Comparison of Receivers for Multi-h Cpm

A noncoherent receiver for the general case of M -ary partial response multi-h continuous phase modulation (CPM) is presented. The receiver operates on the principle of sequence estimation via the Viterbi Algorithm (VA). It offers a significant complexity reduction from the optimal coherent maximum likelihood sequence estimating (MLSE) receiver. The performance of the receiver is evaluated with computer simulations. It performs at a loss of 1–6 dB relative to the MLSE receiver for the CPM schemes considered in the simulations. The receiver shows promise in applications requiring reduced complexity and use of existing hardware. INTRODUCTION The Advanced Range Telemetry (ARTM) Tier II modulation format is multi-h CPM. Some of the attractive features of this modulation format are that it has constant envelope and narrow bandwidth [1]. The optimal MLSE receiver for CPM [2] has complexity that is a function of the modulation indices hi, symbol alphabet size M , and pulse length L. The complexity, i.e. the number of states in the trellis and number of matched filters (MFs), can be large depending on the specific CPM parameters. The MLSE receiver has the additional requirement of coherent detection. Much of the existing hardware intended for use with PCM/FM can be used to demodulate CPM. This approach is a suboptimal method. A receiver of this type is presented which addresses the above difficulties with the MLSE receiver. It is noncoherent, and operates directly on the output of an FM demodulator (FMD). It applies sequence estimation using the VA. Its complexity is independent of the modulation indices, which offers a significant complexity reduction in most cases; the downside of this is that it does not take advantage of the performance gains of multi-h over single-h. The receiver performs at a loss of 1–6 dB relative to the MLSE receiver for the CPM schemes considered in the simulations. These characteristics make the receiver an attractive alternative in applications where sufficient link margin is present and where reduced complexity and use of existing hardware are desired. In the following, the CPM signal is described. The MLSE receiver is summarized and the noncoherent receiver is presented. The relative performance of the two receivers is evaluated with computer simulations. The results are summarized and conclusions are presented. SIGNAL DESCRIPTION CPM refers to a general class of digitally modulated signals in which the phase is constrained to be continuous. The carrier modulated signal may be expressed as s(t) = √ 2E T cos ( 2πfct + φ(t, α) ) (1) where T is the length of the basic signaling interval, E is the signal energy during that interval, and fc is the carrier frequency in Hz. The time-varying phase term φ(t, α) is defined as φ(t, α) = π ∫ t −∞ n ∑ i=−∞ αihig(τ − iT ) dτ nT < t ≤ (n + 1)T (2) φ(t, α) = π n ∑ i=−∞ αihiq(t− iT ) (3) where {αi} is the sequence of M -ary information symbols with possible values ±1,±3, . . . ,±(M − 1), {hi} is the sequence of modulation indices, g(t) is the normalized frequency pulse shape, and q(t) = ∫ g(τ) dτ is the phase pulse. A special case is when hi = h for all i. In this case the modulation index is fixed for all symbols. The general case, when the modulation index varies from symbol to symbol, is called multi-h. In the multi-h case, the modulation index hi cycles through a fixed set of values. The frequency pulse g(t) is non-zero only in the interval 0 ≤ t ≤ LT . When L = 1, the CPM signal is called full response CPM. When L > 1, the CPM signal is called partial response CPM. The phase pulse q(t) is constrained to q(t) =    0 t < 0 ∫ t 0 g(τ) dτ 0 ≤ t ≤ LT 1 t > LT. (4) The information-carrying phase φ(t, α) can be written as φ(t, α) = π n ∑ i=n−L+1 αihiq(t− iT ) + π n−L ∑ i=−∞ αihi mod 2π (5) = θ(t, αn) + θn. (6) The phase φ(t,α) is defined by the correlative state vector αn = (αn−1, αn−2, . . . , αn−L+1) and the phase state θn. The number of correlative phase states is ML−1. For hi = 2ki/p (ki, p integers), the phase state θn takes on p distinct values 0, 2π/p, 2 · 2π/p, . . . , (p− 1)2π/p. The total state is described by the L-tuple σn = (θn, αn−1, αn−2, . . . , αn−L+1) and the number of states is pML−1. MAXIMUM LIKELIHOOD SEQUENCE ESTIMATING RECEIVER The MLSE receiver is described in detail in [2]. The receiver selects as its output the sequence α̂ that maximizes the log likelihood function Λ(α) = ln ( pr(t)|α̂(r(t) | α) ) ∼ − ∫ ∞ −∞ ( r(t)− s(t, α))2 dt (7) = ∫ ∞ −∞ (−r2(t) + 2r(t)s(t, α)− s(t, α)) dt. (8) Since the received signal r(t) is not a function of α̂, and s(t, α̂) is constant-envelope, it is equivalent to maximize the correlation J(α̂) = ∫ ∞ −∞ r(t)s(t, α̂) dt. (9) Computing this correlation for all possible α̂ is not feasible in practice, even for reasonably short data bursts. It is possible to compute J(α̂) recursively, by defining Jn(α̂) = ∫ (n+1)T −∞ r(t)s(t, α̂) dt (10) = Jn−1(α̂) + Zn(α̂n, θ̂n) (11) where Zn(α̂n, θ̂n) = ∫ (n+1)T nT r(t) cos ( 2πfc + θ(t, α̂n) + θ̂n ) dt. (12) Zn(α̂n, θ̂n) can be viewed as the output of a filter sampled at t = (n + 1)T , the input to the filter being r(t). Over each symbol interval, the receiver correlates the received signal with every possible transmitted sequence (there are pM possibilities). By applying trigonometric identities, Zn(α̂n, θ̂n) can be expressed as [2] Zn(α̂n, θ̂n) = cos(θ̂n) ∫ (n+1)T nT Î(t) cos ( θ(t, α̂n) ) dt