Fundamentals of OFDM

Multi-carrier transmission is one of today’s key technologies for communication systems. They use several sinusoidal waves, which are transmitted simultaneously. The basic idea of multicarrier technology is to fragment a frequency-selective channel into narrowband subchannels such that each of these subchannels becomes approximately non-selective. Furthermore, another principle of multi-carrier transmission is to convert a serial high rate data stream on to multiple parallel low rate sub-streams.

Multi Carrier Transmission

OFDM (Orthogonal Frequency Division Multiplexing) and OFDM-based transmission schemes are dominating current wireless communication standards (WLAN, LTE, DAB and DVB). This is due to the following capabilities and benefits of OFDM:

➤ Robustness against frequency selective fading with the division of information symbols in parallel narrowband channels
➤ Efficient use of spectrum due to overlapping transmission of orthogonal parallel narrowband channels
➤ Low-complexity implementation with the use of Fast Fourier Transforms (FFT)
➤ Low-complexity channel equalization compared with single-carrier solutions
➤ Robustness against intersymbol interference (ISI) with the use of cyclic prefix
➤ Robustness against impulsive noise
➤ Simple integration of multiple-input multiple-output (MIMO) systems in the OFDM transmitter/receiver chain
➤ Ability to easily integrate adaptive modulation and coding techniques to efficiently exploit the radio channel
➤ Provision of direct extension to a multiplexing scheme with orthogonal division multiple access (OFDMA) for resource sharing

All in all, compared to single-carrier modulation, OFDM main advantages include high spectral efficiency, robustness against multipath ISI, the simplicity to equalize in frequency domain and the efficiency of applying FFT. Of course, there are also disadvantages like the large Peak to Average Power Ratio, out auf band leakage and the sensitivity to imperfect time and frequency synchronization. The OFDM method is also used for wired transmission and also known as Discrete MultiTone (DMT).

In the following the most important OFDM parameters will be introduced. As an example, all the parameters will be calculated for DVB-T transmission.

OFDM Parameters

Subcarrier (DFT/FFT length)
The total number of subcarriers- or channels $N$ , consists of the number of data-subcarriers $N_{C}$ , pilot-subcarriers $N_{P}$ and null subcarriers $N_{V}$ .

$N = N_{C} + N_{P} + N_{V}$

N Total Number of Subcarriers N_{C} Data subcarriers N_{P} Pilot subcarriers N_{V} Null subcarriers

The number of subcarriers $N$ always occurs in a power of two and expresses also the length of the DFT / FFT.

Data-Subcarrier
The data subcarriers $N_{C}$ represent the number of subchannels used for data transmission. In a simple OFDM structure without virtual subcarriers, the number of subchannels is equal to the FFT length $N$ .

Pilot-Subcarrier
OFDM allows the insertion of so-called pilot tones. Pilot tones are generated by modulating individual subcarriers with specified complex symbols in a fixed time sequence. The complex values of the pilot tones are known in advance to the receiver so that an estimate of the channel inflows or a fine synchronization of the symbol clock can be carried out in the receiver. Pilot subcarriers $N_{P}$ however cannot be used for channel estimation as they are too far apart for interpolation. These Pilot subcarriers only serve the tracking of the carrier synchronization.

Null-Subcarrier
Null subcarriers $N_{V}$ are mandated in most OFDM wireless standards. These subcarriers are not occupied but serve to reduce the PAPR of multi-carrier transmission. This is achieved by reordering the null-subcarriers and data-subcarriers. In addition, they are used to prevent leakage to adjacent bands since OFDM Systems usually do not transmit any data on the subcarriers near the two edges of the assigned band. The unused subcarriers are also known as guard subcarriers or virtual subcarriers. Altogether they are called guard band.

Discrete Length of the Guard Interval
A guard interval, also known as cyclic prefix, is used to prevent certain transmissions from mixing. They increase the immunity to propagation delays, echoes, and reflections, against which digital data tends to be very vulnerable. The length of the guard interval (GI) determines how susceptible a transmission is. The longer such an interval is, the better it protects against interference, but the data rate is reduced.

$N_{G} = \frac{T_{G} \times N}{T_{S}}$

N_{G} Discrete Length of GI T_{G} GI Duration N Total Number of Subcarriers T_{S} Basic OFDM Symbol Duration

To eliminate ISI, a guard interval is usually inserted at the beginning of each OFDM symbol. In addition, it corresponds to a copy of the last seconds of the basic OFDM symbol. The main idea behind this method is to dimension the GI so large, that the signal components delayed by the channel, only disturb the signal component during the GI duration and not the basic OFDM symbol.

For highly frequency selective channels, the cyclic prefix should increase accordingly. In existing standards like LTE with extended prefix or IEEE 802.11, the cyclic prefix is ¼ of the OFDM symbol duration.

Guard Interval Duration
The exact duration of the guard interval results in:

$T_{G} = N_{G} \times t_{s} = \frac{N_{G} \times T_{s}}{N}$

N_{G} Discrete Length of GI T_{G} GI Duration N Total Number of Subcarriers T_{S} Basic OFDM Symbol Duration t_{s} Symbol Duraton

Source Symbol Duration
The source symbol duration $T_{D}$ is referred to as the duration of the symbols to be transmitted between source, channel coder and interleaver. $T_{D}$ of the serial data symbols results after serial-to-parallel conversion in the Total OFDM Symbol Duration.

$T'_{s} = N_{C} \times T_{D}$ $T_{D} = \frac{T'_{s}}{N_{C}}$

N_{C} Data Subcarriers T_{D} Source Symbol Duration T'_{s} Total OFDM Symbol Durtion

Symbol Duration
For the determination of the OFDM time raster, it is very helpful to use the symbol duration of the transmit signal, also called sampling period, before the RF modulator, as a reference. The symbol duration is as follows:

$t_{s} = \frac{T'_{s}}{N + N_{G}} = \frac{T_{s}}{N} = \frac{T_{G}}{N_{G}}$

N Total Number of Subcarriers N_{G} Discrete Length of GI t_{s} Symbol Duration T_{G} GI Duration T_{s} Basic OFDM Symbol Duration T'_{s} Total OFDM Symbol Duration

Basic OFDM Symbol Duration
The long OFDM symbol duration in OFDM systems opens a particularly elegant way to avoid ISI. As already mentioned, it can be reached by prefixing a GI. However, the Basic OFDM symbol duration $T_{s}$ is without a GI and the number of data subcarriers is equal to the total number of subcarriers.

$T_{s} = N \times t_{s}$

T_{s} Basic OFDM Symbol Duration N Total Number of Subcarriers t_{s} Symbol Duration

Total OFDM Symbol Duration
The total OFDM symbol duration $T'_{s}$ is the basic OFDM symbol duration extended by the GI.

$T'_{s} = T_{s} + T_{G} = (N + N_{G}) \times t_{s}$

N Total Number of Subcarriers N_{G} Discrete Length of GI t_{s} Symbol Duration T_{G} GI Duration T_{s} Basic OFDM Symbol Duration T'_{s} Total OFDM Symbol Duration

Bit Rate
The bit rate $R_{B, b r u t t o}$ of the OFDM system measures the amount of transmissible messages within a time interval. If the message set is being quantified with the unit bit, the term bit rate is used. If the same modulation alphabet is used for all subcarriers and a transmission with a GI is assumed, then the bit rate is calculated as follows:

$R_{B, b r u t t o} = \frac{N_{C} \times n_{i}}{T'_{s}}$

$R_{B, n e t t o} = R_{C} \times R_{B, b r u t t o}$

N_{C} Data Subcarriers T'_{s} Total OFDM Symbol Duration n_{i} Bit per Symbol R_{C} Coderate

Bit per Symbol
In a multi-level transmission, a group of $n$ bits are combined into one character (symbol) and transmitted within a signal step of the duration $T_{s}$ . Indeed, there is a correspondance between the number $n$ of bits transmitted per signal step and the required number of steps $M$ of a digital signal.

$M = 2^{n}$ $n = l d (M)$

n Bit per symbol M Number of different modulation symbols

Code Rate
The code rate $R_{C}$ of an optimal, theoretically possible code of infinite length, also referred to as the channel capacity of the binary symmetric channel, can be calculated as follows:

$R_{C} = R_{o p t} = C_{b i n} = 1 + p \times \log_{2} \times p + (1 - p) \times \log_{2} \times (1 - p)$

R_{C} Code Rate p Bit Error Rate C_{b i n} Channel Capacity Binary R_{o p t} Optimal Code Rate

Symbol Rate
The Symbol rate $f_{s}$ is also known as baud rate within terms of digital communications. It is the number of symbol changes, waveform changes, or signaling events, across the transmission medium per time unit using a digitally modulated signal or a line code.

$f_{s} = \frac{1}{t_{s}}$

f_{s} Symbol Rate t_{s} Symbol Duration

Subcarrier Spacing
Subcarriers should only suffer of flat fading. Therefore, subcarrier spacing within OFDM system must be designed carefully. The spacing $F_{s}$ is such that the subcarriers are orthogonal, so they won?t interfere with one another despite the lack of guard bands between them. This comes about by having the subcarrier spacing equal to the reciprocal of basic OFDM symbol duration, which means, that the spacing is directly related to the basic OFDM symbol duration.

$F_{s} = \frac{1}{T_{s}}$

F_{s} Subcarrier Spacing T_{s} Basic OFDM Symbol Duation

Bandwith Efficiency
Bandwidth efficiency $β$ , also known as Spectral Efficiency, is an important piece of communications technology that specifies how many units of information per hertz are transmitted within the available bandwidth. It is thus the ratio of the data transfer rate to the occupied bandwidth, which is given in bit / s / Hz. The spectral efficiency depends on the used modulation method and the coding. Since the available bandwidths cannot be arbitrarily increased, the frequency economy and the modulation method used are decisive for spectral efficiency. The spectral efficiency is limited by the signal-to-noise ratio (SNR). The relationship between bandwidth and signal-to-noise ratio is determined by the Shannon-Hartley law. Thereafter, the channel capacitance increases linearly with the bandwidth and is affected logarithmically by the signal-to-noise ratio. With modern modulation techniques, such as OFDM and complex antenna constellations, such as multiple input multiple output (MIMO), the S / N ratio, the bandwidth efficiency can be improved. The bandwidth efficiency is adversely affected by the GI and reduced proportionately because the channel is occupied during the GI without data being transmitted. On the receiver side, the GI is not used in terms of detection, but the proportion of signal energy is lost.

$β = \frac{T_{s}}{T_{s} + T_{G}}$ $β = \frac{N}{N + N_{G}}$

β Bandwith Efficiency N Total Number of Subcarriers or FFT/DFT Length N_{G} Discrete Length of GI T_{G} GI Duration T_{s} Basic OFDM Symbol Duration

Nyquist Bandwith
The Nyquist bandwidth $B_{N}$ is the bandwidth required for optimal pulse shaping. Although the symbols of the GI do not transmit payload, they proportionately consume transmission bandwidth. The required Nyquist bandwidth thus results from the number of subcarriers and the respective subcarrier spacing.

$B_{N} = (N_{C} + N_{P}) \times F_{s}$ $B_{N} = (N_{C} + N_{P} + N_{D C}) \times F_{s}$ $B_{N} = \frac{1}{β \times T_{D}}$

β Bandwith Efficiency and also S/N Loss B_{N} Nyquist Bandwith F_{s} Subcarrier Spacing N_{C} Data Subcarriers N_{P} Pilot Subcarriers T_{D} Source Symbol Duration N_{D C} Not Occupied DC Subcarriers

Learn more about OFDM and it’s most important parameters by checking out the corresponding experiment!

Calculation of WLAN IEEE 802.11g OFDM Parameters

Determine the missing parameters for IEEE 802.11g (WLAN) in the table below by using the necessary formulas given in the corresponding tutorial to complete the parameter set of this common wireless transmission standard.

Parameter	value for IEEE 802.11g	Comment
Number of Subcarriers $N$	64
Number of Data Subcarriers $N_{c}$	52
Number of Pilot Subcarriers $N_{p}$	4
Number of Null Subcarriers $N_{v}$	$8$	$N_{v} = N - (N_{p} + N_{c})$
Discrete Length of GI $N_{G}$	16
GI Duration $T_{G}$	$0, 8 μ s$	$T_{G} = N_{G} \times t_{s} = \frac{N_{G} \times T_{s}}{N}$
Source Symbol Duration $T_{D}$	$0, 077 μ s$	$T_{D} = \frac{T'_{s}}{N_{c}}$
Symbol Duration $t_{s}$	$0, 05 μ s$	$t_{s} = \frac{T'_{s}}{N + N_{G}}$
Basic OFDM Symbol Duration $T_{s}$	$3, 2 μ s$	$T_{s} = N \times t_{s}$
Total OFDM Symbol Duration $T' s$	$4 μ s$
Bit Rate ${RB}_{brutto}$	$78 \frac{M b i t}{s}$	$R_{B, b r u t t o} = \frac{N_{c} \times n}{T'_{s}}$
Bit per Symbol $n$	6
Code Rate $R_{c}$	0,75
Symbol Rate $f_{s}$	$20 \frac{M S y m}{s}$	$f_{s} = \frac{1}{t_{s}}$
Subcarrier Spacing $F_{s}$	$312, 5 k H z$	$F_{s} = \frac{1}{T_{s}}$
Bandwidth Efficiency $β$	$\frac{4}{5}$	$β = \frac{T_{s}}{T_{s} + T_{G}}$
Nyquist Bandwidth $B_{n}$	$17, 5 M H z$	$B_{N} = (N_{C} + N_{P}) \times F_{s}$
Nyquist Bandwidth with DC Subcarrier $B_{n}$	$17, 8 M H z$	$B_{N} = (N_{C} + N_{P} + N_{D C}) \times F_{s}$
Parameter	value for IEEE 802.11g	Comment