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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
Ender: Radarverfahren- 2 -
RADAR FUNDAMENTALS I
Coherent radar - quadrature modulator and demodulator
The QM transfers the
complex baseband signal to a
real valued RF signal.
Figure: Quadrature modulator Figure: Quadrature demodulator
Ender: Radarverfahren- 3 -
RADAR FUNDAMENTALS I
Coherent radar - complex envelope
Real valued RF signal
Complex envelope,base band signal
Reference frequency (RF)
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- 4 -
RADAR FUNDAMENTALS I
Coherent radar - generic radar system
Traveling time
Traveling distance
Received signal
r Distance to a point scatterer
c0 Velocity of light
t Traveling time
N(t) White noise
s(t;r) Received waveform
a complex amplitude
Point target
Antenna
T/R Switch
r
*
QM
s(t)
f0
QDM
as(t;r)
N(t)
Z(t)
Figure: Radar system with baseband signals
Ender: Radarverfahren- 5 -
RADAR FUNDAMENTALS I
Coherent radar - received waveform
Complex envelope
Received waveform
Wave length and
wave number:
Rk
R
c
Rff
0
0
0
00
2
22
t
0
0
0
00
2
k
f
c
Ender: Radarverfahren- 6 -
RADAR FUNDAMENTALS I
Coherent radar - Fourier transform of the received waveform
Fourier transform:
Baseband frequency
Reference frequency
RF frequency
Wave number in range direction
Ender: Radarverfahren- 7 -
RADAR FUNDAMENTALS I
Coherent radar - optimum receive filter
i.e.
f0
QDM
as(t;r)
N(t)
Z(t)h(t)Y(t)
Ender: Radarverfahren- 8 -
RADAR FUNDAMENTALS I
Coherent radar - matched filter
Received
signal
Replica
Response of
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.
Ender: Radarverfahren- 9 -
RADAR FUNDAMENTALS I
Coherent radar - correlation with the transmit signal
Ender: Radarverfahren- 10 -
RADAR FUNDAMENTALS I
Coherent radar - point spread function
Ender: Radarverfahren- 11 -
RADAR FUNDAMENTALS I
Coherent radar - matched filter, point spread function
Reflectivity of three point targets Output of the 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.
Ender: Radarverfahren- 12 -
RADAR FUNDAMENTALS I
Definitions of resolution
b
cr
tc
r
Rayleigh2
2
Figure: Definitions of resolution
Ender: Radarverfahren- 13 -
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- 14 -
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- 15 -
RADAR FUNDAMENTALS I
Generation of high bandwidth signals
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 DDSAWG principle
Re Im
Memory(writeable,
fast read out)
D/A D/AClock (e.g. 1 GHz)
de-
glitch
de-
glitchFilter
Ender: Radarverfahren- 16 -
DDS principle
D/A D/AClock (e.g. 1 GHz)
de-
glitch
de-glitch
Filter
cos sin
Look-up table(fast read out)
read pointer
fast accu-mulator
(mod 2)
Fast logic(e.g. GaAs)
Memory
slow read out(e.g. 50 MHz)
RADAR FUNDAMENTALS I
Generation of high bandwidth signals
Ender: Radarverfahren- 17 -
The reference signal can be
the designed (wanted) signal
the measured signal in a calibration mode
Receive signal
A/D
FFT
complex conj.
FFT
FFT
ze
Ze
za
S* S referencesignal s
Za
This part is performed only during calibration
RADAR FUNDAMENTALS I
Digital pulse compression in the frequency domain
Ender: Radarverfahren- 18 -
RADAR FUNDAMENTALS I
Pulse compression in the time and in the frequency domain
h(t)=s*(-t)Compression
filter
Tt
Range com-
pressed data
For each range line
Range FFT
T
t Raw data
Tf
For each range line
T
t Raw data
H(f)=S*(f)
Tf
Range IFFT
Tt
Range com-
pressed data
Figure: Range compression with the matched filter.
Left: direct convolution, right: processing in the frequency domain
Ender: Radarverfahren- 19 -
RADAR FUNDAMENTALS I
Pulse compression - Anatomy of a chirp I
2exp)( tjt
trectts
s
ttf )(
Instantaneous frequency:
Frequency span
(= bandwidth for large ):stb
Time-bandwidth product:2
ss tbt
Rectangular chirp
t
R{s(t)}
ts/2-ts/2
t
f
f=t
ts/2-ts/2
Ender: Radarverfahren- 20 -
RADAR FUNDAMENTALS I
Pulse compression - Anatomy of a chirp I
Fourier transform of a rectangular chirp
Ender: Radarverfahren- 21 -
RADAR FUNDAMENTALS I
Pulse compression - Anatomy of a chirp I
Figure: Magnitude of the Fourier transforms of chirps with growing time basis
Ender: Radarverfahren- 22 -
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- 23 -
t
f
t
f
Chirp
F F-1
act
t|.|2
Spectrum
Power
Spectrum
Compression
result
RADAR FUNDAMENTALS I
Pulse compression - Compression of a chirp
Ender: Radarverfahren- 24 -
RADAR FUNDAMENTALS I
De-ramping
Ender: Radarverfahren- 25 -
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- 26 -
RADAR FUNDAMENTALS I
Matched filter / inverse filter / robustified filter
We regard a receive filter with transfer function
Ender: Radarverfahren- 27 -
RADAR FUNDAMENTALS I
Matched filter / inverse filter / robustified filter
Figure: Robustified inverse filter
Ender: Radarverfahren- 28 -
RADAR FUNDAMENTALS I
The k-set
Ender: Radarverfahren- 29 -
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- 30 -
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- 31 -
RADAR FUNDAMENTALS I
Coherent radar
Pulse repetition frequency:
PRF (~ 100 Hz ... 10 kHz)
Intrapulse sampling frequency:
fs (~ 10 MHz ... 1 GHz)
tf
TPRFF
s
s
1
1
T= 1ms: Covered range =150 km
t= 1ns: Range sampling = 15 cm Figure: Two time scales for pulse radar
Ender: Radarverfahren- 32 -
RADAR FUNDAMENTALS I
Pre-processing to the normal form (inverse filter)
Figure: Pre-processing in the k-domain
Ender: Radarverfahren- 33 -
RADAR FUNDAMENTALS I
Doppler effect
Basic component of the Doppler frequency:
Resolution of the waveform in spectral components:
Ender: Radarverfahren- 34 -
RADAR FUNDAMENTALS I
Doppler effect
Object motion negligible during the wave's travelling time (stop and go
approximation):
Ender: Radarverfahren- 35 -
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- 36 -
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- 37 -
RADAR FUNDAMENTALS I
Doppler effect - modulation and time expansion
Ender: Radarverfahren- 38 -
RADAR FUNDAMENTALS I
Doppler effect - in the two-times domain
slow time
fast time
Figure: Doppler-effect for a pulse train
Ender: Radarverfahren- 39 -
RADAR FUNDAMENTALS I
Range-Doppler processing
Range-Doppler
processing with
subsequent pulse
compression and
Doppler filtering
Range-Doppler
processing via the
double frequency
domain
Tk r
Pre-processed
data
Slow-time FFT
Fk r
Double fre-
quency data
Range IFFT
FR
Range-
Doppler data
Ender: Radarverfahren- 40 -
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- 41 -
RADAR FUNDAMENTALS I
Ambiguity function
Figure: Ambiguity function of a chirp with Gaussian envelope
Ender: Radarverfahren- 42 -
RADAR FUNDAMENTALS I
Doppler tolerance of a chirp
time of best fit
Doppler shifted signal
Matched filter
for f0
t0
t
f0t
f
Area of phase match
f0+
t0-ttime shift
Ender: Radarverfahren- 43 -
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- 44 -
RADAR FUNDAMENTALS I
Ambiguity function
Figure: Ambiguity function of a train of 5 rectangular pulses
Ender: Radarverfahren- 45 -
RADAR FUNDAMENTALS I
Ambiguity of range and Doppler
Doppler ambiguity:
PRFT
F
1 PRF modes:
Low PRF: unambiguous range,
ambiguous Doppler
High PRF: ambiguous range,
unambiguous Doppler
Medium PRF: in between
2
0crF
Range ambiguity:
Tc
r 2
0
Area of ambiguity rectangle:
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.