# Intercarrier Interference in OFDM: A General Model for Transmissions in Mobile Environments with Imperfect Synchronization

- Martín García
^{1}and - Christian Oberli
^{1}Email author

**2009**:786040

https://doi.org/10.1155/2009/786040

© M. García and C. Oberli. 2009

**Received: **2 June 2009

**Accepted: **16 September 2009

**Published: **9 November 2009

## Abstract

Intercarrier Interference (ICI) is an impairment well known to degrade performance of Orthogonal Frequency Division Multiplexing (OFDM) transmissions. It arises from carrier frequency offsets (CFOs), from the Doppler spread due to channel time-variation and, to a lesser extent, from sampling frequency offsets (SFOs). Literature reports several models of ICI due to each kind of impairment. Some studies describe ICI due to two of the three impairments, but so far no general model exists to describe the joint effect of all three impairments together. Furthermore, most available models involve some level of approximation, and the diversity of approaches makes it cumbersome to compare power levels of the different kinds of ICI. In this work, we present a general and mathematically exact model for the ICI stemming from the joint effect of the three impairments mentioned. The model allows for a vis-a-vis comparison of signal-to-ICI ratios (SIRs) caused by each impairment. Our result was validated by simulations. An analysis of ICI in IEEE-802.16e-type transmissions shows that during steady-state tracking and at speeds below 150 km/h, SIR due to CFO is typically in the range between 25 dB and 35 dB, SIR due to Doppler spread is larger than 25 dB, and ICI due to SFO is negligible.

## 1. Introduction

Mathematical models of Intercarrier Interference (ICI) in Orthogonal Frequency Division Multiplexing (OFDM) and techniques for mitigating it have been reported by many authors. Studies modeling and dealing with ICI stemming individually from channel variation in time are [1–9]. Likewise, the works of [10–15] address ICI due to Carrier Frequency Offset (CFO) and those of [16, 17] ICI solely due to Sampling Frequency Offset (SFO). Work modeling ICI produced jointly by two of the three impairments is significantly less common. The joint effect of CFO and SFO has been studied in [18, 19], while [20] reports on ICI due to CFO and channel mobility. Despite the attention that the topic has received so far, there is as yet no general model in literature that describes ICI resulting from the joint effect of all three impairments.

Many of the above cited references model ICI by using discrete-time and discrete-frequency signals. Unfortunately, discrete-domain approaches are inaccurate for representing impairments that affect signals outside the time-frequency grid of discrete analysis, such as the SFO, or often restrict the time-frequency properties of the channels for which the approaches are valid. Another limitation of the models in the cited references is the difficulty of combining them in order to make a fair comparison of each kind of ICI under the same conditions.

Our contribution with this paper is the derivation of the general and mathematically exact model of ICI for OFDM transmissions subject to the joint effect of CFO, SFO, and time-varying channels with arbitrary statistics. By including continuous-domain analysis, our derivations yield a model that is general and is devoid of the limitations of purely discrete-domain approaches, and by modeling all three impairments together, we obtain a tool that allows for a direct and clear comparison of the ICI caused by each impairment.

The remainder of this work is organized as follows. Section 2 presents the development of our model of ICI with deterministic signals; Section 3 analyzes the statistical behavior of ICI; in Section 4, the statistical behavior predicted by our model is validated by simulations of IEEE-802.16e-type transmissions ("mobile WiMAX'') [21]. Signal-to-interference ratio curves for a broad range of mobile speeds and CFOs are provided; finally, Section 5 sets out our conclusions.

## 2. Deterministic Model of ICI

In what follows, we derive a mathematical model that includes the effects of CFO, SFO, and channel mobility on OFDM transmissions.

In this equation, is the modulation on subcarrier of a total of subcarriers. The separation between subcarriers is Hertz, where is the sampling period of the transmitter. The cyclic prefix has samples and duration of seconds. Thus, the complete OFM symbol has samples and duration of seconds. The symbol denotes the imaginary unit . Finally, is the rectangular function, equal to 1 when is between and 0 elsewhere.

where and are the respective input and output signals in the time domain. The function is the time-variant impulse response of the system observed at instant due to an impulse at time .

Above, is the impulse function and is AWGN sampled at instants , with . Equation (3) is a continuous-time signal, but its value is 0 at every instant except when , with .

where is the Doppler-variant impulse response defined as the Fourier transform in of [22]. For a fixed delay , and if the channel is static, then is a frequency domain impulse. As channel mobility increases, so does the frequency spread of . The symbol denotes continuous convolution in the frequency domain.

Results (6) through (11) are deterministic and mathematically exact. In (9) (ICI due to mobility) we observe that the interference in subcarrier is the sum of signals from all the other subcarriers, respectively weighted by the integral of . The value of the integral depends on mobility and on the separation between the interfering subcarriers ( ) and the desired subcarrier ( ). This value is relevant only when subcarrier is in the neighborhood of . The size of the neighborhood grows with the mobile's speed, but in any current-day OFDM systems designed for mobility (e.g., DVB-T/H [24], "mobile WiMAX'' [21]), the neighborhood is mainly comprised by subcarriers and . It is to be noted that (9) equals zero if the channel is static, regardless of synchronization. If synchronization is perfect (i.e., and ), then (9) is similar to the description found by many authors [2, 6, 9, 25] for representing interference based on Doppler-variant impulse responses. However, they all use discrete-domain approaches, different from the one employed here, thus capturing the effect of ICI less accurately.

Term (10) (ICI due to imperfect synchronization) has been described by several authors for static channel conditions, either considering CFO and SFO jointly [18, 19], CFO alone [10, 12], or SFO alone [16, 17]. This term equals zero if and only if there are no frequency nor sampling offsets, regardless of mobility.

Finally, of (11) is a new finding and clarifies a frequent misconception that ICI due to mobility and imperfect synchronization is two separate additive terms. There is an ICI enhancement when both impairments are jointly present (we will show in Section 4, however, that this ICI is negligible in practice).

## 3. Statistical Analysis of ICI

where is the scattering function [22, 23]. It is also assumed that the transmitted data symbols are not correlated either in frequency or in time and have an average energy of .

where is the maximum Doppler spread as given by , with denoting the mobile's speed and the carrier wavelength.

Note that (16) implicitly states that the three ICI terms are statistically independent from each other.

In [27], the steps of Appendix D were also followed for determining the covariances of the ICI terms between different subcarriers. The result can be used for generating statistically accurate frequency-correlated ICI from a white Gaussian sequence. (By virtue of the central limit theorem with large enough it is commonly accepted that ICI has Gaussian random properties; see, e.g., [25].) The ICI thus generated might greatly simplify some simulations of imperfectly synchronized OFDM systems in mobile environments.

where follows from (8) and (15).

Strictly speaking, the expected interference powers per subcarrier given by (17), (18), (19), and (20) change with . In practice, however, if the transmission bandwidth is much larger than the Doppler spread bandwidth, these terms are practically constant over frequency. This is so because the time-limited Doppler spread function ( ) takes significant values only in the neighborhood of a subcarrier , thus ensuring statistical homogeneity in most subcarriers other than those at the band edges, which are exposed to less ICI because they have fewer neighboring subcarriers.

In this case, the WiMAX system requires a CFO smaller than 0.2 intercarrier spacing so that less than 10% of energy is lost as ICI (Figure 1(b)).

## 4. Computational Verification of Results

Above, the parameter is defined such that transmissions have an expected signal-to-noise ratio (SNR) of when there is no ICI (i.e., no mobility and perfect synchronization, hence (23) evaluates to ).

Our goal is to compare the theoretical prediction of (23) with values of SINR obtained from simulations by averaging over 300 OFDM symbols, transmitted over the same number of independent realizations of WSSUS time-varying channels with Clarke's statistics, and with receiver-side insertion of CFO and SFO. The time-variant impulse responses were generated using an autoregressive model of order 100 as set out in [28], with an bias to ensure the algorithm's stability. Unit-power QPSK was used for subcarrier modulation.

The parameters used were those of an IEEE-802.16e system [21] with subcarriers, a cyclic prefix of , bandwidth of 5 MHz, and a carrier of 3.5 GHz. Finally, the coherence time was estimated as [29], the average symbol energy was set equal to 1, and the channels' maximum delay spread was restricted to the duration of the cyclic prefix.

Note that evaluating (17), (18), (19), and (20) for (23) requires computing continuous integrals given by (8) and (15) (the latter was computed based on (14)). These were carried out with a sampling density of 100 points between subcarriers. In order to reduce the computational complexity of the resulting calculations, we used the fact that (17) to (20) are essentially invariant in (as noted) and therefore confined ourselves for evaluating the case of .

The top curves of Figures 3 and 4 present a discrepancy between theoretical and simulation results. As discussed by Baddour and Beaulieu [28], the autoregressive approach for simulating a time-varying channel uses ill-conditioned equations, which makes simulating slow-varying channels with Clarke's U-shaped spectral density difficult. As a workaround, they propose a heuristic solution equivalent to adding a very small amount of white noise to the channel's fading process. The effect is also equivalent to a slight enhancement of thermal noise and reveals itself in our simulations when thermal noise dominates over ICI, as in the curves mentioned. When this simulation bias is negligible, however, our theoretical results are well matched by the simulations.

Finally, note that after neglecting SFO the sole parameter remaining in (17), (18), (19), and (20) is the intercarrier spacing . Because all modes of operation specified by the IEEE-802.16e standard use the same intercarrier spacing [21], it follows that the curves in Figures 2 through 7 are in fact valid for any mode of mobile WiMAX transmission.

## 5. Conclusions

A general and mathematically exact model of the power of intercarrier interference (ICI) was derived for OFDM transmissions exposed to the joint impact of sampling frequency offset, carrier frequency offset (CFO), and channel time variation. It was shown that the ICI ensuing from these impairments has three components: one solely caused by Doppler spread, one that depends only on the synchronization offsets, and one that is nonzero only when imperfect synchronization and channel variation happen together. Similar but nevertheless approximate descriptions of the former two components are available in literature. In this paper, besides describing them without approximations, they are presented with the same power scale. This allows for a direct comparison of these two sources of ICI. The third component is a new finding. It was shown to be nonnegligible only in very-high-speed environments of no practical interest at the present.

The new model was validated by computer simulations of OFDM transmissions using IEEE 802.16e parameters (mobile WiMAX).

Signal-to-noise and signal-to-interference ratios (SIR) were used for comparing the different sources of ICI with levels of thermal noise. SIR curves for a broad range of CFOs and mobile speeds were presented. For reference, during steady-state tracking and at speeds below 150 km/h, SIR due to carrier frequency offset is typically in the range between 25 dB and 35 dB, and ICI due to Doppler spread is larger than 25 dB.

## Declarations

### Acknowledgment

This work was supported by Grants FONDECYT 1060718 and ADI-32 2006 from CONICYT Chile.

## Authors’ Affiliations

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