1. Principle introduction:
The FM continuous wave introduced in this article uses a modulation signal in the form of a triangular wave, as shown in the figure below: The radar transmits a detection signal (blue triangle waveform in the figure) to an object with a detection distance of R, and the mathematical formula for the elapsed time is: τ \ tauAfter τ is reflected back (brown triangle waveform in the figure), the radar receiving system interferes (blue trapezoid in the figure).
τ = 2 R c \tau =\frac{2R}{c} t=c2 R
Among them: c is the propagation speed of electromagnetic wave in space;
Where: B is the FM bandwidth of the modulating signal, T is the FM period of the modulating signal; mathematical formula: Δ f \Delta fΔ f is the Doppler frequency shift generated by the moving speed of the measurement target:
Definition: Δ f = 2 v λ \Delta f=\frac{2v}{\lambda}f _=l2 v
2. Calculation formula derivation
Assumption: The functional expressions of the rising phase (first half cycle) and falling phase (second half cycle) of the FM signal are:
fu 1 ( t ) = f 0 + α t f_{u1}(t) =f_{0 }+ \alpha tfin 1(t)=f0+αt
f d 1 ( t ) = f 0 − α （ t − T 2 ) f_{d1}(t) =f_{0}- \alpha （t-\frac{T}{2}) fd 1(t)=f0−a （t−2T)
where:α = BT / 2 = 2 BT \alpha =\frac{B}{T/2}=\frac{2B}{T}a=T/2B=T2 B;
f 0 f_{0} f0is the fundamental frequency of electromagnetic waves.
The function expressions of the detection signal reflected from the target are:
fu 2 ( t ) = f 0 + α ( t − τ ) + Δ f f_{u2}(t) = f_{0}+ \alpha (t -\tau)+\Delta ffu 2(t)=f0+a （t−t )+Δf
f d 2 ( t ) = f 0 − α （ t − τ − T 2 ) + Δ f f_{d2}(t) =f_{0}- \alpha （t-\tau-\frac{T}{2})+\Delta f fd 2(t)=f0−a （t−t−2T)+f _
The frequencies of the obtained interference signals are:
f 1 = ∣ fu 1 ( t ) − fu 2 ( t ) ∣ = α τ − Δ f = 2 BT ∗ 2 R c − 2 v λ f_{1} =|f_{ u1}(t)-f_{u2}(t)|=\alpha \tau -\Delta f=\frac{2B}{T}*\frac{2R}{c}-\frac{2v}{\lambda }f1=∣fin 1(t)−fu 2(t)∣=at−f _=T2 B∗c2 R−l2 v
f 2 = ∣ f d 1 ( t ) − f d 2 ( t ) ∣ = α τ + Δ f = 2 B T ∗ 2 R c + 2 v λ f_{2} =|f_{d1}(t)-f_{d2}(t)|=\alpha \tau +\Delta f=\frac{2B}{T}*\frac{2R}{c}+\frac{2v}{\lambda} f2=∣fd 1(t)−fd 2(t)∣=at+f _=T2 B∗c2 R+l2 v
Combining the above formulas, we can get:
R = c T 8 B ( f 1 + f 2 ) R=\frac{cT}{8B}(f_{1}+f_{2})R=8BcT(f1+f2)
V = λ 4 ( ∣ f 1 − f 2 ∣ ) V=\frac{\lambda}{4}(|f_{1}-f_{2}|) V=4l(∣f1−f2∣)