Digital Transmission through the AWGN Channel ECE460 Spring, 2012

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

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

Example 7.1 4

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

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

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

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 8 DemodulatorDetector Sampler Output Decision

Demodulators Correlation-Type 9 Next, obtain the joint conditional PDF

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 10 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

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. 11

Probability of Error Binary PAM Baseband Signals Consider binary PAM baseband signals where is an arbitrary pulse which is nonzero in the interval and zero elsewhere. This can be pictured geometrically as Assumption: signals are equally likely and that s 1 was transmitted. Then the received signal is Decision Rule: The two conditional PDFs for r are 12 0

Example 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. 13

Probability of Error M-ary PAM Baseband Signals Recall baseband M-ary PAM are geometrically represented in 1- D with signal point values of And, for symmetric signals about the origin, where the distance between adjacent signal points is. Each signal has a different energies. The average is 14

Demodulation and Detection Carrier-Amplitude Modulated Signals Demodulation of bandpass digital PAM signal Transmitted Signal: Received Signal: Crosscorrelation Optimum Detector 15 Oscillator Received Signal r(t)

Two-Dimensional Signal Waveforms Baseband Signals Are these orthogonal? Calculate ξ. Find basis functions of (b). 16

Problem 7.22 In an additive white Gaussian noise channel with noise power- spectral density of, two equiprobable messages are transmitted by 1.Determine the structure of the optimal receiver 2.Determine the probability of error. 17

Two-Dimensional Bandpass Signals Carrier-Phase Modulation 1.Given M-two-dimensional signal waveforms 2.Constrain bandpass waveforms to have same energy 18

Demodulation and Detection Carrier-Phase Modulated Signals The received signal: Giving basis vectors as Outputs of correlators: 19

Two-Dimensional Bandpass Signals Quadrature Amplitude Modulation 20

Multidimensional Signal Waveforms Orthogonal Multidimensional means multiple basis vectors Baseband Signals Overlapping (Hadamard Sequence) Non-Overlapping o Pulse Position Mod. (PPM) 21

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) 22

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. 23

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 24

Demodulation and Detection Carrier-Amplitude Modulated Signals Demodulation of bandpass digital PAM signal Transmitted Signal: Received Signal: Crosscorrelation Optimum Detector 25 Oscillator Received Signal r(t)