The choice of wave form and signal processing methodology in USV radar systems implementation was dependent on the type of mission assigned to the radar and the specific USV mission and function. The complexity and price associated to the development of USV systems constituted the major factor in deciding the type of radar waveform, hardware and software that were suitable for the specific functions that were assigned to the USV. Radar systems have the advantage of using continuous wave forms with or without frequency modulation. The frequency modulation of radar systems can be implemented either through analog techniques or digital methods. An important consideration in the modulation of radar frequencies was the radar range and Doppler frequencies as these had direct associations to the choice of waveform frequency properties implemented in the radar system (Mahafza & Elsherbeni

2010). FMCW radar band pass signal

*x*(

*t*) can be presented as:

$x\left(t\right)=r\left(t\right)cos\left(2\pi {f}_{o}t+{\mathrm{\Phi}}_{x}\left(t\right)\right)$

(1)

Where

*r*(

*t*) represents the radar signal amplitude modulation or envelop, Φ

_{
x
}(

*t*) represents the radar signal phase modulation and

*f*_{
o
} represents the radar carrier frequency. The radar frequency modulation was modelled as:

${f}_{m}\left(t\right)=\frac{1}{2\pi}\phantom{\rule{0.25em}{0ex}}\frac{d}{\mathit{dt}}\phantom{\rule{0.25em}{0ex}}{\mathrm{\Phi}}_{x}\left(t\right)$

(2)

The radar signal instantaneous frequency is modelled as:

${f}_{i}\left(t\right)={f}_{0}+{f}_{m}\left(t\right)\phantom{\rule{0.25em}{0ex}}$

(3)

The radar signal

*x*(

*t*) may also be represented as an analytic signal forming the real part of the complex signal

*ψ*(

*t*) illustrated in equation (

4)

$x\left(t\right)=\mathit{Re}\left\{\psi \left(t\right)\right\}=\mathit{Re}\left\{r\left(t\right){e}^{j{\mathrm{\Phi}}_{x}\left(t\right)}{e}^{j2\pi {f}_{o}t}\right\}$

(4)

The radar analytic signal is then defined as:

$\psi \left(t\right)=v\left(t\right){e}^{j2\pi {f}_{o}t}$

(5)

Where

$v\left(t\right)=r\left(t\right){e}^{j{\mathrm{\Phi}}_{x}\left(t\right)}$

(6)

Implementing a Fourier transform to equation (

5) while keeping the signal under the following conditions illustrated in equation (

7) yields:

$\mathrm{\Psi}\left(\omega \right)=\left[\begin{array}{c}2X\left(\omega \right)\\ 0\end{array}\right]\phantom{\rule{4.75em}{0ex}}\begin{array}{c}\omega \phantom{\rule{0.75em}{0ex}}\ge \phantom{\rule{0.75em}{0ex}}0\\ \omega \phantom{\rule{0.75em}{0ex}}<\phantom{\rule{0.75em}{0ex}}0\end{array}$

(7)

Where Ψ(

*ω*) represents the Fourier transform of

*ψ*(

*t*),

*ω* = 2

*πf*_{0} and

*X*(

*ω*) =

*ψ*(

*t*) is the Fourier transform of

*x*(

*t*). Implementing a step function to equation (

7) yields:

$\mathrm{\Psi}\left(\omega \right)=2U\left(\omega \right)X\left(\omega \right)$

(8)

Where

*U*(

*ω*) represents the step function of the radar signal in frequency domain. From the above models, it can be shown that

*ψ*(

*t*) is represented as:

$\psi \left(t\right)=x\left(t\right)+j\tilde{x}\left(t\right)$

(9)

Where

$\stackrel{\u0303}{x}\left(t\right)$ represents the Hilbert transform of the radar signal

*x*(

*t*). The energy associated with radar signal

*x*(

*t*) can be illustrated using Parseval’s theorem (Mahafza & Elsherbeni

2010) as indicated below,

${E}_{x}=\frac{1}{2}\phantom{\rule{0.25em}{0ex}}\underset{-\infty}{\overset{\infty}{{\displaystyle \int}}}{x}^{2}\left(t\right)\mathit{dt}=\frac{1}{2}\phantom{\rule{0.25em}{0ex}}{E}_{\psi}$

(10)

The exponential form of the radar signal can be modelled as (Zhang et al.

2008):

$i\left({t}_{r}\right)={I}_{o}{e}^{j\left({\omega}_{o}{t}_{r}+\mathit{\alpha \pi}{t}_{r}^{2}\right)}\phantom{\rule{0.75em}{0ex}}-\phantom{\rule{0.75em}{0ex}}\frac{{T}_{r}}{2}\le {t}_{r}<\phantom{\rule{0.5em}{0ex}}\frac{{T}_{r}}{2}$

(11)

Where

*I*_{
o
} represents the current supply to the radar,

*ω*_{
o
} represents the central radian frequency of the radar waveform and

*ω*_{
o
} = 2

*πf*_{0},

*α* represents the radar sweep frequency rate in Hz/s,

*t*_{
r
} represents the time variable and

*T*_{
r
} represents the radar sweep frequency interval. The radar frequency modulation sweep rate is the ratio of the radar frequency bandwidth

*B* and sweep interval. Hence radar frequency rate is modelled as:

$\alpha =\frac{B}{{T}_{r}}$

(12)

The continuous variation in the transmitted frequency of FMCW radar at any given time with the difference in transmitted and received radar signal defines the radar beat frequency

*f*_{
b
}. The radar beat frequency portrays the characteristic property and measure of target range

*R* and it is modelled as (Chan & Judah

1998):

$R=\frac{c{f}_{b}}{2{f}_{m}}$

(13)

Where

*c* represents the speed of light in air,

*f*_{
b
} represents the radar beat frequency and

*f*_{
m
} = ∆

*f/T* represents the rate of change of the transmitted frequency. The sweep frequency of the radar source is denoted by ∆

*f* and

*T* denotes the time taken for each radar signal source sweep (Skolink

1981). The radar echo mixed with some portion of the transmitted signal received after

*τ* seconds produces the beat frequency of the radar. The radar echo is given as (Dorp & Groen

2010):

$\tau =\frac{2{r}_{s}}{c}$

(14)

Where

*r*_{
s
} represents the target slant range and

*c* is the speed of light. The wave form transmitted by FMCW radar in compact form is given as:

$s\left(t\right)={S}_{o}{e}^{j{\varphi}_{t}\left(t\right)}$

(15)

Where

*ϕ*_{
t
} represents the transmitted frequency phase with pulse width

*T*, out power

*S*_{
0
} and bandwidth

*B* having an instantaneous angular frequency of

${\omega}_{t}\left(t\right)=\frac{\delta {\varphi}_{t}\left(t\right)}{\mathit{\delta t}}\phantom{\rule{0.25em}{0ex}}$

(16)

The nonlinear transmitted frequency with instantaneous up-chirp phase is modelled as:

${\varphi}_{t}\left(t\right)=2\pi \left({f}_{o}t+\frac{\mu}{2}{t}^{2}\right)$

(17)

With the corresponding instantaneous frequency given as:

$f\left(t\right)=\frac{1}{2\pi}\phantom{\rule{0.25em}{0ex}}\frac{d}{\mathit{dt}}\phantom{\rule{0.25em}{0ex}}\varphi \left(t\right)={f}_{o}+\mathit{\mu t}$

(18)

Similarly the radar instantaneous down-chirp phase and frequency is given as:

${\varphi}_{t}\left(t\right)=2\pi \left({f}_{o}t-\phantom{\rule{0.5em}{0ex}}\frac{\mu}{2}{t}^{2}\right)$

(19)

$f\left(t\right)=\frac{1}{2\pi}\phantom{\rule{0.25em}{0ex}}\frac{d}{\mathit{dt}}\phantom{\rule{0.25em}{0ex}}\varphi \left(t\right)={f}_{o}-\mathit{\mu t}$

(20)

For

$\phantom{\rule{0.5em}{0ex}}-\frac{\mathrm{{\rm T}}}{2}\le t\le \frac{\mathrm{{\rm T}}}{2}$, where

*f*_{
0
} represents the radar centre frequency and

*μ = (2πB)/T* represents frequency modulation coefficient. The radar return echo with delay

*τ* modelled as

*s*(

*t-τ*) is mixed with the transmitted radar signal to generate the radar beat waveform

*S*_{
b
}.

${s}_{b}\left(t\right)=s\left(t\right){s}^{*}\left(t-\tau \right)$

(21)

Thus the instantaneous transmitted radar signal phase presented as (Dorp & Groen

2010):

${\varphi}_{t}\left(t\right)={a}_{0}+{a}_{1}t+{a}_{2}\phantom{\rule{0.25em}{0ex}}{\left(t-\frac{{T}_{0}}{2}\right)}^{2}$

(22)

for 0 ≤ *t* ≤ *T*_{0}

Having the beat signal phase (Dorp & Groen

2010):

${\varphi}_{b}\left(t\right)={\varphi}_{t}\left(t\right)-{\varphi}_{t}\left(t-\tau \right)$

(23)

$={b}_{0}+{b}_{1}t+{b}_{2}{\left(t-\frac{{T}_{0}}{2}\right)}^{2}$

(24)

for 0 ≤ *t* ≤ *T*_{0}

With

${b}_{0}={a}_{1}\tau +{a}_{2}\tau +2{a}_{2}\tau \frac{{T}_{0}}{2}$

(25)

${b}_{1}=2{a}_{2}\tau $

(26)