Ambiguity function
Figure: Ambiguity function of a chirp with Gaussian envelope
Ender: Radarverfahren - 41 - RADAR FUNDAMENTALS I Doppler tolerance of a chirp f Doppler shifted signal
f0+ time of best fit t Matched filter f0 for f0
Area of phase match
t time shift t0-t t0
Ender: Radarverfahren - 42 - RADAR FUNDAMENTALS I 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 non- overlapping part of the signal spectra. A time shift is effected which is proportional to the Doppler shift.
Ender: Radarverfahren - 43 - RADAR FUNDAMENTALS I Ambiguity function
Figure: Ambiguity function of a train of 5 rectangular pulses
Ender: Radarverfahren - 44 - RADAR FUNDAMENTALS I Ambiguity of range and Doppler Doppler ambiguity: 1 PRF modes: F PRF T Range ambiguity: Low PRF: unambiguous range, c ambiguous Doppler r 0 T 2 High PRF: ambiguous range, unambiguous Doppler Area of ambiguity rectangle: c Medium PRF: in between Fr 0 2 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 c0/2. The product of range ambiguity and radial velocity ambiguity is c0/ = f.
Ender: Radarverfahren - 45 -