PULSE
ANALOG
MODULATION
BDG(44)
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• A Continuous Time (CT) signal can not be
processed by the digital processors.
• Hence to enable digital transmission of CT, it
has to be converted into Discrete Time (DT)
signal. ##1
• Sampling Therom gives the criteria for
spacing Ts between two consecutive
samples.
• Sampling is the process of converting a
signal (for example, a function of continuous
time or space) into a numeric sequence (a
function of discrete time or space).
BDG(44)
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• Sampling Theorem for LP signals –
• A band limited signal of finite energy, which
has no frequency components higher than
W Hz, may be completely recovered from
the knowledge of its samples taken at the
rate of 2W samples per second.
• A band limited signal of finite energy, which
has no frequency components higher than
W Hz, is completely described by specifying
the values of the signal at instants of time
separated by 1/2W seconds
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• In short, a CT signal can be completely
represented in its samples and recovered
back if the sampling frequency (Fs) is twice
the highest frequency component of the
signal.
( Fs >2W.)
• Proof of sampling theorem –
• Can be treated as two parts –
1. Representation of x(t) in terms of its
samples.
2. Reconstruction of x(t) from its samples.
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• Part 1- Representation of x(t) in terms of
its samples
• Step 1- Define xδ(t).
• Step 2- FT of xδ(t) i.e. Xδ(f).
• Step 3- Relation between X(f) and Xδ(f).
• Step 4- Relation between x(t) and x(nTs)
• ##2
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• Part 2 –Reconstruction of x(t) from its
samples• Step 1- Take inverse FT of X(f) which is in
terms of Xδt.
• Step 2- Show that x(t) is obtained back
with the help of interpolation function.
• Step 3- Reconstruction of x(t) by low pass
filter- When interpolated signal of equation
6 is passed through the low pass filter of
BW - W<f <W, the constructed waveform
is obtained.
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• Effect of Under sampling (Aliasing)• Whiling proving sampling theorem it is
considered that fs=2W.
• Let us consider a case fs<2W. Refer figure.
• The spectrum located at X(f),X(f-fs),X(f-2fs)
overlaps each other.
• The high frequencies near W in X(f-fs)
overlap with low frequencies(fs-W)in X(f).
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• Definition of aliasing – When the high
frequency interferes with low frequency
and appears as low frequency, then the
phenomenon is called aliasing.
• Effects of aliasing – (a)Distortion is
generated as high and low frequencies
interfere with each other. (b) The data is
lost and can not be recovered.
• Aliasing can be avoided by (a) sampling
rate fs more than 2W and (b) strictly
bandlimit the signal to W.
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• (a) Sampling rate fs >= 2W – When the
sampling rate is more than 2W, then the
spectrum will not overlap and there will be
sufficient gap between the individual
spectrums.
• (b) – Bandlimiting the signal – If the sampling
rate fs= 2W, there should not be any aliasing.
But there can be few components higher than
2W which may create aliasing. Hence LPF is
used before sampling the signal. Output of LPF
is bandlimited and there is no frequency
component higher than W. Thus avoiding
aliasing. ##
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• Nyquist Rate –
• When the sampling rate becomes exactly
equal to 2W samples/sec, for a given BW
of W hertz, then it is called Nyquist rate.
(2W Hz.)
• Nyquist Interval –
• It is the time interval between any two
adjacent samples when sampling rate is
Nyquist rate. (1/2W seconds)
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• Reconstruction Filter ( Interpolation Filter) –
• The reconstruction of signal is the
succession of sinc pulses weighted by
x(nTs).
• These pulses are interpolated with the help
of LPF. It is called reconstruction or
interpolation filter.
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• Ideal Filter – Refer Figure.
• Figure shows the spectrum of the sampled
signal and frequency response of the
required filter.
• When the sampling frequency is exactly
2W, then the spectrum just touches each
other.
• The spectrum of original signal, X9f) can
be filtered by an ideal filter having
passband from –W</=f </=W.
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• Non Ideal filter –
• An ideal filter of BW W filters out an
original signal.
• But practically ideal filter is not realizable.
It requires some transition band.
• Hence fs must be greater than 2W. It
creates the gap between adjacent
spectrums of Xδ(f). This gap can be used
for the transition band of the
reconstruction filter.
• The spectrumX9f) is then properly filtered
out from Xδ(f).
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• Sampling of Bandpass Signals –
• The bandpass signal x(t) whose maximum
BW is 2W can be completely represented
into and recovered from its samples if it is
sampled at the minimum rate of twice the
BW.
• Thus if the BW is 2W, then minimum
sampling rate for bandpass signal should
be 4W samples per second.
• Spectrum of BP signal.
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• The spectrum is centred around frequency
fc. The BW is 2W.
• The frequencies in the BP signal are from
fc –W to fc +W.
• The highest frequency present in the BP is
fc+W. The centre frequency is greater than
W.
• In-phase and quadrature components –
• xi(t) = In phase component of x(t)
• xq(t) = Quadrature component of x(t)
• x(t) = xi(t)cos(2*pi*fc*t) - xq(t)sin(2*pi*fc*t)
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• Generation of in-phase and quadrature
components- ##
• x(t) is multiplied by cos(2*pi*fc*t) and
sin(2*pi*fc*t). The multiplication produces
sum and difference frequencies at the out
put of the multiplier.
• The LPFs suppress the sum frequency
and pass only difference frequencies.
• Thus xi(t) and xq(t) components contain
only low frequency components.
• Spectrum ##
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• Reconstruction of BP signal from In phase
and quadrature components - ##
• The reconstruction filter generates analog
version of in phase and quadrature
components.
• These xi(t) and xq(t) are multiplied by
cos(2*pi*fc*t) and sin(2*pi*fc*t)
respectively to shift their frequencies in BP
range.
• The two products are added to give final
BP signal. (##8)
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• Uniform Sampling Theorem for BP signals
-##9
• x(t) can be represented into its samples
x(kTs) if sampling rate is fs = 2fxu/m,
where m is largest integer not exceeding
fxu /Bx.
• Examples --- ##10.
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• Pulse Analog Modulation –
• The modulating signal can modulate
amplitude, width(duration) or position of
the pulse.
• Three techniques are
• 1. Pulse Amplitude Modulation (PAM)
• 2. Pulse Width Modulation (PWM)
• 3. Pulse Position Modulation (PPM)
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• Pulse Amplitude Modulation (PAM) –
• The amplitude of the pulse is directly
proportional to amplitude of the modulating
signal at the sampling instant. The width
and position of the pulse remains same.
• Generation of PAM – ##9
• The modulating signal x(t). The low pass
filter performs Bandlimiting.
• The cutoff frequency of LPF is equal to
highest fm present in x(t).
• Aliasing is avoided by using LPF.
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• PAM Waveform –
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• The bandlimited signal is sampled at the
multiplier.
• The multiplier samples x(t) with the help of
a pulse train generator.
• The pulse train generator produces the
pulse train c(t) .
• The multiplication of c(t) and x(t) produces
y(t), the PAM signal.
• The top of the pulses are varied according
to amplitude of x(t).
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• Detection of PAM –
• The PAM signal is passed through a LP
reconstruction filter which reconstructions
the analog signal from PAM pulses.
• Transmission BW of PAM Signal –
• The pulse duration τ(tau) is very very small
as compared to time period Ts between
two samples. If max. frequency is W, then
according to sampling theorem fs should
be more than Nyquist rate i.e. fs>/= 2W.
• Or Ts</= 1/2W & τ(tau) <<Ts </= 1/2W.
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• If On and OFF time of the pulse is same, then
frequency of PAM pulses becomes
f = 1/2
τ(tau) = f max.
• BW required for transmission of PAM signal will
be equal to maximum frequency fmax given by
above equation.
• This BW gives adequate pulse resolution, Bt >/=
f max or Bt >/= 1/τ(tau).
• Since τ(tau) << 1/2W, Bt >/= 1/2τ(tau) >>W.
• Transmission BW of PM signal Bt >> W.
• Thus the BW of PAM is very very large as
compared to highest frequency in the signal x(t).
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• Advantages of PAM –
• 1. PAM forms the basis for other pulse
modulation techniques (PCM,ADM,DM
etc.).
• 2. PAM can be easily generated and
detected.
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• Disadvantages of PAM –
• 1. The amplitude of PAM pulses varies as
per modulating signal. Hence interference
of noise is maximum for PAM and this
noise can not be eliminated easily.
• 2. As amplitude of PAM signal varies, it
also varies the peak power required by the
transmitter with modulating signal.
• 3. The BW needed for transmission of
PAM is very large compared to its
maximum frequency content.
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• Application –
• 1. It is used in A to D converters for
computer interfacing.
• 2. PAM is used in instrumentation system.
• 3. PAM is used for transmitting signals
over a short distance BB channels.
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• Pulse Width Modulation – ( also known
as Pulse Time Modulation)
• Two Types – 1. PWM or Pulse Duration
Modulation. 2. Pulse Position Modulation.
• Principle of PWM (PDM) • The width of the pulse is directly
proportional to amplitude of the modulating
signal at the sampling instant.
• The amplitude and position of the pulse
remains unchanged.
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• PWM Generation - ##10.
• A monostable multivibrator with
modulating in put applied at the control
voltage input.
• Internally the control voltage is adjusted to
2/3 Vcc.
• Externally applied modulating signal
changes the control voltage and hence the
threshold voltage level, giving PWM signal
at the output. (diagram).
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• Demodulation of PWM signal - ##11
• The received PWM signal is applied to the
Schmitt trigger circuit, which removes the
noise in the PWM waveform.
• The regenerated PWM is then applied to
the ramp generator and the synchronization
pulse generator.
• The ramp generator produces ramps for
duration of pulses such that height of the
ramps are proportional to the widths of
PWM pulses. The maximum ramp voltage
is retained till the next pulse.
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• On the other hand, synchronous pulse
generator produces reference pulses with
constant amplitude and pulse width. These
pulses are delayed by specific amount of
delay.
• The delayed reference pulses and the
output of the ramp generator is added in
the adder. The output of the adder is given
to the level shifter.
• The negative offset waveform is clipped by
rectifier and the output is passed through
LPF to recover the modulating signal.
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• Advantages of PWM –
• 1. Does not require synchronization between
transmitter and receiver.
• 2. Separation of signal and noise is easy.
• 3. Unlike PAM, noise is less and hence amplitude
is held constant in PWM
• Disadvantages of PWM –
• 1. Large BW is required for PWM as compared to
PAM.
• 2. Pulses are varying in width and therefore their
power contents are variable. So transmitter must
handle the power contents of the pulse having
maximum PW.
BDG(44)
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• Applications –
• 1. PWM is used to generate PPM.
• 2. PWM is used for synchronous
transmission over noisy channel.
• 3. Motor Control.
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• Pulse Position Modulation – (PPM)
• The amplitude and width of the pulses are
kept constant, while the position of each
pulse with reference to the position of a
reference pulse, is changed according to
the instantaneous sampled value of the
modulating signal.
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• Generation of PPM –
• The sawtooth generator generates the
sawtooth signal of frequency fs (period 1/Ts).
• The saw tooth signal, also called sampling
signal is applied to the inverting input of
comparator.
• The modulating signal x(t) is applied to the
non inverting input of the comparator.
• The out put of the comparator is high only
when instantaneous value of x(t) is higher
than that of sawtooth waveform.
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• Thus the leading edge of the PDM signal
occurs at the fixed time period (kTs)the
trailing edge of the output of comparator
depends on the amplitude of signal x(t).
• When sawtooth waveform voltage is
greater than voltage of x(t) at that instant,
the output of comparator remains zero.
• The trailing edge of the output of the
comparator (PWM) is modulated by signal
x(t).
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• If the sawtooth wave form is reversed,
then the trailing edge will be fixed and
leading edge will be modulated.
• If sawtooth waveform is replaced by
triangular waveform, then both leading and
trailing edges will be modulated.
• The PWM (PDM) signal is nothing but
output of the comparator. The output of
this PWM(PDM) will be positive saturation
of the comparator. (as shown as A in fig.)
• The amplitude is same for all pulses.
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• To generate PPM, PWM (PDM) signal is
used as the trigger input to monostable
multivibrator.
• The output of mono remains zero until it is
triggered.
• The mono is triggered on the trailing(falling)
edge of PWM.
• The output of mono then switches to
positive saturation level.
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• This voltage remains high for the fixed
period and then goes low.
• The mono determines the width of the
pulse.
• The pulse is thus delayed from sampling
time kTs depending on the amplitude of
signal x(t) at kTs.
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• Demodulation of PPM signal –
• In the case of PPM, it is customary to
convert the received pulses that vary in
position to pulses that vary in length. (fig).
• The flip flop circuit is set or turned ‘ON’
(giving high output) when reference pulse
(generated by reference pulse generator
of the receiver with the synchronization
signal from the transmitter) arrives.
• The flip flop is reset or turned ‘OFF’ (giving
low output) at the leading edge of the
position modulated pulse.
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• This repeats and we get PWM pulses at
the output of F-F.
• The PWM pulses are then demodulated by
PWM demodulator to get original message
signal.
• Application – Synchronous communication
of analog pulses over short distances.
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• Advantages – 1. Due to constant pulse
widths and amplitude, transmission power
of each pulse is same. 2. Like PWM, in
PPM amplitude is held constant resulting
less noise interference. 3. Like PWM,
signal and noise separation is easy.
• Disadvantages – 1. Synchronization
between transmitter and receiver is
required. 2. Large BW is required as
compared to PAM.
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• Comparison of Various Pulse Modulation
Methods –
Sr
No.
PAM
PWM/PDM
PPM
1
Amplitude of pulse is
proportional to amplitude
of Modulating signal.
2
BW of channel depends on BW of channel depends
width of the pulse.
on rise time of the pulse.
BW of channel depends
on rise time of the pulse.
3
The instantaneous power
of the transmitter varies.
The instantaneous power
of the transmitter varies.
The instantaneous power
of the transmitter
remains constant.
4
Noise interference is high.
Noise interference is high. Noise interference is high.
5
System is complex.
System is simple.
System is simple.
6
Similar to AM.
Similar to FM.
Similar to PM.
Width of pulse is
Relative position of pulse
proportional to amplitude is proportional to
of Modulating signal.
amplitude of Modulating
signal.
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