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EE4512 Analog and Digital Communications Chapter 3

Chapter 3Chapter 3

Digital Baseband Modulation Digital Baseband Modulation TechniquesTechniques


Chapter 3Chapter 3

Digital Baseband Modulation Digital Baseband Modulation TechniquesTechniques•• Goals in Communication SystemGoals in Communication System

DesignDesign

•• Pages 75Pages 75--7676


•• Communication System DesignCommunication System Design

Source TransmitterSource Transmitter

ChannelChannel

NoiseNoise

ReceiverReceiver

An information An information sourcesource (voice, music, video, text, signals (voice, music, video, text, signals or images) is or images) is modulatedmodulated in the in the transmitter transmitter ..

MS Figure 2.1MS Figure 2.1




ChannelChannel

NoiseNoise

ReceiverReceiver


The output of the transmitter is inputted to the physical The output of the transmitter is inputted to the physical transmission mediumtransmission medium (wires, fiber optics, wireless or (wires, fiber optics, wireless or acoustic) as a acoustic) as a channelchannel with additive with additive noisenoise..




ChannelChannel

NoiseNoise

ReceiverReceiver


The output of the channel is inputted to the The output of the channel is inputted to the receiverreceiverwhich attempts to which attempts to demodulatedemodulate the signal and the signal and recover recover the the data.data.


Chapter 2Chapter 2

Baseband Modulation and Baseband Modulation and DemodulationDemodulation•• Rectangular Pulse AmplitudeRectangular Pulse AmplitudeModulation (PAM)Modulation (PAM)



•• Rectangular Pulse Amplitude Modulation (PAM)Rectangular Pulse Amplitude Modulation (PAM)

Random Integer Generator simulates a binary information Random Integer Generator simulates a binary information source, bit rate source, bit rate rrbb = 1 kb/sec= 1 kb/sec




Random IntegerRandom IntegerGenerator fromGenerator fromthe the CommunicationsCommunicationsBlocksetBlockset, , CommCommSourcesSources



Random IntegerRandom IntegerGenerator block fromGenerator block fromthe the CommunicationsCommunicationsBlocksetBlockset, , CommCommSourcesSources

binarybinaryrandom seedrandom seedrrbb = 1 kb/sec= 1 kb/secTTbb = 1 msec= 1 msec



Binary data [0,1] Binary Polar PAM transmitter Binary data [0,1] Binary Polar PAM transmitter ±± 5 V5 V


Transmit data = ([0,1] Transmit data = ([0,1] –– 0.50.5)10 V = )10 V = ±± 5 V5 V



Additive White Gaussian Noise (AWGN) Channel Additive White Gaussian Noise (AWGN) Channel provides a random noise source, mean = 0 V, varianceprovides a random noise source, mean = 0 V, varianceσσ22 = 0.5 V= 0.5 V




AWGN ChannelAWGN Channelblock from theblock from theCommunicationsCommunicationsBlocksetBlockset, Channels, Channels



AWGN ChannelAWGN Channelblock from theblock from theCommunicationsCommunicationsBlocksetBlockset,,ChannelsChannels

random seedrandom seedvariance variance σσ22




Scope displays Transmit and Receive temporal dataScope displays Transmit and Receive temporal data


•• Output of Binary Polar PAM TransmitterOutput of Binary Polar PAM Transmitter

• Input of Binary PAM Receiver with AWGNInput of Binary PAM Receiver with AWGN



Simple PAM receiver consists of a Sample and Hold Simple PAM receiver consists of a Sample and Hold block with synchronous timing at block with synchronous timing at TTb b /2 /2 provided by a provided by a pulse generator. This induces a delay of pulse generator. This induces a delay of TTb b /2/2.

The Sign and Lookup Table blocks convert the The Sign and Lookup Table blocks convert the received data to the correct format (see MS p. 19). received data to the correct format (see MS p. 19).




The Sign blockThe Sign blockconverts theconverts thevariable amplitudevariable amplitudebipolar receivedbipolar receiveddata from thedata from theSampleSample--andand--Hold block toHold block totrinarytrinary datadata[[––1, 0, 1]1, 0, 1]



The Lookup TableThe Lookup Tableblock converts theblock converts thetrinarytrinary data [data [––1, 0, 1]1, 0, 1]to a replica of theto a replica of thebipolar transmittedbipolar transmittedsignal [signal [±± 5] for5] forcomparison bycomparison bymapping 0 to +5 mapping 0 to +5


•• Input of Binary PAM Receiver with AWGNInput of Binary PAM Receiver with AWGN

• Output of Sample and Hold blockOutput of Sample and Hold blockTTbb / 2 delay/ 2 delay

Tb


•• Output of Sample and Hold blockOutput of Sample and Hold block

• Output of Sign and Lookup Table blocksOutput of Sign and Lookup Table blocks

TTbb / 2 delay/ 2 delay

TTbb / 2 delay/ 2 delay


•• Transmitted PAM signalTransmitted PAM signal

• Received PAM signal with no errorsReceived PAM signal with no errorsTTbb / 2 delay/ 2 delay


Chapter 3Chapter 3

Digital Baseband Modulation Digital Baseband Modulation TechniquesTechniques•• Baseband Modulation UsingBaseband Modulation UsingRectangular Pulses and BinaryRectangular Pulses and BinaryPulse Amplitude ModulationPulse Amplitude Modulation

•• PagesPages 7676--8888


•• PAM generation of data sequence 00, 01, 10 and 11PAM generation of data sequence 00, 01, 10 and 11

0000

0101

1010

1111MS for S&M Figure 3MS for S&M Figure 3--5 5

Fig35.mdlFig35.mdl



Pulse Generator blockPulse Generator blockoutputs a outputs a ––1 V pulse1 V pulsewith a width of 20 msecwith a width of 20 msecand a nominal pulseand a nominal pulseperiod of 200 msecperiod of 200 msec



Transport Delay blockTransport Delay blockdelays the input bydelays the input by20 msec using a sample20 msec using a sampledata buffer automaticallydata buffer automaticallysetset



0000

0101

1010

1111

MS for S&M Figure 3MS for S&M Figure 3--5 5

Fig35.mdlFig35.mdl

TimeTime--based based sampling,sampling,sampling ratesampling rateffSS = 50 kHz,= 50 kHz,sampling periodsampling periodTTSS = 1/ = 1/ ffSS = 20 = 20 µµsecsec


•• Data 00 PAM OutputData 00 PAM Output


ττ 22ττ ττ = 20 msec= 20 msec



•• Spectrum of PAM data sequence 00, 01, 10 and 11Spectrum of PAM data sequence 00, 01, 10 and 11

0000

0101

1010

1111


ESD ESD ≈≈ | FFT || FFT |22

Fig35spectrum.mdlFig35spectrum.mdl



Spectrum Scope requires a discrete or sampled signal Spectrum Scope requires a discrete or sampled signal input. Pulse Generator block changed to sampledinput. Pulse Generator block changed to sampled--based based signal. Integer Delay block used to delay the pulse. signal. Integer Delay block used to delay the pulse.


Fig35spectrum.mdlFig35spectrum.mdl



Pulse Generator blockPulse Generator blockchanged to changed to sampledsampled--based based signal.signal.

Sample time Sample time TTSS = 20= 20 µµsecsecPulse period = 500 000Pulse period = 500 000

samples (10 sec)samples (10 sec)Pulse width = 1000Pulse width = 1000

samples (20 msec)samples (20 msec)



Integer Delay block forInteger Delay block forsampledsampled--based signal.based signal.

Sample time Sample time TTSS = 20= 20 µµsecsecDelay = 1000Delay = 1000

samples (0.2 sec)samples (0.2 sec)



Spectrum Scope simulation for N = 2Spectrum Scope simulation for N = 21818 = 262 144= 262 144samplessamples.

Simulation time Simulation time T = N / T = N / ffSS = 262 144 / 50 000 = 5.24288 = 262 144 / 50 000 = 5.24288 sec sec

Frequency resolution Frequency resolution ∆∆ff = = ffSS / N/ N = 50 000/262 144 = = 50 000/262 144 = 0.1907 Hz 0.1907 Hz

MS for S&M Figure 3MS for S&M Figure 3--5 5 Fig35spectrum.mdlFig35spectrum.mdl


•• Data 00 PAM simulated ESD Output | FFT |Data 00 PAM simulated ESD Output | FFT |22

•• Data 00 PAM simulated ESD Output | FFT |Data 00 PAM simulated ESD Output | FFT |2 2 dBdB

1/21/2ττ 1/1/ττ

50 Hz50 Hz25 Hz25 Hz

ττ = 20 msec= 20 msecS&M Figure 3S&M Figure 3--6a6a




1/1/ττ

50 Hz50 Hz

ττ = 20 msec= 20 msecS&M Figure 3S&M Figure 3--6b6b




1/1/ττ

50 Hz50 Hz

ττ = 20 msec= 20 msecS&M Figure 3S&M Figure 3--6c6c




1/21/2ττ 1/1/ττ

50 Hz50 Hz25 Hz25 Hz

ττ = 20 msec= 20 msecS&M Figure 3S&M Figure 3--6d6d


•• Sum of the Sum of the equally likelyequally likely simulated ESD simulated ESD ≈≈ | FFT || FFT |22 for data for data 00, 01, 10 and 11 00, 01, 10 and 11 unscaledunscaled by probability p = 0.25by probability p = 0.25

1/1/ττ

50 Hz50 Hz

S&M Figure 3.7S&M Figure 3.7

2/2/ττ

100 Hz100 Hz

150 Hz150 Hz

3/3/ττΨΨ(f) = 2A(f) = 2A2 2 ττ2 2 sincsinc22((ππ f f ττ))


•• Power of a Series of PulsesPower of a Series of Pulses

The The powerpower of a series of pulses is the of a series of pulses is the energyenergy in a series in a series of pulses divided by the of pulses divided by the timetime to transmit the series of to transmit the series of pulses (S&M p. 83).pulses (S&M p. 83).

Power = Energy / Transmit TimePower = Energy / Transmit Time

This concept can be extended to find the average This concept can be extended to find the average normalized power spectral densitynormalized power spectral density (PSD) (PSD) G(fG(f) of a series ) of a series of rectangular pulses:of rectangular pulses:

G(fG(f) = n A) = n A2 2 ττ22 sincsinc22((ππ f f ττ) / n ) / n ττ = A= A2 2 ττ sincsinc22((ππ f f ττ) )


•• The total power in a series of rectangular pulses is a The total power in a series of rectangular pulses is a function of the function of the bandwidthbandwidth of the data transmission. of the data transmission.

1/1/ττ90%90%

S&M Figure 3S&M Figure 3--1010

2/2/ττ

95%95%

96.5%96.5%3/3/ττ

ΨΨ(f) = 2A(f) = 2A2 2 ττ2 2 sincsinc22((ππ f f ττ))


• The rectangular pulse widthThe rectangular pulse width ττ is the entire bit time is the entire bit time TTbbhere and is here and is optimumoptimum in the bandwidth sense.in the bandwidth sense.

Table 3Table 3--1 Bandwidth of a Binary Rectangular PAM 1 Bandwidth of a Binary Rectangular PAM Signal as a Percentage of the Total Power (S&M p. 86, Signal as a Percentage of the Total Power (S&M p. 86, MS p. 22) with MS p. 22) with ττ = = TTbb = 1 / = 1 / rrbb

Bandwidth (Hz) Percentage of Total PowerBandwidth (Hz) Percentage of Total Power1/1/TTbb 90%90%1.5/1.5/TTbb 93%93%2/2/TTbb 95%95%3/3/TTbb 96.5%96.5%4/4/TTbb 97.5%97.5%5/5/TTbb 98%98%


Chapter 2Chapter 2

Baseband Modulation and Baseband Modulation and DemodulationDemodulation•• Rectangular PAMRectangular PAMPower Spectral DensityPower Spectral Density



• For an For an ensembleensemble (collection) of pseudo(collection) of pseudo--random random rectangular PAM data signals, the power spectral rectangular PAM data signals, the power spectral density (PSD) is used since the bit period density (PSD) is used since the bit period TTbb is finite is finite and not infinite as for a single pulse with its ESD.and not infinite as for a single pulse with its ESD.

•• The pulse width The pulse width ττ can be less than the bit period can be less than the bit period TTbb but but this is not bandwidth efficient.this is not bandwidth efficient.

•• The The MMATLABATLAB and and SimulinkSimulinksimulation of a binary simulation of a binary rectangular PAM transmitterrectangular PAM transmitter(MS Figure 2.1) is modified(MS Figure 2.1) is modifiedfor a variable pulse widthfor a variable pulse widthττ ≤≤ TTbb to verify theseto verify theseconcepts.concepts.


•• Rectangular PAM with variable pulse width Rectangular PAM with variable pulse width ττ ≤≤ TTb b = 1 = 1 msecmsec

Pulse train, 0 to 1 V, pulse period 1 msec, pulse widthPulse train, 0 to 1 V, pulse period 1 msec, pulse width0.5 msec (50% duty cycle)0.5 msec (50% duty cycle)

Transmitter of MS Figure 2.1 modifiedTransmitter of MS Figure 2.1 modifiedMSFig21mod.mdlMSFig21mod.mdl

MultiplierMultiplier


•• Rectangular PAM withRectangular PAM withvariable pulse widthvariable pulse widthττ ≤≤ TTb b = 1 msec= 1 msec

Pulse train = 0 to 1 V Pulse train = 0 to 1 V Pulse period = 1 msecPulse period = 1 msec

MSFig21mod.mdlMSFig21mod.mdl


•• Rectangular PAM withRectangular PAM withvariable pulse widthvariable pulse widthττ ≤≤ TTb b = 1 msec= 1 msec

Pulse width = 50%Pulse width = 50%(0.5 msec)(0.5 msec)

Phase delay = 0.5 msecPhase delay = 0.5 msec

MSFig21mod.mdlMSFig21mod.mdl


•• Top: Top: Binary data sourceBinary data source. . Middle: Middle: pulse train. pulse train. Bottom: Bottom: rectangular PAM transmit data output with 50% duty cycle. rectangular PAM transmit data output with 50% duty cycle.

TTb b

TTb b /2/2

Product Product with Gain with Gain

DataData

Duty CycleDuty Cycle


•• PSD of rectangular PAMPSD of rectangular PAM

Spectrum scope block calculates the PSD of the output of Spectrum scope block calculates the PSD of the output of the rectangular PAM transmitter with variable duty cyclethe rectangular PAM transmitter with variable duty cycleττ / / TTbb

Fig21modspec.mdlFig21modspec.mdl


•• Scaled PSD of rectangular PAM Scaled PSD of rectangular PAM ττ / / TTbb = 0.5, = 0.5, TTbb = 1 msec= 1 msec

The impulse spectral termsThe impulse spectral termsδδ[(2n[(2n––1) f1) foo] where f] where foo = 1 / = 1 / TTbb = = 1 kHz are due to the 1 kHz are due to the periodicperiodicsignal imposed by the productsignal imposed by the productmodulator with 50% duty cyclemodulator with 50% duty cycle

firstfirst--null bandwidth = 2 kHz =null bandwidth = 2 kHz = 1/1/ττ = 2/T = 2/T


•• Scaled PSD of rectangular PAM Scaled PSD of rectangular PAM ττ / / TTbb = 1, = 1, TTbb = 1 msec= 1 msec

No impulse spectral terms inNo impulse spectral terms inthe PSD for the rectangular the PSD for the rectangular PAM signal because thePAM signal because thesignal is only signal is only randomrandom withwith100% duty cycle (100% duty cycle (ττ = = TTbb)

firstfirst--null bandwidth = 1 kHz = 1/null bandwidth = 1 kHz = 1/ττ = 1/= 1/TTbb


Chapter 2Chapter 2

Baseband Modulation and Baseband Modulation and DemodulationDemodulation•• Performance ofPerformance of RectangularRectangularPAM in a Simple Receiver inPAM in a Simple Receiver inAWGNAWGN



• The performance of rectangular PAM (The performance of rectangular PAM (ττ = = TTbb, , 100% 100% duty cycle) in a simple receiver (single point sampling at duty cycle) in a simple receiver (single point sampling at TTbb / 2) in AWGN is assessed by the / 2) in AWGN is assessed by the bit error ratebit error rate (BER)(BER)

The received binary data is compared The received binary data is compared bitbit--byby--bitbit to the to the transmitted binary data by the Error Rate Calculation transmitted binary data by the Error Rate Calculation blockblock

MS Figure 2.7 MS Figure 2.7


•• Rectangular PAM with AWGN and BER analysisRectangular PAM with AWGN and BER analysis

The Error RateThe Error RateCalculation block delaysCalculation block delaysthe transmitted binarythe transmitted binarydata to correlate it withdata to correlate it withthe received binary datathe received binary data

The BER output isThe BER output isdisplayed numerically displayed numerically


•• Rectangular PAM with AWGN and BER analysisRectangular PAM with AWGN and BER analysis

The computation delayThe computation delayallows the BER analysisallows the BER analysisto be to be delayeddelayed beforebeforestarting if warranted (nostarting if warranted (nodelay is used here)delay is used here)


•• Spectrally white Gaussian noise is added in the channel Spectrally white Gaussian noise is added in the channel with variance with variance σσ22 in volts by the AWGN Channel blockin volts by the AWGN Channel block

The The initial seed initial seed isisthe MATLABthe MATLABvariable variable randseedrandseedso that eachso that eachsimulation issimulation isuniqueunique


•• The signalThe signal--toto--noise ratio (SNR) is the noise ratio (SNR) is the metric metric for the BER for the BER analysis. The signal power in the rectangular PAM signalanalysis. The signal power in the rectangular PAM signalis Ais A22 / R/ RLL and the noise power is and the noise power is σσ22 / R/ RLL. The SNR then . The SNR then is:is:

SNR = (ASNR = (A22 / R/ RLL )/(σσ22 / R/ RLL) = A) = A2 2 / / σσ22

SNR = 10 logSNR = 10 log1010 [A[A22 / / σσ22] dB] dB



•• The signal power in the rectangular PAM signal is AThe signal power in the rectangular PAM signal is A22 / R/ RLLW (watts) and the normalized power (RW (watts) and the normalized power (RLL = 1= 1ΩΩ) is) isAA2 2 VV22 (volts squared).(volts squared).

•• Although it is correct to use the units VAlthough it is correct to use the units V2 2 for normalized for normalized power and then power and then divide divide by Rby RLL to convert to W, the to convert to W, the convention is to use the term convention is to use the term normalized power W normalized power W even even though the units are Vthough the units are V22. Here A = . Here A = ±± 5 V and the 5 V and the normalized power = 25 W.normalized power = 25 W.



•• Rectangular PAM performance in a simple receiver with Rectangular PAM performance in a simple receiver with AWGN (MS p. 24) AWGN (MS p. 24)

Table 2.2 Observed BER as a Function of SNR in an Table 2.2 Observed BER as a Function of SNR in an Unfiltered Rectangular PAM Digital Communication Unfiltered Rectangular PAM Digital Communication System, Normalized Signal Power = 25 W.System, Normalized Signal Power = 25 W.

SNR dB AWGN SNR dB AWGN σσ22 VV22 BERBER∞∞ 00 00

13.9813.98 11 0012.2112.21 1.51.5 0010.9710.97 22 2 2 ×× 1010--44

6.986.98 55 1.24 1.24 ×× 1010--22

3.983.98 1010 5.64 5.64 ×× 1010--22

3.193.19 1212 7.43 7.43 ×× 1010--22


•• The BER performance of a rectangular PAM with AWGN The BER performance of a rectangular PAM with AWGN can be improved with the addition of a lowcan be improved with the addition of a low--pass filter pass filter (LPF) in the receiver. The LPF passes only the bandwidth (LPF) in the receiver. The LPF passes only the bandwidth required for the modulated signal.required for the modulated signal.



•• The lowThe low--pass filter in the rectangular PAM receiver is pass filter in the rectangular PAM receiver is an Analog Filter Design blockan Analog Filter Design block


•• The available simulated analog lowThe available simulated analog low--pass filter types are pass filter types are Bessel, Butterworth, Chebyshev types I and II, and Bessel, Butterworth, Chebyshev types I and II, and Elliptic. The Chebyshev LPF with Elliptic. The Chebyshev LPF with ffcutoffcutoff = 1200 Hz, in= 1200 Hz, in--band band ripple = 0.1 dBripple = 0.1 dB displays the maximum displays the maximum rollroll--offoff attenuation attenuation with a with a tradetrade--off off of the inof the in--band ripple.band ripple.


•• Rectangular PAM performance in a LPF receiver with Rectangular PAM performance in a LPF receiver with AWGN (SVU p. 73) AWGN (SVU p. 73)

Table 2.3 Observed BER as a Function of SNR in an Table 2.3 Observed BER as a Function of SNR in an LPF (9LPF (9--pole Chebyshev, 0.1 dB ripple, pole Chebyshev, 0.1 dB ripple, ffcutoffcutoff = 1.2 kHz) = 1.2 kHz) Binary Rectangular PAM Digital Communication Binary Rectangular PAM Digital Communication System, Normalized Signal Power = 25 W.System, Normalized Signal Power = 25 W.

SNR dB AWGN SNR dB AWGN σσ22 VV22 BERBER∞∞ 00 00

10.0010.00 2.52.5 000.960.96 2020 00

––3.013.01 5050 1 1 ×× 1010--44

––6.026.02 100100 6.3 6.3 ×× 1010--33

––9.03 9.03 200200 3.63 3.63 ×× 1010--22


•• Comparison of BER performance in binary rectangular Comparison of BER performance in binary rectangular PAM in an unfiltered and LPF receiver:PAM in an unfiltered and LPF receiver:

Unfiltered ReceiverUnfiltered ReceiverSNR dB AWGN SNR dB AWGN σσ22 VV22 BERBER

6.986.98 55 1.24 1.24 ×× 1010--22

3.983.98 1010 5.64 5.64 ×× 1010--22

LPF ReceiverLPF ReceiverSNR dB AWGN SNR dB AWGN σσ22 VV22 BERBER––6.026.02 100100 6.3 6.3 ×× 1010--33

––9.03 9.03 200200 3.63 3.63 ×× 1010--22


•• Rectangular PAM Rectangular PAM ±± 5 V, 5 V, 1 kb/sec with AWGN, 1 kb/sec with AWGN, σσ22 = 10 V= 10 V22

Simple receive data sampling occurs at the midSimple receive data sampling occurs at the mid--point point TTbb /2 /2

+ 5V+ 5V

--5 V5 V

TTbb

sampling points at (2n+1)sampling points at (2n+1)TTb b /2/2


Chapter 3Chapter 3

Digital Baseband Modulation Digital Baseband Modulation TechniquesTechniques•• Pulse Shaping to ImprovePulse Shaping to ImproveSpectral Efficiency: Sinc PulsesSpectral Efficiency: Sinc Pulses

•• PagesPages 8989--101101


•• A A practicalpractical sincsinc--shaped pulse has a finite duration T = shaped pulse has a finite duration T = ±± 4 4 TTbb. . ZeroZero--crossings occur at crossings occur at ±± n / n / rrbb = = ±± n n TTbb

binary 0binary 0

binary 1binary 1 rrbb = 1 kb/sec= 1 kb/sec

1/1/rrbb

MS Figures 2.15 and 2.16MS Figures 2.15 and 2.16

2/2/rrbb 3/3/rrbb

4/rb


•• Bandwidth of a sincBandwidth of a sinc--shaped PAM pulse is shaped PAM pulse is rrbb / 2 Hz which / 2 Hz which is 50% of the firstis 50% of the first--null bandwidth of a rectangular PAM null bandwidth of a rectangular PAM pulse (pulse (rrbb Hz).Hz).

binary 0

binary 1binary 1

rrbb = 1 kb/sec= 1 kb/sec

1/1/rrbb 88//rrbb3/3/rrbb

+5 V+5 V

––5 V5 V

00

4/4/rrbb

2/2/rrbb


•• Since the practical sincSince the practical sinc--shaped pulses have shaped pulses have zerozero--crossingscrossings at multiples of 1 / at multiples of 1 / rrbb = = TTbb there is there is no interference no interference between adjacent pulses if sampled at between adjacent pulses if sampled at TTbb..

startupstartupbinary 1binary 1

binary 0binary 0

1/1/rrbb

sampling points at nsampling points at nTTbb



•• Sinc PAM Power Spectral Density (S&M p. 90,Sinc PAM Power Spectral Density (S&M p. 90,MS p. 29MS p. 29--30)30)

The power spectral density (PSD) for a sinc PAM signal is The power spectral density (PSD) for a sinc PAM signal is derived from the energy spectral density (ESD) for a derived from the energy spectral density (ESD) for a single sinc pulse. The ESD single sinc pulse. The ESD ψψ(f) (f) for a sinc pulse with a for a sinc pulse with a peak amplitude A and data rate peak amplitude A and data rate rrbb is:is:

ψψ(f) = (A / (f) = (A / rrbb))2 2 = A= A22 TTbb2 2 –– rrbb/2 /2 ≤≤ f f ≤≤ rrbb/2/2

As for rectangular pulses earlier, As for rectangular pulses earlier, the the normalized normalized PSD PSD G(fG(f) ) of a series of sinc pulses is:of a series of sinc pulses is:

G(fG(f) = n A) = n A22 TTbb22 / n / n TTbb = A= A22 TTbb –– rrbb/2 /2 ≤≤ f f ≤≤ rrbb/2/2

where 1 / where 1 / rrbb = = TTbb


Chapter 2Chapter 2

Baseband Modulation and Baseband Modulation and DemodulationDemodulation•• Sinc Pulse AmplitudeSinc Pulse AmplitudeModulationModulation



• MMATLABATLAB andand Simulink Simulink generation of ageneration of a practical practical (finite (finite duration) sinc PAM signal uses the duration) sinc PAM signal uses the impulse responseimpulse response of of a a filter filter (MS p. 27(MS p. 27--29).29).

impulse trainimpulse traindata sourcedata source

sinc impulse filtersinc impulse filter



• The Random Integer Generator block data has an The Random Integer Generator block data has an amplitude of amplitude of 0, 10, 1 V and a period of 1 msec (1/V and a period of 1 msec (1/rrbb). The ). The output is offset and scaled to provide a output is offset and scaled to provide a ±± 1 V data 1 V data source.source.


• The Pulse GeneratorThe Pulse Generatorblock has an amplitude of block has an amplitude of 1 V, a period of 1 msec or1 V, a period of 1 msec ora frequency of 1 kHza frequency of 1 kHz((rrbb) and a pulse width of 2%) and a pulse width of 2%of the period (20 of the period (20 µµsec)sec)which is the which is the SimulinkSimulinksimulation timesimulation time..


• Simulink Simulink CommComm FiltersFilters in the in the Communications Communications BlocksetBlockset provides the Raised Cosine Transmit Filter provides the Raised Cosine Transmit Filter

which cwhich can generate an generate a sinc a sinc when the when the rolloff factor rolloff factor αα = 0= 0..


• Raised Cosine TransmitRaised Cosine TransmitFilter block generates aFilter block generates asinc with a sinc with a rolloff factorrolloff factorαα = 0. The = 0. The group delaygroup delayof 4 symbols specifiesof 4 symbols specifiesthe the ±± 44TTbb duration.duration.


• Raised Cosine TransmitRaised Cosine TransmitFilter block Filter block upsamplingupsamplingfactor factor 50 specifies the50 specifies theduration of a symbolduration of a symbolTTbb = = 50 x 20 50 x 20 µµsec =sec =1000 1000 µµsec = 1 msec,sec = 1 msec,where where 20 20 µµsec is thesec is theSimulink Simulink simulationsimulationtime. The time. The filter gainfilter gain of 5of 5sets the peak amplitudesets the peak amplitudeto to ±± 5 V.5 V.


• MMATLABATLAB andand Simulink Simulink sinc PAM transmitter can be used sinc PAM transmitter can be used to verify the PSDto verify the PSD..



•• PSD of sinc PAM, PSD of sinc PAM, rrbb = 1 kb/sec = 1 kb/sec

•• PSD of rectangular PAM, PSD of rectangular PAM, rrbb = 1 kb/sec= 1 kb/sec

first null bandwidth first null bandwidth rrb b /2 = 500 Hz/2 = 500 Hz

first null bandwidth first null bandwidth rrbb = 1000 Hz= 1000 Hz

sincsinc22 shaped PSDshaped PSDnonnon--ideal ideal sinc(tsinc(t))


•• PSD of sinc PAM, PSD of sinc PAM, rrbb = 1 kb/sec= 1 kb/sec


relatively flat PSDrelatively flat PSD

sincsinc2 2 shaped PSDshaped PSD


• The performance of practical sinc PAM in a simple The performance of practical sinc PAM in a simple receiver (single point sampling at receiver (single point sampling at TTb b ) in AWGN is ) in AWGN is assessed by the assessed by the bit error ratebit error rate (BER)(BER)

•• The The MMATLABATLAB and and SimulinkSimulink simulation of binary sinc simulation of binary sinc PAM is used to assess performance.PAM is used to assess performance.



• The sinc PAM transmitter is included as a The sinc PAM transmitter is included as a Simulink Simulink subsystemsubsystem..



•• The The MMATLABATLAB and and Simulink Simulink simulation of binary sinc simulation of binary sinc PAM includes a Chebyshev lowPAM includes a Chebyshev low--pass filter (LPF) with pass filter (LPF) with ffcutoffcutoff = 600 Hz to improve BER performance. The = 600 Hz to improve BER performance. The bandwidth is bandwidth is rrb b /2 = 500 Hz,/2 = 500 Hz, rrbb = 1 kb/sec here.= 1 kb/sec here.



• The The signal powersignal power of the practical sinc PAM is of the practical sinc PAM is problematical because of the complex shape and finite problematical because of the complex shape and finite duration. A duration. A SimulinkSimulink simulation can be used to compute simulation can be used to compute the rootthe root--meanmean--square (square (RMSRMS) of the sinc PAM transmit ) of the sinc PAM transmit output. output.



• The The Simulink Simulink RMSRMSblock from theblock from theSignal ProcessingSignal ProcessingBlocksetBlockset,, StatisticsStatisticscomputescomputes the RMS. the RMS.


• The measured normalized power of the practical sinc The measured normalized power of the practical sinc PAM signal is [RMS]PAM signal is [RMS]22 = 4.934= 4.93422 = 24.3 W. The = 24.3 W. The normalized power of the rectangular PAM signal is 25 normalized power of the rectangular PAM signal is 25 W exactly. The sinc PAM power is required for the W exactly. The sinc PAM power is required for the computation of SNR in the BER analysis.computation of SNR in the BER analysis.



•• Sinc PAM performance in a simple receiver with AWGN Sinc PAM performance in a simple receiver with AWGN (MS p. 31(MS p. 31--33) 33)

Table 2.4Table 2.4 Observed BER as a Function of SNR in an Observed BER as a Function of SNR in an LPF (9LPF (9--pole Chebyshev, 0.1 dB ripple, pole Chebyshev, 0.1 dB ripple, ffcutoffcutoff = 600 Hz) = 600 Hz) Binary Sinc PAM Digital Communication System, Binary Sinc PAM Digital Communication System, Normalized Signal Power Normalized Signal Power ≈≈ 24.3 W.24.3 W.

SNR dBSNR dB AWGN AWGN σσ22 VV22 BERBER∞∞ 0 00 0

0.850.85 2020 00−−3.133.13 5050 4 4 ×× 1010--33

−−6.146.14 100100 3.5 3.5 ×× 1010--33

−−9.159.15 200200 2.14 2.14 ×× 1010--22

−−13.1313.13 500500 9.31 9.31 ×× 1010--22


•• Comparison of BER performance in binary sinc and Comparison of BER performance in binary sinc and rectangular PAM with an LPF receiver:rectangular PAM with an LPF receiver:

Sinc PAMSinc PAMSNR dB AWGN SNR dB AWGN σσ22 VV22 BERBER−−6.146.14 100100 3.5 3.5 ×× 1010--22

−−9.159.15 200200 2.14 2.14 ×× 1010--22

Rectangular PAMRectangular PAMSNR dB AWGN SNR dB AWGN σσ22 VV22 BERBER––6.026.02 100100 6.3 6.3 ×× 1010--33

––9.03 9.03 200200 3.62 3.62 ×× 1010--22


Chapter 3Chapter 3

Digital Baseband Modulation Digital Baseband Modulation TechniquesTechniques•• Pulse Shaping to ImprovePulse Shaping to ImproveSpectral Efficiency: Raised CosineSpectral Efficiency: Raised CosinePulsesPulses

•• PagesPages 101101--111111


•• The raised cosine PAM pulse has a finite duration by The raised cosine PAM pulse has a finite duration by definition. Zerodefinition. Zero--crossings occur at crossings occur at ±± n / n / rrbb = = ±± n n TTbb

binary 0binary 0

binary 1binary 1



+5 V+5 V

––5 V5 V

00

4/4/rrbb

2/2/rrbb


•• Bandwidth of a raised cosine pulse is Bandwidth of a raised cosine pulse is rrbb / 2 + / 2 + ββ Hz where Hz where ββ is the damping factoris the damping factor and 0 0 ≤≤ ββ ≤≤ rrbb / 2./ 2.

binary 0binary 0

binary 1binary 1



+5 V+5 V

––5 V5 V

00

4/4/rrbb

2/2/rrbb


•• The rolloff factor The rolloff factor αα of a raised cosine pulse is related to of a raised cosine pulse is related to the damping factor the damping factor ββ: : αα = 2 = 2 ββ / / rrbb 0 0 ≤≤ αα ≤≤ 11

ββ = = αα rrb b / 2/ 2 0 0 ≤≤ ββ ≤≤ rrbb / 2/ 2

binary 0binary 0

binary 1binary 1



+5 V+5 V

––5 V5 V

00

4/4/rrbb

2/2/rrbb


•• Since the raised cosine pulses have Since the raised cosine pulses have zerozero--crossingscrossings at at multiples of 1 / multiples of 1 / rrbb = = TTbb there is there is no interference no interference between between adjacent pulses if sampled at adjacent pulses if sampled at TTbb..


startupstartupbinary 1binary 1

binary 0binary 0

1/1/rrbb

sampling points at nsampling points at nTTbb


•• Raised Cosine PAM Power Spectral Density (S&M p. 102, Raised Cosine PAM Power Spectral Density (S&M p. 102, MS p. 36MS p. 36--38)38)

The normalized power spectral density (PSD) The normalized power spectral density (PSD) G(fG(f) for a ) for a raised cosine PAM signal is derived from the energy raised cosine PAM signal is derived from the energy spectral density (ESD) for a single raised cosine pulse spectral density (ESD) for a single raised cosine pulse with a peak amplitude A and data rate with a peak amplitude A and data rate rrbb = 1 / = 1 / TTbb is: is:

G(fG(f) = A) = A22 / / rrbb | f | | f | ≤≤ rrbb / 2 / 2 –– ββ

G(fG(f) = 4 A) = 4 A22 TTbb [1 + cos [1 + cos ππ / 2 / 2 ββ ( | f |( | f | –– rrbb/2 + /2 + ββ ) ] ) ] 22

rrbb / 2 / 2 –– ββ ≤≤ f f ≤≤ rrbb/2 + /2 + ββ

G(fG(f) = 0) = 0 | f | | f | ≤≤ rrbb / 2 + / 2 + ββ


Chapter 2Chapter 2

Baseband Modulation and Baseband Modulation and DemodulationDemodulation•• Raised Cosine Pulse AmplitudeRaised Cosine Pulse AmplitudeModulationModulation



• MMATLABATLAB andand Simulink Simulink generation of raised cosine PAM generation of raised cosine PAM signal uses the signal uses the impulse responseimpulse response of a of a filter filter (MS p. 33(MS p. 33--36).36).

impulse trainimpulse traindata sourcedata source

raised cosine impulse filterraised cosine impulse filter



• The Random Integer Generator block data has an The Random Integer Generator block data has an amplitude of amplitude of 0, 10, 1 V and a period of 1 msec (1/V and a period of 1 msec (1/rrbb). The ). The output is offset and scaled to provide a output is offset and scaled to provide a ±± 1 V data 1 V data source.source.


• The Pulse GeneratorThe Pulse Generatorblock has an amplitude of block has an amplitude of 1 V, a period of 1 msec or1 V, a period of 1 msec ora frequency of 1 kHza frequency of 1 kHz((rrbb) and a pulse width of 2%) and a pulse width of 2%of the period (20 of the period (20 µµsec)sec)which is the which is the SimulinkSimulinksimulation timesimulation time..


• Simulink Simulink CommComm FiltersFilters in the in the Communications Communications BlocksetBlockset provides the Raised Cosine Transmit Filter provides the Raised Cosine Transmit Filter

which gwhich generates a enerates a raised raised cosine filter cosine filter when thwhen the e rolloff factor rolloff factor 0 < 0 < αα ≤≤ 11 ((αα ≠≠ 0)0)


• Raised Cosine TransmitRaised Cosine TransmitFilter block is generatedFilter block is generatedwith a with a rolloff factorrolloff factor αα = 0.5 = 0.5 The The group delaygroup delay of 4of 4symbols specifies thesymbols specifies the±± 44TTbb duration.duration.


• MMATLABATLAB andand Simulink Simulink raised cosine PAM transmitter raised cosine PAM transmitter can be used to verify the PSDcan be used to verify the PSD..



•• PSD of sinc PAM, PSD of sinc PAM, rrbb = 1 kb/sec, = 1 kb/sec, αα = 0= 0

•• PSD of raised cosine PAM, PSD of raised cosine PAM, rrbb = 1 kb/sec, = 1 kb/sec, αα = 0.5, = 0.5, ββ = 250= 250

first null bandwidth first null bandwidth rrb b /2 = 500 Hz/2 = 500 Hz

first null bandwidth first null bandwidth rrb b /2 + /2 + ββ = 750 Hz= 750 Hzαα = 2 = 2 ββ / / rrbb


•• PSD of raised cosine PAM, PSD of raised cosine PAM, rrbb = 1 kb/sec, = 1 kb/sec, αα = 0.5, = 0.5, ββ = 250= 250


first null bandwidth rfirst null bandwidth rbb = 1000 Hz= 1000 Hz

first null bandwidth first null bandwidth rrb b /2 + /2 + ββ = 750 Hz= 750 Hz


• The performance of raised cosine PAM in a simple The performance of raised cosine PAM in a simple receiver (single point sampling at receiver (single point sampling at TTb b ) in AWGN is ) in AWGN is assessed by the assessed by the bit error ratebit error rate (BER)(BER)

•• The The Simulink Simulink simulation of binary raised cosine PAM simulation of binary raised cosine PAM is used to assess performance.is used to assess performance.



•• The The MMATLABATLAB and and SimulinkSimulink simulation of raised cosine simulation of raised cosine PAM includes a Chebyshev lowPAM includes a Chebyshev low--pass filter (LPF) pass filter (LPF) ffcutoffcutoff = = 900 Hz, in900 Hz, in--band ripple = 0.1 dB to improve BER band ripple = 0.1 dB to improve BER performance. The first null bandwidth is 750 Hz. performance. The first null bandwidth is 750 Hz.



• The computed normalized power of the raised cosine The computed normalized power of the raised cosine PAM signal is [RMS]PAM signal is [RMS]2 2 = 4.67= 4.672 2 = 21.8 W. The normalized = 21.8 W. The normalized power of the sinc and rectangular PAM signals are 24.3 power of the sinc and rectangular PAM signals are 24.3 and 25 W. The raised cosine PAM power is required and 25 W. The raised cosine PAM power is required for the computation of SNR in the BER analysis.for the computation of SNR in the BER analysis.

MS Figure 2.21rcosMS Figure 2.21rcos


•• Raised cosine PAM performance in a simple receiver with Raised cosine PAM performance in a simple receiver with AWGN (MS p. 38AWGN (MS p. 38--40) 40)

Table 2.5Table 2.5 Observed BER as a Function of SNR in an Observed BER as a Function of SNR in an LPF (9LPF (9--pole Chebyshev, 0.01 dB ripple, pole Chebyshev, 0.01 dB ripple, ffcutoffcutoff = 900 Hz) = 900 Hz) Binary Raised Cosine PAM Digital Communication Binary Raised Cosine PAM Digital Communication System, Normalized Signal Power System, Normalized Signal Power ≈≈ 21.8 W.21.8 W.

SNR dBSNR dB AWGN AWGN σσ22 VV22 BERBER∞∞ 00 00

3.383.38 1010 000.370.37 2020 4 4 ×× 1010--44

−−3.613.61 5050 5.8 5.8 ×× 1010--33

−−6.626.62 100100 2.48 2.48 ×× 1010--22

−−9.629.62 200200 6.75 6.75 ×× 1010--22


•• Comparison of BER performance in binary sinc and Comparison of BER performance in binary sinc and raised cosine PAM with an LPF receiver:raised cosine PAM with an LPF receiver:

Sinc PAMSinc PAMSNR dB AWGN SNR dB AWGN σσ22 VV22 BERBER−−3.133.13 5050 4 4 ×× 1010--44

−−9.159.15 200200 2.14 2.14 ×× 1010--22

Raised Cosine PAMRaised Cosine PAMSNR dB AWGN SNR dB AWGN σσ VV BERBER––3.613.61 5050 5.8 5.8 ×× 1010--22

––9.62 9.62 200200 6.75 6.75 ×× 1010--22


End of Chapter 3End of Chapter 3

Digital Baseband Modulation Digital Baseband Modulation TechniquesTechniques

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