Pulse amplitude modulation

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<ul><li> 1. 3F4 Pulse Amplitude Modulation (PAM) Dr. I. J. Wassell</li></ul> <p> 2. Introduction The purpose of the modulator is to convert discrete amplitude serial symbols (bits in a binary system) ak to analogue output pulses which are sent over the channel. The demodulator reverses this process ak Modulator Serial data symbolsChannel analogue channel pulsesDemodulator Recovered data symbols 3. Introduction Possible approaches include Pulse width modulation (PWM) Pulse position modulation (PPM) Pulse amplitude modulation (PAM) We will only be considering PAM in these lectures 4. PAM PAM is a general signalling technique whereby pulse amplitude is used to convey the message For example, the PAM pulses could be the sampled amplitude values of an analogue signal We are interested in digital PAM, where the pulse amplitudes are constrained to chosen from a specific alphabet at the transmitter 5. PAM Scheme Modulator xs (t ) =akk =kPulse generatorSymbol clock Recovered symbolsx(t ) = a (t kT ) a h (t kT )k = k TTransmit filter HT() hT(t)Demodulator y (t ) = a h(t kT ) + v(t )k = Data slicerRecovered clockChannelhC(t)kReceive filter HR(), hR(t)HC()+Noise N() 6. PAM In binary PAM, each symbol ak takes only two values, say {A1 and A2} In a multilevel, i.e., M-ary system, symbols may take M values {A1, A2 ,... AM} Signalling period, T Each transmitted pulse is given byak hT (t kT )Where hT(t) is the time domain pulse shape 7. PAM To generate the PAM output signal, we may choose to represent the input to the transmit filter hT(t) as a train of weighted impulse functions xs (t ) = a (t kT )k = k Consequently, the filter output x(t) is a train of pulses, each with the required shape hT(t) x(t ) = a h (t kT )k = k T 8. xs (t ) =PAM a (t kT )k = x(t ) =k a h (t kT )k = k Tx(t )xs (t ) Transmit FilterhT (t ) Filtering of impulse train in transmit filter 9. PAM Clearly not a practical technique so Use a practical input pulse shape, then filter to realise the desired output pulse shape Store a sampled pulse shape in a ROM and read out through a D/A converter The transmitted signal x(t) passes through the channel HC() and the receive filter HR(). The overall frequency response is H() = HT() HC() HR() 10. PAM Hence the signal at the receiver filter output is y (t ) = a h(t kT ) + v(t )k = kWhere h(t) is the inverse Fourier transform of H() and v(t) is the noise signal at the receive filter output Data detection is now performed by the Data Slicer 11. PAM- Data Detection Sampling y(t), usually at the optimum instant t=nT+td when the pulse magnitude is the greatest yields yn = y (nT + td ) = a h((n k )T + t ) + vk = kdnWhere vn=v(nT+td) is the sampled noise and td is the time delay required for optimum sampling yn is then compared with threshold(s) to determine the recovered data symbols 12. PAM- Data Detection TX data TX symbol, ak Signal at data slicer input, y(t)1 0 0 1 0 +A -A -A +A -A 0tdSample clockIdeal sample instants at t = nT+tdSampled signal, 0 yn= y(nT+td)Data Slicer decision threshold = 0VDetected data1 0 01 0 13. Synchronisation We need to derive an accurate clock signal at the receiver in order that y(t) may be sampled at the correct instant Such a signal may be available directly (usually not because of the waste involved in sending a signal with no information content) Usually, the sample clock has to be derived directly from the received signal. 14. Synchronisation The ability to extract a symbol timing clock usually depends upon the presence of transitions or zero crossings in the received signal. Line coding aims to raise the number of such occurrences to help the extraction process. Unfortunately, simple line coding schemes often do not give rise to transitions when long runs of constant symbols are received. 15. Synchronisation Some line coding schemes give rise to a spectral component at the symbol rate A BPF or PLL can be used to extract this component directly Sometimes the received data has to be nonlinearly processed eg, squaring, to yield a component of the correct frequency. 16. Intersymbol Interference If the system impulse response h(t) extends over more than 1 symbol period, symbols become smeared into adjacent symbol periods Known as intersymbol interference (ISI) The signal at the slicer input may be rewritten as yn = an h(td ) + ak h((n k )T + td ) + vn kn The first term depends only on the current symbol an The summation is an interference term which depends upon the surrounding symbols 17. Intersymbol Interference ExampleModulator input Binary 11.0 0.5 02 4 6 Time (bit periods)Slicer input amplitudeamplitude Response h(t) is Resistor-Capacitor (R-C) first order arrangement- Bit duration is T Binary 11.0 0.5 02 4 6 Time (bit periods) For this example we will assume that a binary 0 is sent as 0V. 18. Intersymbol Interferenceamplitude The received pulse at the slicer now extends over 4 bit periods giving rise to ISI. 1.01 1 0 0 1 0 0 10.5 02 4 6 time (bit periods) The actual received signal is the superposition of the individual pulses 19. Intersymbol Interference amplitude For the assumed data the signal at the slicer input is, 1.01 1 0 0 1 0 0 10.5Decision threshold0246time (bit periods)Note non-zero values at ideal sample instants corresponding with the transmission of binary 0s Clearly the ease in making decisions is data dependant 20. Intersymbol Interference Matlab generated plot showing pulse superposition (accurately) 0.9 0.8amplitude0.7Decision threshold0.6 0.5 0.4 0.3 0.2 0.1 0 01234Received signal5678time (bit periods) Individual pulses 21. Intersymbol Interference Sending a longer data sequence yields the following received waveform at the slicer input 1 09 . 08 . 07 .Decision threshold06 . 05 . 04 . 03 . 02 . 01 . 0 01 02 03 04 05 06 07 0(Also showing individual pulses)1 09 . 08 . 07 .Decision threshold06 . 05 . 04 . 03 . 02 . 01 . 0 01 02 03 04 05 06 07 0 22. Eye Diagrams Worst case error performance in noise can be obtained by calculating the worst case ISI over all possible combinations of input symbols. A convenient way of measuring ISI is the eye diagram Practically, this is done by displaying y(t) on a scope, which is triggered using the symbol clock The overlaid pulses from all the different symbol periods will lead to a criss-crossed display, with an eye in the middle 23. Example R-C response Eye Diagram 1 0.9 0.8h = eye height0.7 0.6h0.5Decision threshold0.4 0.3 0.2 0.1 0 00.10.20.30.40.50.60.70.80.9Optimum sample instant 24. Eye Diagrams The size of the eye opening, h (eye height) determines the probability of making incorrect decisions The instant at which the max eye opening occurs gives the sampling time td The width of the eye indicates the resilience to symbol timing errors For M-ary transmission, there will be M-1 eyes 25. Eye Diagrams The generation of a representative eye assumes the use of random data symbols For simple channel pulse shapes with binary symbols, the eye diagram may be constructed manually by finding the worst case 1 and worst case 0 and superimposing the two 26. Nyquist Pulse Shaping It is possible to eliminate ISI at the sampling instants by ensuring that the received pulses satisfy the Nyquist pulse shaping criterion We will assume that td=0, so the slicer input is yn = an h(0) + ak h((n k )T ) + vn kn If the received pulse is such that 1 h(nT ) = 0for n = 0 for n 0 27. Nyquist Pulse Shaping Theny n = a n + vn and so ISI is avoided This condition is only achieved if k H f + T = T k = That is the pulse spectrum, repeated at intervals of the symbol rate sums to a constant value T for all frequencies 28. Nyquist Pulse Shaping H(f) T f0T 2/ 1/01/2/f 29. Why? Sample h(t) with a train of pulses at times kT hs (t ) = h(t ) (t kT ) k = Consequently the spectrum of hs(t) is 1 H s ( ) = H ( k 2 T ) T k Remember for zero ISI 1 h(nT ) = 0for n = 0 for n 0 30. Why? Consequently hs(t)=(t) The spectrum of (t)=1, therefore 1 H s ( ) = H ( k 2 T ) = 1 T k Substituting f=/2 gives the Nyquist pulse shaping criterion H( f k T) = T k 31. Nyquist Pulse Shaping No pulse bandwidth less than 1/2T can satisfy the criterion, eg, T 2/ 1/01/2/fClearly, the repeated spectra do not sum to a constant value 32. Nyquist Pulse Shaping The minimum bandwidth pulse spectrum H(f), ie, a rectangular spectral shape, has a sinc pulse response in the time domain, T H( f ) = 0for - 1 2T &lt; f &lt; 1 2T elsewhere The sinc pulse shape is very sensitive to errors in the sample timing, owing to its low rate of sidelobe decay 33. Nyquist Pulse Shaping Hard to design practical brick-wall filters, consequently filters with smooth spectral roll-off are preferred Pulses may take values for t 1 2T + 0 With, 0</p>

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