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

Dept. of EE, NDHU 2 Error Probability Performance Error probability function where  is the time cross-correlation coefficient between two signals Antitpodal signal –  equals to -1, then Orthogonal signal –  equals to 0, then

Dept. of EE, NDHU 3 Error Probability of Binary Signaling Unipolar signaling Detection of unipolar baseband signaling

Dept. of EE, NDHU 4 Error Probability of Binary Signaling Bipolar signaling Detection of bipolar baseband signaling

Dept. of EE, NDHU 5 Bit Error Performance of Unipolar and Bipolar Signaling

Dept. of EE, NDHU 6 Intersymbol Interference in the Detection Process

Dept. of EE, NDHU 7 Nyquist Channels for Zero ISI

Dept. of EE, NDHU 8 Pulse Shaping to Reduce ISI Goals and Trade-offs –Compact signaling spectrum is to provide the higher allowable data rate –Time pulse would become spread in time, which induces ISI The Raised-Cosine filter where W is the absolute bandwidth and W 0 =1/2T represents the minimum Nyquist bandwidth and the -6 dB bandwidth

Dept. of EE, NDHU 9 Raised-Cosine Filter Characteristics

Dept. of EE, NDHU 10 Two Types of Error-Perfformance Degradation

Dept. of EE, NDHU 11 Example 3.3 Bandwidth Requirements (a)Find the minimum required bandwidth for the baseband transmission of a four-level PAM pulse sequence having a data rate of R=2400 bits/s if the system transfer characteristic consists of a raised-cosine spectrum with 100% excess bandwidth (r=1) (b)The same 4-ary PAM sequence is modulated onto a carrier wave, so that the baseband spectrum is shifted and centered at frequency f0. Find the minimum required DSB bandwidth for transmitting the modulated PAM sequence

Dept. of EE, NDHU 12 Nyquist Pulse

Dept. of EE, NDHU 13 Square-root Nyquist Pulse and Raised-cosine Pulse

Dept. of EE, NDHU 14 Equalization Maximum-likelihood sequence estimation (MLSE) –Make measurement of channel response and adjust the receiver to the transmission environment –Enable the detector to make good estimates from the distorted pulse sequence (ex. Viterbi equalization) Equalization with filtering –Use filter to compensate the distorted pulse –Linear filter contains only feedforward elements (ex. transversal equalizers) –Non-linear filter contains both feedforward and feedback elements (ex. decision feedback equalizers) –Preset or adaptive filter design –Filter’s resolution and update rate

Dept. of EE, NDHU 15 Receiving / Equalizing Filter The overall transfer function System design goal then H t (f) and H r (f) each have frequency transfer functions that are the square root of the raised cosine. Equalizing filter sometimes not only compensates the channel effect but compensates the ISI brought by the transmitter and receiver (ex. Gaussian filter)

Dept. of EE, NDHU 16 Eye Pattern Eye pattern is a filtering effect

Dept. of EE, NDHU 17 Distorted Pulse Response

Dept. of EE, NDHU 18 Transversal Equalizer A training sequence (like PN sequence) is needed to estimate the channel frequency response A transversal filter is the most popular form of an easily adjustable equalizing filter consisting of a delay line with T-second tapes The main contribution is from a central tap of a transversal filter In practice, a finite-length transversal filter is realized to approximate the ideal filter (infinite-length transversal filter) Consider there are (2N+1) taps with weights c -N, c -N+1, …,c N, the equalizer output samples {z(k)}

Dept. of EE, NDHU 19 Transversal Filter

Dept. of EE, NDHU 20 Zero-Forcing Solution Relationship among {z(k)}, {x(k)}, and {c n } for the transversal filter Disposing the top N the bottom N rows of the matrix X into a square matrix with dimension of 2N+1 and transform Z vector into a vector of 2N+1 Rewrite the relationship Select the weights {c n } so that the equalizer output is

Dept. of EE, NDHU 21 Example: A Zero-Forcing Equalizer Consider a three-taps transversal filter, the given received data {x(k)} are 0.0, 0.2, 0.9, -0.3,0.1. Using the zero-forcing solution to find the weights {c -1, c 0, c 1 } –For the relationship

Dept. of EE, NDHU 22 Minimum MSE Solution Minimize the mean-square error (MSE) of all the ISI terms plus the noise power at the output of the equalizer MSE is defined as the expected value of the squared difference between the desired data symbol and the estimated data symbol MSE solution Minimum MSE solution is superior to zero-forcing solution Minimum MSE is more robust in the presence of noise and large ISI

Dept. of EE, NDHU 23 Decision Feedback Equalizer Limitation of a linear equalizer is that it performs poor on channel having spectral nulls Decision feedback equalizer (DFE) is a non-linear equalizer and uses previous detector decisions to eliminate the ISI on pulse Basic idea is that if the values of the symbols previously detected are known, then the ISI contributed by these symbols can be cancelled out Forward filter and feedback filter are used in the DFE The advantage of DFE is that the feedback filter operates on noiseless quantized levels, and thus its output is free of channel noise

Dept. of EE, NDHU 24 Decision Feedback Equalizer

Dept. of EE, NDHU 25 Preset and Adaptive Equalization The equalizer weights remain fixed during transmission of data, then the equalization is called preset equalization Preset equalization sets the tap weights according to some average knowledge of the channel (Ex. Voice-grade telephone) Adaptive equalization can be implemented to perform tap-weight adjustments periodically or continually Periodic adjustments are accomplished by periodically transmitting a preamble sequence Continually adjustment are performed by the decision directed procedure

Dept. of EE, NDHU 26 Preset and Adaptive Equalization Disadvantages of preset equalization –Require an initial training period –A time-varying channel can degrade system performance If the probability of error exceeds one percent (rule of thumb), decision-directed adaptive equalizer might not converge Common solution to the adaptive equalization –Initialize the equalizer with a preamble to provide good channel-error performance –Then switch to the decision-directed mode –Blind equalization algorithm can be used to form initial channel estimates without a preamble