PAM Communication System

{Ex: -3,-1,1,3}

Binary Impulse Pulse message Mapping sequence Modulator Shaping

π Cos(2 fct) ò Digital data transmission using pulse amplitude modulation (PAM)

ò Bit rate Rd bits/sec ò Input bits are blocked into J-bit words and mapped into

the sequence of symbols mk which are selected from an alphabet of M=2J distinct voltage levels

ò fs=Rd/J is the (baud rate)

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 pulse shaping filter so that the signal is band limited to the channel bandwidth.

1 Pulse Shaping and PAM

ò When the message signal is digital, it must be converted into an analog signal in order to be transmitted. ò “pulse shaping” filter changes the symbol into a suitable analog pulse

ò Each symbol w(kT) initiates an analog pulse that is scaled by the value of the signal

Pulse Shaping

ò Compose the analog pulse train entering the pulse shaping filter as

ò which is w(kT) for t = kT and 0 for t = kT ò Pulse shaping filter output

2 Pulse Shaping

ò 4-PAM symbol sequence triggering baud-spaced rectangular pulse

Intersymbol Interference

ò If the analog pulse is wider than the time between adjacent symbols, the outputs from adjacent symbols may overlap ò A problem called (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

3 Nyquist Pulses

ò The impulse response p(t) is a Nyquist pulse for a T- spaced symbol sequence if there exists a τ such that

ò Rectangular pulse:

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.

4 Frequency 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

5 Eye Diagram

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

Eye Diagrams

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

6 DSP Implementation

ò The input is zero padded. ò Implementing the pulse shaping filter by an Interpolation filter bank to save computation ò For oversampling rate of L, there are L-1 zeros between two consecutive nonzero points.

P (n) 1 x(nT) w(n) x(t) P (n) 2 D/A LPF

PL(n) x(nT+T(L-1)/L)

Interpolation Filter Bank

ò Define L discrete-time interpolation subfilters as

P(0) P0(0)

P(1) P1(0) : :

P(4) P0(1)

P(5) P1(1) : :

P(8) P0(2) : :

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