MCR-IM: Multiple Chirp Rate Index Modulation

• MCR-IM is a LoRa-type low-power modulation scheme that replaces classical chirps with Zadoff–Chu sequences to create a two-dimensional index grid.
• It expands the symbol alphabet by jointly modulating chirp rate and within-rate frequency positions, thereby increasing the bits per symbol without relying on higher-order constellations.
• The scheme employs a non-coherent dechirping receiver with a PD-SIC algorithm to mitigate interference and enhance network scalability in collision scenarios.

Searching arXiv for MCR-IM and closely related chirp-domain index modulation papers. First, I’ll search for the exact MCR-IM paper and its surrounding literature. Multiple Chirp Rate Index Modulation (MCR-IM) is a LoRa-type low-power wide-area signaling scheme in which classical chirps are replaced by Zadoff–Chu (ZC) sequence-based chirps and the information-bearing index space is expanded from a single chirp-rate structure to a joint chirp-rate/frequency grid. In the formulation reported in "Design of A New Multiple-Chirp-Rate Index Modulation for LoRa Networks" (Zhu et al., 17 Jul 2025), a symbol is represented by the selection of $L$ active indices among $2^\nu P$ candidate positions, where $P$ is the number of available chirp rates and $2^\nu$ is the number of within-rate frequency positions. The resulting scheme targets the low transmission rate and large-scale access limitations of classical LoRa, while retaining a non-coherent receiver structure based on dechirping, discrete Fourier transforms, and peak detection (Zhu et al., 17 Jul 2025).

1. Conceptual basis and design objective

MCR-IM is motivated by three limitations attributed to classical LoRa: low transmission rate, spectral-efficiency limits, and difficulty with large-scale access under ALOHA-style collisions. The central design change is to assign multiple chirp rates to a user and to let the selected chirp-rate indices themselves carry information, rather than confining the symbol space to a single chirp-rate family with only frequency-bin indexing (Zhu et al., 17 Jul 2025).

The waveform family is built from Zadoff–Chu sequences rather than classical LoRa chirps. In the reported construction, the ZC root $r$ is interpreted as the chirp rate, while $q$ is an offset. This creates a two-dimensional signaling grid: chirp-rate index and within-rate frequency position. If a transmitter can choose among $P$ chirp rates and among $2^\nu$ frequency positions within each chirp rate, the total index space becomes $2^\nu P$. When $L$ indices are activated, the number of candidate activation patterns is

$2^\nu P$0

and the number of information bits per symbol is

$2^\nu P$1

In this formulation, all symbol bits are carried by index selection; there is no separate conventional PSK or QAM layer (Zhu et al., 17 Jul 2025).

A defining systems tradeoff is also explicit. Assigning multiple chirp rates to each user reduces the number of disjoint parallel channels available from a fixed chirp-rate resource pool. The paper treats this as the main cost of enlarging the symbol alphabet, and addresses it with a peak-detection-based successive interference cancellation (PD-SIC) algorithm at the gateway (Zhu et al., 17 Jul 2025).

2. Signal model and combinatorial mapping

The underlying Zadoff–Chu sequence is

$2^\nu P$2

where $2^\nu P$3 is a prime number, $2^\nu P$4 is the root, $2^\nu P$5 is the offset, and $2^\nu P$6. For dechirping, the receiver uses

$2^\nu P$7

To compare with LoRa, the spread factor is defined as

$2^\nu P$8

The paper uses the one-to-one mapping $2^\nu P$9, $P$0, $P$1, $P$2, $P$3, $P$4, and $P$5, while only the first $P$6 frequency positions are used for activation (Zhu et al., 17 Jul 2025).

The selectable positions are arranged as a $P$7 matrix $P$8, with one row per chirp rate. A symbol is represented by an ordered vector

$P$9

with

$2^\nu$0

Because a full lookup table over $2^\nu$1 possibilities is too large, the mapping uses the combinatorial number system: $2^\nu$2 where $2^\nu$3 is the decimal representation of the index bits (Zhu et al., 17 Jul 2025).

Each selected flattened index $2^\nu$4 is mapped to a chirp rate $2^\nu$5 and an offset $2^\nu$6. The chirp-rate assignment is

$2^\nu$7

and the offset satisfies

$2^\nu$8

The transmitted MCR-IM waveform is then the normalized superposition

$2^\nu$9

with bit energy

$r$0

This construction makes chirp-rate selection and within-rate frequency selection inseparable parts of the symbol alphabet (Zhu et al., 17 Jul 2025).

3. Correlation structure and chirp-rate separability

A major technical claim of MCR-IM is that it inherits the favorable correlation behavior of ZC sequences across different chirp rates. After dechirping with rate $r$1 and taking a DFT, the receiver observes

$r$2

Two cases are central. If $r$3,

$r$4

whereas if $r$5,

$r$6

Thus, matched chirp-rate dechirping collapses the energy into a single DFT bin, while mismatched chirp rates produce uniformly low outputs (Zhu et al., 17 Jul 2025).

The same point is expressed through the normalized cross-correlation

$r$7

The reported properties are: $r$8 if $r$9 and $q$0; $q$1 if $q$2 and $q$3; and

$q$4

if $q$5. This is the core quasi-orthogonality result enabling chirp-rate indexing (Zhu et al., 17 Jul 2025).

The paper contrasts this with classical LoRa chirps, for which different chirp rates do not preserve comparably low cross-correlation. This distinction is the main reason MCR-IM is implemented with ZC sequence modulation rather than with standard LoRa chirps. It also proves a different-spread-factor bound,

$q$6

for two MCR-IM signals of different lengths $q$7. This indicates that inter-SF cross-correlation decays roughly as $q$8, so quasi-orthogonality is preserved across spread factors as well (Zhu et al., 17 Jul 2025).

4. Receiver processing, BER analysis, and PD-SIC

Over Nakagami-$q$9 fading, the received signal is

$P$0

where $P$1 and $P$2 follows a Nakagami-$P$3 distribution. The receiver dechirps against every candidate chirp rate $P$4, $P$5, and computes

$P$6

for $P$7. Stacking these vectors forms

$P$8

Direct detection is non-coherent: the receiver finds the $P$9 strongest peaks in $2^\nu$0, maps them back through

$2^\nu$1

sorts the resulting index set, and inverts the combinatorial mapping to recover the transmitted bits (Zhu et al., 17 Jul 2025).

The BER analysis partitions the receiver outputs into a signal set $2^\nu$2, an interference set $2^\nu$3, and a noise set $2^\nu$4. Using a rule taken from ICS-LoRa, the paper adopts a $2^\nu$5 dB threshold to distinguish interference from noise for non-signal bins. The output magnitudes are modeled as Rician for signal and interference bins, then approximated as Gaussian with mean

$2^\nu$6

and variance

$2^\nu$7

The paper then derives an approximate closed-form BER over Nakagami-$2^\nu$8 fading by combining signal-versus-interference and signal-versus-noise comparison terms and averaging over the fading amplitude through Gauss-Hermite quadrature (Zhu et al., 17 Jul 2025).

The PD-SIC algorithm addresses the loss of directly orthogonal user channels caused by assigning $2^\nu$9 chirp rates per user. Each user is assigned a chirp-rate range of size $2^\nu P$0, and the gateway first detects preambles to determine the number of users, their arrival times, and their chirp-rate ranges. For user $2^\nu P$1, an energy metric is formed as

$2^\nu P$2

and the strongest user is decoded first. The gateway reconstructs that user’s signal, searches over quantized phases

$2^\nu P$3

forms tentative cancellation residuals

$2^\nu P$4

and accepts the cancellation if the residual energy on the detected peak set satisfies

$2^\nu P$5

The paper states that, for $2^\nu P$6 users, the total number of dechirp and DFT operations is

$2^\nu P$7

which it treats as acceptable because the algorithm runs at the gateway (Zhu et al., 17 Jul 2025).

5. Spectral efficiency, throughput, and reported performance

With bandwidth $2^\nu P$8 kHz and sampling interval $2^\nu P$9, the symbol duration is

$L$0

The reported spectral efficiency is

$L$1

and the throughput under $L$2 colliding users is

$L$3

Because $L$4 grows with $L$5, MCR-IM increases rate by enlarging the index alphabet rather than by introducing a higher-order symbol constellation (Zhu et al., 17 Jul 2025).

Several quantitative comparisons are reported. For $L$6, $L$7, and $L$8, MCR-IM carries $L$9 more bits than GCSS and $2^\nu P$00 more bits than FSCSS-IM. For $2^\nu P$01, increasing $2^\nu P$02 to $2^\nu P$03, $2^\nu P$04, and $2^\nu P$05 increases bits per symbol by $2^\nu P$06, $2^\nu P$07, and $2^\nu P$08, respectively, relative to $2^\nu P$09. In BER comparisons at $2^\nu P$10 and $2^\nu P$11, MCR-IM with $2^\nu P$12 and $2^\nu P$13 or $2^\nu P$14 improves on FSCSS-IM by about $2^\nu P$15 dB and $2^\nu P$16 dB; for $2^\nu P$17, MCR-IM with $2^\nu P$18 gains about $2^\nu P$19 dB over FSCSS-IM; and against GCSS, MCR-IM is better above about $2^\nu P$20 dB (Zhu et al., 17 Jul 2025).

The simulation trends are also specific. Increasing $2^\nu P$21 from $2^\nu P$22 to $2^\nu P$23 at $2^\nu P$24, $2^\nu P$25, $2^\nu P$26 increases bits per symbol from $2^\nu P$27 to $2^\nu P$28 with less than $2^\nu P$29 dB BER penalty. Decreasing $2^\nu P$30 from $2^\nu P$31 to $2^\nu P$32 at $2^\nu P$33, $2^\nu P$34, $2^\nu P$35 improves BER by about $2^\nu P$36 dB, but lowers bits per symbol from $2^\nu P$37 to $2^\nu P$38. Increasing $2^\nu P$39 from $2^\nu P$40 to $2^\nu P$41 at $2^\nu P$42, $2^\nu P$43 improves BER by about $2^\nu P$44 dB (Zhu et al., 17 Jul 2025).

In collision scenarios, MCR-IM with PD-SIC is reported to outperform OrthoRa above about $2^\nu P$45 dB. At $2^\nu P$46 dB, the throughput gain over OrthoRa is $2^\nu P$47 for $2^\nu P$48 and $2^\nu P$49 for $2^\nu P$50, both with $2^\nu P$51. These gains summarize the paper’s central systems claim: MCR-IM exchanges some directly orthogonal user multiplicity for a larger per-user symbol space, then partially restores scalability through gateway-side SIC (Zhu et al., 17 Jul 2025).

6. Position within chirp-domain index modulation research

MCR-IM is part of a broader family of chirp-domain index modulation schemes, but its indexed variable is more specific than in most neighboring designs. Several related works index chirp-domain objects without indexing chirp rate itself. The circularly shifted chirp scheme of "Wideband Index Modulation with Circularly-Shifted Chirps" indexes which two circularly shifted chirps are selected from a fixed family, not chirp rate, and links that selection to Golay complementary pairs and low-PMEPR DFT-s-OFDM realization (Hoque et al., 2020). The dual-function radar-communication variant "Index-Modulated Circularly-Shifted Chirps for Dual-Function Radar & Communication Systems" likewise indexes circular shifts, then constrains index spacing through index separation to improve radar estimation without BER degradation (Hoque et al., 2020). In AFDM, "AFDM Chirp-Permutation-Index Modulation with Quantum-Accelerated Codebook Design" indexes permutations of the chirp sequence associated with $2^\nu P$52, rather than multiple chirp-rate values (Rou et al., 2024). "Multiple-Mode Affine Frequency Division Multiplexing with Index Modulation" indexes mode activation patterns and chirp arrangement patterns across AFDM chirps while keeping $2^\nu P$53 and $2^\nu P$54 fixed (Liu et al., 17 Jul 2025). The two AFDM-IM papers (Zhu et al., 2023) and (Tao et al., 2023), together with the CDD-AFDM-IM framework (Tao et al., 2024), index active chirp subcarriers or DAF-domain positions, not chirp-rate states. Finally, "Frequency-Shift Chirp Spread Spectrum Communications with Index Modulation" indexes combinations of active orthogonal frequency-shifted chirps, again without chirp-rate selection (Hanif et al., 2021).

This makes a common misconception easy to state precisely: chirp-domain IM is not synonymous with chirp-rate IM. Circular-shift IM, chirp-permutation IM, AFDM resource activation, multiple-mode AFDM-IM, and FSCSS-IM all operate on chirp-related signaling dimensions, but they do not select among multiple chirp rates in the sense formalized by MCR-IM. A plausible implication is that MCR-IM should be understood as a stricter subclass of chirp-domain index modulation, one in which chirp-rate choice itself becomes part of the discrete signaling alphabet. The most direct antecedents are therefore not the shift-based or AFDM-position-based schemes as such, but the designs that expose a reusable pattern: construct a finite chirp dictionary, map bits to structured subsets of that dictionary, and recover the active indices through non-coherent or low-complexity matched transforms. On that reading, MCR-IM extends the chirp-indexing literature by moving the indexed parameter from chirp identity within a fixed family to chirp rate across multiple families (Zhu et al., 17 Jul 2025).