Orthogonal Frequency-Division Multiplexing (OFDM) 1...

EE456–Digital Communications (Fall 2013) INTRODUCTION TO OFDM

Orthogonal Frequency-Division Multiplexing (OFDM)1

1 Introduction

We have learned that QAM is a combination of amplitude modulation (ASK) and phasemodulation (PSK). On the other hand, FSK uses multiple orthogonal carrier frequenciesto send information bits. Specifically, for FSK modulation with N = 2λ frequencies, onlyone of N frequencies is activated over one symbol duration of Ts = λTb, where Tb is the bitduration. What frequency that is activated over any symbol duration is determined by themapping from λ information bits to the frequency value (or index).

OFDM is essentially a combination of QAM and FSK. Instead of activating only oneout of N carriers as in FSK, in OFDM all the carriers are activated all the time and QAMsymbols are sent over each carrier. This is illustrated in Figure 1.

1f0f 2f 1−Nf

�

f

)( fH

Red: Channel frequency response, H(f)

Blue: Subchannel spectrum in OFDM

)( 0fH )( 1fH )( 2fH )( 1−NfH

1NTf∆ =

Bandwidth N

NB

T

β+=

1

NN TB β+=

Figure 1: Spectrum of OFDM signals.

Referring to the spectrum of OFDM signal, one has the following observations:

• The data rate on each of the subchannels is much less than the total data rate, and thecorresponding subchannel bandwidth is much less than the total system bandwidth.

• The number of subchannels can be chosen so that each subchannel has a bandwidthsmall enough so that the frequency response over each subchannel’s frequency rangeis approximately constant. This ensures that inter-symbol interference (ISI) on eachsubchannel is small.

• The subchannels in OFDM need not be contiguous, so a large continuous block ofspectrum is not needed for high rate transmission.

1This note is largely based on Chapter 12 of textbook “Wireless Communications” (Cambridge Univer-sity Press, 2005), authored by Prof. Andrea Goldsmith, Department of Electrical Engineering, StanfordUniversity).

Dr. Ha Nguyen, University of Saskatchewan


Table 1: Communication services using OFDM.Wireless Wireline

IEEE 802.11a, g, n (WiFi) Wireless LANsADSL and VDSL broadband accessvia POTS copper wiring

IEEE 802.15.3a Ultra Wideband (UWB) Wireless PANMoCA (Multi-media over CoaxAlliance) home networking

IEEE 802.16d, e (WiMAX), WiBro,PLC (Power Line Communication)

and HiperMAN Wireless MANsIEEE 802.20 Mobile Broadband

DOCSIS 3.1 (Data over Cables)Wireless Access (MBWA)DVB (Digital Video Broadcast) terrestrial TVsystems: DVB -T, DVB -H, T-DMB, and ISDB-TDAB (Digital Audio Broadcast) systems:EUREKA 147, Digital Radio Mondiale,HD Radio, T-DMB, and ISDB-TSBFlash-OFDM cellular systems3GPP UMTS & 3GPP@ LTE (Long-Term Evolution),and 4G

• The modulation formats on different subchannels need not be the same. In fact onemay adaptively choose different modulation schemes according to the instantaneousquality of the subchannels.

The most attractive feature of OFDM is that its modulator and demodulator can beefficiently implemented with DSP. As will be seen later, with the DSP implementation, theISI can be completely eliminated through the use of a cyclic prefix. The main technicalissues that impair performance of OFDM are frequency offset and timing jitter, which de-grade the orthogonality of the subchannels. In addition, having QAM signals transmittedsimultaneously over all carriers, the peak-to-average power ratio (PAPR) of the compositeOFDM transmitted signals is significantly higher than that of QAM signals transmitted ona single carrier. This is a serious problem when nonlinear amplifiers are used.

OFDM is currently used in many wired and wireless systems. However, it is not a newtechnique. It was first used for military HF radios in the late 1950’s and early 1960’s. Startingaround 1990, OFDM has been used in many diverse wired and wireless applications, includingdigital audio and video broadcasting in Europe, digital subscriber lines (DSL), and the mostrecent generation of wireless LANs. There are also a number of newly emerging applicationsof OFDM, including fixed wireless broadband services, mobile wireless broadband known asFLASH-OFDM, and data-over-cables systems (DOCSIS 3.1). OFDM has also be adoptedas the air interface in the next-generation cellular systems.



QAM modulator on 0th subchannel

QAM modulator on nth subchannel

QAM modulator on (N-1)th subchannel

Serial-to-

parallel

converter

, ,[ ] [ ] [ ]n I n Q nX k V k jV k= + (2 )e n nj f tπ φ+

bits Infor.

bpsbr OFDM signal

Transmit

shaping

filter

( )p t

QAM

symbol

mapper

Re( )

�

�

Figure 2: OFDM transmitter built as multiple QAM modulators.

2 OFDM Viewed as Multicarrier Modulation with Over-

lapping Subchannels

Figure 2 shows how the OFDM signal is obtained as a sum of N QAM signals, one iscentered at its own carrier frequency. Let TN be the duration of each OFDM symbol. Thenthe relationship between TN and the information bit rate Rb (or the bit duration Tb = 1/rb

can be found as follows. Let Mn = 2λn be the size of the QAM constellation used oncarrier fn, which means that each QAM symbol sent over the nth channel can carry λn bits.The total number of bits sent over all channels in each OFDM symbol is therefore equal to

λ =∑N−1

n=0 λn. It follows that TN = λTb =∑N−1

n=0 λi

rb.

Let Xn[k] = VI,n[k] + jVQ,n[k] represent the QAM symbol that is transmitted on the nthcarrier during the kth OFDM symbol. It then follows that the transmitted OFDM signalcorresponding to the kth OFDM symbol (which carry a total of λ bits) is given as:

s(t) =N−1∑n=0

(VI,n[k]p(t) cos(2πfnt + φn)− VQ,n[k]p(t) sin(2πfnt + φn))

=N−1∑n=0

<{Xn[k]p(t)ej(2πfnt+φn)

}, (k − 1)TN ≤ t ≤ kTN (1)

In the above expression, p(t) is the (bandlimited) baseband shaping filter (e.g., to meet thezero-ISI condition) and φn is the phase of the nth carrier frequency.



Since each QAM symbol lasts over TN seconds, all the inphase/quadrature carriers{cos(2πfnt + φn), sin(2πfnt + φn)}N−1

n=0 can be made noncoherently orthogonal over the du-ration of TN if the minimum carrier spacing is ∆f = 1/TN , i.e., fn = f0 + n/TN , n =0, 1, . . . , N − 1. Now, let β be the roll-off factor of the shaping filter. Then the bandwidthof each subchannel (which is twice the bandwidth of the shaping filter p(t) in baseband) is(1 + β)/TN . Thus, the bandwidth of the OFDM signal in (1) is

B =1 + β

TN

+ (N − 1)∆f =1 + β

TN

+ (N − 1)1

TN

=N + β

TN

(2)

Figure 1 sketches the spectra of individual transmitted QAM signals. Note that thespectra of adjacent QAM signals overlap in frequency, but these signals are still orthogonalin time over the duration of TN as long as the carrier spacing is chosen as ∆f = 1

TN. It

should also be pointed out that, being a summation of N QAM signals, the OFDM signal in(1) occupies a much wider frequency band, from f0 − 1+β

2TNto fN−1 + 1+β

2TN. Thus, the OFDM

signal is transmitted through a wideband channel whose frequency response H(f) is alsodepicted in Figure 1.

For a wideband channel, the frequency response H(f) typically exhibits a noticeablevariation over the frequency, i.e., the channel is frequency-selective. The degree of frequencyselectivity of a channel is usually measured by the so-called channel’s coherence bandwidth,denoted by Bc. Loosely speaking, Bc is the frequency duration over which the channelfrequency response is approximately constant. A related parameter that characterizes thefrequency-selectivity of a channel is the multipath delay spread, which is loosely defined asthe inverse of the coherence bandwidth, i.e., Tm = 1/Bc. The meaning of this parameter isthat, if a short transmitted pulse of duration T is transmitted over the channel, then thereceived signal will have a duration of approximately T +Tm. If Tm is comparable to T , thenthe consequence of delay spread is to cause inter-symbol interference (ISI) of the transmittedsignal at the channel output.

For each QAM subchannel in OFDM, its bandwidth is BN = 1+βTN

. Thus, if BN ismade to be smaller than Bc (by using as many number of carriers as required), then eachQAM subchannel experiences a frequency-flat channel (also called flat-fading channel). Thecondition BN < Bc implies that TN > (1+β)Tm, i.e., ISI can be made negligible. Flat fadingand zero-ISI are desirable and this is achieved with OFDM transmission by using a largenumber of subcarriers.

The structure of the receiver that corresponds to the OFDM demodulator in Figure 2 isillustrated in Figure 3. Note that in the time domain, the wideband channel is representedby an impulse response h(t). However, by the preceding analysis, the complex value (a pairof real values, one for the inphase and one for the quadrature component) at the output ofthe matched filters is related to the complex QAM transmitted symbol as:

X̂n[k] = H(fn) ·Xn[k] + Wn[k] (3)

In (3), Wn[k] = W(I)n [k] + jW

(Q)n [k] is a complex Gaussian random variable. Its real and

imaginary parts are two independent and identically (i.i.d.) Gaussian random variables,each with zero mean and variance N0BN/2. If the channel gain H(fn) in (3) is known2, then

2In practice, this is done by using a channel estimation/sounding algorithm.



QAM demodulator on 0th subchannel

QAM demodulator on nth subchannel

( ) ( ) ( )s t h t t∗ + w

Nt kT=

ˆ [ ]nX k

(2 )e n nj f tπ ϕ− +

Matched

filter

( )p t∗ −

QAM

symbol

demapper

Parallel-

to-serial

converter

QAM demodulator on (N-1)th subchannel

Detected infor. bits

�

�

Figure 3: OFDM receiver built as multiple QAM demodulators.

the detection of the transmitted QAM symbol from X̂n[k] is basically the same as that foran ideal AWGN channel.

It is important to point out that the simple and favourable relationship in (3) is achievedbecause of the following two main reasons: (i) the subchannels are orthogonal (thoughthey overlap in frequency), and (ii) the number of subchannels is selected large enough sothat each subchannel undergoes flat fading, hence there is negligible or zero ISI. There are,however, implementation issues corresponding to these two reasons. First, the orthogonalityof subchannels is compromised by timing and frequency offsets. These offsets, even whenrelatively small, can significantly degrade performance, as they cause subchannels to interferewith each other. Second, when the number of subchannels is large, the requirement formultiple oscillators (both at the transmitter and receiver) makes the system very costly.

The next section presents a different implementation of OFDM systems that is basedon the discrete Fourier transform (DFT) and inverse DFT (IDFT). As will be seen, suchimplementation requires only one oscillator at the transmitter and receiver. Furthermore,though not discussed in this note, the implementation with DFT/IDFT allows one to usemany effective techniques for channel estimation and correction of timing and frequencyoffsets.



3 Implementation of OFDM with DFT/IDFT

3.1 Review of the DFT

Let x[n], 0 ≤ n ≤ N − 1, be a discrete-time sequence. The N -point DFT of x[n] is

DFT{x[n]} = X[i] ,N−1∑n=0

x[n]e−j 2πniN , 0 ≤ i ≤ N − 1. (4)

The DFT is the discrete-time equivalent to the continuous-time Fourier transform, as X[i]characterizes the frequency content of the time samples x[n]. Given X[i], 0 ≤ i ≤ N − 1,the sequence x[n] can be recovered using the IDFT:

IDFT{X[i]} = x[n] , 1

N

N−1∑i=0

X[i]ej 2πniN , 0 ≤ n ≤ N − 1. (5)

When N is a power of two, the DFT and IDFT can be efficiently performed in hardwareusing the fast Fourier transform (FFT) and inverse FFT (IFFT) algorithms.

When the discrete-time sequence x[n] is passed through a discrete-time linear time-invariant system whose impulse response is h[n], the output y[n] is the discrete-time convo-lution of the input and the channel impulse response. That is,

y[n] = h[n] ∗ x[n] = x[n] ∗ h[n] =∞∑

k=−∞h[k]x[n− k] (6)

The N -point circular convolution of x[n] and h[n], both with length N , is defined as

y[n] = h[n]⊗ x[n] = x[n]⊗ h[n] =N−1∑

k=0

h[k]x[(n− k) mod N ]. (7)

From the definition of the DFT, the below derivation shows that circular convolution in time



leads to multiplication in frequency:

DFT{y[n] = x[n]⊗ h[n]}

=N−1∑n=0

(N−1∑

k=0

h[k]x[(n− k) mod N ].

)e−j 2πni

N

=N−1∑

k=0

h[k]

(N−1∑n=0

x[(n− k) mod N ]e−j 2πniN

)

=N−1∑

k=0

h[k]

(N−1−k∑

l=−k

x[l mod N ]e−j2π(l+k)i

N

)

=N−1∑

k=0

h[k]

(N−1−k∑

l=−k

x[l mod N ]e−j 2πliN

)e−j 2πki

N

=N−1∑

k=0

h[k]

(x[N − k]e−j

2π(−k)iN + . . . + x[N − 1]e−j

2π(−1)iN +

N−1−k∑

l=0

x[l]e−j 2πliN

)e−j 2πki

N

=N−1∑

k=0

h[k]

(x[N − k]e−j

2π(N−k)iN + . . . + x[N − 1]e−j

2π(N−1)iN +

N−1−k∑

l=0

x[l]e−j 2πliN

)e−j 2πki

N

=N−1∑

k=0

h[k]

(N−1∑

l=0

x[l]e−j 2πliN

)

︸︷︷︸X[i]

e−j 2πkiN

=N−1∑

k=0

h[k]X[i]e−j 2πkiN = X[i]

(N−1∑

k=0

h[k]e−j 2πkiN

)= X[i]H[i], 0 ≤ i ≤ N − 1 (8)

3.2 OFDM Implementation

The implementation of OFDM transmitter and receiver is shown in Figure 4. The infor-mation bit stream is processed by a QAM mapper, resulting in a complex symbol streamX[0], X[1], . . . , X[N − 1]. This symbol stream is passed through a serial-to-parallel con-verter, whose output is a set of N parallel QAM symbols X[0],. . . ,X[N − 1], each symbol istransmitted over each of the N subcarriers. Thus, the N symbols from the serial-to-parallelconverter correspond to the discrete frequency components of the OFDM signal s(t). Inorder to generate s(t), these frequency components are converted into time samples by theinverse DFT (which could be efficiently implemented using the IFFT algorithm if N is apower of 2). The IFFT yields an OFDM symbol consisting of the complex-valued sequencex[0], . . . , x[N − 1] of length N , where

x[n] =1

N

N−1∑i=0

X[i]ej 2πniN , 0 ≤ n ≤ N − 1. (9)

Next, a very important operation called cyclic prefix (CP) extension is performed on the



st nT=

st nT=

IDFT

or

IFFT

Add

cyclic

prefix,

and

parallel-

to-serial

converter

p(t)

bits/secbr

[0]X

[1]X

[ 1]X N −

[0]x

[1]x

[ 1]x N −

� ( )Qx t�

( )0cos 2 f tπ

( )0cos 2 f tπ

�

Remove

prefix,

and

serial-to-

parallel

converter

[0]y

[1]y

[ 1]y N −

�

[0]Y

[1]Y

[ 1]Y N −

�

FFT

or

DFT

P/S Converte

r

p(t)( )Ix t�

( )0sin 2 f tπ

( )s t

( )tr

( )0sin 2 f tπ

[ ], [ 1], , [ 1]x x x Nµ µ− − + −� � ��

( )p t−

( )p t−

[ ], [ 1], , [ 1]y y y Nµ µ− − + −�

(a) Transmitter

(b) Receiver

bits

M0-QAM

modulator

S/P Converter

M1-QAM

modulator

MN-1-QAM

modulator

M0-QAM

demodulator

M1-QAM

demodulator

MN-1-QAM

demodulator

Figure 4: Implementation of OFDM transmitter and receiver with IDFT/DFT.

OFDM symbol to produce time samples

{x̃[n]} = {x̃[−µ], . . . , x̃[−1], x̃[0], . . . , x̃[N − 1]} = {x[N − µ], . . . , x[N − 1], x[0], . . . , x[N − 1]} .(10)

Let TN be the duration of one OFDM symbol, then the distance between two adjacent timesamples is Ts = TN/N . With µ being the number of time samples in the cyclic prefix, thelength of the cyclic prefix is µTs = µ

NTN . The purpose of this operation as well as what

determines the length µ of the cyclic prefix shall be discussed shortly. The operation of CPextension is graphically illustrated in Figure 5.

[ ], [ 1], , [ 1]x N x N x Nµ µ− − + −� [0], [1], [2], , [ 1]x x x x N µ− −�� [ ], [ 1], , [ 1]x N x N x Nµ µ− − + −�

Append last symbols to the frontµ

Cyclic prefix (CP) of length µ Original signal sequence of length N

≈

Figure 5: Illustration of cyclic prefix extension.



After CP extension, there time samples are ordered by the parallel-to-serial converterand passed through a shaping filter, resulting in the complex baseband OFDM signal x̃(t) =x̃I(t) + jx̃Q(t). The baseband OFDM signal is then upconverted to frequency f0. Thetransmitted signal is filtered by the channel impulse response h(t) and corrupted by additivewhite Gaussian noise w(t). The received signal is r(t) = s(t) ∗ h(t) + w(t). This signalis downconverted to baseband and filtered to remove the high frequency components. Theoutput of the matched filter is sampled every Ts to obtain y[n], −µ ≤ n ≤ N − 1.

It is of interest to obtain a relationship between the set of “received” sample values y[n]and the set of “transmit” time samples x̃[n], −µ ≤ n ≤ N − 1. This is done as follows.Recall that the channel impulse response h(t) is characterized by a coherence bandwidthBc and delay spread Tm. While it is simple to use a large number of carrier N to makeTN > (1 + β)Tm (or equivalently BN < Bc) so that each subchannel in OFDM experiencesflat fading, as far as the duration between adjacent time samples x̃[n] and the samplingperiod used to obtain y[n] is concerned, it is clear that Ts = TN/N can be smaller than Tm,

causing a significant “inter-symbol” interference in each of the value of y[n]. Let µ =⌈

Tm

Ts

⌉,

then the equivalent discrete-time channel that defines the relationship between y[n] and x̃[n]can be represented by the impulse response {h[0], h[1], . . . , h[µ]}. This is illustrated in Figure6.

Equivalent

discrete-time channel

{ }0

[ ]n

h nµ

=

, [ ], [ 1], , [ 1],x x x Nµ µ− − + −� � �� , [ ], [ 1], , [ 1],y y y Nµ µ− − + −� � �

Figure 6: Equivalent discrete-time channel.

By the operation of the CP insertion, one has x̃[n] = x[n mod N ] for −µ ≤ n ≤ N − 1,which also means that x̃[n−k] = x[(n−k) mod N ] for −µ ≤ n−k ≤ N−1. Thus, given x̃[n]is the input of the channel h[n] (i.e., the noise component is ignored), the channel outputbetween 0 ≤ n ≤ N − 1 can be computed as:

y[n] = x̃[n] ∗ h[n] =

µ∑

k=0

h[k]x̃[n− k]

=

µ∑

k=0

h[k]x[(n− k) mod N ] = x[n]⊗ h[n] (11)

where the third equality follows from the fact that, for 0 ≤ k ≤ µ, x̃[n−k] = x[(n−k) mod N ]for 0 ≤ n ≤ N − 1.



The above analysis shows that, by appending a cyclic prefix to the channel input, thelinear convolution associated with the channel impulse response becomes a circular convo-lution. Taking into account AWGN at the channel input, the DFT of the channel output inthe yields

Y [i] = DFT{y[n] = (x[n] + w[n])⊗ h[n]} = X[i]H[i] + W [i], 0 ≤ i ≤ N − 1, (12)

where W [i] is Gaussian noise component. Note that y[n], has length N +µ, yet from (11) thefirst µ samples y[−µ], . . . , y[−1], are not needed. This is due to the redundancy associatedwith the cyclic prefix. Moreover, if we assume that the input x[n] is divided into data blocksof size N with a cyclic prefix appended to each block to form x̃[n], then the first µ samplesof y[n] = h[n] ∗ x̃[n] in a given block are corrupted by ISI associated with the last µ samplesof x[n] in the previous block. The cyclic prefix serves to eliminate ISI between the datablocks (i.e., OFDM symbols) since the first µ samples of the channel output affected by thisISI can be discarded without any loss relative to the original information sequence. This isgraphically illustrated in Figure 7.

[0], , [ 1]y y N −�ISI

µ N

[0], , [ 1]y y N −�ISI

µ N

[0], , [ 1]y y N −�ISI

µ N

CP CP CPData block Data blockData block

��

[0], , [ 1]x x N −�CP [0], , [ 1]x x N −�CP [0], , [ 1]x x N −�CP CP

Figure 7: CP insertion not only turns linear convolution to circular convolution, but alsoeliminates ISI between data blocks.

The benefits of adding a cyclic prefix come at a cost. Since µ symbols are added tothe input data blocks, there is an overhead of µ/N and a resulting data-rate reduction ofN/(µ+N). The transmit power associated with sending the cyclic prefix is also wasted sincethis prefix consists of redundant data. Based on the relationship in (11) and (12), the prefixof y[n] consisting of the first µ samples is removed. The remaining time samples are serial-to-parallel converted and passed through an FFT. This results in scaled versions of the originalQAM symbols, H[i]X[i], where H[i] = H(fi) is the flat-fading channel gain associated withthe ith subchannel. The FFT output is parallel-to-serial converted and passed through aQAM demodulator to recover the original data.

The implementation of OFDM discussed in this section effectively decomposes the wide-band channel H(f) into a set of narrowband orthogonal subchannels with a different QAMsymbol sent over each subchannel. Knowledge of the channel gains H[i], i = 0, . . . , N − 1, isnot needed for this decomposition. However, the demodulator needs the values of these chan-nel gains to recover the original QAM symbols by dividing out these gains: X[i] = Y [i]/H[i].This process is called frequency equalization. It is pointed out that, frequency equalizationleads to noise enhancement, since the noise in the ith subchannel is also scaled by 1/H[i].



4 Application Example of OFDM: The IEEE 802.11a

Wireless LAN Standard

The IEEE 802.11a Wireless LAN standard, which occupies 20 MHz of bandwidth in the5 GHz unlicensed band, is the first version of 802.11 family that is based on OFDM. The802.11g standard is virtually identical, but operates in the smaller and more crowded 2.4GHz unlicensed ISM band. IEEE 802.11ac, released in December 2012, also operates inthe 5 GHz band. It uses a wider RF bandwidth (80 or 160 MHz), multiple-input multiple-output (MIMO) technology (up to 8 MIMO streams), high-density modulation (up to 256QAM) and could deliver a maximum data rate close to 7Gbps (on eight 256-QAM channels,each delivering 866.7Mbps). The latest development of 802.11 standard is 802.11ad, whichoperates in the tri-band 2.5/5.0/60 GHz. This section discusses the properties of OFDMdesign used in 802.11a standard and some of the design choices.

In 802.11a standard, N = 64 subcarriers are generated. However, only 48 carriers areactually used for data transmission, the outer 12 carriers are zeroed in order to reduceadjacent channel interference, and 4 carriers used as pilot symbols for channel estimationand synchronization. The cyclic prefix consists of µ = 16 samples, so the total number ofsamples associated with each OFDM symbol, including both data samples and the cyclicprefix, is 80. The transmitter gets periodic feedback from the receiver about the packet errorrate, and uses this information to pick an appropriate error correction code and modulationscheme. The same code and modulation must be used for all the subcarriers at any giventime. The error correction code is a convolutional code with one of three possible code rates:rc = 1/2, 2/3, or 3/4. The modulation types that can be used on the subchannels are BPSK,QPSK, 16-QAM, or 64-QAM.

Since the 12 outer carriers are not used in order to reduce adjacent channel interference,no pulse shaping is needed and β is set to 0. The bandwidth B (and sampling rate 1/Ts)is 20 MHz, and there are 64 subcarriers evenly spaced over that bandwidth, the subcarrierbandwidth (using β = 0) is:

BN =20 MHz

64= 312.5 KHz. (13)

Since µ = 16 and 1/Ts = 20MHz, the maximum delay spread for which ISI is removed is

Tm < µTs =16

20 MHz= 0.8 µsec, (14)

which corresponds to delay spread in an indoor environment. Including both the data andcyclic prefix, there are 80=64+16 samples per OFDM symbol. Thus the symbol time persubchannel is

T(+CP)N = TN + µTs = (N + µ)Ts = 80Ts =

80

20× 106= 4 µsec (15)

The data rate per subchannel is log2 M/T(+CP)N . Thus, the minimum data rate for this

system, corresponding to BPSK (1 bit/symbol), an r = 1/2 code, and taking into accountthat only 48 subcarriers actually carry information data, is given by

(rb)min = 48 sub.× 1/2 bit

coded bit× 1 code bit

sub. symbol× 1 sub. symbol

4× 10−6= 6Mbps (16)



The maximum data rate corresponds to 64-QAM and r = 3/4 code. It is given by,

(rb)max = 48 sub.× 3/4 bit

coded bit× 6 code bit

sub. symbol× 1 sub. symbol

4× 10−6= 54Mbps (17)

Naturally, a wide range of data rates between these two extremes is possible.

5 Problems

1. Find the data rate of an 802.11a system assuming 16-QAM modulation and rate-2/3channel coding.

2. Now consider a channel with a delay spread of Tm = 20 µsec. You want to designan OFDM system for use with such a channel so that the subchannel bandwidth isBN = Bc/2, where Bc is the coherence bandwidth of the channel.

(a) Suppose that a square-root raised-cosine pulse with β = 0.35 is used and there are128 carriers in your design. Using the approximation of Bc ≈ 1/Tm, determinethe total bandwidth occupied by the OFDM system.

(b) Assuming a constant SNR = Es/N0 = 20 dB on each subchannel, find the max-imum constellation size for M -QAM that can be sent over each subchannel witha target symbol error probability of 10−3. Find the corresponding total data rate(bits/sec) and bandwidth efficiency (bits/sec/Hz) of the system. Assume that Mis restricted to be a power of 2 and use the upper-bound in Equation (8.48) forestimating the symbol error probability of M -QAM.

3. Consider using OFDM technique over an AWGN channel, where there is no multipathproblem and hence no need to use cyclic prefix or zero padding. Let N be the numberof orthogonal subcarriers employed and assume that M -QAM modulation is used foreach subcarrier.

(a) Determine the bandwidth efficiency, i.e., rb/W .

(b) Simply argue (without any derivation) what is the symbol error probability of thesystem. Together with the result in (a) what do you say about the representationof M -QAM OFDM on the bandwidth-power plane?

4. Write a Matlab program to simulate the OFDM design of IEEE 802.11a. The programshould include the OFDM transmitter, equivalent discrete-time channel, AWGN andOFDM demodulator. For the equivalent discrete-time channel, generate the channelfilter coefficients {h[n]}µ

n=0 as i.i.d. zero-mean complex Gaussian random variables,with variance 1/2 for real and imaginary parts. For the OFDM demodulator you canassume that the channel coefficients {h[n]}µ

n=0 are perfectly known. Test your Matlabprogram with 64-QAM modulation on all data subcarriers and obtain a plot of the biterror rate versus received Eb/N0.


Orthogonal Frequency-Division Multiplexing (OFDM) 1...

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