Pulse amplitude modulation

3F4 Pulse Amplitude Modulation (PAM) Dr. I. J. Wassell Introduction • The purpose of the modulator is to convert discrete amplitude serial symbols (bits in a binary system) a k to analogue output pulses which are sent over the channel. • The demodulator reverses this process Modulator Channel Demodulator Serial data symbols a k ‘analogue’ channel pulses Recovered data symbols 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 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 PAM Scheme H C ( ω ) h C (t) Symbol clock H T ( ω ) h T (t) Noise N( ω ) Channel + Pulse generator a k Transmit filter ∑ ∞ −∞= −= k ks kTtatx )()( δ ∑ ∞ −∞= −= k Tk kTthatx )()( Receive filter H R ( ω ), h R (t) Data slicer Recovered symbols Recovered clock )()()( tvkTthaty k k +−= ∑ ∞ −∞= Modulator Demodulator PAM • In binary PAM, each symbol a k takes only two values, say {A 1 and A 2 } • In a multilevel, i.e., M-ary system, symbols may take M values {A 1 , A 2 , A M } • Signalling period, T • Each transmitted pulse is given by )( kTtha Tk − Where h T (t) is the time domain pulse shape PAM • To generate the PAM output signal, we may choose to represent the input to the transmit filter h T (t) as a train of weighted impulse functions ∑ ∞ −∞= −= k ks kTtatx )()( δ • Consequently, the filter output x(t) is a train of pulses, each with the required shape h T (t) ∑ ∞ −∞= −= k Tk kTthatx )()( PAM • Filtering of impulse train in transmit filter Transmit Filter ∑ ∞ −∞= −= k Tk kTthatx )()( ∑ ∞ −∞= −= k ks kTtatx )()( δ )(th T )(tx s )(tx 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 H C ( ω ) and the receive filter H R ( ω ). • The overall frequency response is H( ω ) = H T ( ω ) H C ( ω ) H R ( ω ) PAM • Hence the signal at the receiver filter output is )()()( tvkTthaty k k +−= ∑ ∞ −∞= Where h(t) is the inverse Fourier transform of H( ω ) and v(t) is the noise signal at the receive filter output • Data detection is now performed by the Data Slicer [...]... Nyquist Pulse Shaping • No pulse bandwidth less than 1/2T can satisfy the criterion, eg, T −2/Τ −1/Τ 0 1/Τ 2/Τ f Clearly, the repeated spectra do not sum to a constant value 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 ) =  0 for - 1 2T < f < 1 2T elsewhere • The sinc pulse shape is very sensitive... 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 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... h((n − k )T ) + vn k≠n • If the received pulse is such that 1 h(nT ) =  0 for n = 0 for n ≠ 0 Nyquist Pulse Shaping • Then y 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 Nyquist Pulse Shaping H(f) T f 0 T −2/Τ −1/Τ 0 1/Τ... input Binary ‘1’ 1.0 0.5 0 2 4 6 Time (bit periods) Slicer input amplitude amplitude – Response h(t) is Resistor-Capacitor (R-C) first order arrangement- Bit duration is T Binary ‘1’ 1.0 0.5 0 2 4 6 Time (bit periods) • For this example we will assume that a binary ‘0’ is sent as 0V Intersymbol Interference amplitude • The received pulse at the slicer now extends over 4 bit periods giving rise to... (accurately) 0.9 0.8 amplitude 0.7 Decision threshold 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 Received signal 5 6 7 8 time (bit periods) Individual pulses Intersymbol Interference • Sending a longer data sequence yields the following received waveform at the slicer input 1 09 08 07 Decision threshold 06 05 04 03 02 01 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 (Also showing individual pulses) 1 09 08 07... of δ pulses at times kT ∞ hs (t ) = h(t ) ∑ δ (t − kT ) k = −∞ • Consequently the spectrum of hs(t) is 1 H s (ω ) = ∑ H (ω − k 2π T ) T k • Remember for zero ISI 1 h(nT ) =  0 for n = 0 for n ≠ 0 Why? • Consequently hs(t)=δ(t) • The spectrum of δ(t)=1, therefore 1 H s (ω ) = ∑ H (ω − k 2π T ) = 1 T k • Substituting f=ω/2π gives the Nyquist pulse shaping criterion ∑ H( f − k T) = T k Nyquist Pulse. .. individual pulses Intersymbol Interference amplitude • For the assumed data the signal at the slicer input is, 1.0 ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘0’ ‘0’ ‘1’ 0.5 Decision threshold 0 2 4 6 time (bit periods) Note non-zero values at ideal sample instants corresponding with the transmission of binary ‘0’s • Clearly the ease in making decisions is data dependant Intersymbol Interference • Matlab generated plot showing pulse. .. 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 + β 0  With, 0 . symbols Introduction • Possible approaches include – Pulse width modulation (PWM) – Pulse position modulation (PPM) – Pulse amplitude modulation (PAM) • We will only be considering PAM in these. 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. 3F4 Pulse Amplitude Modulation (PAM) Dr. I. J. Wassell Introduction • The purpose of the modulator is to convert discrete amplitude serial symbols (bits in a