1. Digital Transmission through the AWGN Channel ECE460 Spring, 2012

2. Geometric Representation Orthogonal Basis 1.Orthogonalization (Gram-Schmidt) 2.Pulse Amplitude Modulation a.Baseband b.Bandpass c.Geometric Representation 3.2-D Signals a.Baseband b.Bandpass 1)Carrier Phase Modulation (All have same energy) 1)Phase-Shift Keying 2)Two Quadrature Carriers 2)Quadrature Amplitude Modulation 4.Multidimensional a.Orthogonal 1)Baseband 2)Bandpass b.Biorthogonal 1)Baseband 2)Bandpass 2

3. Geometric Representation Gram-Schmidt Orthogonalization 1.Begin with first waveform, s 1 ( t ) with energy ξ 1: 2.Second waveform a.Determine projection, c 21, onto ψ 1 b.Subtract projection from s 2 (t) c.Normalize 3.Repeat 3

4. Example 7.1 4

5. Pulse Amplitude Modulation Baseband Signals Binary PAM Bit 1 – Amplitude + A Bit 0 – Amplitude - A M-ary PAM 5 Binary PAM M-ary PAM

6. Pulse Amplitude Modulation Bandpass Signals What type of Amplitude Modulation signal does this appear to be? 6 X

7. PAM Signals Geometric Representation M-ary PAM waveforms are one-dimensional where 7 d d d d d 0 d = Euclidean distance between two points

8. Two-Dimensional Signal Waveforms Baseband Signals Are these orthogonal? Calculate ξ. Find basis functions of (b). 8

9. Two-Dimensional Bandpass Signals Carrier-Phase Modulation 1.Given M-two-dimensional signal waveforms 2.Constrain bandpass waveforms to have same energy 9

10. Two-Dimensional Bandpass Signals Quadrature Amplitude Modulation 10

11. Multidimensional Signal Waveforms Orthogonal Multidimensional means multiple basis vectors Baseband Signals Overlapping (Hadamard Sequence) Non-Overlapping o Pulse Position Mod. (PPM) 11

12. Multidimensional Signal Waveforms Orthogonal Bandpass Signals As before, we can create bandpass signals by simply multiplying a baseband signal by a sinusoid: Carrier-frequency modulation: Frequency-Shift Keying (FSK) 12

13. Multidimensional Signal Waveforms Biorthogonal Baseband Begin with M/2 orthogonal vectors in N = M/2 dimensions. Then append their negatives Bandpass As before, multiply the baseband signals by a sinusoid. 13

14. Multidimensional Signal Waveforms Simplex Subtract the average of M orthogonal waveforms In geometric form (e.g., vector) Where the mean-signal vector is Has the effect of moving the origin to reducing the energy per symbol 14

15. Multidimensional Signal Waveforms Binary-Coded M binary code words For example: In vector form: where 15

16. Optimum Receivers Start with the transmission of any one of the M-ary signal waveforms: 1.Demodulators a.Correlation-Type b.Matched-Filter-Type 2.Optimum Detector 3.Special Cases (Demodulation and Detection) a.Carrier-Amplitude Modulated Signals b.Carrier-Phase Modulation Signals c.Quadrature Amplitude Modulated Signals d.Frequency-Modulated Signals 16 DemodulatorDetector Sampler Output Decision

17. Demodulators Correlation-Type 17 Next, obtain the joint conditional PDF

18. Demodulators Matched-Filter Type Instead of using a bank of correlators to generate { r k }, use a bank of N linear filters. The Matched Filter 18 Demodulator Key Property: if a signal s(t) is corrupted by AGWN, the filter with impulse response matched to s(t) maximizes the output SNR

19. Optimum Detector Maximum a Posterior Probabilities (MAP) If equal a priori probabilities, i.e., for all M and the denominator is a constant for all M, this reduces to maximizing called maximum-likelihood (ML) criterion. 19

20. Example 7.5.3 Consider the case of binary PAM signals in which two possible signal points are where is the energy per bit. The prior probabilities are Determine the metrics for the optimum MAP detector when the transmitted signal is corrupted with AWGN. 20