Category: OFDM

Orthogonality of OFDM

OFDM, known as Orthogonal Frequency Division Multiplexing, is a digital modulation technique that divides a wideband signal into several narrowband signals. By doing so, it elongates the symbol duration of each narrowband signal compared to the original wideband signal, effectively minimizing the impact of time dispersion caused by multipath delay spread.

OFDM is categorized as a form of multicarrier modulation (MCM), where multiple user symbols are transmitted simultaneously through distinct subcarriers having overlapping frequency bands, ensuring they remain orthogonal to each other.

OFDM implements the same number of channels as the traditional Frequency Division Multiplexing (FDM). Since the channels (subcarriers) are arranged in overlapping manner, OFDM significantly reduces the bandwidth requirement.

OFDM equation

Consider an OFDM system that transmits a user symbol stream \(s_i\) (rate \(R_u\)) over a set of \(N\) subcarriers. Therefore, the symbol rate of each subcarrier is \(R_s = \frac{R_u}{N}\) and the symbol duration is \(T_s = \frac{N}{R_u}\).

The incoming symbol stream is split into \(N\) symbols streams and each of the symbol stream is multiplied by a function \(\Phi_k\) taken from a family of orthonormal functions \(\Phi_k, k \in \left\{0,1, \cdots, N-1 \right\}\)

In OFDM, these orthogonormal functions are complex exponentials

\[\Phi_k (t) = \begin{cases} e^{j 2 \pi f_k t}, & \quad for \; t \in \left[ 0, T_s\right] \\ 0, & \quad otherwise \end{cases} \quad \quad \quad (1) \]

For simplicity lets assume BPSK modulation for the user symbol \(s_i \in \left\{-1,1 \right\} \) and \(g_i\) is the individual gain of each subchannels. The OFDM symbol is formed by multiplexing the symbols on each subchannels and combining them.

\[S (t) =\frac{1}{N} \sum_{k=0}^{N-1} s_k \cdot g_k \cdot \Phi_k(t) \quad \quad \quad (2)\]

The individual subcarriers are

\[s_n (t) = s_k \cdot g_k \cdot e^{j 2 \pi f_k t} \quad \quad \quad (3)\]

For a consecutive stream of input symbols \(m = 0,1, \cdots\) the OFDM equation is given by

\[S(t) = \sum_{m = 0 }^{ \infty} \left[\frac{1}{N} \sum_{k=0}^{N-1} s_{k,m} \cdot g_{k,m} \cdot \Phi_{k}(t – m T_s) \right] \quad \quad \quad (4)\]

With \(g_{k,m} = 1\), the OFDM equation is given by

\[S(t) = \sum_{m = 0 }^{ \infty} \left[\frac{1}{N} \sum_{k=0}^{N-1} s_{k,m} \cdot e^{j 2 \pi f_k \left(t – m T_s \right)} \right] \quad \quad \quad (5)\]

Orthogonality

The functions \(\Phi\) by which the symbols on the subcarriers are multiplied are orthonormal over the symbol period \(T_s\). That is

\[ \left< \Phi_p (t), \Phi_q (t) \right> = \frac{1}{T_s} \int_{0}^{Ts} \Phi_p (t) \cdot \Phi^*_q (t) dt = \delta_{p,q} \quad \quad \quad (6) \]

where, \(\delta_{p,q}\) is the Kronecker delta given by

\[\delta_{p,q} = \begin{cases} 1, & \quad p=q \\ 0, & \quad otherwise \end{cases} \]

The right hand side of equation (5) will be equal to 0 (satisfying orthogonality) if and only if \(2 \pi \left(f_p−f_q \right)T_s=2 \pi k\) where \(k\) is a non-zero integer. This implies that the distance between the two subcarriers, for them to be orthogonal, must be

\[\Delta f = f_p – f_q = \frac{k}{T_s} \quad \quad (7)\]

Hence, the smallest distance between two subcarriers, for them to be orthogonal, must be

\[ \Delta f = \frac{1}{T_s} \quad \quad (8)\]

This implies that each subcarrier frequency experiences \(k\) additional full cycles per symbol period compared to the previous carrier. For example, in Figure 1 that plots the real and imaginary parts of three OFDM subcarriers (with \(k=1\)), each successive subcarrier contain additional full cycle per symbol period compared to the previous carrier.

orthogonal subcarriers of OFDM generated using 5G 3GPP air interface parameters
Figure 1: Three orthogonal subcarriers of OFDM

With \(N\) subcarriers, the total bandwidth occupied by one OFDM symbol will be \(B \approx N \cdot \Delta f \; (Hz) \)

Figure 2: OFDM spectrum illustrating 12 subcarriers

Benefits of orthogonality

Orthogonality ensures that each subcarrier’s frequency is precisely spaced and aligned with the others. This property prevents interference between subcarriers, even in a multipath channel, which greatly improves the system’s robustness against fading and other channel impairments.

The orthogonality property allows subcarriers to be placed close together without causing mutual interference. As a result, OFDM can efficiently utilize the available spectrum, enabling high data rates and maximizing spectral efficiency, making it ideal for high-speed data transmission in wireless communication systems.

Reference

[1] Chakravarthy, A. S. Nunez, and J. P. Stephens, “TDCS, OFDM, and MC-CDMA: a brief tutorial,” IEEE Radio Commun., vol. 43, pp. 11-16, Sept. 2005.

Cyclic Prefix in OFDM: hands-on demo in Matlab

Synopsis: Cyclic prefix in OFDM, tricks a natural channel to perform circular convolution. This simplifies equalizer design at the receiver. Hands-on demo in Matlab.

Cyclic Prefix-ed OFDM

A cyclic-prefixed OFDM (CP-OFDM) transceiver architecture is typically implemented using inverse discrete Fourier transform (IDFT) and discrete Fourier transform (DFT) blocks (refer Figure 13.3). In an OFDM transmitter, the modulated symbols are assigned to individual subcarriers and sent to an IDFT block. The output of the IDFT block, generally viewed as time domain samples, results in an OFDM symbol. Such OFDM symbols are then transmitted across a channel with certain channel impulse response (CIR). On the other hand, the receiver applies DFT over the received OFDM symbols for further demodulation of the information in the individual subcarriers.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

In reality, a multipath channel or any channel in nature acts as a linear filter on the transmitted OFDM symbols. Mathematically, the transmitted OFDM symbol (denoted as gets linearly convolved with the CIR and gets corrupted with additive white gaussian noise – designated as . Denoting linear convolution as ‘‘, the received signal in discrete-time can be represented as

The idea behind using OFDM is to combat frequency selective fading, where different frequency components of a transmitted signal can undergo different levels of fading. The OFDM divides the available spectrum in to small chunks called sub-channels. Within the subchannels, the fading experienced by the individual modulated symbol can be considered flat. This opens-up the possibility of using a simple frequency domain equalizer to neutralize the channel effects in the individual subchannels.

Circular convolution and designing a simple frequency domain equalizer

From one of the DFT properties, we know that the circular convolution of two sequences in time domain is equivalent to the multiplication of their individual responses in frequency domain.

Let and are two sequences of length with their DFTs denoted as and respectively. Denoting circular convolution as ,

If we ignore channel noise in the OFDM transmission, the received signal is written as

We can note that the channel performs a linear convolution operation on the transmitted signal. Instead, if the channel performs circular convolution (which is not the case in nature) then the equation would have been

By applying the DFT property given in equation 2,

As a consequence, the channel effects can be neutralized at the receiver using a simple frequency domain equalizer (actually this is a zero-forcing equalizer when viewed in time domain) that just inverts the estimated channel response and multiplies it with the frequency response of the received signal to obtain the estimates of the OFDM symbols in the subcarriers as

Demonstrating the role of Cyclic Prefix

The simple frequency domain equalizer shown in equation 6 is possible only if the channel performs circular convolution. But in nature, all channels perform linear convolution. The linear convolution can be converted into circular convolution by adding Cyclic Prefix (CP) in the OFDM architecture. The addition of CP makes the linear convolution imparted by the channel appear as circular convolution to the DFT process at the receiver (Reference [1]).

Let’s understand this by demonstration.

To simply stuffs, we will create two randomized vectors for and . is of length , the channel impulse response is of length and we intend to use -point DFT/IDFT wherever applicable.

N=8; %period of DFT
s = randn(1,8);
h = randn(1,3);

>> s =   -0.0155    2.5770    1.9238   -0.0629   -0.8105    0.6727   -1.5924   -0.8007
>> h =   -0.4878   -1.5351    0.2355

Now, convolve the vectors and linearly and circularly. The outputs are plotted in Figure 1. We note that the linear convolution and the circular convolution does not yield identical results.

lin_s_h = conv(h,s) %linear convolution of h and s
cir_s_h = cconv(h,s,N) % circular convolution of h and s with period N
lin_s_h = 0.0076 -1.2332 -4.8981 -2.3158  0.9449  0.9013 -0.4468  2.9934  0.8542 -0.1885 
cir_s_h = 0.8618 -1.4217 -4.8981 -2.3158  0.9449  0.9013 -0.4468  2.9934
Figure 1: Difference between linear convolution and circular convolution

Let’s append a cyclic prefix to the signal by copying last symbols from and pasting it to its front. Since the channel delay is 3 symbols (CIR of length 3), we need to add atleast 2 CP symbols.

Ncp = 2; %number of symbols to copy and paste for CP
s_cp = [s(end-Ncp+1:end) s]; %copy last Ncp symbols from s and prefix it.
s_cp = -1.5924 -0.8007 -0.0155  2.5770  1.9238 -0.0629 -0.8105  0.6727 -1.5924 -0.8007

Lets assume that we send the cyclic-prefixed OFDM symbol through the channel which perform linear filtering

lin_scp_h = conv(h,s_cp) %linear convolution of CP-OFDM symbol s_cp and the CIR h
lin_scp_h = 0.7767  2.8350  0.8618 -1.4217 -4.8981 -2.3158  0.9449  0.9013 -0.4468  2.9934  0.8542 -0.1885

Compare the outputs due to cir_s_h and lin_scp_h . We can immediately recognize that that the middle section of lin_scp_h is exactly same as the cir_s_h vector. This is shown in the figure 2.

Figure 2: Equivalence of circular convolution and linear convolution with cyclic prefixed signal
cir_s_h   = 0.8618  -1.4217 -4.8981 -2.3158  0.9449  0.9013 -0.4468  2.9934
lin_scp_h = 0.7767   2.8350  0.8618 -1.4217 -4.8981 -2.3158  0.9449  0.9013 -0.4468 2.9934 0.8542 -0.1885

We have just tricked the channel to perform circular convolution by adding a cyclic extension to the OFDM symbol. At the receiver, we are only interested in the middle section of the lin_scp_h which is the channel output. The first block in the OFDM receiver removes the additional symbols from the front and back of the channel output. The resultant vector is the received symbol r after the removal of cyclic prefix in the receiver.

r = lin_scp_h(Ncp+1:N+Ncp)%cut from index Ncp+1 to N+Ncp

Verifying DFT property

The DFT property given in equation 5 can be re-written as

To verify this in Matlab, take N-point DFTs of the received signal and CIR. Then, we can see that the IDFT of product of DFTs of and will be equal to the N-point circular convolution of and 

R = fft(r,N); %frequency response of received signal
H = fft(h,N); %frequency response of CIR
S = fft(s,N); %frequency response of OFDM signal (non CP)

r1 = ifft(S.*H); %IFFT of product of individual DFTs

display(['IFFT(DFT(H)*DFT(S)) : ',num2str(r1)])
display([cconv(s,h): ', numstr(r)])
IFFT(DFT(H)*DFT(S)) : 0.86175  -1.4217  -4.8981  -2.3158  0.94491  0.90128 -0.44682  2.9934
cconv(s,h):           0.86175  -1.4217  -4.8981  -2.3158  0.94491  0.90128 -0.44682  2.9934

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Reference:

[1] H. Sari, G. Karam, and I. Jeanclaude, Transmission Techniques for Digital Terrestrial TV Broadcasting, IEEE Commun. Magazine, Vol. 33, pp. 100-109, Feb. 1995.↗

Topics in this chapter

Orthogonal Frequency Division Multiplexing (OFDM)
● Introduction
● Understanding the role of cyclic prefix in a CP-OFDM system
 □ Circular convolution and designing a simple frequency domain equalizer
 □ Demonstrating the role of cyclic prefix
 □ Verifying DFT property
● Discrete-time implementation of baseband CP-OFDM
● Performance of MPSK-CP-OFDM and MQAM-CP-OFDM on AWGN channel
● Performance of MPSK-CP-OFDM and MQAM-CP-OFDM on frequency selective Rayleigh channel

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Breaking the Bandwidth Barrier – Li-Fi turns LED bulbs into High Speed Internet Hotspots

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Researchers, working on Ultra-Parallel Visible Light Communications (UP-VLC) project, have demonstrated 3.5Gpbs free space data transmission via three separate micro-LEDs that emit red, blue and green colours.  In effect, combining the three colours (that makes up for white colour), researchers say that achieving data rates over 10Gbps is feasible.

The technology enabler here is called – “Li-Fi”, a term coined by Prof. Harold Haas of University of Edinburgh, uses LED lights as medium of data transmission.  Li-Fi, academically referred as “Visible Light Communication”, aims to use existing micro- LED light bulbs for both illumination and communication.

With the advent of this technology, soon you will see all the illuminated spots in offices, houses or any other place turning into internet hotspots streaming high quality data. It has become a hot topic and soon many big corporations will jump on to the Li-Fi bandwagon.

It all started with Prof. Hass’s research group demonstrating the proof-of-concept results that exploited the higher peak-to-average ratio (PAR) property of Orthogonal Frequency Division Multiplexing (OFDM) systems.  PAR, otherwise called “Crest Factor”, is a disadvantage in RF communications. The research group have used this property to turn commercially available LED bulbs to ultra-high speed wireless hotspots. Later, this work was featured in TIME magazine’s “50 Best inventions of the year 2011” and Nobel Laureate Prof. Hänsch listed this work in his book titled “100 ground-breaking ideas” that could shape the next century.

Using OFDM, micro-LED light bulbs are enabled to handing millions of light-intensity-changes per second. Essentially, the micro-LED bulbs are switched on and off depending on the input bits –‘0’ or ‘1’. The switching is done at such a high speed that it is undetectable by human eye. Thus both un-flickering-illumination and high-speed-communication are achievable under this condition.

Li-Fi carries the promise of breaking the bandwidth-crunch suffered by existing wireless systems. The demand for more bandwidth is skyrocketing day by day with the proliferation of wireless enabled devices. The crunch faced by Radio-Frequency (RF) systems, is also compounded by complex government regulations, needs of a growing base of customers and risking of the performance issues due to these challenges.

Li-Fi is essentially a wireless system. It uses the visible light region of the electromagnetic spectrum which is 10,000 times larger than the microwave region of the electromagnetic spectrum. This opens up the utilizable frequencies to the order of terahertz level. It will not interfere with existing devices and it can be used in areas where there is extensive RF noise or in places where RF frequencies are restricted (like in airplanes).

Watch the TED talk – “Wireless data from every light bulb”- where Prof Haas demonstrates the capability of Li-Fi technology system prototype (using a desk-lamp) that steams a HD movie in real-time.

OFDM simulation – performance in AWGN channel

Goal: Simulate discrete-time cyclic-prefixed OFDM communication system. Explain role of IFFT/FFT, cyclic prefix. Simulate M-QPSK / M-QAM based cyclic prefixed OFDM over AWGN channel.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

Discrete-time implementation of baseband CP-OFDM

The schematic diagram of a simplified cyclic-prefixed OFDM (CP-OFDM) data transmission system is shown in Figure 1. The basic parameter to describe an OFDM system is to specify the number of subchannels () required to send the data. The number of subchannels is typically set to powers of 2, such as . The size of inverse discrete Fourier transform (IDFT) and discrete Fourier transform (DFT) need to be set accordingly.

The transmission begins by converting the source information stream into parallel subchannels. For convenience, the information stream is already represented as a symbol from the set . The data symbol in each subchannel is modulated using the chosen modulation technique such as MPSK or MQAM.

Since this is a baseband discrete-time model, where the signals are represented at symbol sampling instants, the information symbol on each parallel stream is assumed to be modulating a single orthogonal carrier. At this juncture, the modulated symbols on the parallel streams can be visualized as coming from different orthogonal subchannels in the frequency domain. The components of the orthogonal subchannels in the frequency domain are converted to time domain using IDFT operation.

Figure 1: Discrete-time simulation model for OFDM transmission and reception

The following generic function implements the modulation mapper (constellation mapping) shown in the Figure 1. The function supports MPSK modulation for and MQAM modulation that has square constellation : . It is built over the mpsk_modulator.m and mqam_modulator.m functions given in sections 5.3.2 and 5.3.3 of chapter 5 (Refer the book Wireless communication systems using Matlab).

Please refer the book Wireless communication systems using Matlab – for full Matlab code

modulation_mapper.m: Implementing the modulation mapper for MPSK and MQAM

function [X,ref]=modulation_mapper(MOD_TYPE,M,d)
%Modulation mapper for OFDM transmitter
% MOD_TYPE - 'MPSK' or 'MQAM' modulation
% M - modulation order, For BPSK M=2, QPSK M=4, 256-QAM M=256 etc..,
% d - data symbols to be modulated drawn from the set {1,2,...,M}
%returns
% X - modulated symbols
% ref -ideal constellation points that could be used by IQ detector
if strcmpi(MOD_TYPE,'MPSK'),
[X,ref]=mpsk_modulator(M,d);%MPSK modulation
else
if strcmpi(MOD_TYPE,'MQAM'),
[X,ref]=mqam_modulator(M,d);%MQAM modulation
else
error('Invalid Modulation specified');
end
end;end

OFDM signal is a composite signal that contains information from subchannels. Since the modulated symbols are visualized to be in frequency domain, it is converted to time-domain using IDFT. In the receiver, the corresponding inverse operation is performed by the DFT block. The IDFT and DFT blocks in the schematic can also be interchanged and it has no impact to the transmission.

In a time-dispersive channel, the orthogonality of the subcarriers cannot be maintained in a perfect state due to delay distortion. This problem is addressed by adding a cyclic extension (also called cyclic prefix) to the OFDM symbol (reference [1]). A cyclic extension is added by copying the last symbols from the vector and pasting it to its front as shown in Figure 2.

Figure 2: Adding a cyclic prefix in CP-OFDM

Cyclic extension of OFDM symbol converts the linear convolution channel to a channel performing cyclic convolution (view demo here) and this ensures orthogonality of subcarriers in a time-dispersive channel. It also completely eliminates the subcarrier interference as long as the impulse response of the channel is shorter than the cyclic prefix. At the receiver, the added cyclic prefix is simply removed from the received OFDM symbol.

On the receiver side, the demapper for demodulating MPSK and MQAM can be implemented by using a simple IQ detector that uses the minimum euclidean distance metric for demodulation. (discussion and function definitions in section 5.4.4 of chapter 5 (Refer the book Wireless communication systems using Matlab).

Performance of MPSK-CP-OFDM and MQAM-CP-OFDM on AWGN channel

The code (given in the book Wireless communication systems using Matlab) puts together all the functional blocks of an OFDM transmission system, that were described here, to simulate the performance of a CP-OFDM system over an AWGN channel. The code supports two types of underlying modulations for OFDM – MPSK or MQAM. It generates random data symbols, modulates them using the chosen modulation type, converts the modulated symbols to frequency domain using IDFT operation and adds cyclic prefix to form an OFDM symbol. The resulting OFDM symbols are then added with AWGN noise vector that corresponds to the specified value (AWGN noise model is described in this article).

On the receiver side, cyclic prefix is removed from the received OFDM symbol, DFT is performed and then the symbols are sent through a demapper for getting an estimate of the source symbols. The demapper is implemented by using a simple IQ detector that uses the minimum euclidean distance metric for demodulation. Finally, the symbol error rates are computed and compared against the theoretical symbol error rate curves for the respective modulations over AWGN. Simulated performance results are plotted in Figure 3.

Figure 3: Performance of M-PSK CP-OFDM and M-QAM CP-OFDM over AWGN channel

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Reference

[1] A. Peled and A. Ruiz, Frequency domain data transmission using reduced computational complexity algorithms, in Proc. IEEE ICASSP- 80, Vol. 5, pp.964 – 967, April 1980.↗

Topics in this chapter

Orthogonal Frequency Division Multiplexing (OFDM)
● Introduction
● Understanding the role of cyclic prefix in a CP-OFDM system
 □ Circular convolution and designing a simple frequency domain equalizer
 □ Demonstrating the role of cyclic prefix
 □ Verifying DFT property
● Discrete-time implementation of baseband CP-OFDM
● Performance of MPSK-CP-OFDM and MQAM-CP-OFDM on AWGN channel
● Performance of MPSK-CP-OFDM and MQAM-CP-OFDM on frequency selective Rayleigh channel

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
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Checkout Added to cart

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Introduction to OFDM – orthogonal Frequency division multiplexing – part 4 – Cyclic Prefix

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Consider a non-ideal channel h(t)≠δ(t), that causes delay dispersion. Delay dispersion manifests itself as Inter Symbol Interference (ISI)on each subcarrier channel due to pulse overlapping. It will also cause ICC (Inter Carrier Interference ) due to the non-orthogonality of the received signal. Adding cyclic prefix to each OFDM symbol mitigates the problems of ISI and ICC by removing them altogether.

Lets say, without cyclic prefix we transmit the following N values (N=Nfft=length of FFT/IFFT) for a single OFDM symbol.

$$ X_0,X_1,X_2,…,X_{N-1} $$

Lets consider a cyclic prefix of length Ncp, ( where Ncp<N ), is formed by copying the last Ncp values from the above vector of X and adding those Ncp values to the front part of the same X vector.With a cyclic prefix length Ncp, ( where Ncp<N ), the following values constitute a single OFDM symbol :

If T is the duration of the an OFDM symbol in secs, due to the addition of cyclic prefix of length Ncp, the total duration of an OFDM symbol becomes T+Tcp, where Tcp=Ncp*T/N. Therefore, the number of samples allocated for cyclic prefix can be calculated as Ncp=Tcp*N/T, where N is the FFT/IFFT length, T is the IFFT/FFT period and Tcp is the duration of cyclic prefix.

The key ideas behind adding cyclic prefix :

1) Convert linear convolution in to circular convolution which eases the process of detecting the received signal by using a simple single tap equalizer
If you wish to know how the addition of cyclic prefix converts linear convolution to circular convolution, visit this link
2) Help combat ISI and ICC.

When a cyclic prefix of length Ncp is added to the OFDM symbol, the output of the channel (r) is given by circular convolution  of channel impulse response (h) and the OFDM symbols with cyclic prefix (x).

$$ r=h \circledast x $$

As we know, for the discrete signals, circular convolution in the time domain translates to multiplication in the frequency domain.Thus, in frequency domain, the above equation translates to

$$ R=HX $$

At the receiver, R is the received signal (in Frequency domain) and our goal is to estimate the transmitted signal (X) from the received signal R. From the above equation, the problem of detecting the transmitted signal at the receiver side translate to a simple equalization equation as follows

$$ \hat{X}= \frac{R}{H} $$

After the FFT performed at the receiver side (i.e. after the FFT block in the receiver side), a single tap equalizer (which essentially implements the above equation) is used to estimate the transmitted OFDM symbol. It also corrects the phase and equalizes the amplitude.

A basic OFDM architecture with Cyclic Prefix is given below. (In the following diagram, symbols represented by small case letters are assumed to be in time domain, whereas the symbols represented by uppercase letters are assumedto be in frequency domain)

An OFDM communication Architecture with Cyclic Prefix

The IEEE specs specify the length of the Cyclic prefix in terms of its duration.
Lets see how to convert the specified duration (Tcp) in to actual number of samples assigned for cyclic prefix (Ncp).

Lets see an example of deriving Ncp from IEEE 802.11 spec [1]

Given parameters in the spec:
N=64;                           %FFT size or total number of subcarriers (used + unused) 64
Nsd = 48;                      %Number of data subcarriers 48
Nsp = 4 ;                        %Number of pilot subcarriers 4
ofdmBW = 20 * 10^6 ;  % OFDM bandwidth

Derived Parameters:
deltaF = ofdmBW/N;      % Bandwidth for each subcarrier - include all used and unused subcarries
Tfft = 1/deltaF;               % IFFT or FFT period = 3.2us
Tgi = Tfft/4;                    % Guard interval duration - duration of cyclic prefix - 1/4th portion of OFDM symbols
Tsignal = Tgi+Tfft;          % Total duration of BPSK-OFDM symbol = Guard time + FFT period
Ncp = N*Tgi/Tfft;           %Number of symbols allocated to cyclic prefix
Nst = Nsd + Nsp;           % Number of total used subcarriers
nBitsPerSym=Nst;           %For BPSK the number of Bits per Symbol is same as number of subcarriers

References:

[1] IEEE 802.11 specification – “Orthogonal frequency division multiplexing (OFDM) PHY specification for the 5 GHz band” – chapter 17

See Also:

(1) Introduction to OFDM – Orthogonal Frequency Division Multiplexing
(2) An OFDM Communication System – Implementation Details
(3) Simulation of OFDM system in Matlab – BER Vs Eb/N0 for OFDM in AWGN channel

Books on OFDM

Introduction to OFDM – orthogonal Frequency division multiplexing – part 3

In the previous article, the architecture of an OFDM transmitter was described using sinusoidal components. Generally, an OFDM signal can be represented as

\[OFDM\; signal = c(t)=\sum_{n=0}^{N-1}s_{n}(t)sin(2\pi f_{n}t )\]

\(s(t)\) = symbols mapped to chosen constellation (BPSK/QPSK/QAM etc..,)

\(f_n\) = orthogonal frequency

This equation can be thought of as an IFFT process ( Inverse Fast Fourier Transform). The Fourier transform breaks a signal into different frequency bins by multiplying the signal with a series of sinusoids. This essentially translates the signal from time domain to frequency domain. But, we always view IFFT as a conversion process from frequency domain to time domain.

FFT is represented by

\[X(k)=\sum_{n=0}^{N-1}x(n) \cdot sin \left(\frac{2\pi kn}{N} \right)+j\sum_{n=0}^{N-1}x(n) \cdot cos \left(\frac{2\pi kn}{N} \right)\]
\[x(n)=\sum_{n=0}^{N-1}X(k) \cdot sin \left(\frac{2\pi kn}{N} \right)-j\sum_{n=0}^{N-1}X(k) \cdots cos \left(\frac{2\pi kn}{N} \right)\]

where as its dual , IFFT is given by

The equation for FFT and IFFT differ by the co-efficients they take and the minus sign. Both equation does the same thing. They multiply the incoming signal with a series of sinusoids and separates them into bins.In fact, FFT and IFFT are dual and behaves in a similar way.IFFT and FFT blocks are interchangeable.

Since the OFDM signal ( c(t) in the equation above ) is in time domain, IFFT is the appropriate choice to use in the transmitter, which can be thought of as converting frequency domain samples to time domain samples. Well, you might ask : s(t) is not in frequency domain and they are already in time domain; so whats the need to convert it into time domain again ? The answer is IFFT/FFT equation comes handy in implementing the conversion process and we can eliminate the individual sinusoidal multipliers required in the transmitter/receiver side. The following figure illustrates, how the use of IFFT in the transmitter eliminates the need for separate sinusoidal converters. Always remember that IFFT and FFT blocks in the transmitter are interchangeable as long as their duals are used in receiver.

OFDM implementation using FFT and IFFT

The entire architecture of a basic OFDM system with both transmitter and receiver will look like this

A Complete OFDM communication system

An OFDM system is defined by IFFT/FFT length – N ,the underlying modulation technique ( BPSK/QPSK/QAM), supported data rate, etc..,. The FFT/IFFT length N defines the number of total subcarriers present in the OFDM system. For example, an OFDM system with N=64 provides 64 subcarriers. In reality, not all the subcarriers are utilized for data transmission. Some subcarriers are reserved for pilot carriers (used for channel estimation/equalization and to combat magnitude and phase errors in the receiver) and some are left unused to act as guard band. OFDM system do not transmitany data on the subcarriers that are near the two ends of the transmission band ( Not necessarily at the ends of the bands, implementation may differ). These subcarriers are collectively called guard band. The reservation of subcarriers to guard bands helps to reduce the out of band radiation and thus eases the requirements on transmitter front-end filters.The subcarriers in the guard band are also called Null subcarriers or virtual subcarriers.

For example IEEE 802.11 standard[1] specifies the following parameters for its OFDM physical layer.

FFT/IFFT size = 64 ( implies 64 subcarriers in total = used + unused = N_{fft})

Number of data subcarriers = 48 (\( N_d\))

Number of pilot subcarriers = 4 (\(N_p\))

Derived parameters from the above specification.

Number of total USED subcarriers = 52 (\( N_u = N_d+ N_p \))

Number of total UNUSED subcarriers = 12 (\( N_{un} = N_{fft} – N_u\)).

According to the spec, these 52 used subcarriers are distributed in the following way. The 52 used subcarriers are named as 1,2,3,…,26 and -1,-2,-3,…,-26. The used subcarriers 1 to 26 are mapped to 1 to 26 of IFFT inputs and the subcarriers -1,-2,..,-26 are mapped to the IFFT inputs 38 to 63. The remaining IFFT inputs 27 to 37 and the input 0 (dc input) are set to 0 .In this manner the 12 null subcarriers are mapped to IFFT inputs.

Check this book for full Matlab code: Wireless Communication Systems in Matlab, ISBN: 978-1720114352 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).

Reference:

[1] IEEE 802.11 specification – “Orthogonal frequency division multiplexing (OFDM) PHY specification for the 5 GHz band” – chapter 17

See also:

(1) Introduction to OFDM – orthogonal Frequency division multiplexing
(2) Role of Cyclic Prefix in OFDM
(3) Simulation of OFDM system in Matlab – BER Vs Eb/N0 for OFDM in AWGN channel

Introduction to OFDM – orthogonal Frequency division multiplexing – part 2

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The article has been consolidated into these following pages. Please refer these links.

If you are looking for Matlab code refer this ebook : Simulation of Digital Communication Systems by Mathuranathan Viswanathan

(1) Introduction to OFDM – Orthogonal Frequency Division Multiplexing
(2) An OFDM Communication System – Implementation Details
(3) Role of Cyclic Prefix in OFDM
(4) Simulation of OFDM system in Matlab – BER Vs Eb/N0 for OFDM in AWGN channel

Books on OFDM

Introduction to OFDM – orthogonal Frequency division multiplexing

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In modulations, information is mapped on to changes in frequency, phase or amplitude (or a combination of them) of a carrier signal. Multiplexing deals with allocation/accommodation of users in a given bandwidth (i.e. it deals with allocation of available resource).
OFDM is a combination of modulation and multiplexing. In this technique, the given resource (bandwidth) is shared among individual modulated data sources. Normal modulation techniques (like AM, PM, FM, BPSK, QPSK, etc.., ) are single carrier modulation techniques, in which the incoming information is modulated over a single carrier. OFDM is a multicarrier modulation technique, which employs several carriers, within the allocated bandwidth, to convey the information from source to destination. Each carrier may employ one of the several available digital modulation techniques (BPSK, QPSK, QAM etc..,).

Why OFDM

OFDM is very effective for communication over channels with frequency selective fading ( different frequency components of the signal experience different fading). It is very difficult to handle frequency selective fading in the receiver , in which case, the design of the receiver is hugely complex. Instead of trying to mitigate frequency selective fading as a whole (which occurs when a huge bandwidth is allocated for the data transmission over a frequency selective fading channel), OFDM mitigates the problem by converting the entire frequency selective fading channel into small flat fading channels (as seen by the individual subcarriers). Flat fading is easier to combat (compared to frequency selective fading) by employing simple error correction and equalization schemes.

Difference between FDM and OFDM:

OFDM is a special case of FDM ( Frequency Division Multiplexing). In FDM, the given bandwidth is subdivided among a set of carriers. There is no relationship between the carrier frequencies in FDM. For example, consider that the given bandwidth has to be divided among 5 carriers (say a,b,c,d,e). There is no relationship between the subcarriers ; a,b,c,d and e can anything within the given bandwidth.

If the carriers are harmonics, say (b=2a,c=3a,d=4a,d=5a , integral multiple of fundamental component a ) then they become orthogonal. This is a special case of FDM, which is called OFDM (as implied by the word – ‘orthogonal’ in OFDM)

Designing OFDM Transmitter:

Consider that we want to send the following data bits using OFDM : D = {d0,d1,d2,…). The first thing that should be considered in designing the OFDM transmitter is the number of subcarriers required to send the given data. As a generic case, lets assume that we have N subcarriers. Each subcarriers are centered at frequencies that are orthogonal to each other (usually multiples of frequencies).

The second design parameter could be the modulation format that we wish to use. An OFDM signal can be constructed using anyone of the following digital modulation techniques namely BPSK, QPSK, QAM etc..,
The data (D) has to be first converted from serial stream into parallel stream depending on the number of sub-carriers (N). Since we assumed that there are N subcarriers allowed for the OFDM transmission, we name the subcarriers from 0 to N-1. Now, the Serial to Parallel converter takes the serial stream of input bits and outputs N parallel streams (indexed from 0 to N-1). These parallel streams are individually converted into the required digital modulation format (BPSK, QPSK, QAM etc..,). Lets call this output S0,S1,..SN. The conversion of parallel data (D) into the digitally modulated data (S) is usually achieved by a constellation mapper, which is essentially a look up table (LUT). Once the data bits are converted to required modulation format, they need to be superimposed on the required orthogonal subcarriers for transmission. This is achieved by a series of N parallel sinusoidal oscillators tuned to N orthogonal frequencies (f0,f1,…fN-1). Finally, the resultant output from the N parallel arms are summed up together to produce the OFDM signal.

The following figure illustrates the basic concept of OFDM transmission (note: In order to give a simple explanation to illustrate the underlying concept,the usual IFFT/FFT blocks that are used in actual OFDM system, are not used in the block diagram) .

OFDM Transmitter

Example:

The first example illustrates the concept of OFDM transmission with BPSK modulation as its underlying modulation format. The second example illustrates the OFDM transmission with pi/4 shifted QPSK modulation. Here 5 orthogonal subcarriers are assumed for the OFDM transmission.

See also:

(1) An OFDM Communication System – Implementation Details
(2) Role of Cyclic Prefix in OFDM
(3) Simulation of OFDM system in Matlab – BER Vs Eb/N0 for OFDM in AWGN channel