---
title: 'MCR-IM: Multiple Chirp Rate Index Modulation'
url: https://www.emergentmind.com/topics/multiple-chirp-rate-index-modulation-mcr-im
type: topic
---

# MCR-IM: Multiple Chirp Rate Index Modulation

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" [2507.14228], 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 [2507.14228].

## 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 [2507.14228].

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
\[
\binom{2^\nu P}{L},
\]
and the number of information bits per symbol is
\[
\eta_b=\left\lfloor \log_2 \binom{2^\nu P}{L} \right\rfloor.
\]
In this formulation, all symbol bits are carried by index selection; there is no separate conventional PSK or QAM layer [2507.14228].

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 [2507.14228].

## 2. Signal model and combinatorial mapping

The underlying Zadoff–Chu sequence is
\[
x\left( N,r,q,n \right) = \frac{1}{\sqrt N }\exp \left( j\frac{\pi }{N}r\left( n + 1 + 2q \right)n \right),
\]
where \(N\) is a prime number, \(r \in \{1,2,\cdots,N-1\}\) is the root, \(q \in \{0,1,\cdots,N-1\}\) is the offset, and \(n=0,1,\cdots,N-1\). For dechirping, the receiver uses
\[
x^*\left( N,r',0,n \right)=\frac{1}{\sqrt N }\exp\left(-j\frac{\pi}{N}r'(n+1)n\right).
\]
To compare with LoRa, the spread factor is defined as
\[
\nu = \left\lfloor \log_2 N \right\rfloor.
\]
The paper uses the one-to-one mapping \(\nu=6 \rightarrow N=67\), \(\nu=7 \rightarrow N=131\), \(\nu=8 \rightarrow N=257\), \(\nu=9 \rightarrow N=521\), \(\nu=10 \rightarrow N=1031\), \(\nu=11 \rightarrow N=2053\), and \(\nu=12 \rightarrow N=4099\), while only the first \(2^\nu\) frequency positions are used for activation [2507.14228].

The selectable positions are arranged as a \(P\times 2^\nu\) matrix \(\mathbf{\Theta}\), with one row per chirp rate. A symbol is represented by an ordered vector
\[
\mathbf{s}=[s_0,s_1,\cdots,s_{L-1}],
\]
with
\[
s_i\in \{0,1,\cdots,2^\nu P-1\},\qquad s_0<s_1<\cdots<s_{L-1}.
\]
Because a full lookup table over \(\binom{2^\nu P}{L}\) possibilities is too large, the mapping uses the combinatorial number system:
\[
D=\binom{s_{L-1}}{L}+\binom{s_{L-2}}{L-1}+\cdots+\binom{s_1}{2}+\binom{s_0}{1},
\]
where \(D\in\left\{0,1,2,\cdots,\binom{2^\nu P}{L}-1\right\}\) is the decimal representation of the index bits [2507.14228].

Each selected flattened index \(s_i\) is mapped to a chirp rate \(r_i\) and an offset \(q_i\). The chirp-rate assignment is
\[
r_i = \left\lfloor \frac{s_i}{2^\nu} \right\rfloor + M_1,
\]
and the offset satisfies
\[
s_i-\left\lfloor \frac{s_i}{2^\nu}\right\rfloor 2^\nu = \bmod(r_i q_i,N).
\]
The transmitted MCR-IM waveform is then the normalized superposition
\[
X(n)=\sqrt{\frac{E_{\rm s}}{NL}} \sum_{i=0}^{L-1} \exp\left(j\frac{\pi}{N}r_i(n+1+2q_i)n\right),
\]
with bit energy
\[
E_{\rm b}=\frac{E_{\rm s}}{\eta_b}.
\]
This construction makes chirp-rate selection and within-rate frequency selection inseparable parts of the symbol alphabet [2507.14228].

## 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 \(r'\) and taking a DFT, the receiver observes
\[
\begin{aligned}
y(k) &= \left| \mathrm{DFT}\left( x(N,r,q,n)x^*(N,r',0,n)\right)\right| \\
&= \frac{1}{N}\left| \sum_{n=0}^{N-1} e^{j\frac{\pi n}{N}\left( (r-r')(n+1)+2(rq-k)\right)} \right|.
\end{aligned}
\]
Two cases are central. If \(r=r'\),
\[
y(k)= \begin{cases} 1, & k=\bmod(rq,N),\\ 0, & \text{otherwise}, \end{cases}
\]
whereas if \(r\neq r'\),
\[
y(k)=\frac{1}{\sqrt N }.
\]
Thus, matched chirp-rate dechirping collapses the energy into a single DFT bin, while mismatched chirp rates produce uniformly low outputs [2507.14228].

The same point is expressed through the normalized cross-correlation
\[
\vartheta_1(r_1,r_2,q_1,q_2) = \sum_{n=0}^{N-1} x_1(N,r_1,q_1,n)x_2^*(N,r_2,q_2,n).
\]
The reported properties are: \(\vartheta_1=1\) if \(r_1=r_2\) and \(q_1=q_2\); \(\vartheta_1=0\) if \(r_1=r_2\) and \(q_1\neq q_2\); and
\[
\left|\vartheta_1(r_1,r_2,q_1,q_2)\right| = \frac{1}{\sqrt N }
\]
if \(r_1\neq r_2\). This is the core quasi-orthogonality result enabling chirp-rate indexing [2507.14228].

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,
\[
\left| \vartheta_2 \right| \le L^2\sqrt{\frac{1+2\varepsilon}{N_1}},
\]
for two MCR-IM signals of different lengths \(N_1>N_2\). This indicates that inter-SF cross-correlation decays roughly as \(1/\sqrt{N_1}\), so quasi-orthogonality is preserved across spread factors as well [2507.14228].

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

Over Nakagami-\(m\) fading, the received signal is
\[
R(n)=hX(n)+\phi(n),
\]
where \(\phi(n)\sim \mathcal{CN}(0,N_0/2)\) and \(|h|\) follows a Nakagami-\(m\) distribution. The receiver dechirps against every candidate chirp rate \(M_\ell\), \(\ell\in\{1,\dots,P\}\), and computes
\[
\mathbf{y}_\ell(k)=\mathrm{DFT}\left(R(n)\frac{1}{\sqrt N}e^{-j\frac{\pi}{N}M_\ell(n+1)n}\right)=h\mathbf{u}_\ell(k)+\Phi_\ell(k),
\]
for \(k=0,1,\cdots,2^\nu-1\). Stacking these vectors forms
\[
\mathbf{Y}=\left[\mathbf{y}_1;\mathbf{y}_2;\cdots;\mathbf{y}_P\right]^{\mathsf T}.
\]
Direct detection is non-coherent: the receiver finds the \(L\) strongest peaks in \(|\mathbf{Y}|\), maps them back through
\[
\hat s_i=(\hat \ell_i-1)2^\nu+\hat k_i-1,
\]
sorts the resulting index set, and inverts the combinatorial mapping to recover the transmitted bits [2507.14228].

The BER analysis partitions the receiver outputs into a signal set \(\mathbb{A}_{\rm s}\), an interference set \(\mathbb{A}_{\rm i}\), and a noise set \(\mathbb{A}_{\rm n}\). Using a rule taken from ICS-LoRa, the paper adopts a \(1.3\) 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
\[
\mu_{\ell,k} = \sigma\sqrt{\frac{\pi}{2} \,{}_1F_1\left(-\frac{1}{2},1,-\kappa_\ell(k)\right)
\]
and variance
\[
\sigma_{\ell,k}^2 = N_0+|h\mathbf u_\ell(k)|^2 - \frac{\pi N_0}{4} \,{}_1F_1^2\left(-\frac{1}{2},1,-\kappa_\ell(k)\right).
\]
The paper then derives an approximate closed-form BER over Nakagami-\(m\) fading by combining signal-versus-interference and signal-versus-noise comparison terms and averaging over the fading amplitude through Gauss-Hermite quadrature [2507.14228].

The PD-SIC algorithm addresses the loss of directly orthogonal user channels caused by assigning \(P\) chirp rates per user. Each user is assigned a chirp-rate range of size \(P\), and the gateway first detects preambles to determine the number of users, their arrival times, and their chirp-rate ranges. For user \(i\), an energy metric is formed as
\[
E_i = \sum_{l=1}^{L} \max^{(l)}\left(|\mathbf{Y}_i|^2\right),
\]
and the strongest user is decoded first. The gateway reconstructs that user’s signal, searches over quantized phases
\[
\theta_d=\frac{2\pi}{K}d,\qquad d\in\{0,1,\cdots,K-1\},
\]
forms tentative cancellation residuals
\[
T_d(n)=R_\chi(n)-\exp(j\theta_d)X_\chi(n),
\]
and accepts the cancellation if the residual energy on the detected peak set satisfies
\[
\hat E_\chi < \beta E_\chi,
\qquad
\beta = 2\left(1-\cos\frac{\pi}{K}\right).
\]
The paper states that, for \(N_u\) users, the total number of dechirp and DFT operations is
\[
\frac{P}{2}\left( (K+5)N_u - K - 3 \right),
\]
which it treats as acceptable because the algorithm runs at the gateway [2507.14228].

## 5. Spectral efficiency, throughput, and reported performance

With bandwidth \(B=125\) kHz and sampling interval \(T_c=1/B\), the symbol duration is
\[
T_s=2^\nu T_c.
\]
The reported spectral efficiency is
\[
SE=\frac{\eta_b(1-P_{\rm err})}{T_s B},
\]
and the throughput under \(N_u\) colliding users is
\[
T_{\text{throughput}} = \frac{\eta_b}{T_s}(1-P_{\rm err}|N_u)N_u.
\]
Because \(\eta_b\) grows with \(\binom{2^\nu P}{L}\), MCR-IM increases rate by enlarging the index alphabet rather than by introducing a higher-order symbol constellation [2507.14228].

Several quantitative comparisons are reported. For \(\nu=7\), \(L=4\), and \(P=4\), MCR-IM carries \(55\%\) more bits than GCSS and \(34\%\) more bits than FSCSS-IM. For \(L=8\), increasing \(P\) to \(2\), \(4\), and \(8\) increases bits per symbol by \(20\%\), \(40\%\), and \(60\%\), respectively, relative to \(P=1\). In BER comparisons at \(\nu=10\) and \(m=3\), MCR-IM with \(L=2\) and \(P=4\) or \(P=12\) improves on FSCSS-IM by about \(0.4\) dB and \(0.6\) dB; for \(L=4\), MCR-IM with \(P=4\) gains about \(0.2\) dB over FSCSS-IM; and against GCSS, MCR-IM is better above about \(6.5\) dB [2507.14228].

The simulation trends are also specific. Increasing \(P\) from \(2\) to \(4\) at \(m=1\), \(\nu=7\), \(L=4\) increases bits per symbol from \(27\) to \(31\) with less than \(0.5\) dB BER penalty. Decreasing \(L\) from \(4\) to \(2\) at \(m=1\), \(P=4\), \(\nu=7\) improves BER by about \(2\) dB, but lowers bits per symbol from \(31\) to \(16\). Increasing \(\nu\) from \(7\) to \(8\) at \(P=L=4\), \(m=1\) improves BER by about \(1\) dB [2507.14228].

In collision scenarios, MCR-IM with PD-SIC is reported to outperform OrthoRa above about \(5\) dB. At \(E_b/N_0=10\) dB, the throughput gain over OrthoRa is \(21\%\) for \(N_u=3\) and \(16\%\) for \(N_u=2\), both with \(P_{\rm frame}=2\). 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 [2507.14228].

## 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 [2010.13220]. 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 [2010.03231]. In AFDM, "AFDM Chirp-Permutation-Index Modulation with Quantum-Accelerated Codebook Design" indexes permutations of the chirp sequence associated with \(c_2\), rather than multiple chirp-rate values [2405.02085]. "Multiple-Mode Affine Frequency Division Multiplexing with Index Modulation" indexes mode activation patterns and chirp arrangement patterns across AFDM chirps while keeping \(c_1\) and \(c_2\) fixed [2507.13037]. The two AFDM-IM papers [2312.01125] and [2310.05475], together with the CDD-AFDM-IM framework [2411.09938], 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 [2102.04642].

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 [2507.14228].

Source: https://www.emergentmind.com/topics/multiple-chirp-rate-index-modulation-mcr-im