Frequency-Domain Modeling of OFDM Transmission with Insufficient Cyclic Prefix using Toeplitz Matrices

Grzegorz Cisek; Tomasz P. Zieli\'nski

REVIEW 3 major objections 5 minor 17 references

Frequency-Domain Modeling of OFDM Transmission with Insufficient Cyclic Prefix using Toeplitz Matrices

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read An exact frequency-domain formula describes OFDM interference from an insufficient cyclic prefix, showing the distortion is localized to nearby subcarriers.

desk verdict Correct Toeplitz-DFT formula and a real banding insight, undercut by two normalization errors in the power analysis that need fixing. read the letter →

arxiv 1908.08900 v1 pith:MPX3LRZF submitted 2019-08-23 eess.SP cs.NI

Grzegorz Cisek , Tomasz P. Zieliński This is my paper · ORCID

classification eess.SPcs.NI

keywords OFDMinsufficientcyclicprefixintersymbolinterferencetriangularToeplitzmatrixfrequency-domainchannelmodelingbandedresidualISInormalitymulti-usersimulation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a closed-form, frequency-domain description of inter-symbol interference (ISI) in OFDM when the cyclic prefix is shorter than the channel delay spread, including the case of no cyclic prefix. It shows that the interference matrix is the DFT of an upper-triangular Toeplitz matrix and proves exact formulas for its entries. From those formulas it follows that the average power of interference falls off as the subcarrier distance grows, so practically only the nearest neighboring subcarriers are affected. This justifies reducing the interference matrix to a narrow band, turning the residual error into near-Gaussian noise, and makes frequency-domain multi-user simulation cheaper than time-domain tapped-delay-line models.

What carries the argument

The central object is the interference matrix $\Phi = F B F^{\dagger}$, where $B$ is the zero-padded upper-triangular Toeplitz matrix built from the channel taps that spill beyond the cyclic prefix. The load-bearing identity is the closed-form DFT of such a matrix, proved in the appendix: off-diagonal entries are $\frac{1}{\sqrt{N}}(1-w^{n-m})^{-1}([\xi]_m-[\xi]_n)$ and diagonal entries are $\frac{1}{\sqrt{N}} F^{\dagger}([N,N-1,\ldots,1]^T \circ \rho)$. This splits $\Phi$ into a time-invariant envelope matrix $E$ with entries $1/(1-w^{n-m})$ and a channel-dependent skew-symmetric matrix $\Gamma$ with entries $[\xi]_m-[\xi]_n$; this split is what makes the banded approximation and its complexity savings work.

What would settle it

Simulate the exact time-domain block transmission in (8) with a fixed channel tap profile, compute the empirical frequency-domain interference matrix by taking F of the received blocks, and compare its off-diagonal entries and their averaged squared magnitudes to (17) and (28) for subcarrier distances up to N/4; any deviation above floating-point precision would show the closed-form DFT proof or the averaging step fails. Alternatively, repeat with a channel whose taps change within one symbol and verify whether nonzero $\Delta H$ and $\Delta B$ terms appear, as the block-fading assumption predicts they should not.

Watch

Extended reading notes

Core claim

Under the block-fading assumption that the channel impulse response is constant during one OFDM symbol, the received signal in the frequency domain is $r_u = G s_u + \Phi(s_{u-1} - W s_u) + \eta_u$. The paper's central discovery is that $\Phi = F B F^{\dagger}$ has fully explicit entries: for $n \neq m$, $[\Phi]_{n,m} = \frac{1}{\sqrt{N}} \frac{1}{1-w^{n-m}} ([\xi]_m - [\xi]_n)$, where $w = e^{-j2\pi/N}$ and $\xi$ is the (reversed-phase) transform of the first row $\rho$ of $B$; the diagonal is the transform of $[N, N-1, \ldots, 1]^T \circ \rho$. The paper also proves an expected-power expression whose dependence on subcarrier distance is governed by $\cot^2(\pi(m-n)/N)+1$ multiplied by a power-delay-profile sum. Because of this decay, the strong interference lives in a band around the diagonal, so only nearby subcarriers need to be tracked in equalization or simulation.

Load-bearing premise

The derivation assumes the channel impulse response is unchanging during a single OFDM symbol; if the channel drifts inside a symbol, the channel matrix is no longer circulant and the simple Toeplitz formulas for Phi stop holding.

Editorial extensions

If this is right

A receiver or precoder that cancels only the $b$ closest subcarriers on each side of the diagonal of $\Phi$ removes most of the ISI; the numerical experiment in the paper raises the signal-to-error ratio by more than 12 dB compared with ignoring $\Phi$ entirely.
For band widths around $N/16$ to $N/8$, the leftover interference is close to Gaussian, so it can be folded into the noise covariance and handled by standard soft-decision decoding.
When the same channel realization is reused over many OFDM symbols and many users share the band, frequency-domain simulation with a banded $\Phi$ needs fewer MAC operations than a time-domain tapped-delay-line simulator.
The framework also covers OFDM without any cyclic prefix, since the formulas are stated for general prefix length $v$, including $v=0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the same leakage structure holds when Doppler-induced ICI is included, one could unify cyclic-prefix ISI and mobility ICI into a single banded matrix inversion; the paper only draws the analogy, it does not develop this.
The envelope-decay formula suggests an adaptive-cyclic-prefix scheduler: the required band width, and hence receiver complexity, is set by the taps that fall just beyond the prefix, so the prefix length could be tuned against a complexity budget.
A natural test is to measure the residual-ISI distribution for alphabets other than 16-QAM; the normality claim was demonstrated for 16-QAM and may change with the constellation.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know upfront. First, the paper's real contribution is a closed-form DFT of a zero-padded upper-triangular Toeplitz matrix (Eqs. 17-18), applied to the OFDM ISI matrix under insufficient cyclic prefix. That derivation is correct; I checked the appendix proof on a small case. Second, the banded-structure insight -- ISI from a subcarrier decays with subcarrier distance -- is real and supported by direct simulation. But the paper's expected-power formulas have normalization errors. Eq. (23) carries a spurious 1/N, and Eq. (28) is missing another 1/N. Together they shift the relative interference power in Fig. 1 by roughly a factor of N^2 (about 54 dB at N=512). That is not a cosmetic typo; the quantitative decay law as printed is unreliable.

What the paper does well: the extension from Huckle's Hermitian Toeplitz preconditioner to the non-Hermitian triangular case is non-obvious and the matrix decomposition into an envelope matrix E and a channel-dependent matrix Γ is clean. The simulations in Section IV-A that use (17) directly confirm the banding behavior, independent of the flawed equations. The residual-ISI normality test is a sensible extra check. The complexity comparison is rough but clearly labeled as a MAC count.

Soft spots, in proportion. The normalization errors are the main problem; they don't touch (17)-(18), but they do invalidate the quantitative power analysis as printed. The authors do not ship code, so reproducibility rests on the formulas. The block-fading assumption is stated explicitly and is standard, but it means the framework does not cover intra-symbol channel variation; that is a scope limit, not a flaw. The complexity comparison is a bit of an apples-to-oranges back-of-envelope; the conclusion just says "when many users and small b", which is fine.

Is it a serious paper? Yes, structurally. The core matrix result is novel and useful for designers of equalizers and channel simulators. The errors are fixable by redoing the algebra in Section III and regenerating Fig. 1. If I were the editor I'd send it to a knowledgeable referee, with a note to check the normalization carefully. After revision, it could be a worthwhile publication.

Referee Report

3 major / 5 minor

Summary. The paper develops a frequency-domain framework for modeling OFDM transmission with insufficient cyclic prefix. The channel is decomposed into a circulant matrix H plus an upper-triangular Toeplitz matrix B, and the paper derives closed-form expressions for the frequency-domain interference matrix Φ = F B F†: off-diagonal entries in Eq. (17), diagonal entries in Eq. (18), with a proof in the appendix. The central claim is that the interference power decays with subcarrier distance, so Φ is effectively banded. The authors derive expected-power formulas (23) and (28), and then validate the banded approximation via SER simulations (Section IV-A), a normality test of residual ISI (Section IV-B), and a complexity comparison (Section IV-C).

Significance. The structural result (17)-(18) is a non-obvious and useful contribution to the OFDM literature; the appendix proof is clear and appears correct (I verified it on a small N=2 example). The paper also provides a convincing demonstration that a banded interference matrix suffices to capture most of the ISI, and the complexity analysis is relevant for frequency-domain channel simulation. The main weakness is that the expected-power analysis contains normalization errors that affect the quantitative values in Fig. 1; the qualitative banding claim survives because it is also supported by direct simulation in Section IV-A. The paper is likely to be of interest to researchers working on interference mitigation and channel simulation for OFDM.

major comments (3)

[Section III, Eq. (28)] Equation (28) is missing a factor 1/N. From (21), for n ≠ m, [Φ]_{n,m} = (1/√N)[E]_{n,m}[Γ]_{n,m}, so E{|[Φ]_{n,m}|^2} = (1/N)|[E]_{n,m}|^2 E{|[Γ]_{n,m}|^2}. Using (22), |[E]_{n,m}|^2 = (1/4)(1+cot^2(π(m-n)/N)), and from (16) and (27), E{|[Γ]_{n,m}|^2} = (4/N) ∑_{k=0}^{N-1} (|[ρ]_k| |sin(π(m-n)k/N)|)^2. The product is (1/N^2)(1+cot^2(π(m-n)/N)) times the sum, so the printed RHS of (28), which divides by only one N, overestimates the interference power by a factor N. This error directly affects the quantitative levels shown in Fig. 1.
[Section III, Eq. (23)] Equation (23) contains a spurious 1/N. With the normalized DFT matrix F defined in Section III and a circulant H under the block-fading assumption, the diagonal of G = F H F† is the unnormalized DFT of the first row of H, so E{|[G]_{n,n}|^2} = ∑_{m=0}^{N-1} E{|h(n,m)|^2}, not (1/N) times that sum. Together with the error in (28), the relative interference powers in Fig. 1 are shifted by a factor of N^2 (about 54 dB at N=512), so Fig. 1 as printed is not quantitatively reliable.
[Section III, Eq. (25)] Equation (25) omits the factor 1/√N from the definition ξ = F†ρ in (16). Since [ξ]_m = (1/√N) ∑_{k=0}^{N-1} [ρ]_k e^{j2πmk/N}, the first equality in (25) should carry a 1/√N factor. The later expression (27) appears to include this factor implicitly, but the intermediate derivation is inconsistent with (16) and should be corrected for clarity and correctness.

minor comments (5)

[Section III, Eq. (28)] Please clarify the fraction in Eq. (28) after correction; the intended form is E{|[Φ]_{n,m}|^2} = (1+cot^2(π(m-n)/N))/N^2 times the sum, and the typesetting should make that unambiguous.
[Section III, Eq. (23)] In Eq. (23), the notation h(n,m) is confusing for a time-invariant block-fading channel, where h(n,m) = h_m; consider writing h_m or h(m) to avoid implying time dependence.
[Fig. 1] The y-axis label in Fig. 1 should explicitly state that the quantity is the interference power relative to the desired-subcarrier power and indicate the normalization used, since the levels are affected by the N-scaling errors discussed above.
[Abstract] There is a typo in the abstract: 'inducted' should be 'induced'.
[Section IV-B, Table I] Some p-values in Table I are reported as exactly 0.000000; please note whether this is a numerical artefact of the finite sample size, or consider reporting them with more significant digits.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Toeplitz-DFT derivation and the banded-interference claim are self-contained; the flagged normalization issues are accuracy concerns, not circular reductions.

full rationale

The paper's central claim is the closed-form DFT of an upper-triangular Toeplitz matrix, Eq. (17)-(18), proved directly in the appendix from the definition Phi = F B F^dagger by summing geometric series (Eqs. (31)-(36)); no fitted parameter or target quantity enters the derivation. The banding/decay claim follows algebraically from Eq. (28) (modulo the independent normalization-error concern) and is separately supported by Monte-Carlo simulations in Sec. IV-A using the external COST259 HT profile and an independent SER comparison. The block-fading assumption in Sec. III is an explicitly stated modeling assumption, not an input secretly equivalent to the output. Self-citations [9] and [10] are used only to position prior frequency-domain simulator work and to borrow a complexity-counting methodology; they do not carry the mathematical derivation or the empirical validation. The normalization errors alleged in Eqs. (23) and (28) are numerical correctness issues, not circular reductions: Eq. (28) is not fitted to the quantities it predicts. No step in the derivation reduces by construction to its own input. Score 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation relies only on standard DFT/Toeplitz algebra plus the block-fading channel assumption; no parameters are fitted to the target result.

free parameters (1)

band width b = N/16 to N/8 (e.g., 32 for N=512)
In (29), b controls the reduced interference matrix; the paper recommends values near N/16..N/8 based on the Jarque-Bera normality test, an empirical choice.

assumptions (4)

domain assumption The channel is constant during one OFDM symbol (block fading), so the ICI matrices Delta-H and Delta-B are zero.
Invoked in Section III before Eq (16); it makes H circulant and Phi the DFT of a Toeplitz matrix. If the channel varies within a symbol, the simplified formulas do not hold.
domain assumption Channel taps are independent, zero-mean complex Gaussian processes, so cross terms vanish in the expectation of squared interference.
Used to convert sums over taps into sums of per-tap powers in Eq (27).
domain assumption Transmit symbols have uniform unit power, E{||s_u[n]||^2}=1.
Stated in Section III before Eq (23); normalizes the power comparisons.
standard math The DFT matrix is unitary with 1/sqrt(N) normalization, and standard geometric series apply.
Used throughout the appendix to derive (17) and (18).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Frequency-Domain Modeling of OFDM Transmission with Insufficient Cyclic Prefix using Toeplitz Matrices." pith.science (2026). https://pith.science/paper/MPX3LRZF

@misc{pith2026190808900,
  author       = {Pith},
  title        = {Pith review of: Frequency-Domain Modeling of OFDM Transmission with Insufficient Cyclic Prefix using Toeplitz Matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MPX3LRZF}},
  note         = {Machine review of arXiv:1908.08900}
}

read the original abstract

A novel mathematical framework is proposed to model Intersymbol Interference (ISI) phenomenon in wireless communication systems based on Orthogonal Frequency Division Multiplexing (OFDM) with or without cyclic prefix. The framework is based on a new formula to calculate the Fast Fourier Transform (FFT) of a triangular Toeplitz matrix, which is derived and proven in this paper. It is shown that distortion inducted by the ISI from a given subcarrier is the most significant for the closest subcarriers and the contribution decays as the distance between subcarriers grows. According to numerical experiments, knowledge of ISI coefficients concentrated around the diagonal of Channel Frequency Response (CFR) matrix improves the receiver's error floor significantly. The potential use of the framework for real-time frequency domain channel simulation was also investigated and demonstrated to be more efficient than conventional time domain Tapped Delay Line (TDL) model when a number of simulated users is high.

Figures

Figures reproduced from arXiv: 1908.08900 by the authors.

**Figure 1.** Power of interference matrix Φ relative to power of non-distorted subcarrier for various FFT sizes N and COST259 HT profile. Moreover, the following trigonometric identity holds [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Accuracy of ISI modeling with reduced interference CFR matrix [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Complexity of frequency domain ISI simulation comparing to time [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

G. R. Parsaee, A. Yarali and H. Ebrahimzad, ”MMSE-DFE Equalizer Design for OFDM Systems with Insufﬁcient Cyclic Preﬁx,” in IEEE 60th V eh. Technol. Conf. (VTC 2004-Spring) , Sept. 2004

work page 2004
[2]

M. D. Nisar et. al. , ”On Channel Estimation and Equalization of OFDM Systems with Insufﬁcient Cyclic Preﬁx,” IEEE 65th V eh. Technol. Conf. (VTC 2007-Spring) , April 2007

work page 2007
[3]

Molisch, M

A. Molisch, M. Toeltsch and S. Vermani, ”Iterative Methods for Cancel- lation of Intercarrier Interference in OFDM Systems,” IEEE Trans. V eh. Technol., vol. 56, no. 4, pp. 2158-2167, July 2007

work page 2007
[4]

T. Pham et. al. , ”Channel Estimation and Data Detection for Insufﬁcient Cyclic Preﬁx MIMO-OFDM,” IEEE Trans. V eh. Technol., vol. 66, no. 6, pp. 4756-4768, June 2017

work page 2017
[5]

Jin and X

Y . Jin and X. Xia, ”An Interference Nulling Based Channel Independent Precoding for MIMO-OFDM Systems with Insufﬁcient Cyclic Preﬁx,” IEEE Trans. Commun. , vol. 61, no. 1, pp. 131-143, Jan. 2013

work page 2013
[6]

N. Kim et. al., ”Precoding Design for Cyclic Preﬁx Overhead Reduction in a MISO-OFDM System,” in IEEE Wireless Commun. Lett. , vol. 6, no. 5, pp. 578-581, Oct. 2017

work page 2017
[7]

Freire et

I. Freire et. al. , ”Low Complexity Precoder and Equalizer for DMT Systems with Insufﬁcient Cyclic Preﬁx,” in IEEE Global Commun. Conf. (GlobeCom), Dec. 2014

work page 2014
[8]

Wu, ”Analysis and Characterization of Intercarrier and Interblock Interferences for Wireless Mobile OFDM Systems,” IEEE Trans

H. Wu, ”Analysis and Characterization of Intercarrier and Interblock Interferences for Wireless Mobile OFDM Systems,” IEEE Trans. Broad- cast., vol. 52, no. 2, pp. 203-210, June 2006

work page 2006

Show all 17 references

[9]

Cisek and T

G. Cisek and T. Zielinski, ”Frequency Domain Multipath Fading Channel Simulator Integrated with OFDM Transmitter for E-UTRAN Baseband Trafﬁc Generator,” in 25th Eur . Signal. Process. Conf. (EUSIPCO) , Aug. 2017

2017
[10]

Cisek and T

G. Cisek and T. Zielinski, ”Frequency-Domain Multi-User OFDMA Fast Fading Channel Simulation in High-Mobility Scenarios,” in 15th Int. Symp. Wireless Commun. Syst. (ISWCS) , Aug. 2018

2018
[11]

M. Zhao, Z. Shi and M. Reed, ”Iterative Turbo Channel Estimation for OFDM System over Rapid Dispersive Fading Channel,” IEEE Trans. Wireless Commun., vol. 7, no. 8, pp. 3174-3184, Aug. 2008

2008
[12]

Huckle, ”Some Aspects of Circulant Preconditioners,” SIAM J

T. Huckle, ”Some Aspects of Circulant Preconditioners,” SIAM J. Sci. Comput., vol. 14, no. 3, pp. 531-541, May 1993

1993
[13]

T. F. Chan, ”An Optimal Circulant Preconditioner for Toeplitz Systems,” SIAM J. Sci. Stat. Comput. , vol. 9, no. 3, pp. 766–771, July 1988

1988
[14]

A Low Complexity ICI Cancellation Method for High Mobility OFDM Systems,

K. Kim and H. Park, “A Low Complexity ICI Cancellation Method for High Mobility OFDM Systems,” in IEEE 63rd V eh. Technol. Conf. (VTC 2006), May 2006

2006
[15]

3GPP TR 25.943, Deployment aspects, version 14.0.0, Release 14, Mar. 2017

2017
[16]

3GPP TS 36.104, Evolved Universal Terrestrial Radio Access (E- UTRA); Base Station (BS) radio transmission and reception , version 15.1.0, Jan. 2017

2017
[17]

Jarque and A

C. Jarque and A. Bera, ”A Test for Normality of Observations and Regression Residuals,” International Statistical Review , vol. 55, no. 2, pp. 163–172, Aug. 1987

1987

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

G. R. Parsaee, A. Yarali and H. Ebrahimzad, ”MMSE-DFE Equalizer Design for OFDM Systems with Insufﬁcient Cyclic Preﬁx,” in IEEE 60th V eh. Technol. Conf. (VTC 2004-Spring) , Sept. 2004

work page 2004

[2] [2]

M. D. Nisar et. al. , ”On Channel Estimation and Equalization of OFDM Systems with Insufﬁcient Cyclic Preﬁx,” IEEE 65th V eh. Technol. Conf. (VTC 2007-Spring) , April 2007

work page 2007

[3] [3]

Molisch, M

A. Molisch, M. Toeltsch and S. Vermani, ”Iterative Methods for Cancel- lation of Intercarrier Interference in OFDM Systems,” IEEE Trans. V eh. Technol., vol. 56, no. 4, pp. 2158-2167, July 2007

work page 2007

[4] [4]

T. Pham et. al. , ”Channel Estimation and Data Detection for Insufﬁcient Cyclic Preﬁx MIMO-OFDM,” IEEE Trans. V eh. Technol., vol. 66, no. 6, pp. 4756-4768, June 2017

work page 2017

[5] [5]

Jin and X

Y . Jin and X. Xia, ”An Interference Nulling Based Channel Independent Precoding for MIMO-OFDM Systems with Insufﬁcient Cyclic Preﬁx,” IEEE Trans. Commun. , vol. 61, no. 1, pp. 131-143, Jan. 2013

work page 2013

[6] [6]

N. Kim et. al., ”Precoding Design for Cyclic Preﬁx Overhead Reduction in a MISO-OFDM System,” in IEEE Wireless Commun. Lett. , vol. 6, no. 5, pp. 578-581, Oct. 2017

work page 2017

[7] [7]

Freire et

I. Freire et. al. , ”Low Complexity Precoder and Equalizer for DMT Systems with Insufﬁcient Cyclic Preﬁx,” in IEEE Global Commun. Conf. (GlobeCom), Dec. 2014

work page 2014

[8] [8]

Wu, ”Analysis and Characterization of Intercarrier and Interblock Interferences for Wireless Mobile OFDM Systems,” IEEE Trans

H. Wu, ”Analysis and Characterization of Intercarrier and Interblock Interferences for Wireless Mobile OFDM Systems,” IEEE Trans. Broad- cast., vol. 52, no. 2, pp. 203-210, June 2006

work page 2006