EE 568 is our first course at the graduate level in Digital Communications. The course has been designed mainly to provide a good understanding of fundamental problems in digital communications over what is known as Additive White Gaussian Noise (AWGN) channels. The course starts with a short review of elements of communication systems and the basic probability and random processes. Linear bandpass signals and systems, orthogonal expansion, vector representation of signals are discussed. Linear modulation techniques with and without memory are presented. This is followed by illustration of different ways of describing memory, e.g., by state diagram, trellis state transmission matrix, etc. Nonlinear modulation with memory; CPFSK, full and partial response CPM are detailed. Power spectral density derivations for all these modulations are presented. General waveform channel synthesis and geometrical representation of signals along with vector recovery by means of a correlator bank will be described. Optimum receiver principles in AWGN channels are discussed; optimum Maximum A-posteriori (MAP) and Maximum Likelihood (ML) decision rules, correlation and matched-filter receivers, SNR, MLSE and its Eucleadian metric are included among other topics. Performance analysis is presented for all the modulation techniques discussed earlier in terms of average bit error and symbol error probability as measure of information quality. Non-coherent and differentially coherent detection methods are discussed. Carrier and symbol synchronization techniques are discussed. Topics covered include: ML criterion, Carrier phase estimation, data and non-data assisted phase estimation techniques. Phase locked loop (PLL), Equivalent PLL with noise, Linearized PLL, Square-law tracking of phase, COSTAS loop, Symbol timing estimation, Timing recovery circuits, data and non-data assisted timing estimation, ML timing estimation and Early late gate synchronization techniques.