# Orthogonal Frequency Division Multiplexing

## Summary


Orthogonal Frequency Division Multiplexing (OFDM) is a key wideband digital communication method used in wireless transmission. Data is split into several streams and transmitted on multiple narrowband channels to reduce interference and crosstalk.

Due to its good spectral efficiency and relatively less complexity, it's one of the popular techniques used in telecommunications. It's used in multiple standards including DAB, HDTV, WLAN (802.11a/g/ac), WiMAX, and LTE. It enables transmission of high data rates (in the order of 1 Gbps) on a wireless channel.

## Discussion

### Could you give an overview of OFDM?

OFDM is a **Multi-Carrier Modulation (MCM)** scheme, which uses closely spaced multiple subcarriers to transmit data. Data to be transmitted is split and transmitted using multiple subcarriers instead of using a single carrier.  The key idea is instead of transmitting at a very high bit rate, the data is transmitted over multiple subchannels each carrying lower bit rates. 

Unlike the traditional Frequency Division Multiplexing (FDM), the OFDM does not use guard bands to separate the various subchannels. One of the key features of OFDM is the orthogonality of the subcarriers used to transmit data. The orthogonality of subcarriers results in more subcarriers in a given bandwidth. This improves spectral efficiency. It also eliminates the interference between subcarriers, often called **Inter-Carrier Interference (ICI)**.


### Why is OFDM used in wireless transmission?

One of the key challenges in wireless transmission as compared to wired transmission is the phenomenon of multipath fading and **Inter-Symbol Interference (ISI)**. OFDM helps in mitigating both these effects, making it one of the key technologies to be used in wireless transmission. OFDM uses spectrum in a more efficient way compared to some of the other techniques used to overcome multipath fading and Inter Symbol Interference. 


### How does OFDM ensure subcarriers do not interfere with each other?

OFDM splits the available spectrum into multiple subbands and transmits data using multiple subcarriers. The subcarriers are chosen such that they are orthogonal to each other. This ensures that data from one subcarrier does not interfere with the data on the other. To maintain orthogonality between subcarriers, the subcarriers are chosen such that they are all integer multiples of the base frequency. If the total bandwidth of the system is B Hz. Then the base frequency (f0) is given by B/N, where N is the number of subcarriers in the system. The subcarriers used are f0, 2f0, 3f0 ... (N-1)f0.

The spectrum of each transmitted subcarrier in OFDM system is a `sinc` function with side-lobes that produce overlapping spectra between subcarriers. Since the carriers are orthogonal, the peak of each subcarrier coincides with nulls of other subcarriers. Even though there's overlap of spectra between subcarriers, there's no interference between subcarriers.


### What's the role of FFT and IFFT in OFDM implementation?

OFDM system involves mapping of symbols onto a set of orthogonal subcarriers that are multiples of the base frequency. This can be implemented in digital domain using Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT). These transforms are important from OFDM perspective as they can be viewed as mapping digital input data onto orthogonal subcarriers.

The IFFT takes frequency-domain input data and converts it to the time-domain output data (analog OFDM symbol waveform). This waveform is transmitted by the OFDM transmitter. The receiver receives the waveform and uses FFT transform to convert the data back from time-domain into frequency domain to recover the data back.


### What's guard band and cyclic prefix in OFDM?

OFDM systems make use of guard band and cyclic prefix (CP) to overcome the issue of ISI. While guard band is not required to achieve orthogonality of subcarriers, it helps in overcoming ISI in a multipath channel. The duration of the guard band should be more than the channel spread of the wireless medium.

The cyclic prefix is transmitted during the guard band interval. After IFFT, some end bits of the OFDM symbol are copied to the guard band before the symbol to form the CP.

Receivers often have a channel equalizer to combat channel distortion. CP simplifies equalizer implementation. Essentially, CP converts a linear convolution to a circular convolution. Circular convolution in time domain is equivalent to a simple multiplication in the frequency domain. The channel equalizer in the receiver multiplies the received symbol by the inverse of the channel coefficients in frequency domain to recover the original transmitted symbol, assuming that fading is constant over the subband.


### How is a typical OFDM transmitter and receiver implemented?

In an OFDM transmitter, the input bits are first grouped into symbols in frequency domain by using a serial-to parallel-converter. These frequency domain symbols are then taken as input by the IFFT block. The IFFT block converts the input symbol into time domain symbol by doing an IFFT operation on the input. The cyclic prefix is added to the output of IFFT block by the cyclic prefix block. This symbol is then converted back to series of bits by the parallel-to-serial converter and transmitted. 

In the OFDM receiver the input signal is passed through the channel equalizer block first, to cancel any impairments introduced by the wireless channel. The output of the equalizer is then input to the prefix extraction block to remove the cyclic prefix. The output of the prefix extraction block is then given to the FFT block. This block converts the input to frequency domain output by doing an FFT operation. Thus the OFDM receiver recovers the original bits back by doing a parallel-to-serial operation. 


### What are the advantages of OFDM?

Here are some advantages: 

    + **High spectral efficiency**: Compared to other schemes like spread-spectrum, OFDM uses the available spectrum in a more efficient way.
    + **Robust against multipath fading and Inter-Symbol Interference**: Due to low data-rate in each subchannel, OFDM is more resilient to inter-symbol interference caused by multipath propagation.
    + **Simpler channel equalizer in receiver**: In case of OFDM the channel equalization can be done in frequency domain and is a multiplication operation of the received symbol with the channel equalizer.
    + **Efficient implementation**: Implementation can be done using IFFT and FFT, thus eliminating the need for multiple mixers in transmitter and receiver.
    + **Robustness against selective fading**: Since the transmission is done using multiple smaller subbands, frequency selective fading appears as flat fading for each subband.
    + **Resilience to narrow band interference**: Due to narrow band interference, contents in some of the subchannels will be lost. It is possible to recover it by using channel coding and interleaving data before transmission.
    + **Tuned subchannel receivers not required**: Unlike conventional FDM, tuned subchannel receivers are not required, thus simplifying the receiver design

### What are the disadvantages of OFDM?

Here are some disadvantages: 

    + **Sensitive to Carrier offset and frequency drift**: In case there is an offset between the transmitter and receiver carrier frequencies, the orthogonal property of the OFDM is lost. Thus OFDM systems are very sensitive to carrier frequency offsets.
    + **High Peak-To-Average power ratio**: Since the output of multiple subbands are combined to get the OFDM signal, the OFDM signal has a very high dynamic range of amplitude. This leads to complex RF design as the amplifiers need to be linear for the entire amplitude range. This also leads to lower efficiency of the RF amplifier.


## Milestones

1870

Alexander Graham Bell is initially funded by his future father-in-law Gardiner Hubbard to work on *harmonic telegraphy*, which is an **FDM** transmission of multiple telegraph channels. With FDM, more than one low rate signal is carried over a relatively wide channel using a separate carrier frequency for each signal.

1957

Collins Radio Company develops the **Kineplex** system to overcome multipath fading. 20 tones are modulated by differential 4-PSK without filtering. Tones can be separable by bank of filters at the receiver. 

1961

Franco and Lachs propose a multitone, code-multiplexing scheme using a 9-point QAM constellation for each carrier. Sine and Cosine waves used to generate orthogonal signals. 

1966

What could be termed as the birth of modern OFDM, Robert W. Chang publishes *Synthesis of Band-Limited Orthogonal Signals for Multichannel Data Transmission*. He uses Fourier transform to make the subcarriers orthogonal. His system is able to transmit signals in parallel without either ISI or ICI. 

1967

B.R. Saltzberg extends Chang's work to complex data, that is, Quadrature Amplitude Modulation (QAM). He shows that I and Q streams should be staggered by T/2, and adjacent channels the other way. Zimmerman and Kirsch publish a paper on the design of an HF (high frequency) radio OFDM transceiver (KATHRYN). This uses 34 subchannels in a 3kHz bandwidth. KATHRYN uses analog hardware to generate orthogonal signals using Discrete Fourier transform (DFT). 

1971

Weinstein and Ebert use **Fast Fourier Transform (FFT)** implementation of DFT. This greatly reduces the cost and complexity of OFDM systems. However, Weinstein notes later Bell Labs didn't show much interest in this. The big applications of OFDM (ADSL, wireless communications, digital audio/video broadcasting) came years later. Weinstein and Ebert also introduce the **guard band** for multipath channels. 

1980

Although earlier work made the subcarriers orthogonal, in a time dispersive channel the orthogonality was lost resulting in ISI. Peled and Ruiz solve this by introducing **Cyclic Extension (CE)**. Today we use the more familiar term **Cyclic Prefix (CP)**. Effective data rates are reduced but the gain in terms of zero ISI is worth it. 

1985

For better channel estimation, L.J. Cimini introduces a pilot-based method to reduce interference from multipath and co-channels. This work becomes important in the context of cellular mobile systems where channels experience fast selective fading. 

Jan  
1993

Amati's prototype of an ADSL modem wins a competition with Carrierless Amplitude-Phase (CAP) Modulation in a Bellcore-sponsored test. The technique used is **Discrete Multi-Tone (DMT)**, which is essentially OFDM. Soon ADSL becomes the first major consumer-oriented application of OFDM. It uses 256-point DFT with subcarriers separated by 4.3125 kHz and a (block) symbol rate of 4000/s. Early deployments using Amati equipment happen with British Telecom in late 1993 and early 1994, offering 2 Mbps downstream. 

1999

**802.11a** WLAN standard is published as an amendment of 802.11 from 1997. It uses OFDM in the physical layer for data transmission. 802.11a's OFDM has 52 subcarriers (4 pilot + 48 data), 64-point FFT, and 312.5 kHz of subcarrier spacing.