COFDM is particularly well-suited to the needs of the terrestrial broadcasting channel. COFDM can cope with high levels of multipath propagation, with a wide spread of delays between the received signals. This leads to the concept of single-frequency networks in which many transmitters send the same signal on the same frequency, generating “artificial multipath”. COFDM also copes well with co-channel narrowband interference, as may be caused by the carriers of existing analogue services.
COFDM has therefore been chosen for two standards for broadcasting – DAB and DVB-T, both of which have been optimized for their respective applications and have options to suit particular needs.
Systems for DAB and DVB-T have been standardized by ETSI for use in Europe and elsewhere in the world.
These systems have been designed in recognition of the circumstances in which they will be used:
• DAB was especially designed to cope with the rigours of reception in moving cars – especially the problem of multipath reception which, in this case, is time-varying;
• For DVB-T, a higher capacity than DAB was essential, mobile reception was not a priority, but multi-path tolerance was still important because of the widespread use of set-top TV antennas.
Defense and Security
COFDM is also well-suited for video links in security and defense applications such as air and ground surveillance.
As just described, the process could be called frequency interleaving. This is all that is needed if the channel only varies slowly with time, and that is why it is used in DVB-T. In mobile operation (a key application for DAB), we may expect the various paths to be subjected to different and significant Doppler shifts, making the frequency response vary with time. Furthermore, a vehicle may drive into shaded areas (such as underpasses) where all the signals are severely attenuated for a period. For this reason, in the DAB system the coded data are also re-distributed over time, to provide time interleaving.
A time-varying channel example. There are two delayed paths, each with different Doppler shift, in addition to the “main” path. The z-axis represents the magnitude of the channel response.
So far we have considered a very special example in order to make it easy to explain – by invoking the close analogy with the use of code puncturing. But what of the other delay values? If the relative delay of the echo is rather shorter than we have just considered, then the notches in the channel’s frequency response will be broader, affecting many adjacent carriers. This means that the coded data we transmit should not simply be assigned to the OFDM carriers in a sequential order, since at the receiver this would cause the Viterbi soft-decision decoder to be fed with clusters of unreliable bits. This is known to cause a serious loss of performance, which we avoid by interleaving the coded data before assigning them to OFDM carriers at the modulator.
A corresponding de-interleaver is used at the receiver before decoding. In this way, the cluster of errors occurring when adjacent carriers fail simultaneously (as when there is a broad notch in the frequency response of the channel) is broken up, enabling the Viterbi decoder to perform better.
Note that received power is shown, to which the SNRs of the carriers will be proportional if the receiver noise is itself flat, as is usual. The “mean power” marked on the diagram is the mean of all carriers. It is equal to the total received power (via both paths), shared equally between all carriers.
Although only a few COFDM carriers are illustrated, the pattern repeats cyclically for all of them. The dotted curve represents the power frequency response of the channel formed by the two paths.
In COFDM, the Viterbi metrics for each bit should be weighted according to the SNR of the carrier by which it travelled. Clearly, the bits from the nulled carriers are effectively flagged as having “no confidence”. This is essentially the same thing as an erasure – the Viterbi decoder in effect just records that it has no information about these bits.
There is another well-known case of regularly-occurring erasures, namely punctured codes. Typically, convolutional codes intrinsically have code rates expressed as simple fractions such as 1/2 or 1/3. When a code having a higher rate (less redundancy) is needed, then one of these lower-rate “mother” codes is punctured, that is to say certain of the coded bits are just not transmitted, according to a regular pattern known to the receiver. At the receiver “dummy bits” are re-inserted to replace the omitted ones, but are marked as erasures – bits having zero confidence – so that the Viterbi decoder treats them accordingly. Punctured codes obviously are less powerful than the mother code, but there is an acceptable steady trade-off between performance and code rate, as the degree of puncturing is increased.
Suppose we take a rate-1/2 code and puncture it by removing 1 bit in every 4. The rate-1/2 code produces 2 coded bits for every 1 uncoded bit, and thus 4 coded bits for every 2 uncoded bits. If we puncture 1 in 4 of these coded bits, then we clearly finish by transmitting 3 coded bits for every 2 uncoded bits. In other words we have generated a rate-2/3 code. Indeed, this is exactly how the rate-2/3 option of DVB-T is made.
Now return to our simple COFDM example in which 1 carrier in 4 is nulled out by the channel– but the corresponding bits are effectively flagged as erasures, thanks to the application of channel-state information. 2 out of 3 of the remaining carriers are received at the same SNR as that of the overall channel, while 1 carrier is actually boosted, having an improved SNR. Suppose that rate-1/2 coding is used for the COFDM signal. It follows that the SNR performance of COFDM with this selective channel should be very slightly better (because 1 carrier in 4 is boosted) than that for a single-carrier (SC) system using the corresponding punctured rate-2/3 code in a flat channel. In other words, the effect of this very selective channel on COFDM can be directly estimated from knowledge of the behaviour of puncturing the same code when used in an SC system through a flat channel. This explains how the penalty in the required CNR, for a COFDM system subject to 0 dB echoes, may be quite small – provided a relatively powerful convolutional code is used together with the application of channel-state information.
We now return to the simple example in which there is a 0 dB echo, of such a delay (and phase relationship) as to cause a complete null on 1 carrier in every 4. The figure illustrates the effect of this selective channel: 1 carrier in every 4 is nulled out, while another carrier in every 4 is actually boosted, and the remaining two are unaffected.
Channel State Information (CSI)
Use of error coding
Why do we need error coding?
In fact, we would expect to use forward error-correction coding in almost any practical digital communication system, in order to be able to deliver an acceptable Bit-Error Ratio (BER) at a reasonably low Signal-to-Noise Ratio (SNR). At a high SNR it might not be necessary – and this is also true for un coded OFDM, but only when the channel is relatively flat. Un coded OFDM does not perform very well in a selective channel. Its performance could be evaluated for any selective channel and for any modulation scheme, by:
• Noting the SNR for each carrier;
• Deducing the corresponding BER for each carrier’s data;
• Obtaining the BER for the whole data signal, by averaging the BERs of all the carriers used.
Very simple examples will show the point. Clearly, if there is a 0 dB echo which is delayed such that every mth carrier is completely extinguished, then the “symbol” error ratio (SER) will be of the order of 1 in m – even at infinite SNR. (Here, “symbol” denotes the group of bits carried by one carrier within one OFDM symbol). An echo delay of say Tu/4– the maximum for which a loss of orthogonality is avoided when the guard-interval fraction is 1/4 (as in DAB and some modes of DVB-T) – would thus cause the SER to be 1 in 4. Similarly, if there is one carrier, amongst N carriers in all, which is badly affected by interference, then the SER will be of the order of 1 in N, even with infinite SNR. This tells us two things:
• Un coded OFDM is not satisfactory for use in such extremely selective channels;
• For any reasonable number of carriers, CW interference that is affecting one carrier is less of a problem than a 0 dB echo.
However, just adding hard-decision-based coding to this uncoded system is not enough, either – it would take a remarkably powerful hard-decision code to cope with an SER of 1 in 4! The solution is to use convolutional coding in conjunction with soft-decision decoding, properly integrated with the OFDM system.
Soft decisions and channel-state information
First let us revise, for simplicity, 2-level modulation of a single carrier: one bit is transmitted per symbol with, say, a “0” being sent by a modulating signal of – 1 V and a “1” by + 1 V. At a receiver, assuming that the gain is correct, we should expect to demodulate a signal always in the vicinity of either – 1 V or + 1 V, depending on whether a “0” or a “1” was transmitted. Any departure from the exact values ± 1 V would have been caused by the inevitable noise added during transmission.
A hard-decision receiver would operate according to the rule that negative signals should be decoded as “0” and positive ones as “1”, with 0 V being the decision boundary. If the instantaneous amplitude of the noise were never to exceed ± 1 V, then this simple receiver would make no mistakes. But noise may occasionally have large amplitude, although with lower probability than for smaller values. Thus if say + 0.5 V is received, it most probably means that a “1” was transmitted, but there is a smaller yet still finite probability that actually “0” was sent. Common sense suggests that when a large-amplitude signal is received we can be more confident in the hard decision, than if the amplitude is small.
This view of a degree of confidence is exploited in soft-decision Viterbi decoders. These maintain a history of many possible transmitted sequences, building up a view of their relative likelihoods and finally selecting the value “0” or “1” for each bit, according to which has the maximum likelihood. For convenience, a Viterbi decoder adds logarithmic likelihoods (rather than multiplying probabilities) to accumulate the likelihood of each possible sequence. It can be shown that, in the case of BPSK or QPSK, the appropriate log-likelihood measure, or metric, of the certainty of each decision is indeed simply proportional to the distance from the decision boundary. The slope of this linear relationship itself also depends directly on the signal to- noise ratio. Thus the Viterbi decoder is fed with a soft decision comprising both the hard decision (the sign of the signal) together with a measure of the amplitude of the received signal.
With other rectangular-constellation modulation systems, such as 16-QAM or 64-QAM, each axis carries more than one bit, usually with Gray coding. At the receiver, a soft decision can be made separately for each received bit. The metric functions are now more complicated than for QPSK, being different for each bit, but the principle – the decoder exploits its knowledge of the expected reliability of each bit – still remains.
Metrics for COFDM are slightly more complicated. We start from the understanding that the soft-decision information is a measure of the confidence to be placed in the accompanying hard decision.
When data are modulated onto a single carrier in a time-invariant system, then a priori all data symbols suffer from the same noise power on average; the soft-decision information simply needs to take note of the random symbol-by-symbol variations that this noise causes.
When data are modulated onto the multiple COFDM carriers, the metrics become slightly more complicated as the various carriers will have different signal-to-noise ratios. For example, a carrier which falls into a notch in the frequency response will comprise mostly noise; one in a peak will suffer much less. Thus, in addition to the symbol-by-symbol variations, there is another factor to take account of in the soft decisions: data conveyed by carriers having a high SNR are a priori more reliable than those conveyed by carriers having low SNR. This extra a priori information is usually known as Channel-State Information (CSI).
The CSI concept can be extended to embrace interference which affects carriers selectively.
The inclusion of channel-state information in the generation of soft decisions is the key to the unique performance of COFDM in the presence of frequency-selective fading and interference.
Coded OFDM, or COFDM, is a term used for a system in which the error control coding and OFDM modulation processes work closely together.
An important step in a COFDM system is to interleave and code the bits prior to the IFFT. This step serves the purpose of taking adjacent bits in the source data and spreading them out across multiple subcarriers.
One or more subcarriers may be lost or impaired due to a frequency null, and this loss would cause a contiguous stream of bit errors. Such a burst of errors would typically be hard to correct. The interleaving at the transmitter spreads out the contiguous bits such that the bit errors become spaced far apart in time. This spacing makes it easier for the decoder to correct the errors.
Another important step in a COFDM system is to use channel information from the equalizer to determine the reliability of the received bits. The values of the equalizer response are used to infer the strength of the received subcarriers.
For example, if the equalizer response had a large value at a certain frequency, it would correspond to a frequency null at that point in the channel. The equalizer response would have a large value at that point because it is trying to compensate for the weak received signal. This reliability information is passed on to the decoding blocks so that they can properly weight the bits when making decoding decisions.
In the case of a frequency null, the bits would be marked as “low confidence” and those bits would not be weighted as heavily as bits from a strong subcarrier.
COFDM systems are able to achieve excellent performance on frequency selective channels because of the combined benefits of multicarrier modulation and coding.