Dept. of EE, NDHU 1 Chapter Three Baseband Demodulation/Detection

Dept. of EE, NDHU 2 Signal and Noise Error performance degradation in communication systems –Filtering effect at the transmitter, channel, and receiver, which causes intersymbol interference (ISI) –Electrical noise and interference produced by a variety sources –Thermal noise (Gaussian distributed, its two-side spectral density is N0/2, which is flat for all frequencies) Demodulation and detection –Demodulation is recovery of a waveform –Detection means the decision-making process of selecting the digital meaning of the waveform

Dept. of EE, NDHU 3 Two basic Steps in the Demodulation / Detection

Dept. of EE, NDHU 4 Receiving Filter and Decision Making The goal of the receiving filter is to recover a baseband pulse with the best possible signal-to-noise ratio (SNR), free of any ISI Matched filter or correlator is the optimum receiver Equalizing filter is only needed for systems where channel-induced ISI can distort the signals Decision making is regarding the digital meaning of the sample Assume the input noise is a Gaussian random process Decision making is performed according to the threshold measurement: hypothesis H 1 is chosen if z(T)> , and hypothesis H 2 is chosen if z(T) < 

Dept. of EE, NDHU 5 Conditional Probability Density

Dept. of EE, NDHU 6 Vectorial Representation of Signal Waveform

Dept. of EE, NDHU 7 Waveform Representation in Orthonormal Functions Assume there are M signal waveforms and N orthonormal basis functions

Dept. of EE, NDHU 8 Signal and Noise in Vector Space

Dept. of EE, NDHU 9 Example: Orthogonal Representation of Waveforms

Dept. of EE, NDHU 10 Representing White Noise In other words, may be thought of the noise that is effectively tuned out by the detector AWGN noise can be partitioned into two components

Dept. of EE, NDHU 11 Detection of Binary Signal in Gaussian Noise

Dept. of EE, NDHU 12 Components of the Decision Theory Problem

Dept. of EE, NDHU 13 Decision Theory Likelihood ratio test Maximum Likelihood Criterion

Dept. of EE, NDHU 14 Maximum Likelihood Binary Decision Binary decision rule (Assume the binary transmitted waveforms are s 1 (t) and s 2 (t) ) where a 1 is the signal component of z(T) when s 1 (t) is transmitted, and a 2 is the signal component of z(T) when s 2 (t) is transmitted. The threshold level  0 is the optimum threshold for minimizing the probability of error. ( Reference to Appendix B.3 : Signal Detection Example )

Dept. of EE, NDHU 15 Error Probability According to Fig.3.2, the error probability can be derived by Where Q(x) is called the complementary error function, and

Dept. of EE, NDHU 16 Matched Filter The goal of the matched filter is to provide the maximum signal-to-noise power ratio Signal-to-noise power ratio Transfer function and impulse response of matched filter Maximum signal-to-noise power ratio

Dept. of EE, NDHU 17 Correlation Realization of the Matched filter

Dept. of EE, NDHU 18 Optimizing Error Performance Minimize the error probability is to maximize where (a 1 -a 2 ) is the difference of the desired signal components at the filter output at time t=T

Dept. of EE, NDHU 19 Signaling Characteristic in Error Probability Error probability function is rewritten by Define a time cross-correlation coefficient  as a measure of similarity between two signals s 1 (t) and s 2 (t)