The radar range equation expresses the value of power received as a function of radar operating and performance parameters. A summary of its derivation applicable to RAASTI, assuming a distributed target is presented here3. The transmitted chirp, in volts can be written as:
st(t) =Vtcos(2πfct+παt2)[V], (2.29)
valid over the chirp duration period (−Tp/2< t < Tp/2), whereVtis the voltage gain of the radar
system, whose variation with time is shown ingure 2.7.
The instantaneous power at the transmit antenna port during generation a single chirp (Tp) is equal 3For a more general description refer toSkolnik[1970], orUlaby et al.[1986a].
2.4. FMCW DISTRIBUTED TARGET RADAR RANGE EQUATION 37
Figure 2.7: Block diagram of the sweep voltage driving the oscillator to output a chirp. The voltage gain of the system leads to a chirp amplitude ofVt.
to:
Pt(t) =st(t)2/R [W], (2.30)
whereR is the input impedance of the antenna (with a coupled antenna this is usually50 ).
This power is convolved with the antenna’s impulse response and radiated into space. As it travels the signal power suffers from spreading loss, with the power density at the distributed target, at a range R equal to:
Sinc=Pt(t)·Gt·4 1
πR2 [W/m
2]. (2.31)
Subsequently this power is scattered by the distributed target. The area from which a single tone will be reected is range-cell limited, with therst range-cell area equal to:
A1 =π·+(R+δR)2−R2, [m2]. (2.32)
Assuming a level of surface roughness the effectiveness of the scattering is determined by the backscattering coefcient,σ0. Consequently, the power scattered by the target is:
Ptar=Sinc· %
A1·σ0
&
[W]. (2.33)
The scattered signal undergoes additional spreading loss on its path towards the receive antenna, and the power density at the receive antenna is:
Sant=Pt(t)·Gtx· 1 4πR2 · % A1·σ0 & ·4πR1 2 [W/m2]. (2.34) This power is then intercepted by the effective area of the receive antenna, Aef f, (dened as:
λ2
4πGrx, seeBalanis, [1977]), and the received power is therefore: Pr(t−τ) =Pt(t)·Gtx· 1 4πR2 · % A1·σ0 & ·4 1 πR2 ·Aef f [W]. (2.35)
Equation 2.35 is the radar range equation for a distributed target for FMCW radar4.
Additional losses to the received power level are due to cable loss, component insertion loss etc. However, these losses were controlled by the inclusion of small valued attenuators between components in the radar to fascilitate coupling, limit feedback and signal reection. Consequently system losses are minimal and not included here.
2.4.1 Signal to Noise Ratio:SN R
In order to calculate the SNR at the output of the FFT, the signal power and noise constituents are traced through the radar components, beginning with the receive antenna.
Figure 2.8 shows the receiver chain with the respective component gains, losses and changing signal bandwidths. After quantisation, an N-point FFT is taken and consequently acoherent power gainof: Gcoh ≡ N22, is applied to the signal, and adc gainof: Gdc ≡ N2, is applied to the noise,
(de Wit, 2005). This results in the power of the frequency peak corresponding to the single tone and the noiseoor at the output of the FFT to be respectively:
PF F T =Gcoh·Grx·Pr[W], (2.36)
NF F T =Gdc·Grx·NT ·βF F T [W]. (2.37)
whereGrxis the gain of the overall receiver chain. 4To paraphraseSkolnik[1970]:
The badge of a novice in the FM-radar eld is a carefully worked out performance appraisal based only on the application of the radar range equation.
Hence, it is acknowledged that the analysis of the inuence of phase noise on the signal noiseoor, the possibility of small signal suppression, and interference is neglected, but is not necessary for a
2.4. FMCW DISTRIBUTED TARGET RADAR RANGE EQUATION 39
Hence, the SNR at the output of the FFT is:
SN RF F T =PF F T/NF F T,
=N· Pr
NT ·βF F T
. (2.38)
Quantisation Noise
The quantisation noise of the ADC may be a limiting factor on the achievable SNR of the system. The signal-to-quantisation-noise ratio of an ideal ADC is dened (Proakis and Manolakis[2004], pp. 756) as: SN RQ= 6.02N + 16.81−20log10( R σx) [dB], (2.39) = 6.02×12 + 16.81−20log10(10/4), (2.40) ≈69[dB], (2.41)
where N is the number of bits, andRis the range of the ADC, andσ2
xis the signal power. It should
be noted here, that although this derivation is for an ideal ADC, it is useful in that it provides an upper bound on the SNR.
2.4.2 Theoretical Minimum Backscattering Coefcient: σ0
The minimum detectable backscattering coefcient can now be approximated by determining conditions for which the SN RF F T satises the probability of detection, and false-alarm rate as
discussed in section 2.3.8. For this to occur, the following relationship must hold:
SN RF F T >6.0[dB]. (2.42)
Using this condition5, and the values provided in tables 2.1, and 2.2, the minimum backscattering
coefcient to which RAASTI is sensitive can be approximated. It is calculated to vary overRmin 5For theSN R
F F T, using an N = 14001 point FFT (as used in the signal processing algorithms
(a) (b) Figure 2.8: Recei ver chain components and their associated noise, gain, and bandwidth values.