Digital Communications Chapeter 3. Baseband Demodulation/Detection Signal Processing Lab

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Block Diagram of a DCS

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Demodulation and Detection  Modeling the received signal

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ.

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Major Sources of Errors  Inter-Symbol Interference (ISI) Due to the filtering effect of transmitter and receiver, symbols are “smeared”.  Thermal noise (AWGN) Disturbs the signal in an additive fashion (Additive) Has flat spectral density for all frequencies interest (White) Modeled by Gaussian random process (Gaussian Noise)

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Demodulation and Detection (cont´d)  Demodulation and Sampling : Waveform recovery and preparing the received signal for detection  Improving SNR using matched filter  Reducing ISI using equalizer  Sampling the recovered waveform  Detection : Estimate the transmitted symbol based on the received sample

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Baseband and Bandpass  Bandpass model of detection process is equivalent to baseband model because: The received bandpass waveform is first transformed to a baseband waveform. Equivalence theorem:  Performing bandpass linear signal processing followed by heterodying the signal to the baseband yields the same results as heterodying the bandpass signal to the baseband followed by a baseband linear signal processing.

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Likelihood

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Signal Space  Inner (scalar) product  Properties of inner product :

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Signal Space (cont´d)  Norm properties :  Euclidean distance between two signals :

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Signal Space (cont´d)  N-dimensional orthogonal signal space is characterized by N linearly independent functions called basis functions. The basis functions must satisfy the orthogonality condition  If all K i =1, the signal space is orthonormal

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Signal Space (cont´d)  Any arbitrary finite set of waveforms where each member of the set is of duration T, can be expressed as a linear combination of N orthonormal waveforms where N≤M

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Vectorial Representation

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Signals and Noise

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. White Noise in Orthonormal Signal Space  AWGN n(t) can be expressed as

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. E b /N o : Figure of Merit in Digital Communications  SNR or S/N is the average signal power to the average noise power. SNR should be modified in terms of bit-energy in DCS because : Signals are transmitted within a symbol duration and hence, are energy signal (zero power) A merit at bit-level facilitates comparison of different DCSs transmitting different number of bits per symbol.

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Bit Error Probability vs E b /N o

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Decision Theory

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. MAP and ML

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Signal Detection Example

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Probability of Bit Error

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Matched Filter Receiver  Problem Design the receiver filter h(t) such that the SNR (signal power to average noise power) is maximized at the sampling time.  Solution The optimum filter is the Matched filter, given by which is the time-reversed and delayed version of the conjugate of the transmitted signal

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Matched Filter (cont´d) The output SNR of a matched filter depends only on the ratio of the signal energy to the PSD of the white noise at the filter input

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Correlator Receiver  The matched filter output at the sampling time can be realized as the correlator output.

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Matched Filter and Correlator

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Implementation of Matched Filter Receiver

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Implementation of correlator receiver

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Statistics of The Vector Signals  AWGN channel model : r = s i + n Signal vector s i =(s i1, s i2, … s iN ) is deterministic. Elements of noise vector n=(n 1, n 2, …, n N ) are i, i.d Gaussian random variables with zero-mean and variance N 0 /2. The noise vector pdf is The elements of observed vector r=(r 1, r 2,….r N ) are independent Gaussian random variables. Its pdf is

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Graphical Example of ML Detection

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Average Probability of Symbol Error  Erroneous decision : For the transmitted symbol m i or equivalently signal vector s i, an error in decision occurs if the observation vector r does not fall inside region Z i. Probability of erroneous decision for a transmitted symbol Probability of correct decision for a transmitted symbol

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Avg. Prob. of Symbol Error (cont´d)  Average probability of symbol error : For equally probable symbols :

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. BER (Bit Error Rate)  Received signal in Additive White Gaussian Noise Channel   After Matched Filtering & Sampling   where,

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Bit Error Probability

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Maximum Likelihood Decision

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. BER versus Eb/No

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Inter-Symbol Interference (ISI)

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Inter-Symbol Interference (ISI)  ISI in the detection process due to the filtering effects of the system  Overall equivalent system transfer function creates echoes and hence time dispersion causes ISI at sampling time

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Inter-Symbol Interference (ISI) (cont’d)  Nyquist pulses: No ISI at the sampling time  Ideal Nyquist pulse:

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Inter-Symbol Interference (ISI) (cont’d)  Nyquist bandwidth constraint  Ideal Nyquist filter is not realizable.  Goals and trade-off in pulse-shaping Reduce ISI Efficient bandwidth utilization Robustness to timing error (small side lobes)

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Inter-Symbol Interference (ISI) (cont’d)  Raised-Cosine Filter A Nyquist pulse (No ISI at the sampling time)

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Inter-Symbol Interference (ISI) (cont’d)

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ Error-Performance Degradation

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Inter-Symbol Interference (ISI) (cont’d)  Square-Root Raised Cosine (SRRC) filter and Equalizer

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Types of Equalizers  Transversal filtering : Zero-forcing equalizer: Neglect the effect of noise Minimum mean square error (MSE) equalizer The basic limitation of a transversal equalizer is that it performs poorly on channels having spectral nulls.  Decision feedback Using the past decisions to remove the ISI contributed by them

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Transversal Equalizer

Signal Processing Lab., Dept. of Elec. and Info. Engr., Korea Univ. Decision Feedback Equalizer