FMCW signal
Frequency modulated continuous wave (frequency modulated continuous wave, FMCW), as the name implies, is a signal that linearly modulates the frequency of the signal .
From the time domain, the frequency modulation is like a continuous sawtooth wave. Each sawtooth is called a chirp, and its duration is called the chirp period (chirp period or chirp repetition time); in actual use, we combine the chirps into one frame (frame) to get the speed of the object (or more Puler frequency deviation) information, is the FMCW signal.
We define the ratio of the sweep bandwidth to the chirp period as the coefficient K, which is the slope of the sawtooth wave.
K = sweep bandwidth chirp cycle = sweep end frequency − sweep start frequency chirp cycle K = \frac{sweep bandwidth}{chirp cycle}= \frac{sweep end frequency-sweep start frequency}{chirp cycle}K=c hi r p cycleSweep bandwidth=c hi r p cycleSweep stop frequency−Sweep start frequency
If we look at the change of the signal amplitude at this time in the time domain, we can find that as time goes by, the frequency corresponding to the sine wave will become higher and higher, that is, it will become denser and denser.
IF signal
The IF signal we use refers to the low intermediate frequency signal (or beat signal, beat signal) obtained after the received echo signal is mixed with the original signal, and then passed through a low-pass filter . Expressed in a simplified diagram:
We only focus on the expressions of 1 transmit signal, 2 receive signals and 3 IF signals.
1 transmit signal
Since the transmitted signal is an FMCW signal, taking one of the chirps, we know that the derivative of the phase with respect to time is the angular frequency (divided by 2 π 2 \pi2 π is the frequency), as follows:
1 2 π d ϕ dt = fo + K t \frac{1}{2\pi}\frac{d \phi}{dt} = f_o+Kt2 p.m1dtdϕ=fo+K t
So, when we integrate both sides of the above formula, we have:
ϕ = 2 π ( fot + 1 2 K t 2 ) + ϕ o = 2 π fot + π K t 2 + ϕ 0 \phi = 2\pi( f_ot + \frac{1}{2}Kt^2)+\phi_o = 2\pi f_ot + \pi Kt^2+\phi_0ϕ=2π(fot+21Kt2)+ϕo=2πfot+π K t2+ϕ0
So the form of the transmitted signal at 1 is:
x T x ( t ) = A sin ( 2 π fot + π K t 2 + ϕ 0 ) x_{\tiny{T}x}(t) = A \sin(2 \pi f_ot +\pi Kt^2+\phi_0)xTx(t)=Asin(2πfot+π K t2+ϕ0)
Note that the above formula is the form of the real signal sent ( all real signals are transmitted in the physical world ), and it is converted into a complex signal (this can be realized by Hilbert transform in signal processing), that is
xr = A ej ( 2 π fot + π K t 2 + ϕ 0 ′ ) x_r = Ae^{j(2 \pi f_ot +\pi Kt^2+\phi_0^{\prime})}xr=Aej(2πfot+πKt2 +ϕ0′)
Among them, K is the slope, ϕ o \phi_oϕois the initial phase of the signal, f 0 f_0f0is the center frequency:
fo = sweep start frequency + sweep end frequency 2 f_o =\frac{sweep start frequency+sweep end frequency}{2}fo=2Sweep start frequency+Sweep end frequency
2 receiving signals
The transmitted signal will be reflected after encountering the target (Target), thus generating the echo signals of the two receiving antennas, assuming that the time delay is τ \tauτ , the attenuation coefficient isaaa , then
x R x ( t ) = ax T x ( t − τ ) = A ′ sin [ 2 π fo ( t − τ ) + π K ( t − τ ) 2 + ϕ 0 ] = A sin [ π K t 2 + 2 π ( fo − K τ ) t + π K τ 2 − 2 π fo τ + ϕ 0 ] x_{\tiny{R}x}(t) = a x_{\tiny{T}x}(t -\tau) = A^{\prime} \sin[2 \pi f_o(t-\tau) +\pi K(t-\tau)^2+\phi_0] \\ =A\sin[\pi K t^2 +2\pi (f_o - K \tau)t+\pi K \tau^2-2 \pi f_o \tau + \phi_0]xRx(t)=axTx(t−t )=A′sin[2πfo(t−t )+πK(t−t )2+ϕ0]=Asin [ π K t2+2π(fo−Kτ)t+π K t2−2πfot+ϕ0]
3 IF signals
According to the product and difference formula in the trigonometric formula, namely:
sin ( α ) sin ( β ) = 1 2 [ cos ( α − β ) − cos ( α + β ) ] \sin(\alpha) \ sin(\beta) = \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta) ]sin ( α )sin ( b )=21[ cos ( a−b )−cos ( a+b )]
So our mixed signal is:
x T x ( t ) × x R x ( t ) = 1 2 AA ′ [ cos ( 2 π K τ t + 2 π fo τ − π K τ 2 ) − cos ( 2 π ( 2 fo − K τ ) t + . . . ) ] x_{\tiny{T}x}(t) \times x_{\tiny{R}x}(t) =\frac{1}{ 2}AA^{\prime}[\cos(2\pi K\tau t+2\pi f_o \tau-\pi K \tau^2 ) \\- \cos(2\pi(2 f_o-K \ tau)t+...)]xTx(t)×xRx(t)=21AA′[cos(2πKτt+2πfot−π K t2)−cos(2π(2fo−Kτ)t+... )]
After passing through the low-pass filter,the sum formula in the result will be filtered out as high-frequency components, leaving only the low-frequency components in the difference formula, and then we will amplify the difference signal through the intermediate frequency amplifier, and finally the Get the intermediate frequency signal at 3 places, namely:
x IF ( t ) = A ′ ′ cos ( 2 π K τ t + 2 π fo τ − π K τ 2 ) x_{\tiny{IF}}(t) = A^{\prime \prime} \cos (2\pi K\tau t+2\pi f_o \tau-\pi K \tau^2 )xIF(t)=A′′cos(2πKτt+2πfot−π K t2)
Let us have a perceptual understanding of the above parameters: Since the electromagnetic wave moves at the speed of light, the time delay τ \tau of the target at 1m aheadThe magnitude of τ is roughly1 0 − 8 10^{-8}10−8， f 0 f_0 f0For millimeter wave radar at 1 0 9 10^9109 magnitude, and the general K is about 1GHz divided by 0.1ms level, that is,1 0 13 10^{13}1013th grade. Compare two additional phases:
fo τ ≈ 10 K τ 2 ≈ 1 0 − 3 f_o \tau \approx 10 \ \ \ K\tau ^2 \approx 10^{-3}fot≈10 K sq 2≈10− 3
Therefore, the additional phase of the last item is almost negligible, that is,the form of the intermediate frequency signal is generally written as:
x IF ( t ) = A ′ ′ cos ( 2 π K τ t + 2 π fo τ ) x_{\tiny{IF}}(t) = A^{\prime \prime} \cos(2\pi K \tau t+2\pi f_o \tau )xIF(t)=A′′cos(2πKτt+2πfot )
The above formula is the final formula to be obtained in this paper.