Robert Calderbank

Orthogonal signaling or Nyquist signaling limits the number of information symbols transmitted in bandwidth $B$ and time $T$ to be $BT$, the time-bandwidth product. Transmitting more than $BT$ symbols leads to loss of orthogonality. The standard approach is to reduce the symbol interval and resolve the resulting inter-symbol interference. This requires changing the sampling frequency and possibly the sampling clock. This paper shows that it is possible to improve spectral efficiency on doubly spread channels without changing the sampling frequency. The idea is to superimpose the information symbols using mutually unbiased bases (MUB) while maintaining the original spacing. We carry this out in the delay-Doppler domain using Zak-transform based orthogonal time frequency space (Zak-OTFS) modulation as it allows construction of MUB. We also construct a precoder that mitigates the effect of the doubly-spread channel. This simplifies receiver processing to detection in Gaussian noise since each basis appears to the other as Gaussian noise. This reduction makes it possible to use trellis coded modulation to further improve the bit-error performance. Numerical results demonstrate that the proposed signaling scheme using MUB achieves good bit-error performance.

Zak-OTFS provides a framework for integrated sensing & communication (ISAC) in high delay and Doppler spread environments. Pulse shaping filter design enables joint optimization of sensing and communication performance. For sensing, a localized pulse shaping filter enables input-output (I/O) relation estimates close to the physical scattering channel. For communication, orthogonality of the pulse shape on the information lattice prevents inter-symbol interference, and no time and bandwidth expansion enables full spectral efficiency. A filter simultaneously meeting all three objectives is ideal for ISAC. Existing filter designs achieve two, but not all three objectives. In this work, we design pulse shaping filters meeting all three objectives via the Isotropic Orthogonal Transform Algorithm. The proposed filters have improved spectral efficiency, data detection and sensing performance over existing filter choices.

There is significant recent interest in designing new modulation schemes for doubly-selective channels with large delay and Doppler spreads, where legacy modulation schemes based on time-frequency signal representations underperform. Multiple modulation schemes, e.g., in the delay-Doppler, chirp, time-sequency, and other domains, have been proposed in the literature for this purpose, with varying implementation details. In this letter, we establish that all previously proposed modulation schemes for doubly-selective signaling are instances of a single family of complex Hadamard-modulated pulse trains. When the delay and Doppler spread of the doubly-selective channel is limited to a certain support, all modulation schemes in this waveform family offer equivalent, full diversity achieving performance with no symbol fading and low channel estimation overhead. The existence of this waveform family also enables flexible, multi-waveform co-existence – allowing a common transceiver architecture to generate multiple waveforms in the family, that may each be flexibly allocated to different users and services.

Linear time-varying (LTV) systems model radar scenes where each reflector/target applies a delay, Doppler shift and complex amplitude scaling to a transmitted waveform. The receiver processes the received signal using the transmitted signal as a reference. The self-ambiguity function of the transmitted signal captures the cross-correlation of delay and Doppler shifts of the transmitted waveform. It acts as a blur that limits resolution, at the receiver, of the delay and Doppler shifts of targets in close proximity. This paper considers resolution of multiple targets and compares performance of traditional chirp waveforms with the Zak-OTFS waveform. The self-ambiguity function of a chirp is a line in the delay-Doppler domain, whereas the self-ambiguity function of the Zak-OTFS waveform is a lattice. The advantage of lattices over lines is better localization, and we show lattices provide superior noise-free estimation of the range and velocity of multiple targets. When the delay spread of the radar scene is less than the delay period of the Zak-OTFS modulation, and the Doppler spread is less than the Doppler period, we describe how to localize targets by calculating cross-ambiguities in the delay-Doppler domain. We show that the signal processing complexity of our approach is superior to the traditional approach of computing cross-ambiguities in the continuous time / frequency domain.

Millimeter-wave (mmWave) communication offers vast bandwidth for next-generation wireless systems but faces severe path loss, Doppler effects, and hardware impairments. Orthogonal Time Frequency Space (OTFS) modulation has emerged as a robust waveform for high-mobility and doubly dispersive channels, outperforming OFDM under strong Doppler. However, the most studied multicarrier OTFS (MC-OTFS) is not easily predictable because the input-output (I$/$O) relation is not given by (twisted) convolution. Recently, the Zak-transform based OTFS (Zak-OTFS or OTFS 2$.$0) was proposed, which provides a single domain delay Doppler (DD) processing framework with predictable I$/$O behavior. This paper presents one of the first over-the-air (OTA) demonstrations of Zak-OTFS at mmWave frequencies. We design a complete Zak-OTFS based mmWave OTA system featuring root-raised-cosine (RRC) filtering for enhanced DD-domain predictability, higher-order modulations up to 16-QAM, and a low-overhead preamble for synchronization. A comprehensive signal model incorporating carrier frequency offset (CFO) and timing impairments is developed, showing these effects can be jointly captured within the effective DD-domain channel. Experimental validation on the COSMOS testbed confirms the feasibility and robustness of Zak-OTFS under realistic mmWave conditions, highlighting its potential for efficient implementations in beyond-5G and 6G systems.

Polarimetry, which is the ability to measure the scattering response of the environment across orthogonal polarizations, is fundamental to enhancing wireless communication and radar system performance. In this paper, we utilize the Zak-OTFS modulation to enable instantaneous polarimetry within a single transmission frame. We transmit a Zak-OTFS carrier waveform and a spread carrier waveform mutually unbiased to it simultaneously over orthogonal polarizations. The mutual unbiasedness of the two waveforms enables the receiver to estimate the full polarimetric response of the scattering environment from a single received frame. Unlike existing methods for instantaneous polarimetry with computational complexity quadratic in the time-bandwidth product, the proposed method enables instantaneous polarimetry at near-linear complexity in the time-bandwidth product. Via numerical simulations, we show ideal polarimetric target detection and parameter estimation results with the proposed method, with improvements in computational complexity and greater clutter resilience over comparable baselines.

Magic state distillation and the Shor factoring algorithm make essential use of logical diagonal gates. We introduce a method of synthesizing CSS codes that realize a target logical diagonal gate at some level l in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of Z-stabilizers, and addition of X-stabilizers. It explicitly tracks the logical gate induced by a diagonal physical gate that preserves a CSS code. The first step is concatenation, where the input is a CSS code and a physical diagonal gate at level l inducing a logical diagonal gate at the same level. The output is a new code for which a physical diagonal gate at level l + 1 induces the original logical gate. The next step is judicious removal of Z-stabilizers to increase the level of the induced logical operator. We identify three ways of climbing the logical Clifford hierarchy from level l to level l + 1 , each built on a recursive relation on the Pauli coefficients of the induced logical operators. Removal of Z-stabilizers may reduce distance, and the purpose of the third basic operation, addition of X-stabilizers, is to compensate for such losses. The approach to logical gate synthesis taken in prior work focuses on the code states, resulting in sufficient conditions for a CSS code to be fixed by a transversal Z-rotation. In contrast, we derive necessary and sufficient conditions by analyzing the action of a transversal diagonal gate on the stabilizer group that determines the code. The power of our approach to logical gate synthesis is demonstrated by two proofs of concept: the [ [ 2 l + 1 − 2 , 2 , 2 ] ] triorthogonal code family and the [ [ 2 m , m r , 2 min { r , m − r } ] ] quantum Reed–Muller code family.