# sampling__pulse_modulation

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```					                  Chapter 11.
Sampling and Pulse Modulation

• The Sampling Theorem
• PAM -- Natural and Flat-Top Sampling
Time-Division Multiplexing (TDM)
Intersymbol Interference (ISI)
• Pulse Width and Pulse Position Modulation
Demodulation
• Digital Modulation
Pulse Code Modulation (PCM)
Delta Modulation (DM)
• Qualitative Comparisons Of Pulse and Digital
Modulation Systems
1
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
The Sampling Theorem

Figure 11-1. Impulse sampling of an analog voltage.
2
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
The Sampling Theorem

• A sampler is a mixer with a train of very narrow
pulses as the local oscillator input.
regular intervals at a rate that is at least twice the
highest analog frequency
fs > 2fa(max)

• then the samples contain all of the information of
the original signal.
3
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
The Sampling Theorem

• The analog signal v(t) has a signal spectrum
represented by the Fourier transform V(f),
• and the sampling signal

s t       t  nT    s
n  

consists of instantaneous impulses every nTs sec,
where n = 0, +1, +2, …
• The Fourier transform of s(t) is

S f              f  nf 
1
s
Ts      n  

4
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
The Sampling Theorem

• The time-domain product performed by the
sampler produces a sampled output
spectrum given by

Vs  f         V  f  nf 
1
s
Ts   n  

• where this spectrum consists of replicas of
the analog signal spectrum V(f), translated
in frequency by each of the sampling
frequency harmonics.
5
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
The Sampling Theorem

• The sampler is a wideband (harmonic) mixer
producing upper and lower sidebands at each
harmonic of the sampling frequency.
• Figure 11-2a illustrates the correct way to sample: if
sampling is done at fs > 2fA(max) the upper and
lower sidebands do not overlap each other,
• and the original information can be recovered by
passing the signal through a low-pass filter (see
Figure 11-2c and d).

6
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
The Sampling Theorem

Figure 11-2. Sample spectra and their outputs. (a) fs > 2fA(max)
Nyquist criteria met. (b) fs < 2fA(max) Frequency foldover of
“aliasing” distortion occurs. (c) fs > 2fA(max) and recovery of
original information with low-pass filter. (d) The original analog   7
signal spectrum following recovery as in (c).
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
The Sampling Theorem

• However, if the sampling rate is less than the
Nyquist rate, fs < 2fA(max) the sidebands overlap,
as shown in Figure 11-2b.
• The result is frequency-folding or aliasing distortion,
which makes it impossible to recover the original
signal without distortion.

8
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling

• The circuit of Figure 11-3 is used to illustrate pulse
amplitude modulation (PAM). The FET is the
switch used as a sampling gate.
• When the FET is on, the analog voltage is shorted to
ground; when off, the FET is essentially open, so
that the analog signal sample appears at the output.
• Op-amp 1 is a noninverting amplifier that isolates
function.
9
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling

Figure 11-3. Pulse amplitude modulator,
natural sampling.
10
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling

• Op-amp 2 is a high input-impedance voltage
follower capable of driving low-impedance loads
(high “fanout”).
• The resistor R is used to limit the output current of
op-amp 1 when the FET is “on” and provides a
voltage division with rd of the FET. (rd, the drain-to-
source resistance, is low but not zero)

11
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling

• The most common technique for sampling voice in
PCM systems is to a sample-and-hold circuit.
• As seen in Figure 11-4, the instantaneous amplitude
of the analog (voice) signal is held as a constant
charge on a capacitor for the duration of the
sampling period Ts.
• This technique is useful for holding the sample
constant while other processing is taking place, but
it alters the frequency spectrum and introduces an
error, called aperture error, resulting in an inability
to recover exactly the original analog signal.
12
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling

• The amount of error depends on how mach the
analog changes during the holding time, called
aperture time.

• To estimate the maximum voltage error possible,
determine the maximum slope of the analog signal
and multiply it by the aperture time DT in Figure
11-4.

13
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Amplitude Modulation –
Natural and Flat-Top Sampling

Figure 11-4. Sample-and-hold circuit and
flat-top sampling.
14
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Time-Division Multiplexing

• In the three-channel multiplexed PAM system of
Figure 11-6, each channel is filtered and sampled
once per revolution (cycle) of the commutator.
• Notice that the commutator is performing both the
sampling and the multiplexing.

• The commutator must operate at a rate that satisfies
the sampling theorem for each channel.
• Consequently, the channel of highest cutoff
frequency determines the commutation rate for the
system of Figure 11-6.
15
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Time-Division Multiplexing

Figure 11-6. Time-division multiplex of three
information sources.
16
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Time-Division Multiplexing

• As an example, suppose the maximum signal
frequency for the three input channels are
fA1(max) = 4 kHz, fA2(max) = 20 kHz,
and fA3(max) = 4 kHz.
• For the TDM system of Figure 11-6,
the multiplexing must proceed at
f > 2fA(max) = 40 kHz
to satisfy the worst-case condition.

17
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Time-Division Multiplexing

• We can calculate the transmission line pulse rate as
follows:
The commutator completes one cycle, called a frame,
every 1/40 kHz = 25 ms.
• Each time around, the commutator picks up a pulse
from each of the three channels. Hence, there are
3 pulses/frame x 40k frames/s = 120k pulses/s.

18
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Time-Division Multiplexing

• The 4 kHz channel is being sampled at five times
the rate required by the sampling theorem. But if
we slow down the commutator, the 20-kHz channel
• One the thought might be to multiplex at 8 k-
frames/sec and sample the 20-kHz channel 5 times
per frame.
• If you sketch this, as is done in Figure 11-7,
you discover that there are
7 pulses/frame x 8k frames/s = 56k pulses/s,
which looks good.                                   19
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Time-Division Multiplexing

Figure 11-7. Possible TDM solution.
20
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Time-Division Multiplexing

• The two missing samples stolen from the 20-kHz
channel results in inadequate sampling and periodic
aliasing distortion.
• For no errors, the commutation rate must be 17.14 kHz,
producing 120k samples/s on the transmission line.
• A better scheme is shown in Figure 11-8 with insertion
of channel 1 and 3 between two samples of channel 2.
• With 12.5 ms/pulse and 7 pulses/frame, the multiplexing
rate can be
(2 pulses/25ms)/(7 pulses/frame) = 11.428k frames/s
and
(11.428k frames/s) x (7 pulses/frame) = 80k pulses/s
with no errors.                                           21
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Time-Division Multiplexing

Figure 11-8. TDM solution for minimum
transmission line pulse rate.
22
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Width and Pulse Position
Modulation

• In pulse width modulation (PWM), the
width of each pulse is made directly
proportional to the amplitude of the
information signal.
• In pulse position modulation, constant-width
pulses are used, and the position or time of
occurrence of each pulse from some
reference time is made directly proportional
to the amplitude of the information signal.
• PWM and PPM are compared and
contrasted to PAM in Figure 11-11.
23
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Width and Pulse Position
Modulation

Figure 11-11. Analog/pulse modulation signals.
24
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Width and Pulse Position
Modulation

• Figure 11-12 shows a PWM modulator. This circuit
is simply a high-gain comparator that is switched
on and off by the sawtooth waveform derived from
a very stable-frequency oscillator.
• Notice that the output will go to +Vcc the instant
the analog signal exceeds the sawtooth voltage.
• The output will go to -Vcc the instant the analog
signal is less than the sawtooth voltage. With this
circuit the average value of both inputs should be
nearly the same.
• This is easily achieved with equal value resistors to
ground. Also the +V and –V values should not
exceed Vcc.                                             25
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Width and Pulse Position
Modulation

Figure 11-12. Pulse width modulator.
26
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Width and Pulse Position
Modulation

• A 710-type IC comparator can be used for positive-
only output pulses that are also TTL compatible.
PWM can also be produced by modulation of
various voltage-controllable multivibrators.
• One example is the popular 555 timer IC. Other
(pulse output) VCOs, like the 566 and that of the
565 phase-locked loop IC, will produce PWM.
• This points out the similarity of PWM to continuous
analog FM. Indeed, PWM has the advantages of
FM---constant amplitude and good noise immunity-
27
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Demodulation

• Since the width of each pulse in the PWM signal
shown in Figure 11-13 is directly proportional to the
amplitude of the modulating voltage.
• The signal can be differentiated as shown in Figure
11-13 (to PPM in part a), then the positive pulses are
used to start a ramp, and the negative clock pulses
stop and reset the ramp.
• This produces frequency-to-amplitude conversion (or
equivalently, pulse width-to-amplitude conversion).
• The variable-amplitude ramp pulses are then time-
averaged (integrated) to recover the analog signal.      28
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Width and Pulse Position
Modulation

Figure 11-13. Pulse position modulator.
29
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Demodulation

• As illustrated in Figure 11-14, a narrow clock pulse
sets an RS flip-flop output high, and the next PPM
pulses resets the output to zero.
• The resulting signal, PWM, has an average voltage
proportional to the time difference between the
PPM pulses and the reference clock pulses.
• Time-averaging (integration) of the output
produces the analog variations.
• PPM has the same disadvantage as continuous
analog phase modulation: a coherent clock
reference signal is necessary for demodulation.
• The reference pulses can be transmitted along with
the PPM signal.                                        30
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Demodulation

• This is achieved by full-wave rectifying the PPM
pulses of Figure 11-13a, which has the effect of
reversing the polarity of the negative (clock-rate)
pulses.
• Then an edge-triggered flipflop (J-K or D-type) can
be used to accomplish the same function as the RS
flip-flop of Figure 11-14, using the clock input.
• The penalty is: more pulses/second will require
greater bandwidth, and the pulse width limit the
pulse deviations for a given pulse period.
31
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Demodulation

Figure 11-14. PPM demodulator.
32
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Code Modulation (PCM)

• Pulse code modulation (PCM) is produced by
analog-to-digital conversion process.
• As in the case of other pulse modulation techniques,
the rate at which samples are taken and encoded
must conform to the Nyquist sampling rate.
• The sampling rate must be greater than, or equal to,
twice the highest frequency in the analog signal,
fs > 2fA(max)

33
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Code Modulation (PCM)

• A simple example to illustrate the pulse code
modulation of an analog signal is shown in Figure
11-15.
digits (bits) in a sequence which represents the
amplitude of the analog sample.
• At time t = 1, the analog signal is 3 V. This voltage
is applied to the encoder for a time long enough that
the three-bit digital "word", 011, is produced.

34
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Code Modulation (PCM)

• The second sample at t = 2 has an amplitude of 6 V,
which is encoded as 110.
• This particular example system is conveniently set
up so that the analog value (decimal) is encoded
with its binary equivalent.

35
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Pulse Code Modulation (PCM)

Figure 11-15. A 3-bit PCM system showing
A/D conversion.
36
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

• Like PCM, a delta modulation system consists of an
encoder and a decoder;
• unlike PCM, however, a delta modulator generates
single-bit words that represent the difference (delta)
between the actual input signal and a quantized
approximation of the preceding input signal sample.

• This is represented in Figure 11-19 with a sample-
and-hold, comparator, up-down counter staircase
generator, and a D-type flip-flop (D-FF) to derive
the digital pulse stream.
37
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

• The continuous analog signal is band-limited in the
low-pass filter (LPF) to prevent aliasing distortion,
as in any sampling system.
• The analog signal VA is then compared to its
discrete approximation VB.

38
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

Figure 11-19. Possible delta modulation encoder.
39
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

• If the amplitude of VA is greater than VB , the
comparator goes high, calling for positive going
steps from the staircase generator.
• if, however, VB exceeds VA, the comparator goes low,
calling for negative-going from the staircase
generator.
• The comparator also sets the D flip-flop (D-FF) and
the output will be properly clocked because the
edge-triggered D-FF can change state only at rising
edges of the input clock.
40
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

• Decoding of the delta modulation (DM) signal
can be accomplished with an up-down staircase
generator and a smoothing filter
• or simply by integrating the DM pulses as shown
in Figure 11-20. The resulting demodulated signal
is illustrated as curve B.
• A practical implementation of a delta modulator is
shown in Figure 11-21, where the up-down counter
and digital-to-analog converter (DAC) comprise the
staircase generator of Figure 11-19.
• The delta modulator of Figure 11-21 is usually
referred to as a tracking or servo analog-to-digital
converter.                                             41
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

Figure 11-20. DM demodulator.     42
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

Fig. 11-21. Up-down staircase generator
for delta modulator.
43
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

• As seen in Figure 11-22, the critical parameters
determining the quality of a system using a constant
step size are the designer’s choice of step size and
sampling period length
• With too small a step size, the analog signal changes
cannot be followed closely enough; this is called
• With too large a step size, two problems arise: poor
signal approximation (resolution) and large
quantization noise (Figure 11-22b). This condition is
called granular noise.
44
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

   Too long a period has the same problem as too small
a step size and poor resolution (Figure 11-22c).
   When the period is too short, too much transmission
bandwidth is required.

Fig. 11-22. Critical design parameters in constant
45
step-size linear delta modulation.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

Example:
   A 5-V pk, 4-kHz sinusoid is to be converted to
a digital signal by delta modulation.
The step size must be 10 mV.
   Determine the minimum clock rate that will
allow the DM system to follow exactly the
fastest input analog signal change,
that is, to avoid slope overload.
46
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

Solution:
   The fastest rate of change of a sinewave, v(t) =
Vsinwt is the slope at the zero crossover points
(t = 0 in Figure 11-23).
slope = dv(t)/dt
= (d/dt).(Vsinwt) = wVcoswt.
   At t = 0, the slope is wV.cos0 = wV or Dv/Dt =
2pfV, where f is the frequency of the analog
sinusoid.
Dv         10 mV
 Dt                           0.079 ms / cycle
2pfV   2p (4000 Hz )5V                      47
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Delta Modulation (DM)

 Dt is half the clock period because steps occur only at
positive transitions of the clock in a practical system.
• Thus, Tclock = 2Dt = 1/fclock, so that fclock = 1/2Dt =
1/(2x0.079x10-6 s/cycle) = 6.3 MHz.

Figure 11-23. Example problem.               48
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Practical DM

• A circuit configuration in present use for telecom-
munication applications involving filters, speech
scramblers, instrumentation, and remote motor
control is shown in Figure 11-24.
• To demodulate the digital signal simple integrate
the pulses. In fact, the integrator has the same RC
time constant as the modulator above.
• The integrated demodulator output is the same as
the B curve of Fig. 11-24. The integrator output is
then put through a sharp cutoff LPF to smooth out
the final gain amplifier (VGA) and decision logic as
indicated in Fig. 11-25 for adaptive DM.               49
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Practical DM

Fig. 11-24. Integrating linear delta modulator
block diagram and signals.           50
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan

• A solution to the tradeoffs and compromises
of the simple delta modulation system above
is to have a variable step size system.
• This could be accomplished as in Fig. 11-25
with a variable-gain amplifier at the output
of the D-type flip-flop.
• A decision circuit that counts the number of
+ and – steps taken over a given period of
time and decides whether the step size
should be increased and by how much.
51
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan

Figure 11-25. Adaptive delta modulator.   52
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan

• As an example of how this circuit might operate,
suppose that the adaptive algorithm (decision
criteria) will be as follows: The preceding four bits
• If an equal number of 1s and 0s occur in this
interval (the last four bits), then the VGA gain will
be (the FET switch will be a short).
• If more 1s than 0s or more 0s than 1s are counted,
the step size is doubled, the input to the integrator
will be doubled.
and compared to the results for the linear DM in
Figs. 11-26 and 11-27.
53
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan

Figure 11-26. ADM with step or double step sizes.
Arrows are shown along the horizontal axis to indicate
where the step size changes. These changes are based
on the number of 1s and 0s of the previous 4 bits.
54
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan

Fig. 11-27. Different DM results depending on
step size used.                55
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Qualitative Comparisons of
Pulse Modulation Systems

• Like PAM, PCM can be time-division multiplexed
because the modulated samples maintain a fixed
position (slot) and duration in time.
• However, PCM is less noise-sensitive than PAM,
and PCM can use digital constant-amplitude
circuitry, unlike PAM, which requires linear
circuits.
• A disadvantage of PCM is its greater bandwidth
requirement. For example, in a simple 3-bit PCM
system, three pulses must be transmitted, whereas
only one is transmitted for the PAM sample (see
Figure 11-30).                                      56
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan

Figure 11-30. Pulse and digital modulation waveforms.
57
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Qualitative Comparisons of
Pulse Modulation Systems

• PWM and PPM are rarely used in
multiplexed communication systems
because of the large bandwidths required;
FDM, which requires a more complex
system than TDM, must also be used.
• While PWM and PPM have better noise
performance than PAM (like FM and PM
over AM), PWM and PPM are not easily
regenerated, and therefore noise
accumulates over long haul networks.
58
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Qualitative Comparisons of
Pulse Modulation Systems

other techniques is that PPM, like continuous-phase
modulation, requires coherent demodulation.
• This usually means that a phase-locked loop and its
acquisition circuitry are required.
pulse modulation techniques, the use of digital
terminal equipment makes PCM more desirable in
today’s communications marketplace.

59
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan

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