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TELE4653 – Lecture 6: Frequency Shift Keying When frequency modulation was studied in analogue systems, we found the analysis to be considerably more complex than that for amplitude modulation, in a large part because FM is a non-linear modulation technique. The same is true in digital systems – the analysis of frequency shift keying is considerably more involved than that of ASK and PSK. It is a powerful and popular modulation technique, however, so it is certainly worth us taking the time to address its key issues thoroughly.

1. Coherent Frequency Shift Keying As we saw in the earlier chapter on signal space concepts, two different frequency pulses;

( ) ( ) ( ) 2,1for 2cos2 == itftgts ii π will be orthogonal if the frequency difference, 21 fff −=∆ is an integer multiple of T21 , where T is the symbol period. Most FSK systems make the frequency choice

such that the different frequencies are orthogonal, and bandwidth considerations naturally dictate that the frequency difference is always minimal. Binary FSK can be defined as involving the transmission, over each symbol interval, of one of the two possible sinusoids described by,

where 1,0for 2

=+

= iTin

f ci , for some fixed integer cn , which essentially defines the

base carrier frequency, Tnf cc 2= . In requiring this frequency to be an integer multiple of the symbol rate, we assure continuous phase FSK, as we know that the sinusoids will always complete a full cycle of phase at the symbol transition point. Note here that we have let the amplitude shaping pulse, ( )tg , be the unit square pulse, ( )tuT , for simplicity. The effects of using more spectral efficient shaping pulses can be accounted for using the same analysis presented in earlier chapters, and we will free ourselves of this additional complication, and will later find that we can further improve bandwidth by introducing shaping pulses to smooth our transitions between the different frequencies – a property that is unique to FSK. Also, using amplitude shaping pulses will mean our transmitted signal will lose the attractive constant envelope property. We can extend this idea to M-FSK by essentially expanding our set of orthogonal frequency signals to M different frequencies,

( ) ( ) ( ) MitinT

tuTE

ts cTs

i ,,1for cos2

K=

+=π

keeping the continuous phase property, and here the integer cn defines the base carrier frequency.

( ) ( ) ( ) 2,1for 2cos2== itftu

TEts iTb

i π

2

The generation of FSK then easy to envisage – merely map symbols to a choice of one of M oscillators, tuned to the set of frequencies we are transmitting. A simple example of a binary FSK transmitter is shown in the diagram below, after we firstly discuss the form of the signal space and hence the optimal receiver.

frequencysynthesizer

1=mf∆

2=mf∆2

Mm =fM∆

symbol clockmaster oscillator

TR 1=

0f

carrier oscillator

)(ts

The above scheme, however, is not particularly practical for the generation of continuous phase FSK, as it would require the phase synchronisation of these different frequency oscillators. This is conceptually an easy idea, but in real hardware this is near impossible. Hence, real FSK transmitters are built using conventional analogue FM modulators, feed by the information sequence represented by a suitable line code and then passed through some frequency pulse shaping circuitry, as we shall soon see. The basis for the signal space of FSK is also trivial to construct. Since each symbol waveform is orthogonal to every other symbol waveform, the basis signals are just normalised versions of the signals themselves.

( ) ( ) ( ) ( ) MitinT

tuTE

tst cT

s

ii ,,1for cos2

K=

+==πψ

FSK is an orthogonal signal constellation, in which each symbol vector is orthogonal to every other symbol vector. The binary case is sketched below. For larger M the constellations are difficult to draw, but the general structure is that each symbol sits on its own axis, orthogonal to every other signal’s axis. Note that FSK has the same attractive property as PSK – that every transmitted symbol carries the same energy. The other important insight that can be gathered from this constellation is that the distance between every pair of signals in the constellation is exactly the same, and equal to sE2 . Note also that every symbol is a nearest neighbour to every other symbol, so as we increase the order of the constellation, M, the number of neighbours to each signal point increases too. This feature is a significant difference between FSK and both QASK and PSK.

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1ψ

2ψ

sE

sE

We thus have the form of the optimal receiver. It consists of a bank of correlators, each matched to one of the M different frequencies that could have been transmitted. This is generic structure is shown below.

Received signal

∫T

dt0

)()()( tntstr +=

)(1 tψ

Discriminatorinputs, rk

∫T

dt0

)(2 tψ

∫T

dt0

)(tKψ

1r

2r

Kr

Naturally, the scheme shown above is a coherent receiver, since we have assumed that the receiver can generate synchronous copies of each of the basis signals. This is practical for small constellations (small M), for which we can use a common oscillator to generate a base frequency and then mix this signal to produce the frequency-shifted versions. This is not practical for large constellations sizes, however. In this case, we are more likely to opt for a non-coherent receiver, discussed shortly, or the trellis-based Maximum Likelihood Sequence Estimator (MLSE), to be discussed in the context of continuous phase modulation.

2. Bit Error Rate Performance The error performance of M-FSK is fairly easy to determine, armed with our understanding of its signal space structure.

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Firstly, for binary FSK, the two signals are separated by a distance bEd 2= . The exact bit error rate can then be calculated as

=

0NE

QP beb

Thus, binary FSK has worse error performance than binary ASK/PSK – a reduction in effective SNR by one half, or 3 dB. This was readily apparent from an inspection of the signal constellations – binary ASK/PSK achieves maximal separation for an M = 2 constellation. For M-FSK, we can determine our upper bound approximation for the bit error rate as follows. From the signal constellation it is clear that all points are equivalent, and so we can equate the average symbol error probability to the probability of error of any one of the symbols. The interesting thing here is that, if we take one specific symbol, all other (M-1) are equally distant, are nearest neighbours, and, as such, are all equally likely to be the result of the symbol error. The symbol error upper-bound formed by summing the pair-wise error probabilities over nearest neighbours is thus,

( )

−=

0

1NE

QMP ses

We cannot use a Gray code here, as we do not have a hierarchy of distances – all pairs of points are equally distant. The bit error rate is then determined as

( )

=

−=

0

2log2

12

NME

QMPMMP b

eseb

or alternatively, since we have MK 2log= bits/symbol, we could write

= −

0

21 log2

NME

QP bKeb

This very different from the results we obtained for M-QASK and M-PSK. The nature of FSK is that as we increase M the error performance actually improves – completely the opposite to QASK and PSK, for which error performance drops substantially as we increase M. This is shown in the diagram below. As we’ll see in the next section the trade-off here is that the spectral efficiency of M-FSK falls markedly with increasing M. Alas, there is no ‘best’ modulation technique. Each has their own advantages and disadvantages.

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0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0NEb

ebP

2=M

4=M

8=M16=M

32=M

3. Spectral Performance The spectral performance of FSK can get quite complicated for large M. However, for the binary case exact formulae can be derived. Let’s begin by calculating the power spectral density for binary continuous phase FSK or CPFSK). This signal can be written as

( ) ( )

±=

Tttftu

TE

ts cTb

22cos

2 ππ

with the sign determined by whether the upper or lower frequency is to be transmitted. cf here represents the average carrier frequency. This can be expanded as

( ) ( ) ( )tfTt

TE

tfTt

TE

ts cb

cb ππππ 2sin

2sin

22cos

2cos

2

= m

From the above expression, it is clear that the In-phase component of the CPFSK signal is constant, and independent of the symbol transmitted in that interval. The Q-phase is equivalent to a binary ASK system with the half-sinusoid shaping pulse. Making the usual approximations of uncorrelated symbol sequence, we can write an expression for the power spectral density of the binary CPFSK signal, in terms of an equivalent baseband PSD,

( ) ( ) ( ){ }cBcB ffSffSfS ++−= 21

where

( ) ( )( )( )222

2

21

cos841

41

2 fT

fTET

fT

fTE

fS bbB

−+

++

−=

π

πδδ

Notice the form of this spectrum, consisting of two strong peaks at the pair of carrier frequencies, and the term that comes from the half-sinusoid behaviour. The half-sinusoidal shaping pulse is extremely spectrally efficient (recall the 99% bandwidth is 1.18/T), so the bandwidth performance of CPFSK is very good. There is a little bit

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more to be said here, though, in comparing the distribution of signal power within the occupied band. It is interesting to note the effect on the PSD resulting from not preserving the continuous phase properties at the symbol transitions. The effect of discontinuities in phase is make the two phases correlated, and the cross-terms in the resulting PSD result in a net decay of 21 f , not as the fourth power of frequency as in CPFSK. This represents a considerable cost in bandwidth efficiency. The equivalent analysis for M-FSK with higher M is quite involved, so we’ll restrict ourselves to making some broad, general comments here. The general picture of M-FSK is that we have a set of frequencies equally spaced by 1/2T, as shown in the diagram below.

f∆

fM∆

f

We could then make a general approximation for the bandwidth required to transmit this signal as

TMB2

=

This will in general be an under-estimate of the true signal bandwidth, as we know that while the different frequency signals are orthogonal they are not spectrally disjoint and there is a significant amount of overlap between the spectral components. Thus, the leakage spectrum at the end of the spectrum shown in the diagram will be significant. However, for large M its contribution to this bandwidth estimate will be relatively small. Using the above result we can then obtain an approximation for the spectral efficiency of M-FSK,

MM

BRb 2log2

==η

While this result is far from being exact, it clearly illustrates that the spectral efficiency of M-FSK drops heavily with increasing M, in contrast once again to QASK and PSK. Naturally, we can think of this as a result of the orthogonal structure of the FSK signal constellation, that increasing M increases the dimensionality of the constellation, resulting in the associated bandwidth increase.

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4. Continuous Phase Frequency Shift Keying We can further exploit the continuous phase structure of FSK to further improve the bandwidth performance and error performance of FSK. The scheme we will describe here is often referred to as Continuous Phase Modulation (CPM), but it is really no more than a variant of CPFSK with smoother phase transitions. The important idea here at play is to exploit the inherent memory in the signal – since the phase transitions are continuous, our knowledge of the previous carrier phase can help is make a decision on the current carrier phase, and hence the current transmitted symbol. Let’s consider binary CPFSK. We can write the transmitted symbol waveform as

( ) ( ) ( )[ ]ttftuTE

ts cTb θπ += 2cos

2

where the carrier phase deviation is determined by the binary symbol transmitted

( ) ( ) TttTht ≤≤±= 0for ,0 πθθ

The parameter h we will define shortly. Notice that the phase changes linearly over a symbol period, which is akin to saying that the frequency of the wave is constant over a symbol period. The quantity h is called the deviation ratio, or alternatively the modulation index (in reference to its similarity to the analogue case), and it is related to the difference in the carrier frequencies,

( )12 ffTTh f −=∆= The minimum frequency separation to maintain orthogonality is 1/2T, which would correspond to 2

1=h . CPFSK with this deviation ratio is called Minimum Shift Keying (MSK), for this reason. The deviation ratio defines the phase change of the carrier over the symbol:

( ) ( )−

=−0 symbolfor ,

1 symbolfor ,0

hh

Tππ

θθ

For MSK this corresponds to a carrier phase change of 2π± each symbol period. The significance of this is that this modulation scheme now has memory, in the sense that the carrier phase state over any one symbol depends not solely on the symbol itself, but also on the previous set of transmitted symbols, encapsulated by the carrier phase at the end of the previous symbol. This is very different from QASK and PSK, as these schemes had no memory, and so our optimal receiver could act on a symbol-by-symbol basis. CPFSK is more like DPSK, and as such our optimal receiver should make decisions by considering more than a symbol at a time. The idea of memory in the transmitted signal is well represented by the ‘phase trellis’. The phase trellis plots the possible evolution of the carrier phase for different possible input sequences. The phase trellis is shown in the diagram below for MSK, for which the carrier phase changes by 2π± each symbol.

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2/3π

π

2/π

0

2/3π

π

2/π

0

2/3π

π

2/π

0

2/3π

π

2/π

0

0=t Tt = Tt 2= Tt 3=

-1

1

-1

1

1

1-1

-1

-1

-1

The important point in this diagram is that at any symbol period not all carrier phase states are possible for the transmitted signal, and knowledge of the previous carrier phase can help us make a decision on the current carrier phase. We will analyse how to use the phase trellis structure to improve the performance of receiver for the case of MSK in the next section. CPM beyond the binary case becomes a form of partial response signalling, which we will touch on a little later. Our final point in CPM is the idea of a frequency shaping pulse. It is instructive to write our sequence of transmitted frequencies, for binary-CPM, which is really the carrier instantaneous frequency as a function of time, as

( ) ( )∑∞

−∞=

−+=n

nci nTtgahftf

where { }1±=na determines whether the frequency transmitted at that symbol is Tff c 411 −= or Tff c 412 += . The function ( )tg represents how we transition

between frequencies, which for the signal we have been discussing, is instantaneously,

( ) ( )tuT

tg T21

=

The instantaneous carrier frequency as a function of time for this frequency shaping pulse is shown in the diagram below.

We can integrate the instantaneous carrier frequency to obtain the carrier phase as a function of time.

( ) ( ) ( )∑∫∞

−∞=∞−

−+==n

n

t

cii nTtqahtfdft ππττπθ 222

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where ( ) ( )dttgtqT

∫=0

is the equivalent phase shaping pulse, equal to the integral of

the frequency shaping pulse. For the square-pulse choice as our frequency shaping pulse, corresponding to instantaneous frequency transitions, the transition pulse is

( )

≥

<<

≤

=

Tt

TtTt

t

tq

,

0,2

0,0

21

)(tg

tT

T21

)(tq

tT

21

For MSK, this corresponds to our familiar phase transitions of 2π± every symbol period, as illustrated below.

tT

πh

1),(2 ]1[]1[ =mm atqahπ

T2 T3

tT

πh

1),(2 ]2[]2[ =− mm aTtqahπ

T2 T3

tTπh−

1),2(2 ]3[]3[ −=− mm aTtqahπ

T2 T3

tT T2 T3

)(tθ

The point is that it is not necessary that we transition the frequencies instantaneously, and can envisage a scheme where we transition between the two carrier frequencies more smoothly. Making the frequency transition smoother makes the overall signal smoother, and this has the important consequence of reducing the bandwidth of the

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signal still further. A common frequency shaping pulse in CPM is the Raised-Cosine (RC) frequency shaping pulse,

( ) ( )tuTt

Ttg T

−=π2cos1

21

The important characteristics being merely that the frequency shaping pulse integrates to be 21 , so that we still maintain the MSK property of inducing carrier phase changes by 2π± each symbol. Note that now we can no longer so much as think of CPM as being us keying the frequency of the carrier between two frequencies – the carrier frequency instead varies continuously over this frequency interval, effectively averaging at one of those two frequencies.

Gaussian MSK (GMSK) is an important practical example of CPM with a non-square frequency shaping pulse, which was adopted as the modulation solution in the GSM global second generation mobile phone communication standard. The scheme to generate GMSK is easy to conceive, and illustrates the general practical way that CPM with non-square frequency shaping pulses can be generated – pass a binary square signal through a suitably designed low-pass filter to produces pulses with the desired form, that can then be used to drive a V.C.O. or another frequency modulator. The Gaussian filter used in GMSK has the following transfer function,

( )2

22log

3

−

= Bf

efH and associated impulse response,

( )22

3

2

2log2

32log2 tB

eBthπ

π −

=

where 3B is the filter 3 dB bandwidth. The output from this filter when a square pulse is the input can easily be shown to be,

( ) ( )

−−

=

TTtB

TtBtg 332

1

2log2erfc

2log2erfc ππ

This is shown in the diagram below.

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The obvious point here is that this frequency pulse does not have finite support, and extends well outside the symbol interval, [ )T,0 . This manifests itself as ISI. The degree of inter-symbol interference is determined by the relative size of the time extent of the resultant shaping pulse relative to the symbol period. The time extent of the shaping pulse is inversely proportional the filter bandwidth, 3B , hence the extent of ISI introduced by the Gaussian filter can be measured and expressed by the 3-dB bandwidth–symbol period product, TB3 . The smaller this value, the larger the pulses are relative to the symbol interval, so the greater the extent of ISI. This then encapsulates the ultimate trade-off in GMSK: a more spectrally efficient scheme is achieved by reducing the bandwidth of the frequency pulse shaping filter to reduce signal bandwidth, but this has the counter-effect of introducing ISI in the signal.

5. Minimum Shift Keying Let’s now turn our attention firmly on MSK, in order to understand how to design a receiver to make use of the phase memory of the transmitted signal. The MSK signal can be written as,

( ) ( )( ) ( ) ( )( ) ( )tftTE

tftTE

ts cb

cb πθπθ 2sinsin

22coscos

2−=

where ( ) ( ) tT

t2

0 πθθ ±= . The phase trellis is as shown below. The observation is that

the carrier phase can take on values of 0 or π at the end of even numbered periods, and values of 2π± at the end of odd numbered periods, taking the reference initial phase as 0.

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2/3π

π

2/π

0

2/3π

π

2/π

0

2/3π

π

2/π

0

2/3π

π

2/π

0

0=t Tt = Tt 2= Tt 3=

-1

1

-1

1

1

1-1

-1

-1

-1

Expressed as above the MSK waveform resembles that of a QASK waveform. Looking at the effective in-phase amplitude, we see that,

( )( ) ( )( ) ( )( )

±−

= t

Tt

Tt

2sin0sin

2cos0coscos πθπθθ

Considering the waveform over the two symbol periods TtT ≤≤− , we’ll take 0=t to be the representative of the even numbered periods, then ( ) 00 =θ or π, and thus

( )( ) 00sin ≡θ . The effective in-phase amplitude can thus be expressed as

( ) TtTTT

Ets b

I ≤≤−

±= ,

2cos

2 π

with the sign determined by the carrier phase at 0=t , positive if ( ) 00 =θ and negative if ( ) πθ =0 . Similarly, it is easy to show that the quadrature amplitude is determined only by the carrier phase state at odd multiples of the sampling period, ( ) 2πθ ±=T , and the quadrature amplitude can be expressed as,

( ) TtTT

Ets b

Q 20,2

sin2

≤≤

±=π

where the amplitude is positive if ( ) 2πθ +=T , and negative if ( ) 2πθ −=T . This implies that we may construct an alternative basis for the MSK signal,

( ) ( ){ } ( ) ( )

= tf

TTtf

TTtt cc ππππψψ 2sin

2sin2,2cos

2cos2, 21

so that the MSK signal is a linear combination of these, ( ) ( ) ( )tststs 2211 ψψ +=

The coefficients can be found as,

( ) ( ) ( )[ ]0cos11 θψ b

T

T

Edtttss == ∫−

and

( ) ( ) ( )[ ]TEdtttss b

T

θψ sin2

2

02 == ∫

Note that here we observe these two amplitudes over two symbol period each, and that these intervals are staggered. This makes sense, in the sense that we cannot make an accurate estimate of say ( )0θ by only observing for Tt ≤≤0 , but we can if we see the resultant change over two symbol periods. Inherently we thus included memory into our optimal receiver, by arriving at this signalling basis. The

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performance gain of MSK comes from allowing ourselves to extend the signal beyond the simple symbol by symbol case. The four combinations of carrier phase states at the end of even numbered and odd numbered symbol periods, expressed here as ( )0θ and ( )Tθ , map to symbols as in the table below:

Transmitted symbol over

Tt ≤≤0

Phase state at even-numbered periods ( )0θ

Phase state at even-numbered periods ( )Tθ

Signal vector, ( )21 , ss

0 0 -π/2 ( )bb EE , 1 π -π/2 ( )bb EE ,− 0 π π/2 ( )bb EE −− , 1 0 π/2 ( )bb EE −,

The signal constellation for MSK, expressed with respect to this basis, is shown in the figure below.

The structure of the optimal receiver is shown in the diagram below. The idea is that the receiver alternatively makes decisions about the carrier phase state at even symbols by looking at the I-phase amplitude over two symbol periods, and for the carrier phase state at odd numbered symbol periods from the Q-phase amplitude over a the two symbol periods offset from the previous pair. From these pair of decisions the receiver reconstructs the original bit sequence.

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It is straightforward to show, from the above signal constellation, that MSK has the same bit error rate performance as BPSK and QPSK, when we utilise the inherent memory in the transmitted signal,

=

0

2NE

QP beb

Our conclusion is that MSK represents then a modulation scheme that combines excellent spectral efficiency with good bit error rate performance.

6. General Continuous Phase Modulation (CPM) We can go further and extend the discussion of the previous two sections to the general case, and then onto partial-response signalling. The most general transmitted signal could be written as

( ) ( )( )0,2cos2

θθπ ++= attfTE

ts cs

where ( )a,tθ is the resultant carrier phase due to the sequence of transmitted symbols, ( )K,, 10 aa=a , where ( ){ }1,,3,1 −±±±∈ Mai K comes from an alphabet of M

symbols. The only thing of interest to us is thus this phase function, ( )a,tθ , as it conveys the transmitted sequence. For a linear frequency modulation scheme, this phase response will be linear combination of the phase response produced by each symbol:

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( ) ( )iTtqahtn

ii −= ∑

=02, πθ a

where h is the modulation index and ( )tq is the phase shaping pulse, defined by the

frequency shaping pulse, ( ) ( )dxxgtqt

∫=0

, as before. Causality implies that

( ) 0,0 <= ttq , and by convention we require that ( ) LTttq ≥= ,21 . Note here that we

allow the current symbol to influence the carrier phase over L consecutive symbols, not simply a single symbol, as did for MSK above requiring implicitly that L = 1. The modulation index, h, is generally chosen to be a rational number, for purposes of producing a finite state trellis, and also typically 3

2≤h , for reasons of spectral efficiency. The data symbol causes a net carrier phase change of πhai radians over the interval of L symbols, ( )[ )TLiiT +, . The two most common choices for frequency shaping pulses used in CPM are rectangular pulses, called L-REC:

( )

≤≤=

otherwise,0

0,2

1 LTtLTtg

and the L-RC, for raised-cosine:

( )

≤≤

−=

otherwise,0

0,2cos12

1 LTtLTt

LTtgπ

Thus, MSK is a special case of CPM with 1-REC and 21=h .

The natural way of visualising CPM is with the phase tree, or phase trellis. A new M-ary symbol influences the phase trajectory every symbol period, T, and its influence last for L symbol intervals. There are thus NM distinct paths or codewords (we will see the relationship of CPM to trellis coding later in the course) when the trellis is viewed over N symbol intervals. The phase trellis is scaled by πh2 . The phase trellis shown in the previous section for MSK had M = 2, with the influence of the symbol lasting only for L = 1 symbol interval. A more general phase trellis, for 3-RC with M = 4 is shown in the diagram below. Note the extreme continuity of the carrier phase, regardless of the precise trellis path, especially when compared to something like M-PSK. This belies the excellent spectral performance of CPM. Note that, in fact, we could incorporate our description of M-DPSK into CPM if we allow discontinuous phase transitions. We get M-DPSK if we set the phase shaping pulse ( )tq to be the step function with height 2

1 and modulation index Mh 1= , so that the phase in each interval changes by one of

( ){ }MMMM πππ 1,,3, −±±± K . This is exactly M-DPSK, aside from a trivial additional phase rotation of Mπ .

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Since CPM is a non-linear modulation technique it’s precise spectral analysis is very complicated. However, a couple of general remarks can be made. The power spectrum generally widens for increasing h, with spectral lines emerging for integer values of this modulation index. Small values of h are not in of themselves the secret to bandwidth efficient communication, and in fact most of the information content of the signal will reside in the low sidelobes (students could think back to the analogue case, for the equivalence of Narrowband FM to AM, so message content comes in the two low power sidelobes). Finally, the phase continuity of the CPM signal implies the spectral sidelobes decay as 8−− cff , or 80 dB/decade. This rapid spectral decay, while limiting out of band interference, is not in of itself the end of the story – it provides no information on how spectral power is distributed within the band. A convenient technique for understanding CPM modulated signals is to think in terms of state diagrams and state space machines. For rational modulation index, rph = , then there are a discrete number of carrier phases that can exist at the end of each symbol interval. Moreover, since

( ) ( ) ( )TntnTnTtqahtn

Lnjjn 1,2,

1

+≤≤−+= ∑+−=

πθθ a

the entire CPM signal could be completely specified each symbol by the carrier phase at the beginning of the symbol interval, and currently ‘active’ symbols (that is, those recent symbols that are still currently influencing the carrier cumulative phase). We

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could thus specify a state of the carrier as ( )11 ,,,, +−− Lnnnn aaa Kθ , of which there are 1−LrM possible values, hence possible states. The next input signal causes the CPM

waveform to change state in a natural way. The state transition diagram for a 2-REC signal with 3

2=h and M = 2 is shown below. We will come back to the power of this state space description of the CPM signal, particularly with regard to practical implementation of the optimal decoder, later when we meet convolutional encoding.

There are two basic implementation of a CPM modulator. The first uses an analogue frequency modulator, driven by a baseband, pulse-shaped signal from the information sequence. While this is fairly simple and the hardware is readily available, it does suffer from the aging and tolerances of components producing drifts in the precise modulation index, h.

( )∑ −

nn nTtga

Baseband, partial-response Pulse-shaping

{ }na FM

modulator (index h)

CPM out

An alternative is to use the fact that CPM can be reduced relative to the QASK basis. As long as we employ an (admittedly complex) signal-mapper with memory, we could realise CPM we could realise CPM in a similar fashion to QASK, as shown in the diagram below:

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( )( )a,cos tθ ( )tfcπ2cos

( )tfcπ2sin

Signal Mapper (with memory)

{ }na

CPM out

( )( )a,sin tθ−

TEs2

Finally, let’s make a few comments about the optimal detector of CPM, which will naturally continue our previous discussion of detectors that operate over a sequence of received symbols, rather than on a symbol-by-symbol basis. Based on our discussion in earlier lectures, the optimal receiver for the sequence of N symbols embedded in white Gaussian noise would correlate the received waveform, r(t), against the LM possible signal trajectories and choose the one that produces the largest correlation, under the assumptions that all sequences are equally probable and that all signals have equal energy. These correlations over the interval NT can be broken into a sum of N correlations each over a T interval. The issue here is we would naturally expect the complexity of such a receiver to grow exponentially with N, as a Maximum Likelihood Sequence Estimator (MLSE). There is an algorithm to greatly reduce this complexity, called the Viterbi algorithm, which exploits the finite state structure of the signal. We will explore this algorithm under a different guise later in this course, when we apply it to convolutional decoding.

Lastly, a comment about the error performance of CPM decoders, implemented using this MLSE structure. An error is caused when a different path through the trellis looks more like the received sequence than the sent path, and so the error performance of CPM is related to the distance between neighbouring trellis paths. Similarly as with the Viterbi algorithm, we will explore this problem under the guise of convolutional encoders in a later lecture. Our reason for this is that the presentation of these ideas in this context is conceptually easier to understand, at least in the opinion of this lecturer.

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7. Partial-Response Signalling Partial Response Signalling is a method for improving our transmission rate to approach or exceed the Nyquist limit (R > 2B) by allowing controlled ISI into the transmitted data sequence. The example of duo-binary signalling for binary PAM will be presented in the hand-written lecture notes. Partial-response signalling achieves a modest improvement in spectral efficiency (increased data rate for a given bandwidth allocation) at moderate cost in BER performance (or essentially the signal power budget).

8. Optimal Non-coherent Receiver We’ll now consider demodulation formally when we have a random carrier phase. The received signal is

( ) ( ) ( ) ( )tntftgatr mcm +++= θγπ2cos2 where θ is assumed to be a random variable, uniformly distributed as

( ) πθππ

θθ ≤<−= ,21p

The set { }mma γ, represent the amplitude and phase modulation that has been given to the carrier to convey the symbol m (be this ASK, PSK, FSK, or whatever). The point is that, no matter what these quantities are, they are known to the receiver. The problem then is to design the optimum receiver for the situation when the carrier phase θ is unknown. To motivate our discussion, let’s consider the issues surrounding the carrier phase determination. Receiver synchronisation ultimately requires the receiver and the transmitter to synchronise their local oscillators at the beginning of communication, and maintain this synchronisation for the duration of transmission. In many cases this is completely impractical. For instance, a movement in the relative distance between the transmitter and the receiver by a fraction of a wavelength will alter the relative phases. For communication at 1 GHz, the wavelength of the associated radiation is around 0.3m, and a multi-pathing radio channel will easily have changes in transmitter-receiver distance of this order. Moreover, consider the effect of a frequency offset in the oscillators. If our 1 GHz has a frequency difference of only 1 Hz, which is a phenomenally high quality of one part in 910 , then the relative phases will drift by 2π in one second. In the next chapter we’ll look at techniques to achieve synchronisation of the receiver’s oscillator using information obtained from the received signal – the techniques of phase locked loops and their digital equivalents. Here we’ll consider the case, which is a design issue, where we can attempt to recover the signal without tracking the carrier phase. Naturally, receivers that have knowledge of the carrier phase must perform better than those that do not, but the may interest here is to develop the tools and analysis for us to be able to formulate this trade-off. The first point is that the variation in the carrier phase means we need to double the dimension of the basis of our signal space. The signal we receive can be expanded as

20

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )tntftgatftga

tntftgatr

mcmmcm

mcm

++−+=

+++=

θγπθγπ

θγπ

sin2sin2cos2cos2

2cos2

With a coherent detector, with θ = 0, we would only need a single basis function for this received signal, ( ) ( ){ }mcg tftgE γπ +2cos2 , but with unknown phase we require

an expanded basis of ( ) ( ) ( ) ( ){ }mcgmcg tftgEtftgE γπγπ +−+ 2sin2,2cos2 . Consequently we must double the number of correlators in the optimal non-coherent receiver. The received signal is then correlated with the In phase and the Quadrature of each basis vector (note that the quadrature of the basis vector, we formally mean the Hilbert transform of the basis vector – the basis vector with its effective phase shifted). Denote the expanded basis as ( ) ( ){ }Kiii tt 1ˆ, =ψψ , where K is the dimension of the signal space, and the outputs of the correlators as

( ) ( )dtttrzT

ici ∫=0

ψ

( ) ( )dtttrzT

isi ∫=0

ψ̂

We could then define the conditional distribution for the received signal vector, ( )θ,| msp r , given the sent message ms and unknown channel phase. Naturally, we

would like to integrate out the unknown channel phase, and determine the probability of us receiving r given ms was sent, regardless of the carrier phase,

( ) ( ) ( ) ( )∫∫−−

==π

π

π

πθ θθ

πθθθ dspdpspsp mmm ,|

21,|| rrr

For the ideal AWGN channel, this probability distribution is determined only by the noise, since the noise is the only reason our received signal differs from the transmitted signal. Thus,

( ) ( )( )( ) θθππ

π

π

dNsrN

sp mii

K

im 0

2

1 0

exp121| −−= ∫∏

− =

r

Expanding the exponent we find,

( )( )

( ) ( )[ ]( ) θθθππ

π

π

dNssreN

spK

imimii

K

i

NrKm

i ∫ ∏∏− ==

− −=1

02

10

2exp211| 0

2

r

or in terms of the vector represented in the expanded signal space,

( )( )

( ) ( )[ ]( ) θθθππ

π

π

dNeN

sp mmN

Km 02

0

2exp211| 0

2

ssrr r −⋅= ∫−

−

Having expressed the probability distribution in terms of the signal vectors allows us to interpret each term more readily. ( ) 2θms is another way of writing the symbol energy, mE , which is independent of θ (and always independent of our choice of basis for the signal space). The first term in the exponent is the correlation between the received signal and the message signal,

21

( ) ( ) ( )dttstr m

T

m θθ ,0∫=⋅ sr

To simplify matters, we’ll consider the case of an orthogonal signal constellation, so that ( ) ( ) ( ) ( ) ( )( )θψθψθ sincos, ttsts mmmm −= , and then

( ) ( )θθθ sincosmm scmm zzs −=⋅ sr

This simplification allows us to express each signal in terms of one basis signal and not as a sum over the basis signals. It is not difficult to expand this analysis into the general situation of a signal constellation that is not orthogonal, however this is rarely needed in practise, and does increase the level of algebra needed substantially. One can immediately conceive of this simplification in terms of the signalling basis. For an orthogonal signal constellation of dimension K the optimum receiver consisted of a bank of K correlators, one for each element signal of the basis. This could alternatively be realised by a bank of K matched filters. In the equivalent non-coherent detector, to account for the unknown carrier phase, we must expand each correlator into a pair, each correlating over the alternative carrier phase. For a non-coherent detector this realisation of the expansion of the signalling basis can be more difficult to envisage. For the case of an orthogonal signalling scheme, we can write the probability distribution as

( )( )

[ ]( ) θθθππ

π

π

dNzzEeeN

spmm

mscm

NENKm 0

0

sincos2exp211| 00

2

−= ∫−

−− rr

This is the key step to allow us to integrate out the angular dependence. Define 22mm scm zzz +=

and

m

m

c

sm z

z1tan −=α

so that

( )( )

( )( ) θαθππ

π

π

dNzEeeN

sp mmmNEN

Kmm

00

cos2exp211| 00

2

+= ∫−

−− rr

Here we have effectively determined the envelope amplitude of the received signal, by taking the correlation with respect to each carrier phase and combining. This integral is actually the zero-order modified Bessel function of the first kind, defined by

( ) [ ] ββπ

π

dxxI ∫=2

00 cosexp

21

Thus,

( )( )

= −−

00

0

21| 002

NEz

IeeN

sp mmNENKm

mrrπ

Note that ( )xI 0 is a monotonic increasing function of x, as shown in the diagram below.

22

We typically work in log-likelihood ratios, when it comes to determining decision thresholds on the transmitted symbol. For the symbol m, the corresponding log-likelihood ratio is, after ignoring terms that are independent of m, the sent symbol, and hence will not help us distinguish between symbols

( ) ( )( )00

0

2log|log

NE

NEz

Ispm mmmm −

== rl

Note that if we followed the analysis through, not making the assumption of an orthogonal signal constellation, the log-likelihood ratio would need to be expressed as a sum of terms dependent on the quadrature outputs from the pairs of in-phase and quadrature correlators, and the associated signal components relative to those basis vectors,

( )01 0

02

logNE

Nsz

Im mK

i

mii −

= ∑

=

l

The added complexity that we do not wish to consider, though, at a fundamental level, will not alter our conclusions. The general form of an orthogonal signal constellation, such as FSK which we are ultimately focussed on in this chapter, though we aim to keep our analysis as general as possible, is that all the symbols have the same energy, say 2

0gEg = . Our choice of message m, given the received vector r, is one that has the highest probability of r being received, over all possible messages sent,

( )mpmm

′=′

|maxarg r

Thanks to the monotonic nature of log(x), this is equivalent to maximising log-likelihood ratio. Thus,

= ′′

′ 00

2logmaxarg

NEz

Im mm

m

As log(x) and ( )xI 0 are monotonic, we could merely maximise mz ′ or 2mz ′ . Thus, our

message is chosen from the rule, m

mzm ′

′= maxarg

with 22mm scm zzz += .

We can conceive the optimal non-coherent detector as a bank of pairs of correlators, consisting of a correlator tuned to each basis signal and its Hilbert transform. The

23

receiver then squares and adds the pairs of amplitudes from each correlator pair, and choses the message that corresponds to the largest,

One interesting point to note is that determining the amplitude mz ′ is equivalent to determining the carrier envelope,

( ) ( ) ( )mcmmmsmmc tfztfztfz απππ +≡− 2cos2sin2cos Thus, each pair of correlators on each basis signal could instead be replaced by an envelope detector, after an appropriate matched filter has been used to select that orthogonal component from the others. The three equivalent forms of receiver implementation for a single basis component are shown in the diagram below.

24

One can see that the non-coherent receiver, when applied to an orthogonal signalling scheme such as FSK, is simple and cheap to implement, as it does away with any need of complex carrier recovery circuits. The important question to then ask is: what is the associated cost in error performance in using a non-coherent detector? For an orthogonal signal constellation, our sent signal vector is

( )0,,0,,0,,0 KK sm E=s since we send one of the orthogonal signals each symbol. Our receiver then forms a received signal vector by passing the received signal r(t) through a pair of correlators, squaring then adding the outputs, and finally square-rooting, to form each component of the received signal vector,

( )Mm zzzz ,,,,, 21 KK=r From the orthogonal nature of the constellation, only one of these components, mz , that corresponding to the sent signal, will depend on the signal energy – the others are just due to the noise. The receiver choses the message corresponding to the largest component in the received signal vector r. The receiver will make an error if one of the other components of the received signal vector is larger than mz , the one on which we sent our signal vector.

( )mkzzPP mkes ≠>= somefor ,

25

To determine this probability we must find the distribution of each of these components. We have already performed most of the analysis for this on pages 15 and 16, when we established the probability distribution for the received signal vector. For the probability that the received amplitude is mz , given the message sent was m, we find a Rician distribution,

( ) ( )( )02

00

0

exp22| NEzNEz

INzmzp sm

smmm +−

=

while the other components of the received signal follow a Rayleigh distribution,

( ) ( )02

0

exp2

| NzNz

kmzp kk

k −=≠

This intuitively makes sense, since for mk ≠ , the resultant distribution is formed by taking the root-squared sum of two zero mean Gaussian random variables, with is known to be a Rayleigh distribution. When the Gaussian random variables do not have zero mean, as in the case for the component the message was sent on, the result of the root-squared sum is a Rician distribution.

To find the probability of a symbol error, it is easiest to find the probability that a correct decision is made. A correct decision is made if, for our received component mz , all other components are less than this, then average over all possible values of

mz , weighted by the likelihood of that value occurring.

( ) ( ) ( ) mmmk dzmzPmmkzzPmP || allfor |correct0∫∞

≠<=

Using the distributions above, and the fact that each component is independent,

26

( ) ( )

[ ] ( )( ) msmsmmMNz

mm

M

z

Nz

dzNEzNEz

INz

e

dzmzPdzeNzmP

m

m

02

00

00

1

1

0 0

exp22

1

|2|correct

02

02

+−

−=

=

∫

∫ ∫∞ −−

−∞ ∞

−

The above integral looks fairly involved, however it is actually quite simple. When the term to the exponent (M-1) using the binomial theorem, it reduces to a sum of integrals of the form ∫ − dxxe x2

, each of which is straightforward to compute. One finds that the symbol error probability is then,

( ) ( )( )

+

−

−+

−=−= ∑

−

=

+

0

1

1

1

1exp

11

1|correct1Nj

jEj

Mj

mPP sM

j

j

es

Being an orthogonal constellation, the associated bit error rate for this non-coherent detection scheme is, with MK 2log= bits per symbol,

( )( )

( )

+

−

−+

−−

= ∑−

=

+

0

21

1

1

1log

exp1

11

12 NjMjE

jM

jMMP b

M

j

j

eb

One should note that, for binary signalling, this has the quite simple form, 0

21 NE

ebbeP −=

9. Non-coherent FSK Having made the general structure of the optimal non-coherent detector for an arbitrary orthogonal signalling scheme, we’ll make a few specific comments for its application in FSK. The first point above that was glossed over was that it was inherently assumed that, while the basis set of signals is orthogonal, so is the expanded basis of the signal basis and the Hilbert transform of each basis signal. This is not trivial, and in the case of FSK, would mean requiring that, for arbitrary frequencies with separation

21 fff −=∆ ,

( ) ( ) 02sin2cos 20

1 =∫ dttftfT

ππ

Performing the integral this requires that ( )

02

2cos1=

∆

∆−

f

f Tππ

which is true if the frequency spacing is an integer multiple of T1 , exactly double the frequency separation needed to maintain orthogonality in coherent FSK. This is the case, that non-coherent detection of FSK involves a reduction in spectral efficiency of one-half. Not maintaining orthogonality would involve a much too greater cost in bit error rate and receiver complexity.

27

Accepting this increase in frequency trade-off, a non-coherent detection scheme could then be implemented for M-FSK, following the principles explained in the earlier section. The form of the receiver is as shown,

Received signal

∫T

dt0

)()()( tntstr +=

tfT

tc 11 2cos2)( πψ =

Discriminatorinputs, rkc,rks∫

T

dt0

∫T

dt0

cr1

sr1

Msr

tfT

ts 11 2sin2)( πψ =

tfT

t MMs πψ 2sin2)( =

or else, the alternative structure using envelope detectors previously described. The bit error rate resulting from using such a detector is,

( )( )

( )

+

−

−+

−−

= ∑−

=

+

0

21

1

1

1log

exp1

11

12 NjMjE

jM

jMMP b

M

j

j

eb

The bit error rate performance of non-coherent FSK is worse than that of coherent FSK, with the same reason as we saw in the case of DPSK, resulting from the effect of squaring or multiplying the noise terms. It is worth comparing the performance of the different schemes in the binary case. For non-coherent detection for binary FSK, the bit error rate is

021FSK n.c. NE

ebbeP −=

For basic CPFSK with coherent detections, we have

=

0

FSK c.

NE

QP beb

Finally, for MSK, with the optimal detector (in turn the same as BPSK), we find

=

0

MSK 2NE

QP beb

These are plotted together in the figure below.

28

10. Case Study - GSM GSM (originally Groupe Speciale Mobile, but later Global Standard for Mobile Communication) was the European solution for the second generation of mobile phone communication networks. The second generation referred to a fully digital network, as the first generation used analogue modulation techniques. The GSM was widely adopted around the world, and was the leading 2G mobile standard worldwide, and as such, is the basis for many of the current 3G mobile phone networks. A summary of the important features of the GSM standard with regard to Radio layer of communications is as follows: Uplink frequency band: 890-915 MHz Downlink frequency band: 935-960 MHz Channel bandwidth: 200 kHz Multiple Access: Time Division (TDMA) – 8 timeslots/frame Modulation: GMSK - Time-Bandwidth product TB3 = 0.3 Data rate: 270.833 kbps (Data rate for individual user varies) Speech coding: Regular Pulse Excited–Long Term Prediction (RPE-LTP, really LPC) Channel Coding: Convolutional Coding, plus block interleaving. Diversity: frequency hoping and time interleaving. The important point for us is the use of GMSK, Gaussian Minimum Shift Keying, chosen for its very good bit error rate performance and excellent bandwidth conservation. Block diagrams illustrating the modulation and demodulation process for a GSM system are shown below:

29