User:Oli Filth/MSK

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Minimum-shift keying (MSK) is a type of continuous-phase frequency-shift keying (CPFSK), but may also be regarded as pulse-shapedoffset quadrature phase-shift keying (OQPSK), and hence is a form of linear modulation.

Contents

[edit] TO-DO

  • Use as a modulation scheme
  • Rationale - constant-modulus
  • Frequency characteristics
  • Comparison of constellation transitions compared to QPSK and OQPSK
    • When filtered with a Nyquist filter
  • Implementation
    • Switched oscillators
    • Continuous-phase VCO modulation
    • Filtered OQPSK
      • Link to other forms, e.g. GMSK
  • Timing diagrams (as a comparison)
    • I/Q form
    • Phase vs. time
    • Modulated vs. time
  • Formula (if relevant)


[edit] MSK as a form of CPFSK

In any binary modulation scheme, the two signalling waveforms, g0(t) and g1(t), are orthogonal if they satisfy:

\int_0^T g_0(t) g_1(t) dt = 0

where T is the symbol period.

In binary FSK, we define each waveform as sinusoidal, e.g.:

\ g_0(t) = \cos ( 2 \pi f_0 t + \theta )
\ g_1(t) = \cos ( 2 \pi f_1 t + \theta )

where θ is some arbitrary phase offset. It can be shown that the minimum frequency separation Δf = f1 − f0 that satisfies the orthogonality constraint is 1 / 2T.

This result is not dependent on the phase offset, θ. Therefore, we may design the system to have continuous phase, as this results in a waveform with lower spectral sidelobes than a waveform with phase discontinuities. In other words, the value of θ in each symbol period is such that the phase continues smoothly from the previous symbol. This is MSK.

Typically, f0 and f1 are defined in terms of a centre frequency, fc, as:

f_0 = f_c - \frac{1}{4 T}
f_1 = f_c + \frac{1}{4 T}

[edit] Implementation

[edit] Transmitter

MSK may be implemented using a voltage-controlled oscillator (VCO), whose control signal is modulated by the data, such that s[n] = + 1 results in f1, and s[n] = − 1 results in f0.

Image:MSK_CPFSK.png

It should be noted that the continuous-phase restriction means that MSK cannot be implemented by on-off keying two separate oscillators.

[edit] Receiver

TODO

[edit] PSK

Remembering that phase is the integral of angular frequency with respect to time, we can calculate the change in phase, Δφ, that occurs over each bit period:

\Delta \phi = 2\pi \int_0^T [ f(t) - f_c ] dt = \pm 2\pi \left(\frac{r_b}{4}\right) T = \pm \frac{\pi}{2}

In other words, during every bit period, the phase advances or recedes by π / 2, which corresponds to quarter of the way around the unit circle in the complex plane.

We can implement this using I/Q modulation, as shown in the diagram:

DIAGRAM

Note that the I/Q modulation is preceded by a differential encoder; this is to ensure that for a given input sequence, the PSK implementation gives the same output as for the FSK implementation.

[edit] Comparison with QPSK, OQPSK and π / 4-QPSK

At this point, we may compare the constellation diagrams of QPSK, OQPSK, π / 4-QPSK, and MSK. As shown in the diagrams, QPSK has transitions that pass through the origin...

Image:Constellations.png

[edit] Constant-envelope considerations

In an electronic communication system, power-efficiency and cost issues often require that the transmitter uses a non-linear power amplifier.

  • Stuff about effect on ISI
  • Spectral regrowth
  • Constellation distortion

Consequently, a constant-envelope modulation scheme is highly desirable.

[edit] The MSK approach

QPSK, OQPSK and MSK all use the same set of constellation points, arranged in a square centred on the origin in the complex plane.

In QPSK, the I and Q components may change simultaneously, allowing transitions through the origin. In a hypothetical system with infinite bandwidth, these transitions occur instantaneously; however, in a practical band-limited system (in particular, a system using a Nyquist filter) these transitions take a finite amount of time. This results in a signal with a non-constant envelope.

OQPSK attempts to reduce the problem by only allowing one component to change at a time; this limits the phase transitions to ±90°. However, the signal is still not constant-envelope.

MSK is a filtered form of OQPSK, performed in such a way that the transitions occur around the unit circle in the complex plane, resulting in a true constant-envelope signal.

NEED CONSTELLATION DIAGRAMS WITH TRANSITIONS SHOWN



[edit] Etymology

The term minimum results from the fact that from the FSK viewpoint, the two frequencies used are at the minimum allowable separation whilst maintaining orthogonality, in the sense that the waveforms corresponding to 1 and 0 satisfy:

\Re \left\{ \int_0^T g_0(t) g_1^*(t) dt \right\} = 0

This allows the receiver to distinguish between them with a pair of correlation filters, although an MSK receiver is seldom implemented in this way.

This orthogonality relationship holds if the frequency separation is any integer multiple of rb / 2.


[edit] Old

Similarly to OQPSK, MSK is encoded with bits alternating between quarternary components, with the Q component delayed by half a bit period. However, instead of square pulses as OQPSK uses, MSK encodes each bit as a half sinusoid. This results in a constant-modulus signal, which reduces problems caused by non-linear distortion.

The resulting signal is represented by the formula

S(t) = a_{I}(t)\cos{(\frac{{\pi}t}{2T})}\cos{(2{\pi}f_{c}t)}+a_{Q}(t)\sin{(\frac{{\pi}t}{2T})}\sin{(2{\pi}f_{c}t)}

where aI(t) and aQ(t) are the square pulses as shown in QPSK.

A similar modulation scheme is Gaussian minimum shift keying, which uses Gaussian instead of sinusoidal pulse shapes.

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