# 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

$$X(k)=\sum_{n=0}^{N-1}x(n)sin(\frac{2\pi kn}{N})+j\sum_{n=0}^{N-1}x(n)cos(\frac{2\pi kn}{N})$$

where as its dual , IFFT is given by

$$ x(n)=\sum_{n=0}^{N-1}X(k)sin(\frac{2\pi kn}{N})-j\sum_{n=0}^{N-1}X(k)cos(\frac{2\pi kn}{N})$$

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.

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

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 = Nfft)

Number of data subcarriers = 48 ( Nd)

Number of pilot subcarriers = 4 (Np)

Derived parameters from the above specification.

Number of total USED subcarriers = 52 ( Nu = Nd+ Np )

Number of total UNUSED subcarriers = 12 ( Nun = Nfft – Nu ).

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.

Full Matlab Code available at Simulation of Digital Communication Systems using Matlab – an ebook by Mathuranathan Viswanathan

### Reference:

### 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

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