COFDM

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 COFDM

The modulation scheme that DAB uses is Coded Orthogonal Frequency Division Multiplexing (COFDM). COFDM uses a very different method of transmission to older digital radio modulation schemes and has been specifically designed to combat the effects of multipath interference for mobile receivers.

 

Multipath

is the term for the different paths that a signal takes in reaching an aerial from the transmitter. For example, one path may be a line-of-sight path from the transmitter to the aerial whereas another path may bounce off a hill or building before reaching the aerial. In this example, the signal that travels along the line-of-sight path arrives at the aerial first followed a short period later by the path that has bounced off the hill or building.

As the different paths travelled are of different length the time taken for the signal to reach the receiver will be different, with the direct path (if there is one) reaching the receiver first, followed by reflected paths. The effect that these multipaths have on the received signal at the antenna is that the amplitude of the received signal fluctuates. The reason for this fluctuation is due to the relative phase angle between the different paths. The received signal is very-high frequency sinusoidal carrier signal with comparatively a very slowly changing information signal that has been modulated onto the carrier. Therefore, a good way to model a carrier signal is to ignore the low frequency modulating signal and just assume that the multipaths are each high frequency sinusoids with different amplitudes due to the different distances covered (the amplitude reduces the further it travels) and relative phase angle due to the different delay. To find out the instantaneous amplitude that is received at the antenna a vector diagram can be drawn on which each multipath is represented by its amplitude (the length of the vector) and its phase angle relative to, say, the phase angle of the direct path (which gives the vector's direction). An example of a vector diagram is given below (ignore the N and E)

Ignoring the labels on the above diagram, the diagram could represent a two-path signal where the direct path is the pink vector and the sky blue vector is the delayed path, and the vector addition produces the red vector, and it is the resultant red vector that the receiver actually "sees".

As a mobile receiver moves relative to the transmitter the distances travelled by the paths also changes and because the wavelength of a radio signal is of the order of 3 metres for VHF FM signals and about 1.5 metres for DAB signals in Band III the relative phase angles between the paths changes rapidly and randomly. For example, if there were two multipaths that are in-phase (zero relative phase difference) then one of the paths only has to travel half a wavelength further than the other (about 75 cm for Band III DAB signals) for the relative phase of the path to change by 180⁰. If you look at the vector diagram above, if the blue vector had a relative phase of 180⁰ and a length equal to the pink vector then it would be facing in the opposite direction to the pink vector so the pink and sky blue vectors would completely cancel one another and the length of the resultant red vector would be zero. As I explained above, the antenna "sees" the red vector, so the amplitude that the antenna sees is also zero. The term for this in physics is "destructive interference" and the signal is said to be in a "deep fade".

Deep fades occur more frequently the faster the mobile is travelling, but the duration that the signal is in a deep fades decreases as the speed of the mobile increases. A typical graph of the amplitude of the carrier signal that the mobile antenna sees as it travels is shown below:

 

 

Wideband & Narrowband Wireless Transmission

The effect of multipath fading in the frequency domain is that wideband signals suffer from "frequency selective fading", which means that different parts of the spectrum are faded more than others. Narrowband signals on the other hand suffer from "flat fading" where the whole signal spectrum fades, so for example, a narrowband signal's spectrum would be multiplied by the above graph, which would mean that for example after travelling about 2.7 metres, destructive interference occurs and the whole spectrum will fade, hence the term 'flat fading'.

Whether a wireless digital communication system is wideband or narrowband depends on the duration of the transmitted symbols over the mobile channel. The mobile channel can be represented by what is called a power delay profile, which shows the received power after the transmission of a very short pulse, called an impulse, and the power of the signal received varies with time due to the different multipaths that arrive at the receiver. The duration from the first received path to the last received path that has significant power gives the maximum delay of the channel. A typical power delay profile varies between approximately 4 µs for urban environments up to about 20 µs for a rural environment.

A wireless digital communication system transmits "symbols" through the channel, for example, for a single-carrier binary phase shift keying (BPSK, which uses either 0⁰ phase angle or 180⁰ phase angles, and a carrier phase of 0⁰ represents a bit value of 0, and a carrier phase of 180⁰ represents a bit value of 1, so each transmitted "symbol" represents one bit of data) modulation scheme then the symbol duration is the duration between when the phase angles can change.

And a wireless digital communication system uses narrowband transmission if the channel symbol duration is greater than the maximum delay of the mobile channel (e.g. 4 µs for urban and about 20 µs for a rural environment) and the system is wideband otherwise.

In a digital wireless communication system, the bit errors are far more likely to occur when the signal is in a deep fade. Therefore these systems must mitigate the negative effects that multipath causes and different systems go about it in different ways. The two best known modern wireless digital communication transmission schemes are CDMA and OFDM. CDMA is used on the new 3G mobile phone system and is a wideband transmission scheme, which means that the channel symbols (which are called chips for CDMA) are far shorter than the maximum delay of the mobile channel. OFDM, as used on DAB and Freeview actually uses narrowband channels (subcarriers), but there are many of these narrowband channels transmitted in parallel, so the overall spectrum is wide (but this doesn't mean that it uses wideband transmission principles).

 

Error Correction Coding

The result of OFDM using a large number of narrowband subcarriers is that each subcarrier suffers from flat fading, as described above. Because the subcarriers are subject to flat fading, DAB uses COFDM (coded OFDM) which means that the data transmitted on the subcarriers is protected by forward error correction (FEC) coding. The type of error correction coding that is used in COFDM is convolutional coding and the effect of convolutional coding is that for every one bit input to the error correction encoder, more than one bit is output depending on the "code rate" being used. For example, a code rate of 1/3 would mean that for every bit input to the error correction encoder, 3 bits will be output and these 3 bits are transmitted. Error correction coding therefore adds redundancy to the signal in order for the receiver to be able to correct any bits that are received in error. The error correction decoder used in COFDM is the Viterbi algorithm which tries to decode what bits were sent depending on the received sampled values.

COFDM also allows different groups of bits to be protected with a different strength code rate because some bits are more important for the correct reproduction of the audio than some of the other bits. For example, important parameters in the MPEG audio stream are the filter parameters, so these are coded with a lower code rate (a lower code rate provides higher protection as more redundancy is added) so that the Viterbi error correction decoder has a higher chance of correcting any errors.

 

Interleaving

Unfortunately, the Viterbi algorithm performs poorly when it is presented with bit errors that are all bunched together in the stream, and because the subcarriers are subject to flat fading bit errors usually do occur in groups when a subcarrier is in a deep fade. To protect against this, DAB uses time interleaving and frequency interleaving.

An example of how time interleaving is used is shown in the above table. The data symbols are written into the interleaving block in column order, then once the block is full, the symbols are read out in row order, so for example the symbols would be read out in the following order: 0, 8, 16, 24, 32, 1, 9, 17 and so on.

At the receiver, the received symbols are written into the same sized interleaving block in row order, and once the block is full, the symbols are read out in column order to return the symbols to the original order.

The effect of this is to spread out symbol errors that occur grouped together. For example, as the first few symbols transmitted in the above table would be 0, 8, 16, 24 and so on, and if a deep fade occurs which makes symbols 8 and 16 to be received in error, then because of the re-ordering carried out in the receiver the errors end up spread out in time, which allows the Viterbi decoder to have a better chance of correcting all of the symbols.

As I explained in the Wideband & Narrowband Wireless Transmission section, wideband wireless signals are subject to frequency selective fading, and because the number of subcarriers used is large (for example DAB transmissions in the UK use Transmission Mode 1, which uses 1536 subcarriers each with a bandwidth of 1kHz), the overall DAB signal spectrum is wideband, so not only are the narrowband subcarriers subject to flat fading, the spectrum as a whole is subject to frequency selective fading. The result of this is that groups of neighbouring subcarriers may all be faded. To mitigate against this, DAB uses frequency interleaving as well as time interleaving so that after the time interleaver, the symbols read out are put on subcarriers that are a certain distance in frequency apart. Again, the receiver reverses this and the overall effect is that the Viterbi decoder sees the data symbols in the original order, but errors are uniformly spread out in the stream.

Interleaving is a powerful method to improve the error correction capabilities of a wireless system that is subject to fading, but of course it cannot perform miracles, and if too many symbols are decoded incorrectly then it will fail and you're then likely to hear the usual "bubbling mud" sound that is characteristic of reception problems.

 

COFDM Transmitter

 

 

After the bit-stream is re-ordered in the time interleaver block, 3072 bits (for Transmission Mode 1 which uses 1536 subcarriers) enter the OFDM modulator. The bit-stream is first split up into 1536 pairs of bits and each pair is mapped to one of four quaternary phase shift keying (QPSK) symbols. 

DAB uses differential QPSK, which means that the bits are mapped to phase changes rather than to an absolute transmitted phase. An example mapping might be as follows:

Data Bits	Phase Change (degrees)
00	0
01	90
11	180
10	270

The above mapping is called a Gray code mapping, because adjacent symbols (or in this case phase changes) only differ by the value of one bit, which lowers the probability of there being two bit errors for one symbol.

After the 1536 pairs of bits have been mapped to one of the four phase changes these phase changes are applied to the 1536 subcarriers. The previously transmitted QPSK symbol on each subcarrier will be placed in memory in the transmitter, and the phase change will then rotate this symbol. For example, if the previous transmitted symbol on a subcarrier was the right-hand point on the horizontal axis in the figure below (called a signal constellation diagram) and the bits that are being mapped onto the subcarrier are '11' then the phase will rotate by 180⁰ so that the left-hand point on the horizontal axis will be transmitted on that subcarrier.

 

 

The QPSK symbols shown in the signal constellation diagram above are represented numerically by their co-ordinates on the diagram. The 'Re' axis is the 'real' axis and the 'Im' axis is the so-called 'imaginary' axis, which are the terms for diagrams that display what are called 'complex numbers'. A complex number consists of the combination of a real plus an imaginary number:

I + j Q

where I is the real part of the complex number and Q is the imaginary part of the complex number, and the 'j' is always multiplied by the imaginary number. The actual meaning of 'j' is that it is equal to the square-root of -1, which doesn't actually exist, and that is why it is called an imaginary number, but complex numbers are a very useful mathematical concept and the fact that the imaginary number doesn't actually exist doesn't matter.

To read an excellent tutorial about complex numbers and their use in digital signal processing download this Acrobat file: http://www.dspguru.com/info/tutor/QuadSignals.pdf (136 KB).

The transmitted symbols have the following (normalized) co-ordinates:

 

Rectangular Co-ordinate	Carrier Phase (degrees)
1 + j0	0
0 + j1	90
-1 + j0	180
0 - j1	270

Complex numbers are used to represent signal points on a constellation diagram because the real and imaginary axes are at 90⁰ apart and a sinewave and a cosine wave (both with the same frequency) are also 90⁰ out of phase. This allows real number co-ordinates represent the amplitude of a cosine wave, and an imaginary number represent the amplitude of the sinewave, then adding the amplitude modulated sinewave and cosine wave together forms a 'quadrature' signal.

For example, COFDM is also used as the transmission scheme for DVB-T (Freeview) which has the option of QPSK, 16-QAM and 64-QAM signal constellations to modulate the subcarriers. QAM stands for quadrature amplitude modulation and to generate one of the signal points on the constellation you amplitude modulate the cosine wave and the sinewave with the co-ordinates of the point on the signal constellation and then add the cosine wave and the sinewave together and the resultant signal is an amplitude and phase modulated signal, which is beneficial because you don't have to phase and amplitude modulate:

 

 

The benefit of using 16-QAM or 64-QAM is that each symbol on each subcarrier can carry more bits of information. The number of bits that each symbol can carry is given by the following equation:

number of bits  =  log₂ M

where log₂ is the logarithm to the base 2 and M is the order of the constellation. So QPSK symbols (M=4) can carry 2 bits of information, 16-QAM symbols (M=16) can carry 4 bits of information, and 64-QAM symbols (M=64) can carry 6 bits of information.

Of course, it is better to use a higher level constellation so that the overall capacity can be higher, but the drawback is that the points are closer together which makes the transmission less robust to errors. As explained earlier, fading alters both the amplitude and phase of a carrier or subcarrier, and in the mobile channel the frequency of the subcarriers are altered by a Doppler shift. Also, thermal noise produced by devices in the receiver such as the RF mixer is added to the received signal, and it is this noise that is used in the signal to noise ratio (SNR) calculations.

The reason why most symbol errors occur when the signal is in a deep fade can be explained using the following diagram which shows how the thermal noise moves the signal point:

 

 

On DAB (using differential QPSK), if a symbol is transmitted and the subcarrier is in a deep fade then the amplitude of the subcarrier is reduced. This moves the received signal point closer to the origin of the diagram (co-ordinates of 0,0) and when noise is added to this in the receiver's RF front end then because the point is already near the origin then it is easy for the noise to move the point to a position where the difference in the angle does not fall within the decision region allowed for a correct decision.

 

OFDM Modulator

 

 

After the symbol mapping is carried out, as explained above, the frequency interleaving will re-order the symbols (not shown in diagram) and then the 1536 complex numbers that represent the symbols to be transmitted on each of the subcarriers will be sent to a serial-to-parallel converter and "placed" on each of the subcarriers. As all of this is done in the digital domain then the above diagram just serves as a way to visualise what happens. In reality the 1536 complex numbers will be stored in two buffers, with one buffer containing the real values of the complex number, and the other buffer containing the imaginary values of the complex numbers.

The OFDM modulator consists of the block in the diagram that is labelled 'IDFT', which stands for inverse discrete Fourier transform. Again, in reality, the actual process carried out is the inverse fast Fourier transform (IFFT), because the IFFT is, as the name suggest, a fast way to calculate the IDFT.

The IDFT calculates the following equation:

x(n) is the n^th output signal complex value (time domain), X(k) is the complex symbol value on the k^th subcarrier (frequency domain), and (for DAB transmission mode TM1) N = 2048 is the number of output signal points calculated, and also the number of input frequency points. 

The equation is a summation from 0 to N-1 for each output value x(n), X(k).e ^{j.2.k.pi.n / N} is summed from k=0 to k=N-1. For example, for x(2) the sum would be:

x(2) = X(0) e ^{j.0.2.pi.2 / N} + X(1) e ^{j.1.2.pi.2 / N} + X(2) e ^{j.2.2.pi.2 / N} + X(3) e ^{j.3.2.pi.2 / N} +  X(4) e ^{j.4.2.pi.2 / N} +    ..........

To understand what the IDFT does, you first need to understand what the discrete Fourier transform (DFT) does for which the IDFT is the inverse. The DFT calculates the discrete frequency spectrum from a block of discrete time samples of the signal (by 'discrete' I mean that a discrete signal or discrete spectrum is only defined at discrete moments of time, e.g. at the sampling instant for a time signal, or at a given frequency for a frequency spectrum). Therefore, the inverse DFT calculates the discrete time samples from a discrete frequency spectrum. This means that the frequency spectrum of the transmitted signal is given by the values of the complex data symbols on the subcarriers.

There are a lot of redundant operations in the DFT, and for an N-point DFT this requires N² complex multiplications, which for example for a 2048 point DFT as would be used for transmission mode 1 this would require 4,194,304 multiplications. The fast Fourier transform (FFT) is, as its name suggest, a fast way to calculate the DFT as many of the redundant operations are discarded, and this allows the FFT to be calculated in (N/2) log₂ N multiplications, which for a 2048 point FFT requires only 11,264 multiplications, which is a massive saving compared to the DFT.

One of the properties of the DFT is what makes it suitable for OFDM, and really what makes OFDM feasible for practicaly implementation in the first place. This property is that the discrete frequency spectrum that is calculated by a DFT from a block of data samples has frequency samples that are all equally spaced in frequency, and this spacing equals 1/T, where T is the total duration of the time samples in the block. For example, for DAB transmission mode 1 (TM1), the "useful" duration of OFDM symbols (not data symbols on the subcarriers, OFDM symbols carry the data symbols on the subcarriers) is 1 ms (i.e. T = 1 ms), so 1/T = 1 kHz, and all the subcarriers are spaced by 1kHz. It is these equally spaced subcarriers that equal the useful symbol duration that gives OFDM its "orthogonal" property in its name orthogonal frequency division multiplexing.

The property of orthogonality for communication signals means that signals that are orthogonal to each other can be transmitted together and they don't interfere with each other. So having the subcarriers all orthogonal to one another (each subcarrier is orthogonal to all the other 1535 subcarriers) means that you can transmit the subcarriers in parallel and they won't interfere with each other. This means that the individual spectra for each of the subcarriers can overlap, and they still won't interfere with one another. A diagram that shows what the frequency spectra of subcarriers looks like for DAB is shown below, and the number of subcarriers for TM1 will be 1536:

:

As you can see from the figure above, for the frequency in red, all the 4 neighbouring spectra are zero where the red spectra is at its peak, and so there is no "intercarrier interference"; this is due to the orthogonality principle.

The reason why the DFT makes OFDM practically feasible is that if you want to transmit 1536 subcarriers that are all orthogonal to each other then you would need 1536 oscillators which are all separated by 1kHz and 1536 filters at the transmitter, and 1536 filters and oscillators in each receiver, which is obviously not practical.

After the IFFT has been calculated, the 1536 output complex numbers are parallel to serial converted (the P/S block in the diagram above), and following this the cyclic prefix (or guard period) is inserted (see diagram at the start of the COFDM transmitter).

The cyclic prefix copies the complex numbers from the end of the block of output values and "pastes" them onto the front of the block (or from the front of the block copied to the end). The reason why the values from the end of the block are copied to the front is to retain orthogonality in the multipath channel.

The reason why the end of the block is copied to the front is so that the delayed paths from the symbol fall within the guard period. To show why this retains orthogonality you have to consider that the OFDM signal consists of the addition of all the subcarrier signals, which are all at different frequencies f₀ and with different values of a_n and b_n as shown in the equation and the waveforms that are added to make the bottom OFDM signal:

 

Then, so long as all the multipaths fall within the cyclic prefix duration and if samples are taken over the "useful" symbol duration (as opposed to the total symbol duration that includes the cyclic prefix) then the DFT equation that is calculated in the receiver "integrates" over an integer number of full sinewave cycles, which is a requirement for orthogonality to hold.

Following the insertion of the cyclic prefix, the values are fed to digital to analogue converters (DAC) and lowpass filters for each of the real and imaginary streams. The real values of the complex numbers are then amplitude modulated onto a cosine RF (radio frequency, i.e. about 210 MHz for Band III) carrier, and the imaginary values of the complex numbers are amplitude modulated onto a sine RF carrier. The sine and cosine carriers are then added together, and sent through a bandpass filter and then sent to the antenna for transmission.

The insertion of the guard period between the useful symbols also enables DAB to use single-frequency networks (SFNs):

Using a cyclic prefix means that receivers can receive signals on the same frequency from different transmitters so long as the delay between the first and last signal to arrive falls within the cyclic prefix duration. So signals from transmitters whose signals are delayed relative to the signals from a closer transmitter are treated as "artificial" multipath.

SFNs allow the same frequency to be used for a given area and this means that a few low power transmitters can be used as opposed to having one very high power transmitter. Overall, the power required using the SFN concept is lower for transmitting to a given area. SFNs are also spectrally efficient when it comes to frequency planning because for example, both the BBC and Digital One use the same frequency right across the UK, so the situation where there are multiple frequencies required is avoided.

I've found that there is a common misconception that only the BBC and Digital One multiplexes use the SFN concept. This is not so, and all DAB multiplexes that have more than one transmitter for a given area use the SFN concept, and this is the vast majority of multiplexes that I'm aware of in the UK.

 

COFDM Receiver

 

 

After the signals are received at the antenna, the signals are I/Q downconverted from RF to generate the real (I) and imaginary (Q) streams, lowpass filtered (LPF) and digitized in the analogue to digital converters (ADC, one ADC for each stream). Following the ADC, the cyclic prefix is stripped off and the remaining sampled values are serial to parallel converted and once there is a full block of samples (1536 for TM1) the DFT is calculated (in reality the FFT is calculated as the FFT requires far fewer multiplications to be carried out than the DFT).

After the FFT (the FFT is the OFDM demodulator), the originally transmitted symbols will be received, but they will be corrupted in that the amplitude and phase will be altered by the channel response for each subcarrier, and noise will be added in the receiver which moves the received point in a random direction and with a random amplitude.

As DAB uses differential modulation, only the difference in phase between the previous and present symbol on each subcarrier needs to be found to decode what was sent (ignoring errors).

The phase angle of a complex number can be found from the following formula:

theta = tan^-1 (I / Q)

To find the phase difference between the previous and present symbol the complex conjugate of the previous received point is multiplied by the present received point, then the angle of the result of this multiplication is the phase change. The complex conjugate of a complex number just changes the sign of the imaginary part, for example, if you have 1 + j2, then its complex conjugate is 1 - j2.

Unfortunately, when DAB was specified in 1991 the engineers decided to use differential modulation instead of coherent (or synchronised) modulation. Synchronised modulation means that the absolute phase of the symbol is transmitted, rather than the difference between phases. In 1991 differential modulation may have been seen to be a good choice, but synchronised modulation will be used in all modern communication systems because it is easy to synchronise the carrier, and differential modulation doubles the number of bit errors compared to synchronised modulation. The reason for why the number of bit errors are doubled is because if one received symbol has been rotated by the channel or by noise to an extent that it causes an error, then because the probability of a bit error is low, there is a very high probability that the following symbol is also received in error.

For example, for a typical probability of error of about 0.0001, if one error occurs then the probability that the following symbol is in error is 1-0.0001 = 0.9999, i.e. virtually certain, so overall differential modulation doubles the number of bit errors.

Following the determination of the change of phase on each of the subcarriers, first the frequency interleaving is reversed and then the time interleaving is reversed, and the values are fed into the Viterbi error correction decoder.

The output bitstream from the Viterbi decoder is then forwarded to software or hardware that goes about splitting the multiplexed data into its constituent streams followed by sending the audio data to the MPEG decoder to generate the PCM bitstream that is sent to the DACs, amplified and sent to the speakers.