4.

Orthogonal Frequency-Division Multiplexing

Orthogonal Frequency-Division Multiplexing (OFDM) forms the basis for 4G, i.e.,
Fourth Generation wireless communication systems. OFDM is used in 4G wireless cellular
standards such as Long-Term Evolution (LTE) and WiMAX (Worldwide Interoperability for
Microwave Access). OFDM is a key broadband wireless technology which supports data rates in
excess of 100 Mbps

Motivation and Multicarrier Basics

Consider a bandwidth B = 2W available for communication, where W is the one-sided
bandwidth, or, in other words, the maximum frequency. For a single carrier communication
system, the symbol time T is given as

1
T=
B
𝟏
basically implying that symbols can be transmitted at intervals of seconds each. Therefore, the
𝑩
symbol rate is given as
Multicarrier Transmission
However, the above MCM system suffers from a significant bottleneck. Implementing the bank
of N modulators and N demodulators with closely spaced subcarrier frequencies is an extremely
challenging task

Observe from the expression above that the sample x (u) is basically the Inverse Discrete Fourier
Transform (IDFT) coefficient of the information symbols X (0) , X (1) , . . . , X (N − 1) at the uth
time point. Thus, the Inverse Fast Fourier Transform (IFFT) can be conveniently employed to
generate the sample MCM signal. Thus, it drastically reduces the complexity of implementing an
OFDM system since it eliminates the need for the bank of modulators corresponding to the
different subcarrier frequencies. This technique, where the MCM signal is generated by
employing the IFFT operation is termed Orthogonal Frequency Division Multiplexing, or
OFDM. At the receiver, to recover the information symbols, one can correspondingly employ an
FFT operation.

Cyclic Prefix in OFDM

Consider a frequency-selective channel modelled with channel taps h (0) , h (1) , . . . ,
h (L − 1). Thus, the received symbol y at a given time instant n can be expressed as Thus, the
received symbol y at a given time instant n can be expressed as

from which it can be seen that the received symbol y (n) at the time instant ‘n’ experiences inter
symbol interference from the previous L − 1 transmitted symbol
 Consider now two OFDM symbols as follows. Let x (0) , x (1) , . . . , x (N − 1) denote
the IFFT samples of the modulated symbols X (0) , X (1) , . . . , X (N − 1), while x̃ (0) , x̃ (1) , . .
. , x̃ (N − 1) denote the IFFT samples of the previous modulated symbol block X ̃ (0) , X
̃ (1) , . . . ,
̃
X (N − 1). Thus, the samples corresponding to these two blocks of OFDM symbols are
transmitted sequentially as

Now, consider the received symbol y (0) corresponding to the transmission of x (0). This can be
expressed as
OFDM Example
The total number of subcarriers N = 256, with a bandwidth of 15.625 kHz per subcarrier.
Therefore,

The raw OFDM symbol time, corresponding to the N = 256 IFFT samples, is 64 μs. WiMAX
employs a cyclic prefix which is 12.5% of the symbol time. Therefore, the duration of the cyclic
prefix is

Thus, the total transmitted OFDM symbol duration with cyclic prefix is 64 μs + 8 μs = 72 μs.
Also, the number of samples in the CP is

Thus, the length of the cyclic prefix Lc = 32 samples and the total number of samples is
256 + 32 = 288. This break-up of the OFDM symbol in terms of the regular samples and the
cyclic prefix is shown in Figure. Finally, the loss in spectral efficiency is
This is the loss in spectral efficiency arising because of the addition of the cyclic prefix.

Bit-Error Rate (BER) for OFDM

Consider the OFDM subcarrier system model given

where N (k) is the subcarrier noise obtained from the FFT of the noise samples at the output
of the receiver as

where N is the number of subcarriers, and n (0) , n (1) , . . . , n (N − 1) are additive noise samples
for each of the output samples y (0) , y (1) , . . . , y (N − 1). We now deduce the statistical
properties of these noise samples N (k), which are required to characterize the BER performance
of the OFDM system. Firstly, observe that the noise N (k) is the linear combination of Gaussian
noise samples n (0) , n (1) , . . . , n (N − 1). Hence, it is Gaussian in nature. Further, the mean or
expected value of N (k) is given as
MIMO-OFDM
MIMO-OFDM is a combination of the Multiple-Input Multiple-Output (MIMO)
wireless technology with that of OFDM, to further increase the rate in broadband multi-antenna
wireless systems. Similar to OFDM, MIMO-OFDM converts a frequency-selective MIMO
channel into multiple parallel flat fading MIMO channels. Hence, MIMO-OFDM significantly
simplifies baseband receive processing by eliminating the need for a complex MIMO equalizer.
We have already seen that the frequency-selective SISO channel is modelled as an FIR channel
filter, with the output y (n) at time instant ‘n’ given as

Therefore, the symbol vector y (n) at the time instant n is affected by inter-symbol
vector interference from x (n − 1) , x (n − 2) , . . . , x (n − L + 1). This is an L-tap frequency-
selective MIMO channel. As can be seen, in a MIMO frequency-selective channel,
the interference occurs between current and previous transmit symbol vectors

In a MIMO-OFDM system, one needs to perform the IFFT operation at each transmit
antenna. employing MIMO-OFDM, the MIMO frequency-selective channel can be converted
into a set of parallel flat-fading MIMO channels. These can be described as

Also, the MMSE receiver for the subcarrier k of the MIMO-OFDM system is given as

where Pd denotes the data power.

Effect of Frequency Offset in OFDM

OFDM divides the available wideband amongst a set of orthogonal overlapping

subcarriers. Hence, the presence of a carrier-frequency offset can introduce severe distortion in
an OFDM system, as it results in a loss of orthogonality amongst the subcarriers. Hence, the
presence of a carrier-frequency offset introduces Inter-Carrier Interference (ICI) in OFDM
systems. In this section, we characterize the effect of frequency offset on the performance of the
OFDM system. Consider a frequency offset Δf such that

where ϵ denotes the normalized frequency offset, normalized with respect to the subcarrier
bandwidth B/N. Corresponding to the frequency offset e, the baseband received samples y(n)
are given as
Thus, in the absence of a carrier frequency offset, i.e., = 0, the system y(n) reduces to the earlier
flat-fading OFDM system across each subcarrier, i.e.,
Now consider the received symbols y (n) in the presence of a carrier-frequency offset
OFDM–Peak-to-Average Power Ratio (PAPR)

The Peak-to-Average Power Ratio (PAPR) is a critical problem in OFDM systems, which needs
to be handled effectively in order to limit the distortion at the receiver. Consider a non-OFDM or
single-carrier system with BPSK modulated symbols. For example, let the symbol stream x (0) ,
x (1) , x (2) , . . . be given as +a, −a, +a, . . . and so on. The power in each symbol equals a2
Further, also observe that this is the peak power at any given instant of time.
Therefore, we have

Thus, since the peak and average power are equal, the peak-to-average power ratio, or PAPR,
is given as
OFDM subcarrier loading
Peak Power:

Therefore, the peak power is given as a2 . Hence, the peak-to-average power ratio in an OFDM
system is given as

it can be seen that the peak-to-average power ratio in an OFDM system is N , which is
significantly higher compared to that of the single-carrier system, which is 1. Further,
interestingly, this PAPR rises with N , i.e., the number of subcarriers. Larger the number of
subcarriers, larger is the PAPR. This high PAPR of the OFDM arises because of the IFFT
operation

OFDM PAPR for various number of subcarriers N

 The amplifier operates around a bias point, as shown in Figure below which is
roughly around the average power of the signal. As long as the signal amplitude is restricted to
the dynamic range of the amplifier around this bias point, for which the amplifier characteristic is
linear, there is no nonlinear distortion at the output. However, in the case of OFDM, since the
peak power deviates significantly from the average power, there is a high chance that the signal
crosses into the voltage region outside the dynamic range of the amplifier, thus resulting in a
nonlinear distortion of the received signal. This nonlinear effect, arising out of amplifier
saturation, leads to loss of orthogonality of the subcarriers and inter-carrier interference. The net
result is a poor decoding performance and a rise in the bit-error rate. A slightly modified OFDM
technique, which can significantly reduce the PAPR, is SC-FDMA

Nonlinear amplifier characteristic

SC-FDMA
SC-FDMA, which stands for Single-Carrier Frequency Division for Multiple Access, can
be employed to reduce the peak-to-average power ratio in an OFDM system by the insertion of
an N -point FFT block before the N -point IFFT block. It can then be seen that the FFT and the
IFFT cancel the effect of each other and the net output is the exact input symbol stream, i.e.,
corresponding to a single-carrier system. This drastically reduces the PAPR, since, as seen
previously, the PAPR of a single-carrier system is 0 dB.
The SC-FDMA transmitter
However, instead of using an N -point FFT, one can use an M-point FFT, where M < N ,
to reduce the PAPR, while still retaining the properties of the OFDM system. Hence,
introduction of the M-point FFT in SC-FDMA significantly reduces the PAPR of the system.
This is the central principle of SC-FDMA.

Subcarrier Mapping in SC-FDMA

Subcarrier mapping, in which the M samples at the output of the M-point FFT are
mapped to the N subcarriers, is a key operation in SC-FDMA, Consider M = 4 SC-FDMA
symbols and N = 12 subcarriers. Let x (0) , x (1) , x (2) , x (3) denote the symbols and X (0) , X
(1) , X (2) , X (3) denote the corresponding M = 4-point FFT samples which are to be loaded
onto the subcarriers. Let the number of subcarriers be N = 12. In Interleaved FDMA (IFDMA),
the samples X (i) are interleaved with zeros. In Localized FDMA(LFDMA), which is also
employed in the uplink of the 4G mobile standard LTE, the samples are loaded as a block onto
the subcarriers, with appropriate zero padding.
The SC-FDMA receiver
SC-FDMA receiver incorporates two new blocks compared to the OFDM receiver. The
purpose of these additional blocks can be described as follows. After the N -point FFT operation
at the receiver, the signals are equalized across all the subcarrires, to remove the effect of the
fading-channel coefficient across the subcarriers. Following the above operation, they are
demapped from the subcarriers, which are N in number, to the original FFT block size of M.
Finally, the M-point FFT is performed on these samples to generate the symbol stream.