The following signal examples have been tested for MIMO SONAR system applications...
Further information can be found here:
Bandpassfiltered noise
Bandpassfiltered Noise
The reference wave form for the following analysis is bandlimited noise because of its great correlation properties. Nevertheless in a real application these signal types are not optimal because of the channels influence and restrictions on the hardware.
TimeDomain Analysis
FrequencyDomain Analysis
TimeFrequency Analysis
Auto and CrossCorrelation Analysis
Auot and CrossAmbiguity Functions
Barker
Modulated Barker Sequences
For Barker coding the signal is devided into several subpulses. Each element of the code belongs to one subpulse. The elements of the Barker Code consist of \(m\,\in[1,1]\).There are seven known Barker codes:
\begin{eqnarray} {\bf b}_2 \,\, &=& \,\, [1,\, 1]^{\textrm{T}}, \nonumber \\[1mm] {\bf b}_3 \,\, &=& \,\, [1,\, 1,\, 1]^{\textrm{T}}, \nonumber \\[1mm] {\bf b}_4 \,\, &=& \,\, [1,\, 1,\, 1,\, 1]^{\textrm{T}}, \nonumber \\[1mm] {\bf b}_5 \,\,&=& \,\, [1,\, 1,\, 1,\, 1,\, 1]^{\textrm{T}}, \nonumber \\[1mm] {\bf b}_7 \,\, &=& \,\, [1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1]^{\textrm{T}}, \nonumber \\[1mm] {\bf b}_{11} \,\, &=& \,\, [1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1]^{\textrm{T}}, \nonumber \\[1mm] {\bf b}_{13} \,\, &=& \,\, [1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1,\, 1]^{\textrm{T}}. \nonumber \end{eqnarray}
The longest Barker code only contains thirteen elements. For getting longer codes the seven different Barker codes can be combined. The combining principle is to take one of the seven Barker codes and to replace each of its elements with another Barker code multiplied by the element of the first one. This approach can be repeated until the desired code length is met.
There were two options tested to use the Barker Codes for signal generation.

The first one is to use the Barker codes as phase coding. Every subpulse consists of a sinusoid with a phase depending on the belonging code element. It is defined:
\begin{equation*} s(t)\,=\,\text{sin}\big(2\pi\,t\,f_c\,+\,\phi(t)\big),\,\,\,\,\,\,\,\text{with}\,\,\,\, \phi(t)\,=\, \begin{cases} 0, & \text{for}\,1\\[2mm] \pi & \text{else}. \end{cases}\end{equation*}
 The other possibility is to use Barker Codes to distinguish between up and downchirps. Every subpulse now consists of a chirp and the chirps characteristic depends on the belonging code element. In this example a HFMchirp is used:
\begin{equation*}
s(t)\,=\,\text{sin}\left(2\pi\,\int_{t'=0}^tf(t')\,dt'\right)\,,\,\,\,\,\,\,\,
f(t)\, = \,
\begin{cases}
f_1\,\cdot\,\frac{f_{0}}{f_1}^\frac{t}{t_{1}} & \text{for } 1 \\
f_0\,\cdot\,\frac{f_{1}}{f_0}^\frac{t}{t_{1}} & \text{for } +1
\end{cases}
\end{equation*}
To generate different signals the elements of the Barker Codes will be shifted.
The following combination seems to yield good results and is used in for the SONAR signal:
\begin{equation} {\bf b}_{13} \,\ast\, {\bf b}_{13} \,\ast\, {\bf b}_{13}. \nonumber \end{equation}
References
[Bar53]  R. H. Barker: Group Synchronizing of Binary Digital Systems, Communication Theory, England, London, Butterworth Scientific Publications, pp. 273287, 1953 
TimeDomain Analysis
FrequencyDomain Analysis
TimeFrequency Analysis
Auto and CrossCorrelation Analysis
Auot and CrossAmbiguity Functions
BPSK
Modulated Binary Phase Shift Keying (BPSK)
The good correlation properties of noise can be exploited by using noise as random phase coding for a signal that merges a number of subpulses.
These subpulses contain of sine signals that have the phase of their belonging code elements. Noise is generated and its magnitude is normalized
to the magnitudes maximum to get values between zero and one. All values that are greater than the mean are said to be the phase \(\phi=\pi\) and all
values that are less than the mean are said to be the phase \(\phi=0\):
\begin{equation*}
s(t)\,=\,\text{sin}(2\pi\,t\,f_c\,+\,\phi(t)),\,\,\,\,\,\,\,
\phi(t)\,=\,
\begin{cases}
0 & \text{for}\,n(t)\,\leq\,0.5\\
\pi & \text{for}\,n(t)\,>\,0.5.
\end{cases}
\end{equation*}
\(f_c\) is the carrier frequency and \(n(t)\) the noise sequence.
TimeDomain Analysis
FrequencyDomain Analysis
TimeFrequency Analysis
Auto and CrossCorrelation Analysis
Auot and CrossAmbiguity Functions
Chirp
Chirp
A simple chirp is a frequency modulated sines signal and is defined as:
\begin{equation*}
s(t)\,=\,\text{sin}\Big(2{\pi}\,\int_{t'=0}^tf(t')\,dt'\Big).
\end{equation*}
The frequency modulation can be chosen as linear and hyperbolic. For the linear case f(t) can be determined by:
\begin{equation*}
f(t)\,=\,f_0\,+\,k\,t.\end{equation*}
\(f_0\) is the start frequency. The chirprate \(k\) is defined by the chirps bandwidth \(BW\) and the chirps duration \(T\):
\begin{equation*}
k\,=\,\frac{BW}{T}.\end{equation*}
In this example the frequency modulation is rising. For a descending frequency modulation the sign of the chirprate has to become negative and \(f_0\) has to be the stop frequency. For the hyperbolic frequency modulation \(f(t)\) is calculated as the following:
\begin{equation*}
f(t)\,=\,f_0\,\cdot\,\beta^t.\end{equation*}
The constant \(\beta\) for a rising hyperbolic frequency modulation can be determined by:
\begin{equation*}
\beta\,=\,\frac{f_1}{f_0}^{\frac{1}{t_1}}.\end{equation*}
\(f_1\) is the stop frequency and \(t_1\) is the end time. For a descending frequency modulation \(\beta\) would be calculated as:
\begin{equation*}
\beta\,=\,\frac{f_0}{f_1}^{\frac{1}{t_1}}.\end{equation*}
In this example the band range of each chirp depends on the number of chirps you want to realize since the bandwidth of the whole system is fix.
TimeDomain Analysis
FrequencyDomain Analysis
TimeFrequency Analysis
Auto and CrossCorrelation Analysis
Auot and CrossAmbiguity Functions
Costas
Costas Codes
The principle of the coding by Costa is based on primitive roots. Every potency of the primitiv root can be an element oft the prime residue class.
This property is exploited by Costas Coding, that consists of elements that are calculated by a modulo operation:
\begin{equation*}
\textbf{a}\,=\,g^{\textbf{b}}\,\text{mod}\,p.
\end{equation*}
Vector \(a\) includes a sequence of numbers that are used for coding. \(g\) is an arbritary number and \(\textbf{b}\) is a vector conaisting
of \(\textbf{b}\in[1,...,N]\). Modulo operator \(p\) is set to \(P=N+1\) with \(N\) being the number of subpulses. Every element of \(a\) gets assigned
to one subpulse.
There are two ways to use Costas Coding. One option is to apply it to a CW signal.
Therefore every element of \(a\) stands for a certain carrier frequency. These carrier frequencies will be assigned to their belonging subpulses, which
consist of a sinusoid with that certain frequency. The carrier frequencies are generated by:
\begin{equation*}
\textbf{f}\,=\,f_0\,+\textbf{a}\,\cdot\,{\Delta}f.
\end{equation*}
\(f_0\) is the lowest carrier frequency of the system. It is determined by the substraction of the center frequency \(f_c\) with
half of the bandwidth \(BW\) of the system. \({\Delta}f\) is defined by:
\begin{equation*}
{\Delta}f\,=\,\frac{BW}{N}.
\end{equation*}
By using the modulo operation the carrier frequencies don't increase linearly and therefore the correlation properties will improve.
The second option is to use Costas Coding for FM signals. Instead of using vector \(a\) for a mixed positioning of carrier frequencies it is now used
for a mixed positioning of frequency ranges. The previous calculated carrier frequencies are now assigned to be the start frequencies of the chirps.
The end frequencies can be determined by:
\begin{equation*}
\textbf{f}_e\,=\,\textbf{f}\,+{\Delta}f\,\,x
\end{equation*}
\(x\) is a sefety distance to avoid frequency overlapping. This approach is defined for upchirps. For generating downchirps the start and stop
frequencies have to be changed.
For generating more signals \(g\) has to get individual values for each signal.
TimeDomain Analysis
FrequencyDomain Analysis
TimeFrequency Analysis
Auto and CrossCorrelation Analysis
Auot and CrossAmbiguity Functions
Cut FM
Cut FM
The CutFM signal consits of a multiplication of a LFMChirp with a cutter signal. The so called cutter signal cuts out particular parts of the LFMChirp to achieve a higher doppler sensitivity by the gaps in the frequency spectrum. The CutFM is defined as:
\begin{equation*}s(t)\,=\,s_{LFM}(t)\,\cdot\,s_{cut}(t).\end{equation*}
\(s_{LFM}(t)\) is a windowed version of a LFM chirp that can be determined by:
\begin{equation*} s_{LFM}(t)\,=\,w(t)\,\cdot\,\text{exp} \bigg(j\,2{\pi}\Big(f_0\,t\,+\,\frac{BW\,t^2}{2\,T}\Big)\bigg).\end{equation*}
Constant \(BW\) is the bandwidth, \(f_0\) is the start frequency, \(T\) is the pulse duration and \(w(t)\) is a window function.
For this example a Hannwindow is used.
The cutter signal \(s_{cut}\) is a periodic signal:
\begin{equation*}
s_{cut}(t)\,=\,\sum_{m=1}^Ms_s\Big(t\,\,m\,T_r\Big).
\end{equation*}
\(s_s(t)\) are subpulses with a duration of \(T_r\). \(M\) is the number of subpulses and is determined by \(M=T/T_r\). The subpulses contain of a part that holds a window and a part that is set to be zero.
\begin{equation*}s_s(t)\,=\,
\begin{cases} c(t), &\text{for}\,\,0\,\le\,t\,\le\,T_c,\\[2mm] 0, &\text{else.} \end{cases}
\end{equation*}
\(c(t)\) is a window function for those parts of the signal that shall consist. For this examples a Hannwindow is chosen again. \(T_c\) is the duration of the window. In this example the choice is \(T_c=T_r/2\).
References
[Noe15]  M. Noemm, P. A. Hoeher: CutFM SONAR Signal Design, Applied Acoustics, vol. 90, pp. 95110, April 2015 
TimeDomain Analysis
FrequencyDomain Analysis
TimeFrequency Analysis
Auto and CrossCorrelation Analysis
Auot and CrossAmbiguity Functions
CW
Constant Wave (CW)
This signal is a simple sine sequence with its common definition:
\begin{equation} s(t) \,\,=\,\, A\,\sin(2\pi \, f_c \, t). \label{eq_mimo_sonar_signals_cw_01} \nonumber \end{equation}
\(f_c\) depends on the center frequency of the SONAR system and the number of sine components you want to generate. For the creation a signal with just one sine component \(f_c\) will be the center frequency. For more than one sine component
\begin{equation} s(t) \,\,=\,\, \sum\limits_{i\,=\,0}^{N1}A_i\,\sin(2\pi \, f_{c,i} \, t) \label{eq_mimo_sonar_signals_cw_02} \nonumber \end{equation}
the frequencies will be shifted from the center frequency depending on the band width and number of sine sequences \(N\) to be generated. The more sine components you create, the worse the cross correlation between different signals will get.
TimeDomain Analysis
FrequencyDomain Analysis
TimeFrequency Analysis
Auto and CrossCorrelation Analysis
Auot and CrossAmbiguity Functions
Micro Chirps
Micro Series
The micro sequences use a random coding to get good correlation properties. They received their name because they consist of
very short subpulses of the duration \(T_P\). The principle of the micro sequences is to generate signals that consist of subpulses
that have different frequency ranges. The bandwidth of each subpulse \(BW_P\) is defined by:
\begin{equation}
\begin{aligned}
T_P&=\,\frac{T}{N}\\
BW_P&=\,\frac{1}{T_P}\\
N_{PBW}&=\,\frac{BW}{BW_P}
\end{aligned}
\end{equation}
To make sure that the bandwidth of all subpulses is completely exploited there will be \(N_{PBW}\) different frequency ranges that will be repeated
and then assigned to the \(N\) subpulses.
Micro sines series
For the micro sines the subpulses contain sinusoids with different carrier frequencies. There are \(N_{PBW}\) possible carrier frequencies that
are repeated, randomly mixed and then assigned to the subpulses. The distance between these carrier frequencies is \({\Delta}f=BW_P\). The
carrier frequencies \(f_n\) can be frequencies in the range of \(f_n\,\in[f_cBW/2,f_c+BW/2]\). \(F_c\) is the center frequency and \(BW\) is the bandwidth
of the system.
Micro chirp series
Instead of using subpulses that contain sinesoids with different center frequencies chirps with different frequency ranges can be used.
\(N_{PBW}\) possible up and down chirps of frequency ranges with the bandwidth \(BW_P\) are repeated, randomly mixed and then assigned to
the \(N\) subpulses.
The definitions of the sinusoids can be looked up in the description of the CW signal and the definition of the chirps can be found in the
description of the Chirp.
TimeDomain Analysis
FrequencyDomain Analysis
TimeFrequency Analysis
Auto and CrossCorrelation Analysis
Auot and CrossAmbiguity Functions
Cox Comb
Cox Comb
The cox comb signal is a superposition of multiple sinusoids and is defined as:
\begin{equation*}
s(t)\,=\,\sum_{m=1}^{M}\text{exp}\Big(j\,2{\pi}\,f_m(t\,+\,\alpha)\Big).
\end{equation*}
The changing carrier frequency \(f_m\) is generated by:
\begin{equation*}
f_m\,=\,
\begin{cases}
f_c\,\,BW/2,\,&\text{for}\,m\,=\,1,\\
f_{m1}\,+r^{m2}\,{\Delta}f,\,&\text{for}\,m\,\in\{2,...,M\}
\end{cases}
\end{equation*}
\(M\) ist the number of superpositions and \(\alpha\) is an arbitrary, very small parameter for achieving a better PeaktoAveragePowerRatio. For these
examples it is set to \(\alpha=0.1\). \(r\) is slightly greater than one to get unequal frequency steps between the carrier frequencies.
TimeDomain Analysis
FrequencyDomain Analysis
TimeFrequency Analysis
Auto and CrossCorrelation Analysis
Auot and CrossAmbiguity Functions