Generalize from CPFSK to continuous phase modulation (CPM) by allowing: non-rectangular frequency pulses frequency pulses with durations LT, L > 1

Transcription

1 8.3 Generalize to CPM [P4.3.3] CPM Signals Generalize from CPFSK to continuous phase modulation (CPM) by allowing: non-rectangular frequency pulses frequency pulses with durations LT, L > o Example: 2REC and we can go to L-REC: o Example: 2RC (raised cosine) and we can go to L-RC: 8.3-

2 o Example: Gaussian MSK (GMSK), with parameter BT : The frequency pulse is a Gaussian filtered rectangular pulse: The frequency pulse has area /2, as usual: Why bother with CPM? o Smoother transitions, so lower bandwidth; o Slower rise of phase pulse (takes L symbols to get to ½) means smaller frequency shift, so lower bandwidth; o Longer rise of phase pulse to ½ means a chance of increasing d min between alternative signals 8.3-2

3 The generic CPM transmitted signal is then φ (, t I) = 2 πh I q( t kt) + φ k k o st (, I) = Pe φ j (, t I ) where φ o is usually taken to be zero with coherent detection. Typical CPM signal trajectories: o First, the total phase φ(, t I) o Next, the complex signal sti (, ): 8.3-3

4 8.3.2 CPM States and Trellis The state description of general CPM is more complex than for the CPFSK special case. Consider the effect of the past on φ () t in interval n, nt t < ( n + ) T, shown below for 3RC, which has L = 3. All symbols are taken as + here. n φ (, t I) =π h I + 2 πh I q( t kt) + 2 πhi q( t nt) k k n k n L k= n L+ current pulse phase state θ correlative state I,, I n n n L+ the state σ = ( θ, I,, I ) n n n n L+ o The phase state θ is not the same as the signal phase φ (nt). It is the n sum of all saturated phase pulses, up to and including. In L o The correlative state comprises all past symbols for which the phase pulse is still changing across interval n. o And, of course, the remaining influence is the current symbol I n

5 So the state affecting interval n, nt t < ( n + ) T is σ = θ, I, I,, I, summarizing relevant history, n n n n 2 n L+ phase correlative state state and the state update is ( ) θ n+ =θ n +πhin L+ mod 2 π (, I, I,, I ) σ n+ = θ n+ n n n L+ 2 The allowable values of the phase state θ n are the same as in full response ( L =), like CPFSK. Example: binary transmission with 2REC The phase pulse is o The phase in interval n, nt t < ( n + ) T is determined by (a) the new bit I n ; (b) the legacy from the past: In, which is still causing the phase to change; and πh I =θn, which is static, from saturated pulses. k n 2 So the state is (, ) σ = θ. k n n In 8.3-5

6 8.3-6 o The phase state is determined by θn n 2 I and previous bits. The possible values of θ n are determined by h alone, not L or M. For h = 23: o Transitions: Since θ n+ =θ n +πhin, the In part of the state n ( n, In ) σ = θ tells you whether θ n+ will increase from θ n or not. For example, (0,+) must go to (, x) (0,-) must go to (2, x) o Sketch a trellis using state σ = ( θ, ) n n In 8.3-6

7 Caution: Unlike CPFSK, the phase φ (nt) at the start of interval n is not sufficient itself as a state variable o If it were, then the phase trajectory φ () t in nt t < ( n + ) T would be determined solely by φ (nt) and the data I n. o But φ () t in nt t < ( n + ) T is affected by previous symbol values in the phase state and the correlative state, and φ (nt) cannot unambiguously represent both components of the state. o Example: 2REC, h = 23, again. We have π φ ( nt ) =θ + 2 π hq( T ) I =θ + I 3 The phase n n n n φ (nt) can take on 3 values: Solid dots: phase state θn Open dots: the 3 possible values of starting phase φ ( nt) These phases cannot unambiguously represent θ n and In. For example, phase 0 can arise from 0 ( I = ). n π + ( n 3 I = + ) or from 2 π π