Continuous Phase Modulation (cpm)

In order to further improve bandwidth efficiency, some of the mobile radio systems use special kind of frequency modulation schemes called continuous phase modulation-CPM. The schemes are particularly attractive because they have constant envelope and excellent spectral characteristics resulting from phase changes in a continuous manner. The complex envelope of any CPM signal can be represented as:) () (2) (t j d nT h x k j Ae Ae t v t n f n f Φ − = ∫ ∑ = ∞ − τ τ π (1) where, A is the amplitude, kf is the peak frequency deviation, hf(t) is the frequency shaping pulse, {xn} where xn∈{±1, ±3, ..., ±(M-1)} is the source symbol sequence, M is the symbol alphabet size, and T is the symbol duration. The phase term Φ(t) for the time interval kT ≤ t ≤ (k+1)T can be rewritten as, ∫ ∑ ∫ ∞ − − −∞ = − + − = Φ kT K n t kT f k f f n f d kT h x k d nT h x k t 1) (2) (2) (τ τ π τ τ π (2) where the first term corresponds to accumulated phase and the second term to current phase from the beginning of current symbol period up to time t. Sometimes, Φ(t) is called excess phase. The phase is therefore continuous as long as the frequency shaping function hf(t) does not contain impulses.