A Note on the Orthogonality of MSK Signals

A Note on the Orthogonality of MSK Signals J.A. Schoonees and R.M. Braun Department of Electrical Engineering University of Cape Town Rondebosch 7700 South Africa

Abstract — Minimum shift keying (MSK) got its

name from its description as the coherent FSK scheme with ‘the minimum frequency spacing which allows the two FSK signals to be coherently orthogonal’. This notion is not well defined in the literature. MSK signals are not strictly orthogonal over any bit interval for all frequencies, as a simple calculation shows. The MSK waveform should be observed over two bit intervals instead of one, as Massey showed with his optimal coherent MSK receiver. Even then, a naive implementation may show degradation due to non-orthogonality at low signal frequencies. We derive expressions for the optimal coherent MSK receiver which show that the signals are orthogonal only for certain values of the centre frequency and initial phase. This is important for demodulator designs which work at low signal frequencies, such as sampled implementations.

I

Introduction

Minimum shift keying (MSK) [1, 2] is a digital modulation scheme using binary frequency shift keying (FSK) signals. The two signals are constrained to have continuous phase at the bit boundaries, and to differ in frequency by half the bit rate (in other words, the modulation index h = ∆f T = 12 , where T is the bit period). A practical way of generating MSK signals is to use a phase continuous voltage controlled oscillator (VCO) driven by a rectangular (NRZ) data waveform (figure 1). The amplitude of this waveform is controlled by a level shifting circuit so that the right modulation index is obtained. The differential encoder (mod 2) causes the ‘standard’ Type I MSK to be generated [3]. Several definitions of MSK exist, differing only in the coding applied to the data [4]. This determines the mapping between input data and signal frequency. Although the scheme of figure 1 does not guarantee precisely controlled frequency or phase, its simplicity is

0011101001 data ∈{0,1}

010011101

q T- j 6

-

Level shift

-

²¯ MSK ±°

-

VCO

Differential encoder

Figure 1: VCO implementation of MSK modulator preferable to that of other options in applications such as satellite links. Such an implementation will require a robust demodulator. Another way of describing the MSK waveform is with a phase tree diagram as shown in figure 2(a). It shows how the waveform phase changes relative to the phase of the signal with the lower frequency, depending on the choice of data. Assuming that we start with a particular phase at a bit boundary, it can be seen that the two possible signals diverge by half a cycle (a phase difference of π radians), over the next bit interval. This is an important characteristic of MSK and follows from the condition that the modulation index h = 21 . We can denote the two signals by s0 (t) s1 (t)

= A sin(2πf0 t + θ) and = A sin(2πf1 t + θ).

(1)

A is an arbitrary amplitude, f0 and f1 are the two signalling frequencies, and θ is an arbitrary phase offset. The average carrier frequency for random data is fc = (f0 + f1 )/2. If we ‘wrap around’ the phase back to zero when it reaches 2π radians, we obtain a more schematic version of the phase tree. The state trellis (figure 2(b)) identifies phase 0 with state 0, and phase π with state 1. This state is in fact the state of the differential encoder

s0 (t)

6

θ(t)−φ

? R - j -

4π 3π 2π

q q

T

kT

Xk

+ q T Σj + 6

π

0

T

2T

3T

4T

r(t)

t

(a)

π

:state 1 e

−s0 (t)

e

s1 (t)

e

−s0 (t)

−s0 (t)

e

s1 (t)

−s1 (t)

+ ? j Σ− 6

pr

- j 6

Y ? pq T - Σj q q-k j− kT 6 +

R

T

s1 (t) −s1 (t)

0

:state 0 e

s0 (t)

(k−1)T

s1 (t)

e

ck ∈ {−1, +1}

−s1 (t)

e

s0 (t)

kT

- dˆ

s0 (t)

(k+1)T

e

Figure 4: Massey’s MSK demodulator

(k+2)T

(b)

So far we have viewed MSK as simple CPFSK. A more sophisticated approach recognises that MSK is a quadrature modulation scheme with orthonormal basis functions which are orthogonal over two bit intervals [7]:

Figure 2: MSK phase tree and state trellis φ1 (t) = of figure 1, being the contents of its memory (delay) element. Each branch of the trellis is labelled with the signal being transmitted during that bit interval. Note that a signal sometimes has to be inverted in order to maintain phase continuity.

II

φ2 (t) =

kT

must yield I01 (fc , θ) = 0 for all fc and θ during any bit interval k. But substituting (1) into (2) and integrating, we obtain (3)

where ωc = 2πfc . This is certainly not zero (figure 3), because zero correlation is only obtained between such signals when the centre frequency fc =

nc 2T

where nc is some fixed positive integer.

r

2 π sin( t) sin(2πfc t) ; 0 ≤ t < 2T. T 2T

fc =

MSK is described as the coherent FSK scheme with ‘the minimum frequency spacing which allows the two FSK signals to be coherently orthogonal’ [1, 5, 6]. At first glance, this may be taken to mean that the test for orthogonality Z (k+1)T s0 (t) · s1 (t) dt (2) I01 (fc , θ) =

A2 T sin(2θ) − sin(2θ + 2ωc T ) 2 2ωc T

π 2 cos( t) cos(2πfc t) ; − T ≤ t < T T 2T

(4)

(5)

(6)

Even these are strictly orthogonal for

MSK’s ‘orthogonality’

I01 (fc , θ) =

r

nc 4T

(7)

only, where nc is some fixed positive integer.

III

Massey’s optimal receiver

We digress for a moment to look at Massey’s MSK receiver, which gives another insight into MSK’s supposed orthogonality. Massey was the first to prove that MSK can be demodulated optimally over only two symbol intervals. His demodulator is shown in figure 4. He gives the proof of the optimality of the receiver in [8], based on the matched filter and maximum likelihood detection. The state trellis of figure 2(b) is time-variant, because even and odd bit intervals have differently labelled branches. We can hide this by introducing a multiplier ck = −1, +1, −1, +1 . . . as shown in figure 5. Now the branch labelling is the same in every bit interval. Massey then shows that the maximum likelihood decision is made when Xk + Xk+1 ≥ −ck Yk + ck+1 Yk+1

(8)

0.5

3π 8

π 4 I01 A2 T

0 π 8

π 2

0

-0.5 0

0.25

0.5

0.75

1 fc T

1.25

1.5

Figure 3: I01 versus centre frequency fc for various starting phases θ

1.75

2

π

:state 1 e

e

−s0 (t)

e

−s0 (t)

ck s1 (t)

−s0 (t)

e

figure 6. The correlation I11 (when the lower correlator in figure 4 is matched to the input) is the same, within a frequency offset:

ck s1 (t)

ck s1 (t)

I11 (fc , θ) = I00 (fc + −ck s1 (t)

0

:state 0 e

−ck s1 (t)

e

s0 (t)

s0 (t)

kT

(k−1)T

s0 (t)

e

It is apparent in both sampled autocorrelations that the graphs for the various starting phases all pass through the nominal value of A2 T /2 when the centre frequency is no fc′ = no odd, no ≥ 3. (12) 4T It is only for this specific frequency choice that I00 and I11 assume their nominal value of A2 T /2 irrespective of the phase offset. I01 is still a function of θ, but the offsets cancel cleverly in Massey’s decoder structure for all θ.

(k+2)T

Figure 5: Time-invariant MSK state trellis where Xk Yk

= =

(k+1)T

kT Z (k+1)T

r(t)s0 (t)dt and r(t)s1 (t)dt.

(9)

kT

The illustrated receiver structure follows directly from (9). The important conclusion to be drawn from this is that in general there are no restrictions on the choice of the FSK signals s0 (t) and s1 (t), nor any requirement of orthogonality, except for the following two conditions [8]: 1. s0 (t) and s1 (t) must have the same energy in any given bit interval; and 2. The data inputs must be equally likely. Point 1 follows from the fact that it is a matched filter (correlator) implementation, which seeks to maximise the signal-to-noise ratio, and point 2 follows from the fact that it is a maximum-likelihood detector (as opposed to a maximum a posteriori probability detector [9, page 214]) which is optimum only for a random input with equal data probabilities. Of course, for different s0 (t) and s1 (t), we do not have MSK any more. By making s0 (t) = s1 (t) (identical , not orthogonal) and ck = 1 in figure 5 for all k we get BPSK, and Massey’s receiver will demodulate that optimally as well. The sampled autocorrelation over one bit interval of s0 (t) is the sampled output of the integrate-and-dump circuit when matched to the input signal: I00 (fc , θ)

(11)

−ck s1 (t)

e (k+1)T

Z

1 , θ) = |Yk |. 2T

= |Xk | Z (k+1)T s20 (t) dt = kT · ¸ sin 2θ + sin(2θ + 2ωc T ) A2 T 1+ (10) . = 2 2ω0 T

The only effect that the sign of the carrier can have is to change the sign of this expression. I00 is plotted in

At these frequencies the input to the limiter, in the absence of noise, as shown in figure 4 is an ideal rectangular function. At other frequencies, point 1 above is violated and the receiver performance is degraded. When the centre frequency is much higher that the bit rate, the effect is negligible, but at low values such as fc = T1 simulations clearly show a distortion in the nominally rectangular shape of the decoder output, before a hard bit decision is made.

IV

Discussion

MSK is ‘orthogonal’ only as long as its signals s0 (t) and s1 (t) have equal energy in any given bit interval. This condition is satisfied only for certain values of the centre frequency, given by (12). The effect is most noticeable at low frequencies, relative to the bit rate, which has implications for demodulators which are designed to work at the lowest possible frequencies.

Acknowledgements We thank Dr. Stephen Hodgart for first drawing our attention to the issue of MSK’s orthogonality, and for his initial analysis. This work was supported by a grant from the Foundation for Research Development.

References [1] M.L. Doelz and E.H. Heald. Minimum-shift data communication system, March 1961. U.S. Patent 2977417. [2] R. de Buda. Coherent demodulation of frequencyshift keying with low deviation ratio. IEEE Trans. Commun., COM-20:429–435, June 1972. [3] P.Z. Peebles. Digital Communication Systems. Prentic-Hall, 1987.

1

π 2 3π 8 I00 A2 T

0.5

π 4 π 8

0

0 0

0.25

0.5

0.75

1 fc T

1.25

1.5

Figure 6: I00 versus centre frequency fc for various starting phases θ

1.75

2

[4] B. Bohm. An investigation into a generalised formulation of minimum shift keying. Dept. Electrical Engineering, University of Cape Town, November 1992. Final year thesis. [5] S. Pasupathy. Minimum shift keying: A spectrally efficient modulation. IEEE Commun. Mag., 17(4):14– 22, July 1979. [6] F.G. Stremler. Introduction to Commnunication Systems, third edition. Addison-Wesley, 1990. ISBN 0201-51651-9. [7] S. Haykin. Digital Communications. John Wiley and Sons, 1988. ISBN 0-471-62947-2. [8] J.L. Massey. A generalized formulation of minimum shift keying modulation. In Conf. Rec. Int. Conf. Commun. ICC’80, volume 2, pages 26.5.1–26.5.4. IEEE, June 1980. [9] J.M. Wozencraft and I.M. Jacobs. Principles of Communication Engineering. John Wiley and Sons, 1965.