MSK is a Linear Modulation More on MSK Instructor: M.A. Ingram ECE4823 Recall BFSK The Phase for MSK Recall the phase trajectories for binary FSK when rectangular pulses are used for m(t): φ (t ) = 2πk f ∑ I n q (t − nTS ) φ (t ) = π ∑ I n q (t − nTS ) n n π π /2 TS 2TS 3TS −π / 2 −π − 3π / 2 − 2π − 5π / 2 4TS Y-intercept Form In MSK, kf=0.5 5π / 2 2π 3π / 2 5πk f 4π k f 3πk f 2πk f πk f − πk f − 2πk f − 3πk f − 4πk f − 5πk f MSK is an unusual type of FSK because it can be represented as a linear modulation It looks like OQPSK with pulses that are twice as wide The “symbols” weighting the pulses are not the original symbol sequence [Simon et al., ’94] The phase can be expressed πt φ (t ) = I n 2TS + xn where xn is the Y-intercept for the nth symbol interval [xn mod( 2π )]∈ {0, π } TS 2TS 3TS 4TS Example Consider the symbol sequence Io=1, I1=-1, I2=-1, I3=1, I4=1, I5=1, I6=-1 5π / 2 2π 3π / 2 π π /2 −π / 2 −π − 3π / 2 − 2π − 5π / 2 TS 2TS 3TS 4TS 1 For Io=1 and 0<t<TS πt 2TS φ (t ) = I 0 πt + x0 = 1 ⋅ 2TS For I1=-1 and TS<t<2TS + 0 x0 = 0 5π / 2 2π 3π / 2 2TS 3TS 4TS πt + x2 = −1 ⋅ 2TS πt 2TS φ (t ) = I 2 + π x2 = π 2TS 3TS 4TS πt 2TS φ (t ) = I 3 πt + x3 = 1 ⋅ 2TS − 2π x3 = −2π =0 5π / 2 2π 3π / 2 π π /2 TS 2TS 3TS 4TS −π / 2 −π − 3π / 2 − 2π − 5π / 2 I/Q Formulation of MSK It can be shown that where x1 = π For I3=1 and 3TS<t<4TS π π /2 s (t ) = TS −π / 2 −π − 3π / 2 − 2π − 5π / 2 5π / 2 2π 3π / 2 + π π π /2 TS For I2=-1 and 2TS<t<3TS −π / 2 −π − 3π / 2 − 2π − 5π / 2 πt + x1 = −1 ⋅ 2TS 5π / 2 2π 3π / 2 π π /2 −π / 2 −π − 3π / 2 − 2π − 5π / 2 πt 2TS φ (t ) = I1 2ε b ∑ cn p (t − ( 2n − 1)TS ) cos (2πf c t ) TS n − ∑ d n p (t − 2 nTS ) sin (2πf c t ) n πt sin p (t ) = 2TS 0 0 ≤ t ≤ 2TS otherwise c n = (− 1) cos x 2 n −1 n d n = (− 1) I 2 n cos x 2 n n TS 2TS 3TS 4TS Example, cont’d n In 0 1 xnmod2π cosxn 0 cn 1 -1 π -1 2 -1 π -1 3 1 0 1 4 1 0 1 5 1 0 1 6 -1 0 1 dn d0=1 1 c1=1 d1=-1 c2=1 d2=1 c3=-1 d3=1 2 Baseband Waveform for the Example c1 = 1 c3 = − 1 c2 = 1 MSK Resembles OQPSK I The basic pulse shape is half a sinusoid that is 2TS wide t TS 2TS 3TS 4TS 5TS 6TS 7TS c1 = 1 c3 = 1 c2 = 1 I d0 = 1 d1 = − 1 d3 = 1 d2 = 1 TS 2T S 3T S Q 2TS 3TS 4TS 5TS 6TS The MSK receiver can be implemented coherently like an OQPSK receiver ( 2 n +1)TS r(t) 2 n −1)TS (• )dt ( 2 n + 2 ) TS ∫( πt sin 2TS 2 n )TS Logic Circuit (• )dt t d4 = 1 Decision Circuit 2T S 3T S 4T S 5T S 6TS 7T S Summary Decision Circuit πt cos (2πf c t ) cos 2TS 7T S t Quadrature Detection ∫( 6TS Q 7TS TS 5T S d3 = 1 d 2 = −1 d1 = 1 t TS 4T S MSK is actually a linear modulation method The basic pulse shape is half a sinusoid that is two bit periods wide Resembles OQPSK with this basic pulse shape sin (2π f c t ) References [Rapp, ’02] T.S. Rappaport, Wireless Communications, Prentice Hall, 2002 [Simon et al., ’94] M.K. Simon, S.M. Hinedi and W.C. Lindsey, Digital Communication Techniques, Signal Design and Detection, PTR Prentice Hall, 1994. 3