Non-ideal effects in an OFDM system

This section will examine the effects of non-idealities in an OFDM system. These effects will include impairments and receiver offsets. Because the fourier transform is a fundamental operation in OFDM, the effects of several offsets can be intuitively understood by applying fourier transform theory.

Local oscillator frequency offset

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At start-up, the local oscillator (LO) frequency at the receiver is typically different from the LO frequency at the transmitter. A carrier tracking loop is used to adjust the receiver's LO frequency in order to match the transmitter's LO frequency as closely as possible. The effect of having an LO frequency offset can be explained by Fourier Transform theory. The LO offset can be expressed mathematically by multiplying the received time-domain signal by a complex exponential whose frequency is equal to the LO offset amount. Recall from Fourier Transform theory that multiplication by a complex exponential in time is equivalent to a shift in frequency. The LO offset results in a frequency shift of the received signal spectrum. This shift causes a condition called "loss of orthogonality" to occur. The frequency shift causes the OFDM subcarriers to no longer be orthogonal. The orthogonality of the subcarriers is lost because the bins of the FFT will no longer line up with the peaks of the received signal's since pulses. The result is a distortion called inter-bin interference or IBI. IBI occurs when energy from one bin spills over into adjacent bins and this energy distorts the affected subcarriers. In Fourier Transform theory this effect is called DFT leakage.

The left plot of Figure 8 shows the spectrum of a received OFDM signal with no LO offset. For the purpose of clarity, only one non-zero subcarrier was transmitted. Note that this subcarrier is not interfering with its adjacent subcarriers. The spectrum of the non-zero subcarrier actually extends over the entire range of the FFT, however, due to the orthogonal nature of the signal, the zero-crossings of the spectrum exactly line up with the other FFT bins. The right plot of Figure 8 shows the received spectrum of the same signal with one non-zero subcarrier, however, in this case there is an LO offset. This offset has resulted in a loss of orthogonality, and the zero-crossings of the non-zero subcarrier's spectrum no longer line up with the FFT bins. The result is that energy from the non-zero subcarrier is spread out among all of the other subcarriers, with those subcarriers closest to the non-zero subcarrier receiving the most interference. This simple example was for the case of only one non-zero subcarrier. In a practical system, almost all of the subcarriers would be actively used for transmitting data. A given subcarrier would experience IBI due to energy from all of the other active subcarriers in the system. The central limit theorem states that the sum of a large number of random processes will result in a signal that has a Gaussian distribution. Because of this property, the IBI will manifest itself as additive Gaussian noise, thus lowering the effective SNR of the system.

The effect of an LO frequency offset can be corrected by multiplying the signal by a correction factor. The correction factor would be a sinusoid with a frequency that is ideally equal to the amount of the LO frequency offset. Various carrier tracking algorithms exist that can adaptively determine the frequency that will correct for the offset.

LO phase offset

It is also possible to have an LO phase offset, separate from an LO frequency offset. The two offsets can occur in conjunction or one or the other can be present by itself. As the name suggests, an LO phase offset occurs when there is a difference between the phase of the LO output and the phase of the received signal. This effect can be represented mathematically by multiplying the time-domain signal by a complex exponential with a constant phase. The result is a constant phase rotation for all of the subcarriers in the frequency-domain. The constellation points for each subcarrier experience the same degree of rotation. If the phase rotation is small, the frequency-domain equalizer can correct this effect. Each filter coefficient in a frequency-domain equalizer multiplies its corresponding subcarrier by a complex gain (i.e., amplitude scaling and phase rotation). The equalizer's coefficients can be used to correct for a small phase rotation as long as the rotation doesn't cause the constellation points to rotate beyond the symbol decision regions. Larger phase rotations are corrected by a carrier tracking loop.

FFT window location offset

Another non-ideal effect that can occur in a real-world OFDM system is an FFT window location offset. An N-point FFT at the receiver processes data in blocks of N samples at a time. Ideally, the N samples taken in by the FFT will correspond to the N samples of a single transmitted OFDM symbol. In practice, a correlation is often used with a known preamble sequence located at the beginning of the transmission. This correlation operation aids the receiver in synchronizing itself with the received signal's OFDM symbol boundaries. However, inaccuracies still remain, and they manifest themselves as an offset in the FFT window location. The result is that the N samples sent to the FFT will not line up exactly with the corresponding OFDM symbol. If the offset is very large, part of the N samples will be from one OFDM symbol, and the rest of samples will be from another OFDM symbol. Such a situation would result in a severe distortion of the received subcarrier's constellations. Fortunately, such a large offset does not typically occur if a robust synchronization algorithm is used. More likely, an FFT window location offset of just a few samples will occur. The presence of the cyclic prefix gives enough headroom to enable a small offset to be present without taking samples from more than one OFDM symbol. However, even an offset of just one sample will cause some degree of distortion. Again, the effect can be understood from Fourier Transform theory. The offset can be viewed as a shift in time. As long as the FFT window location offset does not go beyond an OFDM symbol boundary, this shift in time is equivalent to a linearly-increasing phase rotation in the frequency-domain constellations. Constellations on subcarriers corresponding to low frequencies will be rotated slightly, whereas constellations on higher-frequency subcarriers will experience a larger rotation. The amount of rotation increases linearly as the subcarrier's FFT bin location increases. Examples of the effects of different degrees of FFT window location offsets are shown in Figure 9.