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Home , Pulse shaping, Symbol rate

PAM Communication System {Ex: -3,-1,1,3}

Binary Upsampling Pulse message Mapping sequence (Impulse Mod) Shaping

π Cos(2 fct) ò Modulate M = 2J discrete messages or J of into amplitude of signal

ò If amplitude mapping changes at of fsym then rate is Rb=Jfsym ò Conventional mapping of discrete messages to M

uniformly space amplitudes ∞ M M = δ − = − =− + s(t) ƒ ak (t k Tsym ) ai d(2i 1) i 1 , ..., 0 , ... , 2 2 k =−∞ No pulses overlap in time: requires infinite Impulse modulator

ò Represent the symbol sequence by the Dirac impulse train

ò The impulse modulator block forms this function. This impulse train is applied to a transmit filter so that the signal is band limited to the channel bandwidth. Pulse Shaping Block Diagram

DSP mk Transmit Source Mapping L pTsym[m] D/A Filter symbol sampling sampling analog analog rate rate rate

ò Upsampling by L denoted as L ò Outputs input sample followed by L-1 zeros ò Upsampling by converts symbol rate to sampling rate

ò Pulse shaping (FIR) filter pTsym[m] ò Fills in zero values generated by upsampler ò Multiplies by zero most of time (L-1 out of every L times) DSP Implementation

ò Random bit generation …001100100111010…. Source

2

1.5

1 ò Mapping bits onto symbols, Mapping 1‰1, 0‰-1 0.5 0 symbol -0.5

-1 rate

-1.5

2 ò Upsampling to match the -2 40 50 60 70 80 90 100 110 120 130 140 150 L sampling sampling rate 1.5 1 rate

0.5

0

-0.5 ò Pulse shaping filter -1 pTsym[m] sampling -1.5 2 rate -2 1.5 30 40 50 60 70 80 90 100

1 ò Send the output samples 0.5 D/A through D/A 0 analog -0.5

-1

-1.5

-2 30 40 50 60 70 80 90 100

ò If the analog pulse is wider than the time between adjacent symbols, the outputs from adjacent symbols may overlap ò A problem called intersymbol interference (ISI) ò What kind of pulses minimize the ISI? ò Choose a shape that is one at time kT and zero at mT for all m≠k ò Then, the analog waveform contains only the value from the desired input symbol and no interference from other nearby input symbols. ò These are called Nyquist Pulses Nyquist Pulses

ò Sinc Pulse

ò where f0 = 1/T . Sinc is Nyquist pulse because pS(0) = 1 and pS(kT) = sin( k)/ k = 0. ò Sinc envelope decays at 1/t. ò Raised-cosine pulse:

ò with roll-off factor = f∆/f0.

ò T = 1/2f0 because pRC has a sinc factor ò sin( k)/ k which is zero for all nonzero integers k. ò Raised-cosine envelope decays at 1/|t3|. ò As 0, raised-cosine sinc. Domain

ò Fourier transform

ò where ò B is the absolute bandwidth,

ò f0 is the 6db bandwidth,

ò f∆ = B − f0,

ò f1 = f0 − f∆, and

ò = (|f|−f1)/2f∆ Spectrum

ò Spectral comparison of rectangular and raised-cosine pulses ò Note the band-limitation of raised-cosine shaping Eye Diagram

ò Eye diagram is a popular robustness evaluation tool. ò For 4-PAM, single--wide Hamming blip with additive broadband channel noise, retriggering after every 2 baud intervals produces

eye

Optimum sampling locations Eye Diagrams

ò Eye diagrams with raised-cosine pulse shaping with 2-PAM and 4-PAM systems