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# Inter Carrier Interference (ICI) in OFDM due to frequency offset

by on August 8, 2009

In this post, let us evaluate the impact of frequency offset resulting in Inter Carrier Interference (ICI) while receiving an OFDM modulated symbol. We will first discuss the OFDM transmission and reception, the effect of frequency offset and later we will define the loss of orthogonality and resulting signal to noise ratio (SNR) loss due to the presence of frequency offset. The analysis is accompanied by Matlab/Octave simulation scripts.

## OFDM transmission

As discussed in the post on Understanding an OFDM transmission,  for sending an OFDM modulated symbol, we use multiple sinusoidals with frequency separation $\frac{1}{T}$ is used, where $T$ is the symbol period. The information $a_k$ to be send on each subcarrier $k$ is multiplied by the corresponding carrier $g_k(t)=e^{\frac{j2\pi kt}{T}}$ and the sum of such modulated sinusoidals form the transmit signal. Mathematically, the transmit signal is,

$\begin{eqnarray}s(t) &= &a_0g_0(t) + a_1g_1(t)+\ldots+a_{K-1}g_{K-1}(t)\\ & = &\sum_{0}^{K-1}a_kg_k(t) \\&=&\frac{1}{\sqrt{T}}\underbrace{\sum_{0}^{K-1}a_ke^{\frac{j2\pi kt}{T}}}\ w(t) \end{eqnarray}$

The interpretation of the above equation is as follows:
(a) Each information signal $a_k$ multiplies the sinusoidal having frequency of $\frac{k}{T}$.
(b) Sum of all such modulated sinusoidals are added and the resultant signal is sent out as $s(t)$.

## OFDM reception

In an OFDM receiver, we will multiply the received signal with a bank of correlators and integrate over the period $T$. The correlator to extract information send on subcarrier  $k$ is$c_m(t)=\frac{1}{\sqrt{T}}e^{-\frac{j2\pi mt}{T}}$.

The integral,
$\begin{eqnarray}\frac{1}{\sqrt{T}}\int_Ts(t)e^{-\frac{j2\pi mt}{T}} & = & a_k,& m=k\\ & = & 0, & m\ne k\end{eqnarray}$,

where
$m$ takes values from $0$ till $K-1$.

## Frequency offset

In a typical wireless communication system, the signal to be transmitted is upconverted to a carrier frequency prior to transmission. The receiver is expected to tune to the same carrier frequency for down-converting the signal to baseband, prior to demodulation.

Figure: Up/down conversion

However, due to device impairments the carrier frequency of the receiver need not be same as the carrier frequency of the transmitter. When this happens, the received baseband signal, instead of being centered at DC (0MHz), will be centered at a frequency $f_{\delta}$, where
$f_{\delta} = f_{Tx} - f_{Rx}$.

The baseband representation is (ignoring noise),

$y(t) = s(t)e^{j2\pi f_{\delta}t}$, where

$y(t)$ is the received signal

$s(t)$ is the transmitted signal and

$f_{\delta}$ is the frequency offset.

## Effect of frequency offset in OFDM receiver

Let us assume that the frequency offset $f_{\delta}$ is a fraction of subcarrier spacing $1/T$ i.e.
$f_{\delta}=\frac{\delta}{T}$.

Also, for simplifying the equations, lets us assume that the transmitted symbols on all subcarriers,  $a_k=1$

$y(t) = s(t)e^{\frac{j2\pi\delta }{T}t}$.

The output of the correlator for sub-carrier $m$ is,

$\begin{eqnarray}\frac{1}{\sqrt{T}}\int_Ty(t)e^{-\frac{j2\pi mt}{T}} & = & \frac{1}{T}\int_{T}e^{\frac{j2\pi (k+\delta-m)}{T}t},& m=\in \{0,...,K-1\}\\ & = & \frac{1}{j2\pi (k+\delta-m)}(e^{j2\pi (k+\delta-m)}-1) & \end{eqnarray}$.

For  $\delta=0$,

$\begin{array}{rrr}\frac{1}{j2\pi(k+\delta-m)}(e^{j2\pi (k+\delta-m)}-1)&=1, &m=k\\ \frac{1}{j2\pi(k+\delta-m)}(e^{j2\pi (k+\delta-m)}-1)&=0, &m \ne k\\\end{array}$

The integral reduces to the OFDM receiver with no impairments case.

However for non zero values of $\delta$, we can see that the amplitude of the correlation with subcarrier $m$ includes

• distortion due to frequency offset between actual frequency $\frac{m+\delta}{T}$ and the desired frequency $\frac{m}{T}$.
• distortion due to interference with other subcarriers with with desired frequency $\frac{m}{T}$. This term is also known as Inter Carrier Interference (ICI).

## Simulation Model

1. Generates an OFDM symbol with all subcarriers modulated with $a_k=1$.

2. Introduce frequency offset and add noise to result in $\frac{E_b}{N_0}$=30dB.

3. Perform demodulation at the receiver.

4. Find the difference between the desired and actual constellation.

5. Compute the rms value of error across all subcarriers.

6. Repeat this for different values of frequency offset.

Figure: Error Magnitude vs frequency offset for OFDM

## Observations

The theoretical results are computed by $\sum_m\frac{1}{j2\pi (k+\delta-m)}e^{j2\pi (k+\delta-m)}$.

Quite likely the simulated results are slightly better than theoretical results because the simulated results are computed using average error for all subcarriers (and the subcarriers at the edge undergo lower distortion).

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