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Nachrichtentechnik (SS2011) Matlab Project 2

Matlab Project 2: OFDM

1 Introduction

In this exercise we take a look at Orthogonal Frequency Division Multiplexing (OFDM).This document is structured as follows:

First we give a short introduction to OFDM, its applications, advantages and draw-backs. Section 2 gives an overview of a mathematical description of an OFDM trans-mission system. Then the details of your MATLAB implementation tasks are describedin Section 3, and Section 4 contains questions that are to be answered by you.

OFDM is known as an ecient technique for high data rate transmission systems.

The main advantages of OFDM are:

For longer delay spreads, OFDM is an ecient scheme to cope with multipathpropagation.

In relatively slow timevarying channels, it is possible to enhance the throughputby adapting the data rate per subcarrier according to the signalto noise ratioof that particular subcarrier. The drawback of this procedure is the need of areverse channel that yields the information about the signalto noise ratio of therespective subcarrier.

OFDM is quite robust against narrowband interference, because this interferenceaects only a few subcarriers. This makes it a suitable transmission scheme forpowerline communications, which suer from many narrowband interferers.

OFDM allows singlefrequency networking, which is especially attractive forbroadcasting applications like DVBT or DAB.

There are also some drawbacks compared to singlecarrier modulation:

OFDM is more sensitive to frequency oset and phase noise. OFDM has a relatively large peaktoaverage power ratio, which tends to reduce

the power eciency of the RF amplier.

OFDM displays large outofband power, which requires very good lters toincrease the stopband attenuation.

In several practical applications the multipath spread experienced in the realsystem is longer than the specied maximum multipath spread the OFDM systemis designed to cope with, which leads to a severe performance degradation ofconventional detectors.

OFDM also loses the subchannel orthogonality if the channel impulse responsechanges during one OFDM block.

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Matlab Project 2 Nachrichtentechnik (SS2011)

2 Overview and system description

The following gure shows an uncoded OFDM transmission over a linear distortingtimeinvariant channel hK(t) in the equivalent lowpass domain:

A digital data source emits symbols x[i], taken from the symbol alphabet Ax. A serialtoparallel converter puts together blocks of NF symbols x[i] to obtain the columnvectors

x[k] = (x1[k], x2[k], . . . , xNF [k])T

where k is the block index and NF denotes the number of OFDM subcarriers. Theelements xi[k] are taken from the symbol alphabet Ax. Source and channel codingmay be included in x[k], but are not considered in this exercise. Each data block x[k]is modulated onto NF subcarriers by a normalized inverse discrete Fourier transform(IDFT), which can be described by a multiplication with an inverse Fourier matrix F1

as follows:s[k] = F1x[k], with F1mn =

1NF

ej 2NF

(m1)(n1) (1)

The last NG elements of the resulting vector s[k] are copied and put as cyclic prexto form the OFDM block s[k], a column vector with NF + NG elements, which can beexpressed as

s[k] = (sNFNG+1[k], . . . , sNF [k], s1[k], s2[k], . . . , sNF [k])T (2)

After paralleltoserial conversion the diracsampled transmit signal is ltered by thetransmit lter hTf(t) that is bandlimited to the cuto frequency fg = 12t . The re-sulting signal s(t) is passed through the frequency selective and time invariant channelhK(t). After adding complex white gaussian noise with twosided power spectral densityN02 , the signal g(t) is passed through the receive lter hRf(t) that is again bandlimited

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to the cuto frequency fg = 12t . By sampling at it and serialtoparallel convertingwe obtain the vector sequence y[k].

In this exercise we will describe the concatenation of the blocks hTf(t), hK(t), andhRf(t) with the equivalent discretetime channel representation

[i] =14

hTf(t) hK(t) hRf(t)t=it

. (3)

The receiver removes the rst NG elements from the vector y[k], yielding y[k], and per-forms the demodulation by the normalized discrete Fourier transform (DFT), describedby

x0[k] = F y[k], with Fmn =1NF

ej 2

NF(m1)(n1) (4)

The zeroforcing (ZF) detection can be described as an elementwise division of x0[k]by the transfer function of the discretetime channel, which is computed by a DFT ofthe zeropadded discretetime channel impulse response

:= DFT([0], . . . , [L1], 01(NFL)

)T

=

NF F ([0], . . . , [L1], 01(NFL)

)T(5)

where L denotes the length of [i]. After a paralleltoserial conversion we obtain theestimated symbols x[i].

3 MATLAB tasks

Write a script OFDMsim.m that simulates an OFDM transmission. Consider the stepsdescribed in the following subsections 3.1 and 3.2.

3.1 Basic transmission chain

Generate a (N F N Block) matrix q mat of random bits, using the functionRANDBITS from exercise 1. This matrix describes our packet of N Block OFDM RANDBITSblocks in the frequency domain. Use the parameters N F = 16 and N Block = 3.(Remark: Such kinds of packets are often called frames in practice.)

Map the matrix q mat of random bits to the matrix x mat of transmit symbolsfrom the BPSK symbol alphabet Ax = {1,+1} using the function BIT2SYM from BIT2SYMexercise 1.

Write the functions FFTNORM and IFFTNORM which perform columnwise a normal-ized (inverse) fourier transform according to Equations (4) and (1), using thefunctions FFT and IFFT. FFT, IFFT


Matlab Project 2 Nachrichtentechnik (SS2011)

Perform the normalized IFFT (function IFFTNORM) over the matrix x mat, result- IFFTNORMing in the matrix s mat.

Add a cyclic prex of N G elements to each column of s mat according to Equation(2), resulting in the matrix s mat guard. Choose N G = 3.

Execute a paralleltoserial conversion of the matrix s mat guard and performa convolution of the resulting vector with the discretetime channel impulse re-sponse [i]. The vector after this convolution shall be named ch out.In this part of the exercise, use the threepath, fourtap channel with [i] =0.7454 [i] + 0.5963 [i 2] + 0.2981 [i 3].

Generate a vector n vec of the same size as ch out with samples from complexvalued Gaussian noise of variance (sigma n)2 with sigma n as parameter. Usethe function RANDN. In this part of the exercise, choose sigma n = 0.5.RANDNAdd ch out and n vec, resulting in g vec.

RESHAPE the vector g vec to a ((N F+ N G) N Block) matrix g mat, hereby ex-RESHAPEtracting the (N F+N G)N Block elements of g mat corresponding to s mat guard.

Remove the rst N G rows from g mat and perform the normalized FFT (functionFFTNORM) over the remaining matrix, resulting in the matrix x schl.FFTNORM

Compute the frequency response alpha vec of the discretetime channel impulseresponse [i] according to Equation (5). In order to avoid numerical problemsin the ZF detection, dene a parameter epsilon = 0.01 and set the elements ofalpha vec with an absolute value smaller than epsilon to 1.

Perform the ZF detection using the modied vector alpha vec, execute a signdecision, yielding the matrix x hut of decided symbols. Compute the number ofbit errors in this packet.

3.2 BER simulations

In Section 3.1 we have transmitted one packet of N Block = 3 OFDM blocks. In orderto get BER graphs, we will now transmit N Packet = 1000 packets for each of the EbN0values from 0 dB to 23 dB in steps of 2,5 dB (to be called EbN0dB in the MATLAB le).

Compute sigma n from the parameters channel energy and EbN0dB. Do notforget to consider the increase in transmit energy by the N G symbols of the cyclicprex.

Create an outer EbN0 loop and an inner packet loop. This inner packet loop sumsup the packets bit errors.

Carry out simulations for case 1: N G = 3

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case 2: N G = 0

and the following normalized channels:

ideal AWGN channel [i] = [i]

twopath channel [i] = 0.7071 [i] + 0.7071 [i 1] threepath, fourtap channel [i] = 0.7454[i]+0.5963[i2]+0.2981[i3]

Create three BER graphs (one for each channel model), each displaying the detec-tion results for N G = 3, N G = 0, and the analytical curve for the AWGN channel(compare exercise 1).

4 Questions

1. Compare the transfer functions of the three simulated channels.

2. Classify the simulated channels by the resulting BERs. Which channel shows theworst performance?

3. What is the BER for the twopath channel [i] = 0.7071 [i] + 0.7071 [i 1]in case of EbN0 ? Give reasons.

4. Explain the degradation of the BER for NG = 3 in the case of the threepath,fourtap channel [i] = 0.7454[i]+0.5963[i2]+0.2981[i3] in comparisonwith the AWGN BER.

5. What is the reason for the dierence of the BERs at the threepath, fourtapchannel [i] = 0.7454 [i] + 0.5963 [i 2] + 0.2981 [i 3] for NG = 0 andNG = 3? Is the used ZF detector the appropriate detector in case of NG = 0?


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