Integrated Communication and Imaging: Design, Analysis, and Performances of COSMIC Waveforms (2024)

The signals must be orthogonal to let the receiver able to separate the single transmitted waveforms at each receiving antenna. In this way, when the range compression¹¹1Range compression and matched filtering are two different jargon used by the radar community and the communication community to indicate the filtering operation that maximize the SNR at the receiver. occurs, each receiver antenna can separate the contribution for each of the $N$ transmitted signals. In the case of an extended scattering scenario, where an infinite number of infinitesimal scatterers are distributed at many different distances from the radar, it is not sufficient imposing that

Indeed, the condition in Eq. (2) is equivalent to enforcing the spatial covariance matrix of the transmitted waveforms to be the identity matrix [49]. Such a condition allows for perfect separation of the scattered waveforms from a single point target; however, as demonstrated in Appendix A, it does not allow for perfect signal separation in the case of spatially extended scattering scenarios, which is a necessary condition for imaging purposes [11]. In this latter scenario, the orthogonality condition must be satisfied for all the possible differences of delays induced by the targets in the scene, or, in other words, the cross-correlation of all the signals must be zero within a specific interval limited by the maximum range of the device itself.

This orthogonality condition can be written as:

As stated before, the cross-correlation must be strictly zero only within a range of delays that depends on the maximum distance traveled by the signal in the whole path from the transmitter to the receiver. As an example, if the maximum range for a target is 100 m and TX/RX are co-located, it means that the maximum delay that the signal will experience is:

Thus, the cross-correlation of the waveforms must be zero between $-\tau_{\mathrm{s}}/2$ and $+\tau_{\mathrm{s}}/2$, while no constraints are enforced in the region of $t<-\tau_{\mathrm{s}}/2$ and $t>+\tau_{\mathrm{s}}/2$ since, by definition, there are no targets at these delays, the orthogonality is not required.

From now on, all the signals are sampled at a sampling frequency $f_{s}$ equal to the bandwidth. The sampling interval is denoted by $T_{s}=1/f_{s}$. The discrete-time signal has $K$ samples:

For the sake of simplicity, we assume that all the transmitting antennas send an informative signal. This assumption can be relaxed to improve sensing efficiency as explained in Section VI. We start the waveform design procedure by enforcing that each signal must be written as a linear combination of orthogonal basis, in other words:

where $\bm{C}_{n}\in\mathbb{C}^{K\times K_{s}}$ is an orthogonal matrix, such that

where the superscript ^H indicates the Hermitian product, $\bm{I}$ is the identity matrix, and $\bm{0}$ is the zero matrix. The number of columns of $\bm{C}_{n}$ is lower or equal to the number of rows ($K_{s}\leq K$). The vector $\bm{\alpha}_{n}\in\mathbb{C}^{K_{s}\times 1}$ encodes the information in a way that will be clear in the following.

At the receiver, either the communication or sensing one, the antennas record the coherent superposition of all the signals. By ignoring the delays, it can be written as

Equation (8) refers to the special case where all the transmitting antennas are used to transmit information and perform imaging simultaneously. Each transmission has its own key; therefore, $N$ keys are derived from the master key.For the first waveform $\bm{s}_{1}$, the vector $\bm{\alpha}_{1}$ is already informative, and it contains $K_{s}$ complex symbols drawn from a modulation scheme as QPSK, M-QAM, etc. The key $\bm{C}_{1}$ projects the information into the signal space. For example, if by assuming a 50 $\mu s$ signal sampled with $f_{s}=300$ MHz, it has a length of $K=15000$ samples. While with three transmitting antennas and by assigning to the antennas keys the same size ($K_{s}=5000$), the first waveform carries 5000 communication symbols.

When deriving $\bm{s}_{2}$, however, it is not sufficient to have it built as a linear combination of vectors $\bm{C}_{2}$, orthogonal to $\bm{C}_{1}$, i.e., $\bm{C}_{1}^{H}\bm{C}_{2}=\bm{0}$. The transmitted signals must also be sensing-wise orthogonal as described in Eq. (3). The cross-correlation as a matrix-vector product becomes

where $\bm{r}_{12}\in\mathbb{C}^{(2K-1)\times 1}$ is the vector containing the linear cross-correlation between the two discrete-time signals, $\bm{s}_{2}\in\mathbb{C}^{K\times 1}$ is the unknown signal to be derived and $\bm{S}_{1}\in\mathbb{C}^{(2K-1)\times K}$ is the cross-correlation matrix generated in the same way as a convolution matrix, but with the time-reversed and complex conjugate entries of $\bm{s}_{1}$:

It is important to notice that in Eq. (3) the cross-correlation must be zero only within a time interval of length $\tau_{s}$, centered around zero, not for all the $2K-1$ samples of the auto-correlation. We can force this condition by computing the cross-correlation only within a set of possible delays, which is implemented in the discrete-time cross-correlation by reducing the number of rows of $\bm{S}_{1}$. The discrete-time version of Eq. 3 is then:

where $\tilde{\bm{S}_{1}}\in\mathbb{C}^{K_{z}\times K}$ is obtained from $\bm{S}_{1}$ just by selecting the $K_{z}$ rows corresponding to the delays that must be at zero correlation. It is essential to notice that $K_{z}$ is much lower than $K$ since the region at zero correlation is much shorter than the pulse length.

By plugging $\bm{s}_{2}=\bm{C}_{2}\bm{\alpha}_{2}$ into Eq. (11)

with $\bm{W}_{1}=\tilde{\bm{S}_{1}}\bm{C}_{2}\in\mathbb{C}^{K_{z}\times K_{s}}$.The size of the second key $\bm{C}_{2}$ is assumed equals to $K_{s}$ as for the first key. Equation (12) suggests that the vector $\bm{\alpha}_{2}$ must be searched in the null space of the matrix $\bm{W}_{1}$

In light of this, we can define the matrix $\bm{N}_{1}\in\mathbb{C}^{K_{s}\times(K_{s}-K_{z}+1)}$ which contains the orthonormal basis spanning the null space of $\bm{W}_{1}$. Notice that $\bm{W}_{1}$ is not full-rank. $\bm{N}_{1}$ can be found from the Singular Value Decomposition (SVD) of $\bm{W}_{1}$, and a linear combination of its columns can be used to generate a suitable $\bm{\alpha}_{2}$ as:

where $\bm{\beta}_{2}\in\mathbb{C}^{(K_{s}-K_{z}+1)\times 1}$ is again a vector containing symbols drawn from a communication scheme such as a M-QAM, QPSK, etc. The matrix $\bm{N}_{1}$ can be seen as the operator projecting (or encoding) the information into the null space of $\bm{W}_{1}$. The signal to be transmitted by the second antenna can be easily derived as $\bm{s}_{2}=\bm{C}_{2}\bm{\alpha}_{2}$. Following the previous example, if the interval at zero correlation must be 100m, it follows that $K_{z}=200$, thus the second waveform will carry 4801 symbols.

We can now proceed by finding a third signal $\bm{s}_{3}$, which must be orthogonal to both $\bm{s}_{1}$ and $\bm{s}_{2}$, thus we can work in the same way, enforcing zero correlation for both:

Which can be written in compact form as

The vector $\bm{\alpha}_{3}$, this time, must be searched in the null space of $[\bm{W}_{2}^{H},\bm{W}_{3}^{H}]^{H}$.
The size of $[\bm{W}_{2}^{H},\bm{W}_{3}^{H}]^{H}$ is now $2K_{z}\times K_{s}$, thus the size of its null space is reduced by a factor $K_{z}$. Once again, in this special case, the size of the third key $\bm{C}_{3}$ is equal to the size of the first and the second key. The basis spanning the null space of $[\bm{W}_{2}^{H},\bm{W}_{3}^{H}]^{H}$ are collected in the columns of $\bm{N}_{2}\in\mathbb{C}^{K_{s}\times(K_{s}-2K_{z}+2)}$. The vector $\bm{\alpha}_{3}$ is found as a linear combination of the columns of $\bm{N}_{2}$:

where again $\bm{\beta}_{3}\in\mathbb{C}^{(K_{s}-2K_{z}+2)\times 1}$ is the vector containing the communication symbols. Then, as in the previous case, the signal $\bm{s}_{3}$ can be found as $\bm{s}_{3}=\bm{C}_{3}\bm{\alpha}_{3}$. Following again the previous example, the third waveform will carry 4602 symbols.

First of all, the receiver derives an estimate of all the $\hat{\bm{\alpha}}_{n}$ vectors using a simple matched filter as

We have $\hat{\bm{\alpha}}_{n}=\bm{\alpha}_{n}$ if $\bm{C}_{n}$ is correctly signaled and in case of an ideal transmission. With the aim of describing the main features of the COSMIC system, the decoding procedure in the noiseless scenario is here highlighted, while an analysis of the effect of the noise is presented in Section VII-B.Each transmitted waveform can be reconstructed at the receiver as

At this point, the receiver follows the same procedure as the transmitter to derive the basis of the null space. For the first antenna, $\hat{\bm{\alpha}}_{1}$ is already informative, containing the symbols drawn from the given constellation. Then, by knowing $\hat{\bm{s}}_{1}$ and $\bm{C}_{1}$, it is possible to derive $\hat{\bm{W}}_{1}$ and from that the bases of the null space $\hat{\bm{N}}_{1}$. Once these bases are known, the information in $\bm{\beta}_{2}$ can be decoded by simply performing a projection:

In the same way, by knowing $\hat{\bm{s}}_{1}$, $\hat{\bm{s}}_{2}$ and $\bm{C}_{3}$, the matrices $\hat{\bm{W}}_{2}$ and $\hat{\bm{W}}_{3}$ can be calculated and from those the matrix $\hat{\bm{N}}_{2}$. Once again, the information in $\bm{\beta}_{3}$ can be estimated as:

The process is performed sequentially up to the final transmitted signal, thus estimating all the informative $\hat{\bm{\beta}}_{n}$ transmitted. An assessment of the performance of the communication system is presented in Section VII-B.

The imaging receiver is co-located with the transmitter and has the perfect knowledge of the set of signals $\{\bm{s}_{n}\}$. Each one of the $M$ receiving antenna records the delayed echo of the coherent superposition of all the transmitted signals:

For the sake of simplicity, we omit the delay induced by the radar-to-target distance. The signals are separated at the receiver by a simple correlation or, in radar jargon, range compression:

where the last equality holds due to the orthogonality between signals in Eq. 11 that can be generalized as

The signal $\tilde{\bm{S}}_{n}\bm{s}_{n}$ is the auto-correlation of the $n-$th signal calculated just within the zero correlation interval. Each receiving antenna performs the correlation with all the transmitting signals, leading to an equivalent number of virtual channels equal to $N\times M$. If TX and RX antenna elements are appropriately spaced, the elements of the generated monostatic virtual array will be spaced by $\lambda/4$, leading to the unambiguous imaging of the whole field of view [50].An analysis of the imaging performances of the system is presented in Section VII-A