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University of Manchester CS3282: Digital Communications 2007

Section 8: Carrier modulated digital transmission (new)

8.1. Introduction:

In designing digital transmission systems, the objective is to convert binary data into a form which

is best suited to the characteristics of the channel. These characteristics include the usable

frequency range and the gain and phase distortion caused within this usable range. With a ‘base-

band’ channel, the usable frequency range extends down to, or almost down to, 0 Hz. This would be

the case with direct transmission over a pair of wires. With base-band transmission, suitably shaped

pulses, for example rectangular shaped pulses, are the symbols which represent the data.

8.1.1 Modulation of single sine-wave carriers: This section concerns the transmission of data over

channels which may not be considered base-band: for example a channel of bandwidth 200 kHz

centred on 900 MHz. Such transmission may be achieved by a ‘modulating’ a single sine-wave, of

suitable frequency, by base-band symbols. Modulation means that some aspect of the carrier, such

as its amplitude or its frequency, is varied in sympathy with the base-band signal. This variation

must be detectable at the receiver. Before modulation, a pure sine-wave exists at only one

frequency and therefore has an infinitesimally narrow bandwidth. The modulation spreads the

energy about the nominal frequency according to the spectrum of modulating signal. The

modulation thus shifts the base-band spectrum up in frequency to a ‘pass-band’ range of frequencies

centred on the carrier frequency. This must be within the usable frequency range offered by the

channel. The modulation may be achieved by multiplying (in the time-domain) the carrier by the

base-band signal, though this is by no means the only possible way of modulating a carrier by a

base-band signal.

8.1.2. Spread spectrum modulation: The use of a single sine-wave as a carrier is common, but is not

the only possible choice. A widely used alternative is a ‘pseudo-random’ signal whose

characteristics need be known only at the transmitter and receiver. The bandwidth of the modulated

pseudo-random carrier signal is generally much wider than that of a modulated sine-wave, typically

by a factor of about 50. This may appear very wasteful of bandwidth. However, because the carrier

is randomised, it will appear, even when modulated, as noise to receivers not tuned to its exact

characteristics. It is as though the transmission is ‘coded’ by the pseudo-random carrier, and

security is an added bonus. Two transmitter-receiver systems using different pseudo-random

carriers can co-exist in the same channel, each experiencing a small degree of background noise

from the other transmitter. The noise from the other transmitter may increase the bit-error rate but

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by an amount that is tolerable. More transmitter-receivers may be accommodated until the bit-error

rate introduced by the accumulated background noise becomes too severe.

This ‘single carrier approach is known as ‘spread spectrum multiplexed access’ (SSMA). It is

based on the use of a ‘direct sequence’ of pseudo-random bits to produce the pseudo-random

carrier, and is therefore referred to as ‘direct sequence’ SSMA (or DS-SSMA). It is also referred to

as ‘code division multiplexed access’ (CDMA) and is the basis of most 2G mobile phone systems in

the USA. Third generation mobile telephony will be based on an enhanced form of CDMA.

8.1.3. Multi-carrier modulation: Great advances have been made in recent years with the

simultaneous use of multiple carriers rather than just a single carrier. Multi-carrier modulation

techniques use a set of ‘sub-carriers’ rather than just one carrier. The sub-carriers are currently

sinusoidal, though the use of multiple CDMA carries has been the subject of some research papers.

As well as being efficient in its use of bandwidth, multi-carrier modulation is well suited to radio

channels subject to frequency-selective fading. Frequency-selective fading occurs due to reflections

of radio signals from buildings and walls interfering with each other and cancelling each other out

at certain frequencies. It becomes more and more serious as we widen the band-widths of channels

to achieve higher bit-rates. The final part of this section introduces ‘orthogonal frequency division

multiplexing’ (OFDM) as a means of implementing a highly bandwidth-efficient 'multi-carrier'

modulation scheme. OFDM has many advantages for wireless computer networks (WLANs), and is

also used for digital radio and TV broadcasting. . Digital radio and TV broadcasting typically use

1024 or more ‘sub-carriers’ and current wireless LAN standards use 64. Fortunately the modulation

process is achieved on all 1024 or 64 sub-carriers by means of a single fast Fourier transform FFT

computation. The use of the FFT and a ‘cyclic’ extension has a profound effect on the pulse-

shaping, matched filtering and equalisation problems arising with single carrier systems. It will be

seen that these processes are greatly simplified.

8.2. Modulation

8.2.1: Amplitude & frequency modulation: With analogue. signals, the most well known

modulation techniques are amplitude modulation (‘am’) and frequency modulation (‘fm’) as used

for analogue radio and TV broadcasting. Double sideband 'am' modulation is achieved by

straightforward multiplication of a sine-wave carrier by a base-band signal. In the case of radio

broadcasts, the base-band signal is speech or music. With digital transmission, the same approach

may be used where the base-band signal is a sequence of symbols or pulses. Multiplication of a

carrier by the base-band signal makes the amplitude of the carrier variable and imposes an envelope

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shape determined by the base-band signal. The spectrum of the modulated carrier will have a lower

and upper side-band. Therefore 'am' doubles the bandwidth of the base-band signal or pulse train as

illustrated below.

Effect of amplitude modulation on spectrum of a baseband signal

To understand why a lower and an upper sideband are produced, consider what happens to a single

sinusoid, cos(ωMt), say within the base-band. When this is multiplied by the carrier A cos(ωCt),

(with ωC = 2πfC ), we obtain:

A cos(ωCt) . cos(ωMt) = 0.5A cos(ωCt + ωMt) + 0.5A cos(ωCt - ωMt) = 0.5A cos( (ωC + ωM) t ) + 0.5 A cos((ωC - ωM)t) So now we have two cosine waves, one at ωC + ωM within an upper sideband and one at ωC − ωM

within a lower side-band.

Demodulation of the ‘double side-band’ signal is made easier if the base-band modulating signal is

never allowed to become negative. The amplitude of a sine-wave is by definition a positive

quantity (or zero). You cannot have a sine wave of amplitude say -2 volts, though, of course, the

value of the sine-wave could be -2 volts at certain times. In the calculation above, cos(ωMt) does

become negative so, strictly, cos(ωMt) would not be suitable as a base-band signal for ‘amplitude

modulation’. However, if a constant voltage is added to the base-band signal, it can be made purely

positive. This is always done with broadcast ‘am’ radio stations. If we add 1 volt to cos(ωMt) in

the calculation above, the modulating signal becomes non-negative and the modulated output

becomes

Acos (ωC t ) + 0.5A cos( (ωC + ωM) t ) + 0.5 A cos((ωC - ωM)t)

This is a ‘large carrier’ double sideband ‘am’ signal. Its generation is illustrated below:

carrier

frequency

Power spectral density

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Demodulation of am can be carried out in various ways to regenerate the original base-band signal.

The simplest demodulation technique for ‘am’ (large carrier DSB), as used in simple radio

receivers, is known as ‘envelope detection’. A simple envelope detector uses a diode circuit to

‘rectify’ the ‘am’ signal so that its envelope may be obtained by low-pass filtering. The process is

illustrated below.

Envelope detection is referred to as ‘non-coherent’ demodulation and it has the great advantage of

simplicity. Coherent demodulation is more complex as it requires the receiver to generate in some

way a ‘local carrier’ which is an exact copy of the sine-wave that carries the received modulated

signal. It must be exact in frequency and phase and it must be derived from the received signal.

Rectify

Low-pass filter

t

V

t

V

V

t

V

t V

Multiply

V

t

t

1+cos(ωMt)

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Local carrier generation at the receiver will be discussed in the next section. Assuming we have

done this, a coherent demodulator is shown below

If the received signal is A cos(ωCt) .(1+cos(ωMt) ), multiplying by the local carrier gives

A cos2 (ωCt) . cos(ωMt) = 0.5 A(1 + cos(2ωCt)) .(1 + cos(ωMt) )

= 0.5 A . (1+cos(ωMt)) + 0.5A cos(2ωCt) . (1+cos(ωMt) )

The low-pass filter removes the component 0.5A cos(2ωCt) . (1+cos(ωMt) ) since it is a high

frequency signal centred on 2ωC , i.e. twice the carrier frequency. What remains after low-pass

filtering is the required base-band signal (1+cos(ωMt) ).

Coherent detection no longer requires the modulating signal to be purely positive, and it would still

work if the modulating signal were cos(ωMt) rather than (1+cos(ωMt) ). This would no longer be

large carrier double sideband ‘am’. It would be double sideband suppressed carrier modulation.

This does not mean that carrier sine-wave amplitudes must be considered negative when the

modulating signal is negative. What happens is that. a negative modulating amplitude causes a180

degrees phase shift in the carrier since cos (θ + 180o) = − cos (θ). So we are effectively modulating

the phase of the carrier as well as its amplitude.

With a digital transmission, the modulating signal is no longer a raised cosine or music signal but is

instead a pulse sequence with shaping applied, i.e. a base-band signal as seen in previous sections.

So pulse-shaping is carried out before modulation and matched filtering and equalisation may

carried out after demodulation. Therefore all the theory developed in previous sections about pulse-

V

t

V

Multiply

V

t

t 1+cos(ωMt)

Low-pass filter

Local carrier

Received signal

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shaping, matched filtering and equalisation applied to base-band transmission is equally applicable

when the base-band is transmitted by modulating a carrier.

8.2.2 Vector modulator & complex base-band: With carrier modulated transmissions, we can do

something that is not possible with base-band transmission. That is to use a 'vector-modulator' to

independently modulate a 'cosine' carrier and a 'sine' carrier of the same frequency. The cosine and

sine carriers are summed to produce a ‘single carrier’ transmission . The receiver is able to recover

the modulation carried by the cosine component of this summed transmission while being ‘blind’ to

the modulation carried by the sine component. Also it can simultaneously recover the modulation

carried by the sine component while being ‘blind’ to the cosine component. The simultaneous

demodulation of the cosine and sine components is achieved by a ‘vector demodulator which relies

on the ‘orthogonality of sine and cosine functions. The vector-demodulator must be 'coherent'

which means that precise sine and cosine versions of the carrier must be generated at the receiver. .

The use of a vector-modulator doubles the 'bandwidth efficiency' (bits/second per Hz) and can

therefore compensate for the doubling of bandwidth that occurs with carrier-modulation.

A convenient mathematical representation of a carrier modulation/demodulation scheme is to define

a 'complex base-band'. Instead of being a real signal, the base-band becomes complex; the real part

(or 'in-phase' part) modulates cos(2πfCt) and the imaginary part (or 'in quadrature' part) modulates

sin(2πfCt) where fC is the carrier frequency in Hertz.. The real and imaginary parts can carry data

independently and can be considered independent channels, even though the frequencies of the

'cosine' version and the 'sine' version of the carrier are the same. More about 'complex base-band'

and 'vector-modulation' later.

8.2.3 Modulation for digital transmission: To frequency-shift the energy of a base-band symbol and

center it at a given frequency, e.g. 900 MHz for a typical cellular mobile radio link, we can

modulate the amplitude, frequency and/or phase of a carrier cos(2πƒC t). These three basic forms of

modulation when used independently give us

(a) amplitude shift keying (ASK)

(b) frequency shift keying (FSK) and

(c) phase shift keying (PSK).

There are many versions of each of these three forms of digital modulation which are illustrated below, and it is possible to use a combination of more than one form..

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Binary FSK

Binary ASK

Binary PSK

Modulate carrier

Map to base-band

10110

tvolts

t

Map to base-band

10110

tvolts

Multiply

Map to base-band 10110

tvolts

Multiply

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Multi-level (4-ary) amplitude shift keying (ASK)

Combined multi-level ASK & PSK

The ‘map to baseband’ process takes a sequence of bits and produces the required analogue

baseband signal. Rectangular pulse shapes are shown in the diagrams above, but in practice sinc-

like pulses with appropriate zero-crossings would be used. Notice that the difference between

binary ASK and binary PSK lies only in the base-band signal which is unipolar (& always positive)

for binary ASK and bipolar (positive and negative) for PSK.

Now we first consider ASK in more detail.

8.3. Amplitude-shift keying (ASK)

Map to base-band

10110

t

volt

Multiply

tvo

volt

Map to base-band

10110

tvolts

Multiply

t

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8.3.1 Generation of ASK: Binary ASK may be generated by applying a unipolar base-band

signalling waveform b(t) to a multiplier as shown below. This assumes rectangular pulses for ease

of drawing.

With such rectangular pulses, the spectrum for ASK would be infinitely wide. If the rectangular

pulses were replaced by non-time-limited ‘50%’ RRC pulses, the resulting spectrum would have

magnitude approximately as shown below:

The maximum obtainable band-width efficiency with RRC pulses is 1/T symbols in 1-sided

bandwidth of 1/T Hz. This is a 'bandwidth efficiency' of '1 symbol / second / Hz' or '1 bit/s per Hz' if

binary signalling is used. Remember the maximum bandwidth efficiency was 2 bits/second/Hz for

base-band binary signaling, obtained with a 0% RRC filter. The doubling of bandwidth is due to the

‘double side-band’ modulation, i.e. multiplication by cos(2πfCt) . To achieve the band-width

efficiency of 1 b/s/Hz with binary ASK, a 0% RRC spectrum (corresponding to a ±1/(2T) Hz brick-

wall shaping filter) base-band pulse shape would be needed. The resulting time-domain base-band

shape, in theory of infinite duration with appropriate zero-crossings, becomes the envelope of the

modulated signalling waveform.

ƒ

|R(ƒ)|

-ƒ0 ƒ0 ƒ0-1/T ƒ0+1/T ƒ0+ 1/2T

t

r(t)

cos(2πƒC t)

b(t)

t

b(t) r(t)

Map ..1 0 1 …

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8.3.2. Non-coherent detection of ASK: Non-coherent detection means that demodulation is carried

out without the aid of a clock which is locked in frequency and phase with that of the carrier

cos(2πƒC t). A possible method for non-coherent detection is shown below. It is a form of

'envelope detector. The diode and resistor allow current to flow only in one direction to produce the

'half-wave rectified' voltage waveform shown (centre) when the input voltage is the ASK waveform

shown on the left. This is a type of rectifier referred to as a ‘half-wave rectifier’. When the half-

wave rectified waveform is smoothed by a low-pass filter (sometimes a simple capacitor will

suffice) the voltage waveform shown on the right is obtained. This waveform may be sampled at

appropriate points in time to recover the original bit-stream

t

Low-pass filter (smoother)

t t t

V V

Sample Diode

Resistor

r(t) t

cos(2πƒct)

Shaping

Map ..1 0 1..

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8.3.3 Constellation diagram for ASK: Constellation diagrams show ‘in phase’ and ‘quadrature’

components of a complex base-band signal as a graph as illustrated below for two examples. The

'in-phase' component modulates cos(2πƒCt) and the quadrature component modulates sin(2πƒCt).

This graph can be considered an 'Argand diagram' with real axis labeled 'in-phase' and imaginary

axis labeled 'in-quadrature'. So far with ASK we have only modulated cos(2πƒC t) so the complex

base-band signal has zero imaginary part.

Binary ASK with symbols 0 & Acos(..)

In phase with carrier

Quadrature to carrier Q

I

4-ary ASK with symbols 0, Acos(..), 2Acos(..), 3Acos(..)

0 A 2A 3A

8.3.4 Coherent detection of ASK

Multiply by local carrier locked in frequency and phase with the carrier received. If received signal

is s(t)cos (2πƒct) and the local carrier is cos (2πƒct) the output from the multiplier is

)4cos()()(5.0)(cos)( 2cc ftststfts ππ +=

Filtering by a low-pass filter will remove cos(4πƒct) to leave us with 0.5s(t).

8.3.5 Coherent versus non-coherent detection for ASK

Let the signal be: b(t)cos(2πƒct). Additive noise n(t) may be expressed as: N(t)cos(2πƒct + θ(t))

where N(t) is a random envelope and θ(t) is random phase. This equals:

)2sin()(sin)( )2cos()(cos)( tfttNtfttN cc πθπθ +

Half the noise power is in phase with cos(2πƒct ) and half of it is in phase with sin(2πƒct ) the latter

half being in quadrature with cos(2πƒct ).

Low Pass Filter

cos(2πƒC t)

s(t)cos(2πƒC t)

Threshold detector Generate local carrier

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Non-coherent detection measures the envelope of the signal plus noise and is affected by the full

power of the noise. Coherent detection multiplies by cos(2πƒct ) low-pass filters and thus eliminates

half the noise power i.e. the part in phase with sin(2πƒct ). The gain is a 3dB reduction in effective

noise power as seen by the detector. Therefore coherent detection can tolerate 3dB more noise than

non-coherent detection to achieve the same bit-error rate (BER). There is another advantage of

coherent detection as will be seen.

8.4 Complex base-band and the vector-modulator/demodulator

8.4.1 Vector modulator : The block diagram below shows two ASK modulators (they could be

PSK), one applied to a cosine carrier, the other applied to a sine carrier of the same frequency. The

outputs of the two modulators are summed and transmitted.

The mappings to baseband signals bR(t) and bI(t) from independent bit-streams produce an appropriately shaped analogue pulse for each bit.. The block shown with a dashed edge is a "vector modulator" multiplying bR(t) by the ‘in-phase carrier component’ cos(2πfCt), bI(t) by the ‘quadrature carrier component’ sin(2πfCt), and adding the two resulting signals together. In complex number notation, this is equivalent to multiplying bR(t) + jbI(t) by exp(-2πj fC t) and taking the real part of the resulting complex number. Hence, with j = √(-1), bR(t) + jbI(t) is referred to as a ‘complex base-band’ signal b(t). To verify this, remembering that exp(jθ) = cos(θ) + j sin(θ), [bR(t) + jbI(t)] exp(-2πj fC t) = [bR(t) + jbI(t)] (cos(2π fC t) - j sin(2πfC t) ) The real part of this expression is bR(t) cos(2π fC t) + bI(t) sin(2πfC t) ) as required.

..11010.. Map

sin(2πfCt)

cos(2πfCt)

bI(t)

bR(t)

bR(t)cos(2πfCt) + bI(t)sin(2πfCt)

Map ..10010..

Mult Map

exp(-2πfCt)

b(t) 10110

11011

Complex signal. Take real part. Complx

base-band

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8.4.2: Vector-demodulator: The value of the concept of ‘complex base-band’ may be demonstrated by considering a corresponding vector-demodulator which is able to receive bR(t)cos(2πfCt) + bI(t)sin(2πfCt) and recover bR(t) and bI(t) separately. This means that bR(t) and bI(t) can each transmit data, and may be considered independent digital channels. If each transmits data at the maximum theoretically possible rate of 1 bits/second per Hz, then we achieve a total of 2 bits/second/per Hz. The vector-demodulator is shown below. Clearly the vector-demodulator is capable of receiving "two ASK signals for the price of one". The vector-demodulator has two coherent detectors, one based on the ‘in phase’ carrier component cos(2πfCt) and the other based on the ‘quadrature’ component sin(2πfCt). The ‘in-phase’ detector is exactly as seen earlier, but we can now show that it is ‘blind’ to the modulation carried by sin(2πfCt). Similarly the quadrature detector is blind to the modulation carried by cos(2πfCt). To verify this, let r(t) = bR(t) cos(2π fC t) + bI(t) sin(2πfC t) ) Then r(t) cos(2π fC t) = bR(t) cos2(2π fC t) + bI(t) sin(2πfC t) )cos(2π fC t) = 0.5 bR(t)[1 + cos(4π fC t)] + 0.5 bI(t) sin(4πfC t) ) If the low-pass filter removes the high frequency components centred on twice fC,, i.e. 0.5 bR(t) cos(4π fC t) and 0.5 bI(t) sin(4πfC t) ) we are left with 0.5 bR(t) as required. Similarly, r(t) sin(2π fC t) = bR(t) cos(2π fC t)sin(2πfC t) + bI(t) sin2(2πfC t) ) = 0.5 bR(t) sin(4π fC t) + 0.5 bI(t) [1-cos(4πfC t) ] If the low-pass filter removes the high frequency components centred on twice fC,, i.e. 0.5 bR(t) sin(4π fC t) and 0.5 bI(t) cos(4πfC t) ) we are left with 0.5 bI(t) as required. This operation and the blindness of the cosine to the sine channel & vice versa arise from the fact that cos 2 (θ) and sin 2 (θ) have a constant (or DC) component 0.5 whereas sin(θ) cos (θ) does not. The relevant trig formulae are:

cos 2 (θ) = 0.5 + 0.5 cos(2θ) sin 2 (θ) = 0.5 - 0.5 cos(2θ) sin(θ) cos (θ) = 0.5sin(2θ)

..10010.. Mult

Mult Threshold Detector

Threshold Detector

Cos(2πfCt)

Sin(2πfCt)

bR(t)

bI(t) Low pass

Low pass

..11010..

Derive local carrier (cos & sin)

Received signal r(t)

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So if we multiply cos (θ) by cos (θ) or sin(θ) by sin (θ) and remove the cos(2θ) component (by lowpass filtering) we are left with 0.5. However, if we multiply sin(θ) by cos (θ) and remove the sin(2θ) component, we are left with zero. This is a form of orthogonality displayed by sin(θ) and cos (θ) . With complex base-band notation, the vector demodulator may be represented as follows, 8.4.3: Constellation diagrams for ASK with complex base-band: A constellation diagram as shown earlier for simple ASK becomes more interesting where we have an in-phase component bR(t) and a quadrature component bI(t).

Binary ASK for i(t) & q(t)

In phase with carrier

Quadrature to carrier

Q

I

4-ary ASK for i(t) and for q(t)

0 A 2A 3A

8.5 Frequency Shift Keying (FSK)

8.5.1 Introduction & FSK generation: FSK can be a very straightforward form of digital

modulation, simple to generate and detect and, being of constant amplitude throughout a

transmission, insensitive to fluctuations of the channel attenuation. It is effectively frequency-

modulation, but uses a set of distinct frequencies to represent the required symbols. The principle is

to transmit a constant amplitude sine-wave whose frequency varies between the frequencies

assigned to each symbol. For binary signalling there would be two frequencies, ƒ0 and ƒ1 say. We

may consider three generation methods.

Mult Threshold Detector

(constellatn)

exp(2πjfCt)

Low pass

(complx)

..11010..

Derive local carrier (cos & sin)

Received signal r(t)

..10011

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The first method, illustrated above is the “Voltage controlled oscillator(VCO)” or FM modulator

method. It involves simply applying a base-band pulse shape (e.g. a rectangular pulse) to control the

frequency of the output. Although a rectangular pulse shape is easiest to visualise, it is better to

have a more smoothly changing pulse. A 'raised cosine' or RRC shape would be satisfactory but

note that here it is the frequency change rather than the amplitude variation that is being affected by

the pulse shaping. A 'Gaussian' pulse shape is more commonly used as will be explained later. The

result is a transition between the 2 frequencies which is gradual rather than sudden. It is without

phase discontinuity and is therefore a “continuous phase" form of FSK i.e. CPFSK. Further, with

the Gaussian pulse shaping, it does not have sudden changes in instantaneous frequency therefore it

is spectrally efficient in its bandwidth requirement.

The second method is a simpler “switched oscillator” method of generating FSK as illustrated

below. This will not produce a continuous phase output unless the waveforms are chosen such that

the end points and starting points are equal in voltage and in this case the frequency variation may

not be continuous.

A variation of the switched oscillator method is to read the two waveforms from 'look-up' tables and

this introduces the possibility of achieving a continuous and smooth frequency trajectory as well as

a continuous phase trajectory. For binary signalling, an interesting idea is to have four versions of

each pulse i.e. eight pulse shapes in total: (i) for a 0 preceded by 0 and followed by 0, (ii) for 0

between 0 and 1, (iii) for 0 between 1 and 0, (iv) for 0 between 1 and 1, (v) for 1 between 0 and 0,

etc. Each pulse would achieve the appropriate frequency at its centre and drift in some controlled

way from the nominal frequency of the previous pulse towards that of the next pulse so that there

are not sudden instantaneous frequency or phase transitions at the boundaries. It would be

interesting to analyse the spectral efficiency and practicality of this 'look up' scheme as an exercise.

FM Modulator (VCO) 1

0 0 1 0

1 0

FSK

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A third way of generating FSK is to use a “vector-modulator” or “quadrature-modulator” “which is

illustrated below: This vector-modulator can generate many forms of modulation including ASK.

ASK would be achieved by simply applying zero to the ‘Q’ input and a unipolar M-ary signal to the

‘I’ input.. To generate binary FSK with symbol frequencies ƒc+ƒ1 and ƒc-ƒ1, apply cos (2πƒ1t) to

the ‘Q’ input and ±sin(2πƒ1t) to the ‘I’ input, the sign determining the symbol. Pulse-shaping may

be applied to smooth the transition between +sin(2πƒ1t) and -sin(2πƒ1t) and vice-versa to improve

spectral efficiency by modifying the amplitudes of the FSK symbols during transitions. This is a

little different from applying pulse-shaping to the voltage input to an FM modulator thus modifying

frequency transitions. It is probably not as spectrally efficient and does not produce a precisely

constant envelope during the change from one frequency to another. But in practice it is an

effective approach which is commonly used.

Exercise 8.1: Check that this generates the required binary FSK frequencies.

Solution: When I=+sin(2πƒ1t) the output is:

sin(2πƒ1t)cos(2πƒct)+cos(2πƒ1t)sin(2πƒct)=sin(2π(ƒc+ƒ1)t)

When I=-sin(2πf1t) the output is:

-sin(2πƒ1t)cos(2πƒct)+cos(2πƒ1t)sin(2πƒct)=sin(2π(ƒc–ƒ1)t)

8.5.2 Non-coherent detection of FSK at the receiver (low bit-rates):

Non-coherent detection of FSK at low data-rates, for example for 200 bits/second over a 300 to

3400 Hz telephone channel, may be achieved by any of the following three methods:

(a) A set of band-pass filters followed by envelope detectors; one set for each signalling frequency.

A decision is made as to which band-pass filter is producing the largest output power.

“Q” input

“I” input

Sin(2πƒct)

Cos(2πƒct)

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(b) A “discriminator” (i.e. a single filter whose gain is different for each of the signalling

frequencies) followed by an envelope-detector and finally a decision block which decides what

has been transmitted on the basis of the voltage produced by the discriminator. The

discriminator turns frequency variations into amplitude variations; i.e. it turns FSK into ASK

which allows easier detection.

(c) A 'phase locked loop' (PLL) detector. A PLL is an extremely useful device with many

applications in communications and other areas. It may be thought of as a 'black box' with one

input and two useful outputs as illustrated below:

8.5.3 Details of phase locked loop(PLL): The PLL circuit contains a voltage controlled oscillator

(VCO) whose output is a periodic waveform (say a sine-wave). The frequency of the 'VCO output'

is controlled by the 'VCO input' voltage: the higher the voltage the higher the frequency. The PLL

circuitry adapts the VCO input voltage such that the VCO output matches the frequency modulated

input signal in frequency and phase which means that it must change as the frequency of the input

signal changes.

Discriminator

Low-pass filter (smoother)

t t t V

t

f

Gain Resistor

f1 f0

f1 f0

V

PLL

t V

t

t

VCO input (Voltage ∝ input frequency)

VCO outputFrequency modulated input

CS3282 8.18 11/03/07 / BMGC

The VCO input voltage is generated by measuring the phase difference between the VCO output

and the incoming signal. This phase difference is measured simply by multiplying the two signals

together (remembering that cos(ωt)cos(ωt+φ) = 0.5cosφ+0.5cos(2ωt+φ)) which becomes 0.5cos(φ)

after low pass filtering. As well as eliminating 0.5cos (2ωt+φ), the low pass filter smoothes the

variations in 0.5cos (φ) and thus generates the control signal to the VCO.

Block-diagram of phase-locked loop (PLL) circuit

This 'VCO input' voltage will be proportional to the input frequency and can therefore be used for

detecting the data bits transmitted by FSK. Further, the 'VCO output' voltage which matches the

input in frequency and phase, but not necessarily amplitude, is often useful for producing a 'local

carrier' or clock signal as required for coherent detection, as we shall see.

8.5.4 Non-coherent detection of FSK at receiver (higher bit-rates)

A non-coherent FSK detector for higher data rates, say up to 1.2kb/s with binary signalling, over a

300 to 3400Hz telephone channel would be considerably more complicated. Consider the following

“zero crossing counter” type of detector which counts the number of clock pulses between each

zero-crossing of the FSK input waveform. The limiting amplifier converts the FSK waveform into

a rectangular waveform VL as shown which may be considered a logic signal. An 'AND' gate lets

through a sequence of very fast clock pulses when VL is at logic one and blocks then when VL is a

logic zero. The number of fast clock pulses while VL = '1' may be counted and the number used as

a measure of the time-duration between zero-crossings of the FSK signal and hence of the FSK

frequency. So the final counter output (which must be reset whenever VL goes low) increases when

the FSK frequency decreases and vice-versa. A threshold may be set for this counter output to

determine (digitally) whether a 1 or a 0 was transmitted. This is a non-coherent technique because

there is not clock matched in frequency and phase to either or any of the FSK frequencies.

Low-pass filter

VCO

VCO input voltage

VCO output voltage

tt

V

V

CS3282 8.19 11/03/07 / BMGC

8.5.5 Coherent FSK detection: This is similar to the technique used for coherent ASK detection.

We must have carrier sine-waves generated at the receiver which match exactly in frequency and

phase the FSK symbols being received. For binary transmission there would be two locally

generated sine-waves of frequency ƒ0 and ƒ1 respectively. The incoming signal is multiplied by both

sine waves and the two signals which result are low-pass filtered. A comparator then has to decide

which frequency ƒ0 or ƒ1 produced the larger output, and that determines the symbol.

8.5.6 Spectrum of FSK: The spectrum of FSK is not so easy to derive exactly for the VCO or FM

modulator generation method, but we can get an idea from the FSK spectrum that would be

obtained with an AM method such as the 'vector-modulator' approach described earlier. Let the

signalling be binary at 1/T symbols/second. When a base-band signal is modulated to form FSK

with signalling frequencies ƒ1 and ƒ0, all the ‘one’s may be considered to form a double side-band

ASK spectrum centred on ƒ1Hz and all the ‘zero’s may be considered to form a double side-band

ASK spectrum centred on ƒ0. The resulting spectrum is the sum of these two spectra. This is an

approximation when the amplitude remains constant as the frequency is shifted in a smooth and

controlled way from f0 to f1 and vice-versa (as with the VCO or FM method) because with ASK the

amplitude variation rather than the frequency variation is controlled by pulse shaping to achieve the

required limited bandwidth. The dual ASK spectrum for binary signalling will be as follows with

100% RC or RRC spectral shaping applied to the amplitudes of the base-band pulses:

PSD

ƒ

ƒ0-1/T ƒ0 ƒ0+1/T

PSD

ƒ

ƒ1-1/T ƒ1 ƒ1+1/T

Limiting Amplifier

Clock

Decide Counter

Reset

Data FSK

AND

t

VL t

+ =

CS3282 8.20 11/03/07 / BMGC

The bandwidth required around f0 and f1 is determined by the amplitude pulse-shaping or how

smoothly the frequency shift is made with the VCO (FM) method. A good idea to place ƒ0 at

ƒ1±1/T and ƒ1 at ƒ0±1/T and arrange that the transmission at f0 has an envelope which is zero at f =

f1. Similary the transmission at f1 becomes zero at f = f0 . This is known as “Sunde’s FSK method”

which is a form of non-coherently detectable 'orthogonal' FSK signalling since

0)()( 00 1 =−∫ dttstsT

τ

when s1(t) and s2(t) are the two FSK symbols transmitted. This means that s1(t) is 'orthogonal to

s0(t) even when τ ≠ 0 which means that s1(t) may be advanced or delayed with respect to s2(t)

without affecting the orthogonality. This is because

τany for /1 when 0 ))(2cos()2cos( 010 01 TffdttftfT

±==−∫ τππ

as may be proved with a bit of effort (see Sklar 2nd edition (2001) page 202). Without pulse

shaping, i.e. with rectangular time-domain pulses of duration T it produces a spectrum which

switches between the solid and dotted waveform shown below and is therefore approximately

constant amplitude. It can be decoded non-coherently at the receiver.

PSD

ƒ0-1/T ƒ0 ƒ1

ƒ

ƒ1+1/T

f f0 f1

CS3282 8.21 11/03/07 / BMGC

8.5.7 Minimum shift keying (MSK) is a form of FSK where the difference between ƒ0 and ƒ1 is not

1/T but 1/(2T) Hz. This narrow spacing makes MSK very efficient in its spectral utilisation, but the

price to be paid is increased complexity in the generation and detection process. Non-coherent

detection is difficult for MSK since

0.τ only when )2/(1 when 0 ))(2cos()2cos( 010 01 =±==−∫ TffdttftfT

τππ

.

With rectangular shaping, the spectra are as shown above, and it may be seen that the spectrum of

one signal does not become zero at the frequency of the other. But the binary FSK signalling

remains orthogonal in that

0)()( 00 1 =∫ dttstsT

i.e. there is no interference between s1(t) and s0(t) at the receiver if the detection is coherent i.e. non-

delayed or phase shifted copies of s1(t) and s0(t) are available at the receiver. (Sklar p152).

As with base-band signalling and ASK, the spectrum of FSK can be reduced by a pulse-shaping

filter placed just before the FSK modulator to control how the frequency changes from ƒ0 to ƒ1 and

vice-versa. In GSM based digital cellular mobile phone systems the shaping is not 100r% raised

cosine but something similar called a Gaussian frequency-response. The latter has a Gaussian

shaped gain response which is similar to 100r% RRC. In fact GSM phones use Gaussian shaping in

conjunction with MSK. This form of FSK is known as “Gaussian MSK” generated as follows:

8.5.8 Advantages and disadvantages of FSK

Advantages are:

1. Constant envelope hence not too sensitive to varying attenuation on the channel.

f0 f1

f

FIR Gaussian shaping filter

VCO

Map to impulses

..10110.. GMSK

CS3282 8.22 11/03/07 / BMGC

2. Detection based on frequency changes, therefore not very sensitive to frequency shifts of

channel, Doppler shifts in mobile systems, etc.

Disadvantages of FSK:

1. Less bandwidth efficient than ASK or PSK (except MSK)

2. Bit error rate performance in AWGN worse than PSK.

8.6 Phase shift keying (PSK)

8.6.1 Introduction & generation method: Transmit a sinusoidal carrier with phase changes

determined by the data. Consider a simple example of binary PSK with one bit per cycle and 00

phase shift for “0” and 1800 for “1”.

PSK ±sin(…)

Data +1V for “1” -1V for “0”

Sin(2πƒct) Carrier

t

V

1 1 0 0 1 1 0

CS3282 8.23 11/03/07 / BMGC

8.6.2 Coherent Detector for binary PSK

The low-pass filter must allow its output to change at the data rate and eliminate ±cos(4πƒct). A

matched filter will achieve this because of the orthogonality of ±cos(4πƒct) to sin(2πƒct). The local

carrier must be generated accurately to be of the correct frequency and exactly in phase with the

carrier being received. It must do this from the data itself. One method is to square the incoming

signal and then divide the frequency of the resulting signal by two.

The spectrum of PSK is similar to that of ASK noting that we are effectively multiplying a carrier

by a bipolar … +V, -V, -V, +V, +V, … base-band pulse train. With ASK it is a unipolar .. +V,

0, 0, +V, +V, … pulse train. The multiplication shifts up the base-band spectrum producing a

double side-band spectrum centred on the carrier frequency.

PSK is less susceptible to the effects of group delay and attenuation than ASK and FSK It may be

shown that with binary signalling, the error probability for coherently detected PSK is lower than

for coherently detected FSK and ASK in the presence of channel noise as shown by the graph on

page 137 of “Digital Communications” by A Bateman and page 135 (or 160) of Sklar (no ASK

here).

8.6.3. Differential PSK: Coherently detected PSK has the problem of maintaining long-term carrier

phase relationship at the receiver. Minor frequency shifts arising from frequency division

multiplexing and de-multiplexing appear as continually changing phase shifts whose effect is

accumulated over time. Instead, differential PSK (DPSK) may be used where the phase shift of the

carrier with respect to the previous bit transmitted indicates the current bit: say 00 shift for “1” and

1800 for “0” as indicated below assuming 1bit/cycle for simplicity.

LPF (MF)

sin(2πƒct) Locally generated

±sin(2πƒct) + for “1” - for “0”

±sin2(2πƒct) =±1/2(1-cos4πƒct)

Decide

Data +1/2:”1” -1/2:”0”

±1/2

CS3282 8.24 11/03/07 / BMGC

Exercise 8.2: Repeat this for 1 bit per 1.5 cycle.

Insert diagram

In practice 900 and 2700 phase shifts are often preferred so that phase changes occur at the data rate

even when a long succession of “0”s or “1”s are sent. This allows the receiver to keep track of the

number of bits being sent more easily. Otherwise the received signal would have no component at

the data rate which may not be fully synchronised with carrier frequency. A typical waveform is as

follows, assuming 1 bit/sample and 900 for “1”.

Exercise 8.3: Repeat for 1 bit per 1.5 cycles.

8.6.4 Differential detection of DPSK:

Considering firstly the case where the phase shifts are 00 and 1800 to indicate “0” or “1” respectively the following detection techniques may be used. Instead of generating a local carrier, the detector takes the previous symbol as the required carrier segment. Whether this makes the technique classifiable as “coherent” or “non-coherent” is a matter for discussion. There is certainly some penalty to be paid for this form of detection of differentially encoded PSK as compared with a fully coherent technique, but the penalty is not believed to be great. We assume that pulse-shaping has been applied to produce s(t).

t

V

1 0 1 0 0 0

1 1 0 1 1 0

t

V

CS3282 8.25 11/03/07 / BMGC

The LPF output is +s(t)/2 if the carrier has been subject to 00 phase shift (logic “1” say) and –s(t)/2

for 1800 (logic “0”). This non-coherent technique may be improved by stabilizing the delayed

carrier with respect to a locally generated coherent carrier, otherwise channel noise affects both

data and delayed data used as carrier. This technique used to be used for modem data over

telephone lines, 1200 bits/s being possible over worst case lines. Wow! This could be increased to

2400bits/s using quaternary phase shift keying (QPSK).

8.6.5 Detector for binary DSPK with 90O & 270O phase shifts rather than 0 and 180O.

8.6.6 Quaternary phase-shift keying (QPSK):

There are two ways to look at QPSK. The first way is to realise that it transmits two bits at once

using symbols which are sine-wave segments with phase shifted by 45O (for 00, say), 135O (for

01), -135O (for 11) and –45O (for 10) with respect to the carrier sine-wave. Assuming the carrier

frequency to be ΩC = 2 π fC , the ±45O, ±135O QPSK modulation is easily achieved by means of a

vector-modulator as shown below.

LPF (MF)

±s(t)cos(2πƒct)

±s(t)cos2(2πƒct) =±1/2s(t)(1+cos4πƒct)

Decide

Delay by T (Delay for 1 bit)

LPF (MF)

±s(t)cos(2πƒct) or ±s(t)sin(2πƒct)

±s(t)cos2(2πƒct) or ±s(t)sin2(2πƒct) =±1/2s(t)(1±cos4πƒct)

Decide

Delay by T (Delay for 1 bit)

900 phase shift

CS3282 8.26 11/03/07 / BMGC

This diagram does not show pulse shaping that would normally be applied to the 'in-phase' and

'quadrature' signals i(t) and q(t) which modulate cos(2πfCt) and sin(2πfCt) respectively.

The second way to look at QPSK is to realise that because of the orthogonality of sine and cosine

when coherent detection is employed, as we shall see, it is possible to send two independent binary

PSK signals using a single carrier of frequency fC say. One of the binary PSK transmissions

modulates cos(2πfCt) and the other modulates sin(2πfCt). At the receiver, a coherent PSK detector

using a local carrier synchronised in frequency and phase to cos(2πfCt) will be blind to the

transmission on sin(2πfCt) and vice-versa.

It is convenient to refer to i(t) + jq(t) as a 'complex base-band' signal. The transmitted QPSK signal

is then equal to the real part of [i(t) +jq(t)] exp(-j2πfCt) . The way to understand the phase shifts

is to construct the following table:

−Ω=Ω−Ω−

++Ω=Ω+Ω−

+−Ω=Ω−Ω

++=++Ω=Ω+Ω

=

jV- V- 11for )135cos(2sincos

VV,- 10for )135cos(2sincos

jV- V 01for )45cos(2sincos

V Vq(t) ji(t) 00for )45cos(2sincos

0

0

0

0

tVtVtV

jtVtVtV

tVtVtV

jtVtVtV

QPSK

ccc

ccc

ccc

ccc

±V (Bit 2)

±V (Bit 1)

SinΩct

CosΩct

QPSK

Transmit i(t)

q(t)

CS3282 8.27 11/03/07 / BMGC

Exercise 8.4: Alternatively phase shifts of 0o, 90O, 180O and 270O may be used for QPSK but this is

less convenient in some ways. Show how the 0o, 90O, 180O and 270O QPSK modulation and

demodulation may be achieved

8.6.7 QPSK Detector:

Coherent detection is achieved using a vector-demodulator fed with locally generated versions of cos(2πfCt) and sin (2πfCt).

To check:

Transmission of 0,0 by )45sin(2sincos 0+Ω=Ω+Ω tVtVtV ccc

Multiply by cosΩct to obtain

tVtVttVtV ccccc Ω+

Ω+=ΩΩ+Ω 2sin

212cos

21

21cossincos2

Low pass filter removes 2Ωc frequency components, leaving V/2 signifying that the bit sent on the

cosine, i.e. bit 1, is 0. Multiplying by sin Ωct, we get

Ω−+Ω=Ω+ΩΩ tVtVtVttV cccCc 2cos

21

212sin

21sinsincos 2

After low pass filtering to remove frequency components at 2Ωc we obtain V/2 to signify that bit 2

is 0. 0,0)45sin(2 0 →+Ω∴ tV c

Exercise 8.5: Now check that the same demodulation procedure applied to VcosΩCt −VsinΩCt =

1,0)45sin(2 0 →−Ω tV c .

±V/2 (Bit 1)

±V/2 (Bit 2)

Ωc=2πƒc

Input)

Local carrier generator

Local carrier generator

LPF (MF)

LPF (MF)

SinΩct

CosΩct

CS3282 8.28 11/03/07 / BMGC

A constellation diagram for ±45o, ±135o QPSK is shown below (left):

The constellation diagram on the right (above) uses 0o, 90o, 180o and 270o.

QPSK may be called 4-PSK and it is possible to extend the concept to 8-PSK sending 3 bits per

symbol and 16-PSK sending 4 bits per symbol. Constellation diagrams for 8-SPK and 16-PSK are

shown below. These may be considered multi--level signalling techniques since each symbol sends

two or more bits rather than just one as with binary signalling. These techniques will be considered

again in the next section.

Differential forms of QPSK and M-PSK are often used where changes in phase signify the data

rather than the phases themselves. The principle is similar to that used with differential PSK

discussed earlier.

Exercise 8.6: Consider how the symbols for 8-PSK and 16-PSK may be associated with sequences

of 3 or 4 bits, i.e. label the constellation diagrams. Use a form of 'Gray coding' and explain why

this is a good idea.

Exercise 8.7: Show how a vector-modulator may be used to generate the 8 or 16 symbols of 8-PSK

and 16-PSK. The inputs to the 'in-phase' and 'in-quadrature' channels of the vector-modulator will

Imag pt (In quadrature with cos)

1,1

1,0

0,1

0,0

In phase with cos

(real pt) 45o

Imag pt

Real

V

V

-V

0,1

1,1

ReReal pt

8-PSK 16-PSK

Imag pt

CS3282 8.29 11/03/07 / BMGC

no longer be ±V as with QPSK, but combinations of other voltages which may be deduced from the

appropriate constellation diagram..

Exercise 8.8: If the radius of the constellation diagram circle is V volts for QPSK, 8-PSK and 16-

PSK calculate the energy per bit for each of these schemes assuming rectangular pulses. Taking the

'noise immunity' as the minimum distance between each symbol on the constellation diagram and

the nearest one to it, estimate the noise immunity for QPSK, 8-PSK and 16-PSK when the radius is

V in each case.

Exercise 8.9: How will pulse-shaping be applied to QPSK, 8-PSK and 16-PSK? With 100% RRC

pulse shaping and symbol duration T seconds, what would be the band-with efficiency (in

bits/second per Hz) for each of these techniques. What would be the theoretical maximum

bandwidth efficiency in each case?

8.7. Multi-carrier modulation and OFDM 8.7.1. Introduction This section introduces the concept of multi-carrier modulation and compares it with single carrier modulation to determine some of its advantages and disadvantages. Orthogonal frequency division multiplexing (OFDM) is a highly efficient form of multi-carrier modulation which is widely used in broadcasting, ADSL and wireless LAN technology. OFDM will be explained and the means of implementing it using the FFT and inverse FFT will be developed. The parameters of the OFDM implementation used by IEEE802.11 WLAN equipment are investigated. Before introducing multi-carrier modulation, we revise some important aspects of single carrier modulation. 8.7.2 Matched filtering & equalization in single carrier modulation systems Pulse shaping within the ‘map to base-band’ function of a transmitter may be achieved by means of a ‘pulse shaping filter’. In response to a bit, or a series of bits, the pulse shaping filter generates a ‘sinc like’ pulse of the correct shape, amplitude and polarity, at the right time. This pulse is added to the ongoing effects of previous pulses, and the resulting base-band waveform is modulated onto a carrier for transmission. The diagrams below illustrate first the generation and modulation of a single pulse and secondly the sort of PSK modulated wave-shape that will be transmitted when there are several pulses. With ASK, the ‘sinc like’ pulse shape becomes the ‘envelope’ of the modulated carrier.

Pulse shaping filter

Excite Pulse-s filter

b(t) ..11101.. Multiply

t Volts

t

volts

Map to base-band

Volts

t

CS3282 8.30 11/03/07 / BMGC

Modulation of a single ASK sinc-like pulse

CS3282 8.31 11/03/07 / BMGC

The receiver must first demodulate the signal obtained from the channel to obtain a base-band signal b(t) containing the pulse shapes produced at the transmitter. These pulse shapes will have been distorted in shape and affected by additive noise. If the distortion is not too serious and the noise is not too high in amplitude, the ‘sample and detect’ techniques discussed at the end of lecture B6 for rectangular pulses may then be employed. This simplistic approach may work satisfactorily provided b(t) is sampled at the correct point in time. In practice the channel distortion and added noise will be too much to allow it to work, especially as, in mobile equipment, the transmit power must be minimised to preserve battery life. Reducing the transmit power decreases the signal-to-noise ratio at the receiver. The performance

Output of multiplier when there are two PSK pulses

Volts

t

Demodulator

Sample & detect

..1100b(t) Channelsignal + AWGN

t Volts Derive

local carrier

Pulse shaping filter

Excite Pulse-s filter

b(t) ..11101. Multiply

t Volts

t

volts

Map to base-band

Volt

t

envelope

Modulation of a single PSK ‘sinc-like’ pulse

CS3282 8.32 11/03/07 / BMGC

of the simplistic approach can be considerably improved by the introduction of (i) a matched filter and (ii) an equalizer at the receiver as illustrated below. The matched filter is optimally tuned to the shape of the transmitted pulses to minimise the effect of additive white Gaussian noise (AWGN). The channel equaliser aims to cancel out distortion to the shape of the pulses which have been introduced by the frequency selective fading channel. Where vector-modulation and demodulation (e.g. QPSK) are employed, the matched filter and channel equaliser may be considered to have complex valued input signals corresponding to the ‘in-phase’ and the ‘quadrature’ channels. Multi-level pulses may be used in preference to binary and this only complicates the ‘sample & detect’ block by requiring multiple thresholds rather than just one threshold. After the effects of the modulation, channel, demodulation, matched filtering and channel equalization, the pulse shapes seen at the input to the ‘sample and detect’ block must have the ‘Nyquist’ property; i.e. the centre of each pulse must coincide with the zero-crossings of all others. With ‘raised cosine’ (RC) pulse shaping, the received pulses must have as close as possible to the true ‘sinc-like’ shape at this point, e.g. 50% RC. To make this possible when a matched filter is employed, the pulse shaping filter at the transmitter must be a ‘root raised cosine’(RRC) pulse shaping filter. Instead of ‘raised cosine’ (RC) pulses the transmitter generates ‘root raised cosine’ (RRC) pulse shapes. At first sight, they look very similar, and both may be described as ‘sinc-like’ pulses. The use of RRC pulse shapes is necessary because the receiver’s matched filter is ‘tuned’ to listen for particular pulse shapes to optimally distinguish them from noise. To do this, the matched filter must have the same magnitude spectrum as the transmitter’s pulse shaping filter. So the matched filter effectively multiplies the received pulse shape by a copy of itself. The received pulse is therefore ‘squared’ by the matched filter. If the transmitter sent ‘raised cosine’ pulses and the matched filter were tuned to these, the detector would see squared ‘raised cosine’ pulses. These would not have zero-crossings in the right places for eliminating inter-symbol interference. Therefore RRC pulse shapes must be transmitted and matched at the receiver. The ‘channel equaliser’ is a filter also. In fact it is an ‘adaptive filter’ which is programmed to correct any differences between the pulses seen at the output of the matched filter and the ideal raised cosine pulses required by the detector. It aims to cancel out the effect of the channel, in particular the effects of frequency selective fading. Since frequency selective fading reduces the received signal strength at some frequencies and reinforces it at others, the equalizer must do the opposite of this to cancel out the distorting effect of the channel. The equaliser must be adaptive so that it can automatically adjust to changes that will be constantly occurring to the fading channel characteristics. This is a demanding filtering task, and it cannot always be successful since if there

Matched filter

Demodulator

Channel equaliser

Sample & detect

..1100b(t) Channel

signal + AWGN

tDerive local carrier

V

CS3282 8.33 11/03/07 / BMGC

is virtually complete cancellation at a certain frequency, i.e. a very deep fade, it will just not be possible to reverse it. Trying to do so will just emphasize the noise at that frequency. Single carrier sine-wave modulation techniques have been used since the beginning of radio and are still successfully used for analogue and digital communications. 8.7.3. Spread spectrum modulation: The use of a single sine-wave as a carrier is common, but is not the only possible choice. A widely used alternative is a ‘pseudo-random’ signal whose characteristics need be known only at the transmitter and receiver. The bandwidth of the modulated pseudo-random carrier signal is generally much wider than that of a modulated sine-wave, typically by a factor of about 50. This may appear very wasteful of bandwidth. However, because the carrier is randomised, it will appear, even when modulated, as noise to receivers not tuned to its exact characteristics. It is as though the transmission is ‘coded’ by the pseudo-random carrier, and security is an added bonus. Two transmitter-receiver systems using different pseudo-random carriers can co-exist in the same channel, each experiencing a small degree of background noise from the other transmitter. The noise from the other transmitter may increase the bit-error rate but by an amount that is tolerable. More transmitter-receivers may be accommodated until the bit-error rate introduced by the accumulated background noise becomes too severe. This alternative ‘single carrier’ approach is known as ‘spread spectrum multiplexed access’ (SSMA). It is based on the use of a ‘direct sequence’ of pseudo-random bits to produce the pseudo-random carrier, and is therefore referred to as ‘direct sequence’ SSMA (or DS-SSMA). It is also referred to as ‘code division multiplexed access’ (CDMA) and is the basis of most 2G mobile phone systems in the USA. Third generation mobile telephony will be based on an enhanced form of CDMA. 8.7.4. Introduction to multi-carrier modulation Assume we have a 20 MHz radio channel available, centred on 2.457 GHz, i.e. ‘channel 10’ in the 2.4 GHz ISM band. One option is to apply single carrier modulation to a sinusoidal carrier placed at 2.457 GHz. With QPSK modulation, the maximum achievable bandwidth efficiency of 2 b/s per Hz would allow 40 Mb/s to be transmitted. This would require 0% RRC pulses (pure sinc functions) to be used. Adopting 50% RRC pulses would reduce the bandwidth efficiency to 1.33 b/s per Hz allowing only about 26.7 Mbits/s could be transmitted. However, generating reasonable approximations to the ‘sinc-like’ pulses would become much easier. In both cases the whole 20 MHz would be used by the single carrier modulated signal. An alternative to single carrier modulation is to divide the 20 MHz band into a number of sub-bands and to introduce a sinusoidal ‘sub-carrier’ into the centre of each band. Instead of one carrier we now have many carriers which we call sub-carriers. IEEE802.11g and 802.11a divides a 20 MHz channel into 64 sub-bands each of bandwidth 312.5 kHz. There are now 64 sinusoidal sub-carriers at frequencies F + f0, F + f1, …, F+f63 Hz where F is close to the lowest frequency of the 20MHz band and f0 = 156.25 Hz, f1 = 468.75 Hz, …, f63 = 19843.75 Hz. Modulating each of these sub-carriers with QPSK with 0% or 50% RRC pulse shaping would achieve 625 (= 312.5 x 2) or 417.2 (= 312.5 x 1.33) kb/s per sub-band respectively. Multiplying by 64, we obtain the total bit-rate of 40 Mb/s or 26.7 Mb/s respectively, which are the same bit-rates as were obtained with single carrier modulation with 0% or 50% RRC pulse-shaping respectively. But now the bits are divided into 64 parallel sub-streams. The bit-rate of each sub-stream is 1/64 of the original, and each sub-stream modulates its own sub-carrier. This is multi-carrier modulation.

CS3282 8.34 11/03/07 / BMGC

To see the advantage of multi-carrier modulation, look again at the demands of pulse shaping which for single carrier modulation is necessary to have a band-limited spectrum. As shown below, a ‘sinc’ pulse with zero-crossings at t=±T, ±2T, ±3T, etc. has a rectangular and therefore strictly band-limited spectrum. A rectangular pulse of duration T has a ‘sinc-like’ frequency spectrum with zero-crossings at f =±1/T, ±2/T, ±3/T, etc. With single carrier systems, we must transmit close approximations to ‘sinc-like’ pulses. To make them easier to approximate, R% raised cosine pulses are used, but this is at the expense of increasing the required bandwidth and therefore decreasing the band-width efficiency. With multi-carrier modulation, pulse shapes very close to rectangular pulses may be used for each sub-carrier. This means that their spectra are ‘sinc-like’ and of very wide bandwidth, in theory infinite. As we have to send 64 adjacent sub-bands at the same time, there is clearly a danger of one sub-band interfering with the next and many others besides. So the danger now is of inter spectrum interference, often called ‘inter-sub-carrier interference (ICI). There is also a danger of the multi-carrier spectrum leaking outside the allowed 20 MHz bandwidth. Both these dangers may be avoided. Close to rectangular pulses may be used provided it is ensured that the peak of the spectrum for each sub-band corresponds to zero crossings for all the other modulated sub-carriers. This eliminates ICI in a way that reminds us of how inter-symbol interference is avoided in single carrier modulation. However interference is now avoided in the frequency-domain rather than the time-domain. Looking at the lower of the previous two graphs, ICI is avoided if adjacent sub-carriers are spaced exactly 1/T Hz apart when the sub-band bit rate is 1/T bits/second.

f

1/(2T) -1/(2T)

T

T.rect1/T(f) sincT(t)

t

1

T -T

2T -2T

3T -3T

4T -4T

Fourier transform

Real pt shown Imag pt = 0

T.sinc1/T(f)

t

T/2 -T/2

1

rectT(t)

f

T

1/T -1/T

2/T -2/T

3/T -3/T

4/T -4/T

Fourier transform

Real part shown Imag part = 0

CS3282 8.35 11/03/07 / BMGC

This form of multi-carrier modulation is called orthogonal frequency division multiplexing (OFDM) and is highly efficient because the sub-carriers are as close together as they can possibly be without introducing spectral interference. Each modulated sub-carrier is ‘orthogonal’ to all others which means that they do not interfere with each other. The principle of OFDM is further illustrated in the diagrams on the following pages.

CS3282 8.36 11/03/07 / BMGC

t

T/2 -T/2

1

rectT(t)

Modulate F

t

T/2 -T/2

1

rectT(t)

Modulate F+2/T

t

T/2 -

1

rectT(t

Modulate F+1/T

Fourier transform SUM

T.sinc1/T(..)

f

T

F+2/T

F

T.sinc1/T(f-F)

f

T

F+1/T

F

T.sinc1/T(..)

f

F+2/T

F+1/T F+3/T

f

1/T 3/T

Combine real spectra

Assume purely real spectrum

Assume purely real spectrum

Assume purely real spectra

CS3282 8.37 11/03/07 / BMGC

The previous page shows purely real spectra. Here, more realistically, are modulus spectra. The idea being illustrated is exactly the same.

t

T/2 -T/2

1

rectT(t)

Modulate F

t

T/2 -T/2

1

rectT(t)

Modulate F+2/T

t

T/2 -

1

rectT(t

Modulate F+1/T

Fourier transform SUM

|T.sinc1/T(..)|

f

T F+2/T

F

|T.sinc1/T(f-F)|

f

T

F+1/T

F

|T.sinc1/T(..)|

f F+2/T

F+1/T F+3/T

f

1/T 3/T

|SUM(f)|

Modulus spectra shown

CS3282 8.38 11/03/07 / BMGC

Because the pulse-rate (1/T) for each sub-channel is 1/64 times what is would have been for single carrier modulation, the zero-crossings of the sinc spectra (at ±1/T Hz, ±2/T, ±3/T, …are much closer together than they would be with single carrier modulation by rectangular pulses. So the sinc spectra ‘die away’ must faster. The ones in the centre of the 20 MHz band will have died away almost completely at the edges. But this cannot be said for the ones near the edges, so we simply do not modulate them. Out of the 64 available sub-carriers, there are reasons for deciding not modulate the first six, the last five and number 32. If four other sub-carriers are reserved as ‘pilot sub-carriers’, this leaves 48 sub-carriers that can be modulated with data. In the IEEE802.11 standard it is specified that OFDM sub-carriers 0 and 27 to37 are not modulated and that four others are designated as pilots. Again this leaves 48 sub-carriers for data. Depending on how the processing is carried out, the two approaches just mentioned are similar. Modulation of sub-carriers: With IEEE802.11g and 802.11a, each OFDM sub-carrier modulated by choice of:

binary-PSK, (1 bit per pulse) QPSK, (2 bits per pulse) 16-QAM (4 bits per pulse) 64-QAM (6 bits per pulse)

‘16-QAM’ & ‘64-QAM’ are multi-level schemes. Each sub-carrier modulation is implemented by a vector-modulator according to a ‘constellation’ as illustrated below for QPSK & 16-QAM. ‘Gray coding’ makes the nearest dots differ in just 1 bit so that a small amount of noise causing one pulse to be mistaken for an adjacent one will only cause one bit error. With natural binary order, 0111 and 1000 would be adjacent in a 16-QAM constellation so that a small amount of noise could cause four bit-errors. Differential PSK, QPSK & QAM is used where the difference between the current and the previous pulse specifies the bit pattern. So it is phase differences rather than actual phases that determine the bit-pattern.

modulating cos

0,0

0,1

1,0

1,1

Bit1 Bit2 bR bI 0 0 A A 0 1 A -A 1 0 -A A 1 1 -A -A

Modulating sin

QPSK constellation

CS3282 8.39 11/03/07 / BMGC

A

3A

-A

-3A

A 3A Real

Imag

(0000)

-A

(0001)

(0010)

(0011)

(0100) (1000)

(1001)

(1010)

(1011)

(1100)

(1101)

(1110)

(1111)

(0110)

(0101)

(0111)

(modulates cos)

(modulates sin)

A ‘16-QAM constellation

Mult

Mult

ADD

Map

Cos(2πfCt)

Sin(2πfCt)

3A,-3A,..

-3A,-A,..

1011 1101..

t

V

t

V

Re..

Vector-modulator as used for 16-QAM

Mult Map

exp(-2πjfCt)

b(t) 1011 1101..

Complx base-band

Take real pt

Vector modulator in complex notation Sometimes people make this exp(2πjfCt).

Makes little difference as long as they are consistent.

CS3282 8.40 11/03/07 / BMGC

The Fast Fourier Transform & its inverse FFT : x[n]0,N-1 → X[k]0,N-1 Inverse FFT: X[k]0,N-1 → x[n]0,N-1 Both are ‘fast’ in that they can be programmed or implemented in hardware very efficiently especially when N is a power of 2, e.g. 64, 128, 512, 1024 8.7.5. Implementation of OFDM Take 64 sub-carrier frequencies over range F to F + 20 MHz:

fC + 0, fC + fD, fC + 2fD, … , fC + 63fD with fD = 20MHz / 64 = 312.5 kHz and fC = F + 176.25 kHz

As seen in the diagrams above, for orthogonality (correct frequency-domain zero crossings) the sub-carriers must be 1/T Hz apart. T is the duration of the rectangular pulse applied to each sub-carrier, therefore 1/T is the number of pulses per second on each sub-carrier. So fD = 1/T and the pulse duration T = 1 / 312.5k = 3.2 x 10-6 s = 3.2 µs on each sub-carrier. This is the key point! The separation between sub-carriers determines the value of T if orthogonality is to be achieved. If we tried to vary T, for example to transmit more pulses per second on each sub-carrier with the same fD, orthogonality would be lost and ICI would occur. With fD = 312.5 kHz, we could transmit 312.5 k pulses per second on each sub-carrier, each pulse being of duration 3.2µs. But we don’t quite do this. Instead, we extend each pulse to 4 µs with a 0.8 µs ‘guard-interval’. So we transmit 250 k ‘extended pulses’ per second on each sub-carrier. The guard-interval’ extension could be 0.8 µs of zero voltage. But it’s not. It’s a ‘cyclic extension’ as we will see later. When all 64 modulated sub-carriers are added together over a single period of T seconds, a highly complicated waveform segment, of duration T, is produced. This is referred to as a single ‘OFDM symbol’. When T = 3.2 µs and each OFDM symbol is extended to 4 µs, we can transmit 250 k extended OFDM symbols per second. Each extended OFDM symbol is the sum of 64 extended modulated sub-carrier segments. Bandwidth efficiency of IEEE802.11 OFDM The theoretical maximum bandwidth efficiency of a multi-carrier modulation scheme such as OFDM is 1 pulse/s per Hz in each sub-band for each of the 64 sub-carriers. So, overall, the theoretical maximum for OFDM is 1 symbol/s per Hz. This is the same as for single carrier modulation. However, using only 48 out of 64 sub-channels loses 25% of total capacity. We lose another 20% (=0.8/4) because of the guard-interval (cyclic extension). The maximum bandwidth

[ ] [ ] 1-N....., 2, 1, 0, =k for /2 where k

1

0NkenxkX

N

n

nj k πωω == ∑−

=

−

[ ] [ ] 1-N....., 2, 1, 0, =n for /2 where1 k

1

0NkekX

Nnx

N

k

nj k πωω == ∑−

=

CS3282 8.41 11/03/07 / BMGC

efficiency for 802.11a and 802.11g OFDM is therefore 60% (=3/4 x 4/5) of 1 symbols/s per Hz= 0.6 symbols per second per Hz. This allows 1.2 b/s per Hz, if QPSK is used for all 48 sub-carriers. With QPSK, the bit-rate in 20 MHz will be 24 Mb/s. With 64-QAM, the bit-rate achieved is 72 Mb/s. This would be reduced to 36 Mb/s by a half rate convolutional coder. However, IEEE802.11 specifies a ¾ rate ‘punctured coder’ for 64-QAM which gives a bit-rate of 72 x3/4 = 54 Mb/s. A ¾ rate punctured convolutional coder is a half rate coder with 2 out of every 6 bits strategically deleted (erased) to reduce the bit-rate to 4/3 times the original. Erasing these bits reduces the error correcting power of the convolutional coding. But they are erased in such a way that the original bit-stream is recoverable through the power of the Viterbi decoder if the number of bit-errors introduced by the channel is not too high. OFDM modulation in principle For each of the 64 sub-bands, apply a PSK, QPSK, 16-QAM or 64-QAM mapping by reading a complex number from the appropriate constellation diagram and producing a pair of rectangular pulses expressed as a complex voltage. Let the complex voltages be: X0(t), X1(t), ..., X63(t). These complex voltages must remain constant for each ‘pulse (symbol) period’ T. Then vector-modulate these complex volytages onto the 64 'sub-carriers' of frequencies: fC , fC + fD, fC + 2fD , … , fC + 63fD OFDM modulation in practice: In practice the modulation described above is performed in two stages:

Mult

Map

exp(2πjfCt)

X0(t) 10110..

Mult

Map

exp(2πj(fC+fD)t)

X1(t) 11001..

Mult

Map

exp(2πj(fC+63fD)t)

XN-1(t) 11001..Multi-carrier modulation in principle

CS3282 8.42 11/03/07 / BMGC

Stage 1: Having produced the complex voltages X0(t), X1(t), ..., X63(t), vector-modulate these onto 'sub-carriers' of frequencies: 0 , fD, 2fD , …, 63fD . Stage 2: Vector-modulate the high frequency carrier fC (e.g. 2.457 GHz) with the result of performing Stage 1. Stage 1 is illustrated in the diagram below. There is no mention of fc here: This is Stage 2: For stage 1 we obtain: 63 x(t) = Σ Xm(t) exp (2πjmfD t ) with fD = 1/T m=0

Map

X0(t) 10110..

Mult

Map

exp(2πjfDt)

X1(t) 11001..

Mult

Map

exp(2πj63fDt)

X63(t) 11001..

Stage 1

∑=

63

0

2)(m

tjmfm

DetX π

t

X0(t)

exp(2πjfCt)

∑=

63

0

2)(m

tjmfm

DetX π∑=

+63

0

)(2)(m

tmffjm

DCetX π

Complex multiplication.

= x(t) (complex) (complex but need only real part)

OFDM

CS3282 8.43 11/03/07 / BMGC

Take 64 samples of the pulse x(t) which is of duration T. Let τ = T/64 and denote x(nτ) by x[n] for n = 0, 1, ..., 63. Set Xm(nτ) =Xm : constant for 0 < n < 63. Then, 63 x(nτ) = x[n] = Σ Xm exp (2πjm nτ /T ) m=0 Therefore,

) ( 64

64/2 where

630for ))64/2(exp( ][

63,0

63

0

63

0

m

mm

jnm

mm

XIFFT

meX

nnjmXnx

m

×=

==

<<=

∑

∑

=

=

πω

π

ω

This formula generates a set x[0], x[1], …, x[63] of complex numbers which is a ‘base-band OFDM symbol’ lasting 3.2µs. The formula is identical to the inverse FFT formula given earlier apart from a scaling factor 1/64. So x[0], x[1], …, x[63] can be calculated very efficiently by applying the inverse FFT formula to the set of complex numbers X0, X1, …, X63. So now there are no voltage pulses or complex multipliers, and the OFDM modulation is done completely numerically. There are 64 samples in 3.2 µs which means that the sampling rate is 20 MHz for both the real part and the imaginary part. In principle each part could be ‘digital-to-analogue’ converted to an analogue signal ready to be applied to an analogue implementation of Stage 2. Repeat for the next set of X0, X1, ..., X63 values to get another pulse & so on. In Stage 2, the real part of x(t) multiplies cos(2πfCt) and the imaginary part multiplies sin(2πfCt). Taking the real part of the output generates an OFDM signal starting at fC Hz rather than zero. It is convenient to implement Stage 2 digitally and this means that exp(2πjfCt) must be sampled and that x(t) must be ‘up-sampled’ to the same sampling rate. Assume the carrier frequency fc = 100 MHz and that cos(2πfCt) and sin(2πfCt) have been sampled at 400 MHz. So far, x(t) is available sampled only at 20 MHz. So we have to increase the sampling rate of x(t) by a factor of 20. This is straightforward and is achieved by increasing the inverse FFT order by a factor 20. Instead of a 64 point inverse FFT, we need a 1280 point inverse FFT. Although 1280 is not a power of 2, there are convenient algorithms for performing such an inverse FFT efficiently. The formula now required is 63 x[n] = Σ Xm exp(jm(2π/1280)n) : 0 < n < 1279 m=0 which is more conveniently written as

CS3282 8.44 11/03/07 / BMGC

) (1280

1280/2 where

12790for ))1280/2(exp( ][

1279,0

1279

0

1279

0

m

mm

jnm

mm

YIFFT

neY

nnjmYnx

m

×=

==

<<=

∑

∑

=

=

πω

π

ω

where

≤≤≤≤

=127964 : 0

630 : Xm

mm

Ym

Applying a 1280 point inverse FFT to Ym0,1279 which is a ‘zero-padded’ version of Xm0,63 gives a version of x(t) which is sampled at 400 MHz rather than 20 MHz. Since exp(2πjfCt) is also sampled at 400 MHz, it is now a simple matter to implement ‘Stage 2’ digitally by multiplying x(t) by exp(2πjfCt) sample by sample. Taking the real part of result we obtain a sinusoidal carrier of frequency 100 MHz modulated by a base-band OFDM signal of bandwidth 20 MHz, the result being sampled at 400 MHz. Converting to analogue and removing all frequencies above about 130 MHz leaves an analogue version of the required OFDM signal. The shape of OFDM symbol conveys the bit sequence. With QPSK on 48 carriers, 296 ≈ 8 x 1028 different symbol shapes. The shape must be accurately represented and processed by highly linear circuits. Highly linear amplifiers (Class A) are very power inefficient. The cyclic extension Each 3.2 µs pulse is extended to 4 µs by prefixing a 0.8 µs ‘guard interval’. The prefix is made to be a copy of the final 0.8 µs (16 samples) of the pulse. It is called a ‘cyclic prefix’ or ‘cyclic extension’. We now generate 80 time-domain complex numbers for each ‘extended pulse’. Each extended pulse takes 4 us, so we send 250 k extended pulses/second OFDM receiver The receiver consists of a coherent demodulator followed by a sampler, synchronisation and ‘base-band extended OFDM symbol’ extraction block. Applying an FFT to 3.2µs of the extended baseband OFDM symbol recovers the complex number sequence X0, X1, …, X63 for each symbol. Distortion introduced by the channel may be cancelled out by an equaliser which is applied to the FFT output.

Realx[n]

n

80 160-80

Similarly for imaginary part.

16

Cyclic prefix 3.2µs pulse

Cyclic prefix 3.2µs pulse

3.2µs pulse

Complex multiplication.

OFDM

Sample & extract 4µs ext

FFT

Detector

Detector20 kHz lowpass

Equaliser

CS3282 8.45 11/03/07 / BMGC

Since x[n]0,63 is the inverse FFT of X0, X1, …., X63, the FFT of x[n]0,63 gets back exactly to X0, X1, …., X63, i.e. a set of 64 complex numbers. For each complex number, depending on the sub-carrier modulation used at the transmitter (B-PSK, QPSK, 60-QAM or 64-QAM), the bit or sequence of 2, 4 or 6 bits may be detected by finding the nearest dot on the appropriate constellation diagram. A ‘nearest dot’ detector for each complex number generated by the FFT is therefore required. Effect of the cyclic extension As a guard interval, the cyclic extension eliminates inter-symbol interference between 3.2µs OFDM symbols. The 0.8µs duration was chosen to be longer than any delay likely to occur between a direct path and any reflected paths within a reasonable sized building. As the speed of radio wave propagation is about 300×106 m/s, a 0.8µs guard-interval will allow for a path-length difference of 0.8 × 300 = 250 m. Any reflected path up to 250 m longer than the direct path will not cause one 3.2µs OFDM symbol to interfere with the next 3.2µs OFDM pulse. However the multipath propagation may still distort the structure of individual OFDM symbols and an equaliser is required to reverse this distortion. The function of a guard interval as described in the previous paragraph could have been fulfilled by 0.8 µs of zero voltage. However, the cyclic extension is more than just a guard interval. In combination with the FFT, it greatly simplifies the equalisation process. The effect of multi-path propagation is to cause the radio channel to act like a ‘filter’ in attenuating and delaying certain frequency components in comparison to others. Remember that a single carrier demodulator employs an adaptive filter to cancel out this effect. Filtering, especially adaptive filtering, is a computationally intensive operation. One way of performing a filtering operation is to apply an FFT and then to multiply each output by an equalisation constant. Filtering in the time-domain becomes multiplication in the frequency-domain. Multiplication, even complex multiplication, is much easier than filtering. It is fortunate that the FFT is part of the OFDM demodulation process, so the equalisation, using multiplication rather than filtering, can be applied directly to the FFT output. There is a small difference between filtering performed using an FFT and normal filtering. The former may be described as ‘cyclic’ filtering and the latter as ‘linear’ filtering. Fortunately, the difference disappears when the input to the FFT is the result of applying a cyclically extended signal to a linear filter i.e. the channel. This is the real purpose of the cyclic extension to the OFDM symbol. It allows equalisation to be carried out at the receiver by ‘cyclic’ filtering as implemented by an FFT and complex multiplication.

CS3282 8.46 11/03/07 / BMGC

The cyclic extension is also useful for carrier & symbol synchronisation at the receiver since, if the first 16 samples of an extended pulse are the same as last 16, we are synchronised. Exercise: generation of OFDM with 4 sub-carriers Given 8-bits, 00011011, show how one OFDM symbol may be generated by a 4-point inverse FFT. Use QPSK to modulate the 4 sub-carriers. Extend to 6 samples x[n]0,6 by cyclic extension and explain how a high frequency carrier would be modulated by the samples of x. Show how the original data can be recovered by 4-point FFT. Solution: Data is: 00 01 10 11 Then X0 =1+j, X1 = 1- j, X2 = -1+j, X3 = -1-j X = [ 1+j 1-j -1+j -1-j ]; % array of 4 complex numbers Perform 4 point IFFT on X to obtain array x x=ifft(X) % This does it in MATLAB Array x now contains the 4 samples of the required symbol: [ 0 0.5 + 0.5j j 0.5 - 0.5j ] Including the cyclic extension, this becomes: [ j 0.5 - 0.5 j 0 0.5 + 0.5j j 0.5 - 0.5j ] 8.7.6 Advantages and disadvantages of OFDM OFDM is spectrally efficient because of the orthogonality of the 64 carriers. It is very good for channels affected by frequency selective fading for several reasons: (i) The effects of fading, affecting a small range of frequencies, can be spread out using ‘interleaving’ so that FEC can more easily correct any bit-errors. Interleaving means that the bits to be transmitted are reordered so that adjacent bits are not sent on the same or adjacent sub-carriers within an OFDM symbol. So fading affecting adjacent subcarriers or the same sub-carriers in consecutive symbols should not cause a ‘burst’ of bit-errors which would be difficult to correct. (ii) The cyclic extension, acting as a guard-interval, eliminates inter-symbol interference (ISI) caused by multi-path propagation as explained above. This is a simpler way of eliminating ISI than the time-domain pulse-shaping mechanism used in single carrier systems. With OFDM, the pulses do not run on into each other, whereas with single carrier they do, and we eliminate ISI by aligning the zero-crossings correctly. (iii) Equalisation is much easier than with single carrier systems which use adaptive filtering. OFDM equalisation is done in the frequency-domain after the FFT by multiplying the FFT spectrum by a complex weighting function. The OFDM receiver can examine the output from the FFT and amplify the real and imaginary parts of all sub-carriers such that they have same amplitude. This is possible because of the cyclic extension as explained above. A disadvantage of OFMD is the "peak to mean" ratio of the symbols which can be very large by the nature of the DFT and its inverse. The shape of each OFDM symbol (and there are very many of them, remember) is very complex and must be sent and received accurately. Amplitudes can become very large in comparison to the mean. This is definitely not "constant envelope". The

CS3282 8.47 11/03/07 / BMGC

transmitter and receiver must be linear to preserve the shape, and this necessitates the use of amplifiers that are "class A" and less efficient in terms of power consumption than those used for constant envelope transmissions such as FSK. A lot of power is lost in the amplifiers. This is not really ideal for mobile equipment with small batteries especially not mobile phones that have to be on for a long time. The situation is not so bad for mobile computers with bigger batteries that are not sending data continuously. A further disadvantage of OFDM is that it is sensitive to frequency shifts as may occur due to the Doppler effect in rapidly moving mobile equipment. So it is not ideal for a mobile device, e.g. a PDA or lap-top when used on a high speed train. 8.7.7. Some details about IEEE 802.11a/g OFDM as used for wireless LANs With IEEE802.11a and g, OFDM symbols are transmitted in 4 µs giving a maximum throughput of 250 k symbols/second. Each symbol can carry between 1 and 6 bits per sub-carrier using BPSK, QPSK, 16-QAM or 64-QAM. The highest bit-rate with 64-QAM & 3/4 rate convolutional coder is 48 x 6 x (3/4) x 250 kb/s = 54 Mb/s. The distances over which this bit-rate is achievable in practice will be restricted by transmission loss and interference. Lower bit-rates (48, 36, 24, 18, 12, 9 and 6Mb/s) are available. The two lowest bit-rates (9 & 6 Mb/s) use binary PSK & 3/4 or 1/2 rate convolutional FEC coding :

48 x (3/4) x 250kb/s = 9 Mb/s 48 x (1/2) x 250 kb/s = 6 Mb/s.

For 18 & 12 Mb/s, QPSK is used on each of 48 data carriers. For 36 & 24 Mb/s use 16-QAM. With a 1/2 rate coder, 64-QAM would give 36 Mb/s, so use 2/3 rate for 48 Mb/s. 8.7.8. Conclusions and learning outcomes Matched filtering affects pulse-shaping in single carrier modulation. Channel equalisation, required to cancel the effects of frequency selective fading, is a computationally expensive adaptive filtering task. OFDM is highly efficient form of multi-carrier modulation. OFDM has been compared with single carrier modulation. Single carrier modulation uses ‘sinc-like’ pulses in the time domain with zero-crossings arranged to eliminate inter-symbol interference. OFDM uses ‘sinc-like’ pulses in the frequency domain with zero-crossings arranged to eliminate inter-spectral interference. In the time-domain OFDM pulse shapes are rectangular, perhaps with some ‘raised cosine’ smoothing to zero at the beginning and the end. Among the many advantages of OFDM are that it virtually eliminates the need for pulse shaping and makes equalisation much easier to implement. Channel equalisation much easier to implement - no adaptive filter needed. The use of the inverse FFT and FFT for implementing OFDM make it a convenient and hardware/software efficient modulation technique, though the need for highly linear amplification and the wide range of peak-to-mean ratios to be anticipated cause practical problems especially for battery powered mobile equipment . The parameters of the OFDM implementation used by IEEE802.11 equipment have been analysed. 8.8 Problems and discussion points 1. What is the maximum bit-rate that can be transmitted without ISI on a 1 MHz channel using (i) B-PSK, (ii) QPSK, (iii) 16-QAM. 2. What is the maximum bit-rate that can be transmitted with arbitrarily low bit-errors over a noise-less channel of 1 MHz bandwidth [Ans: ∞] 3. Repeat Q.2 for a noisy channel where the SNR is 30 dB. 4. How does spectrum of a 50% RC pulse differ from that of a pure sinc pulse. 5. Why are RRC rather than RC pulses used in single carrier transmissions.

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6. How many different OFDM symbol shapes are there with 64-QAM? 7. Why are the first & last few sub-carriers left unmodulated? 8. With 16-QAM, why are the 4-bit numbers arranged in ‘Gray coder’ order? 9. Derive a constellation for 64-QAM. 10. Why are interleaving & FEC very important with OFDM? 11.Given that their bandwidth was 30 kHz and typical coherence bandwidths in cities is about 30 kHz, why was an equaliser not needed in a ‘1G’ mobile phone. Why is an equaliser definitely needed in a WLAN receiver when single carrier modulation is used? 12. Explain why the bandwidth efficiency of IEEE802.11 OFDM is 0.6 symbols per Hz without FEC. What is the bandwidth efficiency when a ¾ rate convolutional coder is employed? 13. If a single carrier modulation scheme is used with R% RRC pulse shaping, what value of R would give a bandwidth efficiency of 0.6 pulses (symbols) per Hz ? 14. How are 24 and 36 Mb/s achieved over an IEEE802.11g WLAN? 15. Some non-standard versions of IEEE802.11 claim to achieve 108 Mb/s. How is this done? 16. IEEE 802.11g claims a maximum bit-rate of 54Mb/s for the OFDM payload. But the cost of sending synchronising preamble and headers for each packet reduces this bit-rate considerably even in the most ideal conditions. Assuming ideal conditions with a single transmitter and receiver close together, estimate the maximum average bit-rate (i) where close to maximum length packets (assume ≈ 2000 byte payload) are always sent and (ii) where packets contain only 160 bytes of payload (20 ms of G711 encoded speech).

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Appendix: Explain why modulation doubles the bandwidth of a base-band signal: Multiplying a base-band signal by a sine-wave shifts it up in frequency and also doubles its bandwidth by creating two copies of it: an upper sideband and a lower side-band as illustrated below. We only need one of these copies, but filtering out the other would be complicated and unnecessary when we discover of vector-modulation.

Effect of amplitude modulation on spectrum of a baseband signal To understand why a lower and an upper sideband are produced, consider what happens to a single sinusoid, cos(ωMt), say within the base-band. When this is multiplied by the carrier A cos(ωCt), (with ωC = 2πfC ), we obtain: A cos(ωCt) . cos(ωMt) = 0.5A cos(ωCt + ωMt) + 0.5A cos(ωCt - ωMt) = 0.5A cos( (ωC + ωM) t ) + 0.5 A cos((ωC - ωM)t) So now we have two cosine waves, one at ωC + ωM within an upper sideband and one at ωC − ωM within a lower side-band.

carrier

frequency

Power spectral density