Identification capacity has been established as a relevant performance metric for various goal-/task-oriented applications, where the receiver may be interested in only a particular message that represents an event or a task. For example, in olfactory molecular communications (MCs), odors or pheromones, which are often a mixture of various molecule types, may signal nearby danger, food, or a mate. In this paper, we examine the identification capacity with deterministic encoder for the discrete affine Poisson channel which can be used to model MC systems with molecule counting receivers. We establish lower and upper bounds on the identification capacity in terms of features of the affinity matrix between the released molecules and receptors at the receiver. As a key finding, we show that even when the number of receptor types scales sub-linearly in the number of molecule types N, the number of reliably identifiable messages can grow super-exponentially with the rank of the affinity matrix, T, i.e., ~ 2(T log T)R, where R denotes the coding rate. We further derive lower and upper bounds on R, and show that the proposed capacity theorem includes several known results in the literature as its special cases.

Deterministic K-Identification (DKI) for the binary symmetric channel (BSC) is developed. A full characterization of the DKI capacity for such a channel, with and without the Hamming weight constraint, is established. As a key finding, we find that for deterministic encoding the number of identifiable messages$K$may grow exponentially with the codeword length$n$, i.e.,$K\ =\ 2^{\kappa n}$, where$\kappa$is the target identification rate. Furthermore, the eligible region for$\kappa$as a function of the channel statistics, i.e., the crossover probability, is determined.

Several applications of molecular communications (MC) feature an alarm-prompt behavior for which the prevalent Shannon capacity may not be the appropriate performance metric. The identification capacity as an alternative measure for such systems has been motivated and established in the literature. In this paper, we study deterministic identification (DI) for the discrete-time Poisson channel (DTPC) with intersymbol interference (ISI) where the transmitter is restricted to an average and a peak molecule release rate constraint. Such a channel serves as a model for diffusive MC systems featuring long channel impulse responses and employing molecule counting receivers. We derive lower and upper bounds on the DI capacity of the DTPC with ISI when the number of ISI channel taps$K$may grow with the codeword length$n$(e.g., due to increasing symbol rate). As a key finding, we establish that for deterministic encoding, the codebook size scales as$2^{(n\log n)R}$assuming that the number of ISI channel taps scales as$K=2^{\kappa\log n}$, where$R$is the coding rate and$\kappa$is the ISI rate. Moreover, we show that optimizing$\kappa$leads to an effective identification rate [bits/s] that scales linearly with$n$, which is in contrast to the typical transmission rate [bits/s] that is independent of$n$.

The Binomial channel serves as a fundamental model for molecular communication (MC) systems employing molecule-counting receivers. Here, deterministic identification (DI) is addressed for the discrete-time Binomial channels (DTBC), subject to an average and a peak constraint on the molecule release rate. We establish that the number of different messages that can be reliably identified for the DTBC scales as 2(n log n)R, where n and R are the codeword length and coding rate, respectively. Lower and upper bounds on the DI capacity of the DTBC are developed.

Deterministic K-identification (DKI) is addressed for Gaussian channels with slow fading (GSF), where the transmitter is restricted to an average power constraint and channel side information is available at the decoder. We derive lower and upper bounds on the DKI capacity when the number of identifiable messages K may grow sub-linearly with the codeword length n. As a key finding, we establish that for deterministic encoding, assuming that the number of identifiable messages K = 2κ log nwith κ ∈ [0, 1) being the identification target rate, the codebook size scales as 2(n log n)R, where R is the coding rate.

The deterministic identification (DI) capacity is developed in multiple settings of channels with power constraints. A full characterization is established for the DI capacity of the discrete memoryless channel (DMC) with and without input constraints. Originally, Ahlswede and Dueck established the identification capacity with local randomness at the encoder, resulting in a double exponential number of messages in the block length $n$ . In the deterministic setup, the number of messages scales exponentially, as in Shannon's transmission paradigm, but the achievable identification rates are higher. An explicit proof was not provided for the deterministic setting. In this paper, a detailed proof is presented for the DMC. Furthermore, Gaussian channels with fast and slow fading are considered, when channel side information is available at the decoder. A new phenomenon is observed as we establish that the number of messages scales as $2^{n\log (n)R}$ by deriving lower and upper bounds on the DI capacity on this scale. Consequently, the DI capacity of the Gaussian channel is infinite in the exponential scale and zero in the double exponential scale, regardless of the channel noise.

Identification capacity is developed without randomization at neither the encoder nor the decoder. In particular, full characterization is established for the deterministic identification (DI) capacity for the Gaussian channel and for the general discrete memoryless channel (DMC) with and without constraints. Originally, Ahlswede and Dueck established the identification capacity with local randomness given at the encoder, resulting in a double exponential number of messages. In the deterministic setup, the number of messages scales exponentially, as in Shannon’s transmission paradigm, but the achievable identification rates can be significantly higher than those of transmission. Ahlswede and Dueck further stated a capacity result for the deterministic setting of a DMC, but did not provide an explicit proof. In this paper, a detailed proof is given for both the Gaussian channel and the general DMC. The DI capacity of a Gaussian channel is infinite regardless of the noise.

Deterministic identification (DI) is addressed for Gaussian channels with fast and slow fading, where channel side information is available at the decoder. In particular, it is established that the number of messages scales as 2nlog(n)R, where n is the block length and R is the coding rate. Lower and upper bounds on the DI capacity are developed in this scale for fast and slow fading. Consequently, the DI capacity is infinite in the exponential scale and zero in the double-exponential scale, regardless of the channel noise.