PAM Communication System
{Ex: -3,-1,1,3}
Binary Impulse Pulse message Mapping Modulation 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 symbol rate (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 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
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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