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The simplest way to describe an orthogonal frequency-division multiplexing (OFDM) signal is as a set of closely spaced frequency-division multiplexed (FDM) carriers. While this is a good starting point for those unfamiliar with the technology, it falls short as a model for analyzing the effects of signal impairments.

The reason it falls short is that the carriers are more than closely spaced; they are heavily overlapped. In a perfect OFDM signal, the orthogonality property prevents interference between overlapping carriers. This is different from the FDM systems we’re all familiar with. In FDM systems, any overlap in the spectrums of adjacent signals will result in interference. In OFDM systems, the carriers will interfere with each other only if there is a loss of orthogonality. So long as orthogonality can be maintained, the carriers can be heavily overlapped, allowing increased spectral efficiency.

This article will address modulator and demodulator impairments. It will discuss how these impairments affect OFDM systems and, where appropriate, how this is different from the effect on single-carrier modulation formats.Many theoretical papers already exist that discuss impairments and their impact on bit error rate (BER). This article will instead attempt to discuss how an impairment introduces error into a signal — something that could be infinitely more useful when performing real-world RF design and troubleshooting.

Table 1 lists a variety of common analog signal impairments and their effects on both OFDM signals and the more familiar single-carrier modulations such as quadrature phase-shift keying (QPSK) or 64-QAM (quadrature amplitude modulation). Most of these impairments can occur in either the transmitter or the receiver.

IQ imperfections

For cost reasons, analog in-phase and quadrature (I/Q) modulators and demodulators are often used in transceivers — especially for wide bandwidth signals. Being analog, these I/Q modulators and demodulators usually have imperfections that result in an imperfect match between the two baseband analog signals, I and Q, which represent the complex carrier. For example, gain mismatch might cause the I signal to be slightly smaller than the Q. In a single-carrier modulation system, this results in a visible distortion in the constellation— the square constellation of a 64-QAM signal would become rectangular.

To better understand how gain imbalance will affect an OFDM signal, look at the equations describing each individual subcarrier. In the following analysis, it’s important to keep in mind that, while we are analyzing individual subcarriers,the IQ gain imbalance error is on the signal that is the composite of all subcarriers. In Equation 1, Ck,m is a complex number representing the location of the symbol within the constellation for the kth subcarrier at the mth symbol time. For example, if subcarrier k is binary-phaseshift-keying (BPSK) modulated, then Ck,m might take on values of ±1+j0. The complex exponential portion of Equation 1 represents the kth subcarrier,which is amplitude- and phase-modulated by the symbol Ck,m. Therefore:(1)Using Euler’s relation, Equation 1 can be rewrittenas:(2)Now add the term “β” to represent gain imbalance.For a perfect signal, set β = 0. As shown, the gain imbalance term will also produce a gain change.This was done to simplify the analysis. Therefore:

The equation can be rearranged and this can be rewritten as the sum of aperfect signal and an error signal:(4) Finally, converting back into complex exponential notation, we get: (5) In words, Equation 5 shows that again imbalance produces two error terms. The first error term is at the frequency of the kth subcarrier. The second error term is at the frequency of the–kth subcarrier. The phase and magnitude of the error terms are proportional to the symbol being transmitted on the kth subcarrier. Another way of saying this is that I/Q gain imbalance will result in each subcarrier being interfered with by its frequency mirror image subcarrier. If you’re familiar with sideband modulation, you’ll instantly recognize this as imperfect sideband cancelation. Equation 5 has several implications.

First, it is generally true that for subcarriers used to carry data (as opposed to pilots), the symbol being transmitted at any given time on the kth subcarrier is uncorrelated to the symbol on the–kth subcarrier.

For a given subcarrier, the lack of correlation from the mirror-image subcarrier implies a certain randomness to the error. This results in a spreading of the subcarrier’s constellation states in a noise-like fashion. This is especially true for higher-order modulations such as 64-QAM. For lower-order modulations,such as BPSK, the error term from the mirror-image carrier has fewer states.

This can result in constellations as shown in Figure 2, where the BPSKpilot carriers of an 802.11a signal exhibit spreading that does not appear noise like.Also, as the BPSK pilots do nothave an imaginary component; the error terms associated with the pilot subcarriers are real — so the spreading is only along the real (I) axis. Note that the phase relationships between the pilot carriers in an 802.11a system are highly correlated, so the errors introduced by quadrature errors are not random.

(for the remainder of this article, including diagrams and charts, please download the PDF above).