Communication Systems, 5e
Chapter 6: Sampling and pulse modulation
A. Bruce CarlsonPaul B. Crilly
© 2010 The McGraw-Hill Companies
Chapter 6: Sampling and pulse modulation
• Sampling theory and practice• Pulse-amplitude modulation• Pulse-time modulation
© 2010 The McGraw-Hill Companies
Sampling Theory and Practice
• The spectrum of a sampled signal– The time domain and spectrum of a sampling waveform
• Minimum sampling frequency– Based on the maximum allowable aliasing error,
message BW, LPF characteristics, etc.– The Nyquist Rate
• Practical sampling versus ideal sampling• Signal reconstruction• PAM, PDM and PPM
Sampling
• Multiplicative, periodic sampling waveform tptxtxs
tpFtxFtxF s
tpTnttp pulsen
s
tpT
nfT
tp pulsen ss
11
sT2sT
sT 2sT
tppulse
Convolution with Sampling Spectrum
• Spectral replication of F[x(t)]– If x(t) not band limited, there will be spectral aliasing– The shape of the sampling pulse may change the
magnitude and phase of the spectral replicas!
tpFtxFtxF s
n sspulses T
nfT
tptxtx 11
Spectral Replication
• Reduce the sampling rate to the minimum txtstxs
tXfStXs
tXTnf
T1tX
ns
sffreq
W sf2sfsf2 W
Replication Interval
sffreq
W sf2sfsf2 W
fX
fX s
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 6.1-3, Spectra for switching sampling (a) message (b) sampled message (c) sampled message
Nyquist Rate
WWf min,s
W2fs
W2fs
Aliasing by an ADC
• The desired baseband signal spectrum prior to sampling, typical definitions– Filter Design for baseband signals or complex signals– Center frequency is zero
sF2Fs
sF2Fs
Practical Filter Construction
Wffff passbandtransitionpassbands 2
• For practical applications, we usually sample at greater than the Nyquist rate. This allows for a guard band around the signal of interest (SOI)
• If ADC systems are involved, many times we use 4 x W sampling, 2 times the Nyquist rate or even higher.
Wfff stopbandpassbands 2
Wf passband
stopbandtransitionpassband fff
Bandpass Aliasing by an ADC• The desired baseband signal spectrum prior to
sampling.– Filter design for bandpass signal sampling.– Signal-of-interest center frequency is fs/4
Sampling in Matlab
• Using the interpolated message from before …• A sampling rate of fs/4 can be used
0 1 2 3 4 5 6
x 104
-150
-100
-50
0Sequenctial FFTs of the message
Frequency (Hz)
Pow
er (d
B)
0 2 4 6 8 10 12 14
x 104
-160
-140
-120
-100
-80
-60
-40
-20Sequenctial FFTs of the Sampled Waveform
Frequency (Hz)
Pow
er (d
B)
Question, what if I wanted to shift a baseband signal to a higher frequency … sample(?) and BPF!
Aliasing (1)
• The frequency domain response of the perfect sampling function is:
• Convolve with the input signal spectrum …
sF sF2 sF3 sFsF3 sF2
kn T
k2jjT2TnjexpjP
0
Aliasing (2)
• The frequency bands that “could” be aliased when sampled are
sF sF2 sF3 sFsF3 sF2
0
k
Tk
p jkjGT1
Tk2jjG
T1jG
Aliasing Example (1)
• Predict the aliasing result for
sF2Fs
2F3 s
sF2 sF2Fs
Aliasing Example (2)
Original spectrum
sF2Fs
2F3 s
sF2 sF2Fs
2Fs
2Fs
2F
f2F3 ss
2F
f2F ss
2F3
f2F ss
2F5f
2F3 ss
2Fs
2Fs 0
Aliased Baseband Spectrum
Interpolation-Filter Example
• 2-tone test signal (30&60 Hz, fs = 1000 Hz)– fft scaled to maintain power
• interpolation x4 (upsample function)– scaled by 4 so interpolated signal has the same power
• interp-filter (interp function)– filter includes “interpolation gain” of x4
Continuous Time Reconstruction
• Reconstruction of a discrete time waveform into a continuous time waveform.– If it’s digitized, it probably needs to be restored to an analog
waveform at sometime.
• The sampled representation– Shown as a spectrum from –fs/2 to + fs/2
or for DSP people from 0 to fs (it’s shown aliased!?)– It theoretically contains all frequency replicas!
sF sF2 sF3 sFsF3 sF2
0
k
Tk
p jkjGT1
Tk2jjG
T1jG
Continuous Time Reconstruction
• Impulse outputs at the “sample times”• Perfect Reconstruction Filter to eliminate replicas
– Derived from rect in the frequency domain– The convolution of the samples with a sinc function
delayttBBKth 2sinc2
delaytf2jxpeB2frectKfH
Ttt
nTtnTxtx delay
nsincˆ
thtxtx̂ s
BKLet 21
Reconstruction (2)
• Perfect Reconstruction Filter– Derived from rect in the frequency domain– The convolution of the samples with a sinc function
Ttt
sincnTtnTxtx̂ delay
n
TtnTt
sincnTxtx̂ delay
n
0delaytLet
nTtnTxtx
nsincˆ
Reconstruction (3)
• Perfect Reconstruction Filter– Each sample causes a time offset sinc– All the sinc’s are summed to form the continuous signal
nTtnTxtx
nsincˆ
-5 -4 -3 -2 -1 0 1 2 3 4 5
-0.2
0
0.2
0.4
0.6
0.8
1
Individual sinc functions with T delays
Reconstruction (4)
• Perfect Reconstruction Filter
nTtnTxtx
nsincˆ
-80 -60 -40 -20 0 20 40 60 80-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-80 -60 -40 -20 0 20 40 60 80-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Individual sinc’s Resulting summation
Alternate Functions Used for Reconstruction
• Zero Order Hold (like a Digital to analog converter)– Should be followed by LPF
• First Order Hold (Triangle)
TnTtectrnTxtx̂
nrect
TnTttrinTxtx̂
nrect
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Signal reconstruction from sample (a) ZOH
Figure 6.1-8
ZOH
-80 -60 -40 -20 0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-80 -60 -40 -20 0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Signal reconstruction from sample (b) FOH
Figure 6.1-8
FOH
-80 -60 -40 -20 0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-80 -60 -40 -20 0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Aliasing Error in Reconstruction
• Estimated as the amount of filter power that comes from the first adjacent spectral replica.– 1st order Butterworth LPF and spectral replica– Integrate the aliased portion of the filter in the passband– 1st order estimate: the aliasing filter power at the passband bandedge
Figure 6.1-9
(a) output of RC filter, (b) after sampling
n
ww
jwH 2
0
2
1
1
Butterworth LPF
Aliasing Computation
• Dr. Bazuin’s assumptions:– Passband has negligible
attenuation (book use W with B at the 3 dB point)
– Aliasing at passband W not B (book uses B)
– A digital filter will be used to “clean-up” the transition band.
Figure 6.1-9: (a) output of RC filter, (b) after sampling
ns
voltageVoltage
BWf
errorAtten2
1
1
ns
power
BWf
error 2
1
1
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Telephony Example
• Maximum voice W=3.4 kHz, find the sample rate– Passband 0.5 dB ripple, N=4th order Butterworth filter– Find the filter cutoff (3 dB) frequency– Stopband Attenuation -40 dB, find the stopband frequency– Compute the sample rate
105.0
22 10
11
nBH
n21
105.0
110
7556.0,7687.0 Let
kHzkHzkHzB 423.47556.04.34.3
1040
22 10
11
nBH
n21
1040
110
16.3,16.3 Let
kHzkHzWfB
Wfs
s 985.1316.3423.4
kHzkHzkHzfs 385.174.3985.13
BWfs
Telephony Example (2)• Maximum voice 3.4 kHz, find the sample rate
– Passband 0.5 dB ripple, N=1st order Butterworth filter– Find the filter cutoff (3 dB) frequency– Stopband Attenuation -40 dB, find the stopband frequency– Compute the sample rate
105.0
22 10
11
nBH
n21
105.0
110
3487.0,3493.0 Let
kHzkHzkHzB 733.93487.04.34.3
1040
22 10
11
nBH
n21
1040
110
995.99,995.99 Let
kHzkHz
kHzBWfs
295.973995.99733.975.9
kHzkHzkHzBWfs 695.9764.3295.973
Pulse-Based Modulation
• As long as sampling is performed at appropriate sample rates, any communications signal that conveys the sampled value during the sample time interval can communicate a continuous waveform.– The carrier doesn’t have to be continuous, it can be
different … as long as the sample value can be recovered.
• Pulse Communications– Pulse with amplitude (PAM)– Pulse with duration (PDM or PWM)– Pulse with a position in the sample time frame (PPM)
Pulse-Amplitude Modulation
• The pulse output from an instantaneous sampler
nTtpnTxtxn
PAM
TnttpTntp
TntnTxtpTnttpnTxtxnn
PAM
txtptxPAM
fXfPfXPAM
Aperture Effects result from the “time aperture” p(t)
PAM Pulses
• For unipolar signals the pulses may appear as an AM modulation amplitude– AM with pulses instead of a carrier
tm1txp
nTtpnTm1txn
PAM
Analog signal and corresponding PAM signal
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 6.2-1
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) sample & hold circuit (b) waveformsFigure 6.2-2
Flat-top sampling
• PAM time delayed by ½ the PAM width– If transmitted, does the time delay matter? …
… probably not
PAM Waveforms
• The modulation is dependent upon the pulse amplitude.– Signals near zero may be hard to detect– Can the receiver detect positive and negative pulses– Therefore, use an AM like offset for the amplitude
“Called uni-polar flat top in the text”
k
ssp TktpTkxAtx 10
01 sTkxwhere
PAM Spectral Content
• Convolution of the pulse and the AM-like waveform
• Spectrum based on “widest: frequency element, typically the symbol period or sample pulse.– Use sinc null-to-null as Bandpass Bandwidth, BT
tpFtxFtxF s
fMfAfXtxFtmAtx 00 1
sn ss Tc
Tnf
TtpF sin11
WTs
21 WBT
21
PAM Applications
• Rarely used for single channel communication systems, but …
• used in conjunction with instrumentation, data telemetry, and instrumentation systems
• One element of aTime-division multiplexing (TDM) systems
• A basis for other digital modulation systems
© 2010 The McGraw-Hill Companies
Pulse-Time Modulation
• PAM receivers require amplitude to be determined for brief pulses, can we translate the sampled signals into a form that might be easier to receive?– One not dependent upon amplitude?
• Pulse-Duration Modulation , PDM(also called Pulse-Width Modulation, PWM)– The length of the pulse width– Nominally centered on the periodicity
• Pulse-Position Modulation, PPM– The position of the pulse relative to the pulse period– Pulse widths are fixed
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 6.3-1
Types of pulse-time modulation
© 2010 The McGrawHill Companies
(a) Generation of PDM and PM signals, (b) waveforms
Generation of PWM and PPM
PWM and PPM Bandwidth
• Based on the minimum pulse width, but …– The value is dependent upon accurately measuring
time. Therefore, the faster the receiver rise-time in tracking the signal, the more accurate the analog measurement.
– Note 1: For PPM, if the pulse locations are describe using probability, the power spectral density can be computed as the product of the PSD and sinc.
– Note 2: For PWM, the pulse center and width should both be described probabilistically.
WTt sr
21 W
tB
rT
21
Conversion of PDM or PPM into PAM for Demodulation
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 6.3-3
Desired PAM Amplitude
Top Related