UNIT-III Waveform generation: Sine wave generation - Wein bridge, phase shift oscillators;
Multivibrators, triangular wave generators, sawtooth wave generators, voltage to frequency
and frequency to voltage converters, voltage controlled oscillators. Multiplier: Analog
multipliers, Applications of multipliers - Division, Square, square root, frequency doubler,
rectifier and Phase shift detector circuits; Amplitude, Frequency, Pulse width modulation
circuits, Demodulation.
WAVEFORM GENERATION
Sine wave
The graphs of the sine and cosine functions are sinusoids of different phases.
The sine wave or sinusoid is a mathematical curve that describes a smooth repetitive
oscillation. It is named after the function sine, of which it is the graph. It occurs often in pure
and applied mathematics, as well as physics, engineering, signal processing and many other
fields. Its most basic form as a function of time (t) is:
where:

A = the amplitude, the peak deviation of the function from zero.

f = the ordinary frequency, the number of oscillations (cycles) that occur each second
of time.
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
ω = 2πf, the angular frequency, the rate of change of the function argument in units of
radians per second

= the phase, specifies (in radians) where in its cycle the oscillation is at t = 0.
o
When
amount
is non-zero, the entire waveform appears to be shifted in time by the
/ω seconds. A negative value represents a delay, and a positive
value represents an advance.
The oscillation of an undamped spring-mass system around the equilibrium is a sine wave
The sine wave is important in physics because it retains its wave shape when added to
another sine wave of the same frequency and arbitrary phase and magnitude. It is the only
periodic waveform that has this property. This property leads to its importance in Fourier
analysis and makes it acoustically unique.
General form
In general, the function may also have:

a spatial variable x that represents the position on the dimension on which the wave
propagates, and a characteristic parameter k called wave number (or angular wave
number), which represents the proportionality between the angular frequency ω and
the linear speed (speed of propagation) ν

a non-zero center amplitude, D
which is
, if the wave is moving to the right
, if the wave is moving to the left
The wavenumber is related to the angular frequency by:.
where λ is the wavelength, f is the frequency, and v is the linear speed.
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This equation gives a sine wave for a single dimension; thus the generalized equation given
above gives the displacement of the wave at a position x at time t along a single line. This
could, for example, be considered the value of a wave along a wire.
In two or three spatial dimensions, the same equation describes a travelling plane wave if
position x and wavenumber k are interpreted as vectors, and their product as a dot product.
For more complex waves such as the height of a water wave in a pond after a stone has been
dropped in, more complex equations are needed.
Phase-shift oscillator
A phase-shift oscillator is a linear electronic oscillator circuit that produces a sine wave
output. It consists of an inverting amplifier element such as a transistor or op amp with its
output fed back to its input through a phase-shift network consisting of resistors and
capacitors. The feedback network 'shifts' the phase of the amplifier output by 180 degrees at
the oscillation frequency to give positive feedback.[1] Phase-shift oscillators are often used at
audio frequency as audio oscillators.
The filter produces a phase shift that increases with frequency. It must have a maximum
phase shift of more than 180 degrees at high frequencies so the phase shift at the desired
oscillation frequency can be 180 degrees. The most common phase-shift network cascades
three identical resistor-capacitor stages that produce a phase shift of zero at low frequencies
and 270° at high frequencies.
Op-amp implementation
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A simple example of a phase-shift oscillator
One of the simplest implementations for this type of oscillator uses an operational amplifier
(op-amp), three capacitors and four resistors, as shown in the diagram.
The mathematics for calculating oscillation frequency and oscillation criterion for this circuit
are surprisingly complex, due to each RC stage loading the previous ones. The calculations
are greatly simplified by setting all the resistors (except the negative feedback resistor) and
all the capacitors to the same values. In the diagram, if R1=R2=R3=R, and C1=C2=C3=C, then:
and the oscillation criterion is:
Without the simplification of all the resistors and capacitors having the same values, the
calculations become more complex:
Oscillation
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4
As with other feedback oscillators, when the power is applied to the circuit, thermal electrical
noise in the circuit or the turn-on transient provides an initial signal to start oscillations. The
oscillations grow rapidly in amplitude until saturation of the op-amp or transistor limits the
gain, and they stabilize at a constant amplitude at which the loop gain of the circuit is unity.
One potential problem with the single op-amp circuit is the high gain required to maintain the
oscillation. If it is assumed that each RC segment does not affect the other, a gain of about 8
to 10 will be sufficient to enable oscillation. As mentioned previously, each RC section loads
the next section, and a larger gain (about 27 to 30) is required to keep the circuit in
oscillation. An improved version of this circuit can be made by putting an op-amp buffer
between each RC stage. (This also simplifies the calculations.) Тhe voltage gain of the
inverting channel is always unity.
When the oscillation frequency is high enough to be near the amplifier's cutoff frequency, the
amplifier will contribute significant phase shift itself, which will add to the phase shift of the
feedback network. Therefore the circuit will oscillate at a frequency at which the phase shift
of the feedback filter is less than 180 degrees.
Multivibrator
A multivibrator is an electronic circuit used to implement a variety of simple two-state
systems such as oscillators, timers and flip-flops. It is characterized by two amplifying
devices (transistors, electron tubes or other devices) cross-coupled by resistors or capacitors.
The name "multivibrator" was initially applied to the free-running oscillator version of the
circuit because its output waveform was rich in harmonics.[1] There are three types of
multivibrator circuits depending on the circuit operation:

astable, in which the circuit is not stable in either state —it continually switches from
one state to the other. It functions as a relaxation oscillator.
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
monostable, in which one of the states is stable, but the other state is unstable
(transient). A trigger pulse causes the circuit to enter the unstable state. After entering
the unstable state, the circuit will return to the stable state after a set time. Such a
circuit is useful for creating a timing period of fixed duration in response to some
external event. This circuit is also known as a one shot.

bistable, in which the circuit is stable in either state. It can be flipped from one state
to the other by an external trigger pulse. This circuit is also known as a flip flop. It can
be used to store one bit of information.
Multivibrators find applications in a variety of systems where square waves or timed intervals
are required. For example, before the advent of low-cost integrated circuits, chains of
multivibrators found use as frequency dividers. A free-running multivibrator with a frequency
of one-half to one-tenth of the reference frequency would accurately lock to the reference
frequency. This technique was used in early electronic organs, to keep notes of different
octaves accurately in tune. Other applications included early television systems, where the
various line and frame frequencies were kept synchronized by pulses included in the video
signal.
The terminology of multivibrators has been somewhat variable, historically. For example:
Astable multivibrator
An astable multivibrator consists of two amplifying stages connected in a positive feedback
loop by two capacitive-resistive coupling networks. The amplifying elements may be junction
or field-effect transistors, vacuum tubes, operational amplifiers, or other types of amplifier.
The example diagram shows bipolar junction transistors.
The circuit is usually drawn in a symmetric form as a cross-coupled pair. Two output
terminals can be defined at the active devices, which will have complementary states; one
will have high voltage while the other has low voltage, (except during the brief transitions
from one state to the other).
Operation
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Figure 1: Basic BJT astable multivibrator
The circuit has two astable (unstable) states that change alternatively with maximum
transition rate because of the "accelerating" positive feedback. It is implemented by the
coupling capacitors that instantly transfer voltage changes because the voltage across a
capacitor cannot suddenly change. In each state, one transistor is switched on and the other is
switched off. Accordingly, one fully charged capacitor discharges (reverse charges) slowly
thus converting the time into an exponentially changing voltage. At the same time, the other
empty capacitor quickly charges thus restoring its charge (the first capacitor acts as a timesetting capacitor and the second prepares to play this role in the next state). The circuit
operation is based on the fact that the forward-biased base-emitter junction of the switchedon bipolar transistor can provide a path for the capacitor restoration.
State
1
(Q1
is
switched
on,
Q2
is
switched
off):
In the beginning, the capacitor C1 is fully charged (in the previous State 2) to the power
supply voltage V with the polarity shown in Figure 1. Q1 is on and connects the left-hand
positive plate of C1 to ground. As its right-hand negative plate is connected to Q2 base, a
maximum negative voltage (-V) is applied to Q2 base that keeps Q2 firmly off. C1 begins
discharging (reverse charging) via the high-value base resistor R2, so that the voltage of its
right-hand plate (and at the base of Q2) is rising from below ground (-V) toward +V. As Q2
base-emitter junction is reverse-biased, it does not conduct, so all the current from R2 goes
into C1. Simultaneously, C2 that is fully discharged and even slightly charged to 0.6 V (in the
previous State 2) quickly charges via the low-value collector resistor R4 and Q1 forwardbiased base-emitter junction (because R4 is less than R2, C2 charges faster than C1). Thus C2
restores its charge and prepares for the next State C2 when it will act as a time-setting
capacitor. Q1 is firmly saturated in the beginning by the "forcing" C2 charging current added
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to R3 current; in the end, only R3 provides the needed input base current. The resistance R3
is chosen small enough to keep Q1 (not deeply) saturated after C2 is fully charged.
When the voltage of C1 right-hand plate (Q2 base voltage) becomes positive and reaches 0.6
V, Q2 base-emitter junction begins diverting a part of R2 charging current. Q2 begins
conducting and this starts the avalanche-like positive feedback process as follows. Q2
collector voltage begins falling; this change transfers through the fully charged C2 to Q1 base
and Q1 begins cutting off. Its collector voltage begins rising; this change transfers back
through the almost empty C1 to Q2 base and makes Q2 conduct more thus sustaining the
initial input impact on Q2 base. Thus the initial input change circulates along the feedback
loop and grows in an avalanche-like manner until finally Q1 switches off and Q2 switches on.
The forward-biased Q2 base-emitter junction fixes the voltage of C1 right-hand plate at 0.6 V
and does not allow it to continue rising toward +V.
State
2
(Q1
is
switched
off,
Q2
is
switched
on):
Now, the capacitor C2 is fully charged (in the previous State 1) to the power supply voltage V
with the polarity shown in Figure 1. Q2 is on and connects the right-hand positive plate of C2
to ground. As its left-hand negative plate is connected to Q1 base, a maximum negative
voltage (-V) is applied to Q1 base that keeps Q1 firmly off. C2 begins discharging (reverse
charging) via the high-value base resistor R3, so that the voltage of its left-hand plate (and at
the base of Q1) is rising from below ground (-V) toward +V. Simultaneously, C1 that is fully
discharged and even slightly charged to 0.6 V (in the previous State 1) quickly charges via
the low-value collector resistor R1 and Q2 forward-biased base-emitter junction (because R1
is less than R3, C1 charges faster than C2). Thus C1 restores its charge and prepares for the
next State 1 when it will act again as a time-setting capacitor...and so on... (the next
explanations are a mirror copy of the second part of Step 1).
Multivibrator frequency
The duration of state 1 (low output) will be related to the time constant R2C1 as it depends on
the charging of C1, and the duration of state 2 (high output) will be related to the time
constant R3C2 as it depends on the charging of C2. Because they do not need to be the same,
an asymmetric duty cycle is easily achieved.
The voltage on a capacitor with non-zero initial charge is:
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Looking at C2, just before Q2 turns on, the left terminal of C2 is at the base-emitter voltage
of Q1 (VBE_Q1) and the right terminal is at VCC ("VCC" is used here instead of "+V" to ease
notation). The voltage across C2 is VCC minus VBE_Q1 . The moment after Q2 turns on, the
right terminal of C2 is now at 0 V which drives the left terminal of C2 to 0 V minus (VCC VBE_Q1) or VBE_Q1 - VCC. From this instant in time, the left terminal of C2 must be charged
back up to VBE_Q1. How long this takes is half our multivibrator switching time (the other half
comes from C1). In the charging capacitor equation above, substituting:
VBE_Q1 for
(VBE_Q1 - VCC) for
VCC for
results in:
Solving for t results in:
For this circuit to work, VCC>>VBE_Q1 (for example: VCC=5 V, VBE_Q1=0.6 V), therefore the
equation can be simplified to:
or
or
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The period of each half of the multivibrator is therefore given by t = ln(2)RC.
The total period of oscillation is given by:
T = t1 + t2 = ln(2)R2 C1 + ln(2)R3 C2
where...

f is frequency in hertz.

R2 and R3 are resistor values in ohms.

C1 and C2 are capacitor values in farads.

T is the period (In this case, the sum of two period durations).
For the special case where

t1 = t2 (50% duty cycle)

R2 = R3

C1 = C2
[8]
Output pulse shape
The output voltage has a shape that approximates a square waveform. It is considered below
for the transistor Q1.
During State 1, Q2 base-emitter junction is reverse-biased and the capacitor C1 is
"unhooked" from ground. The output voltage of the switched-on transistor Q1 changes
rapidly from high to low since this low-resistive output is loaded by a high impedance load
(the series connected capacitor C1 and the high-resistive base resistor R2).
During State 2, Q2 base-emitter junction is forward-biased and the capacitor C1 is "hooked"
to ground. The output voltage of the switched-off transistor Q1 changes exponentially from
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low to high since this relatively high resistive output is loaded by a low impedance load (the
capacitance C1). This is the output voltage of R1C1 integrating circuit.
To approach the needed square waveform, the collector resistors have to be low resistance.
The base resistors have to be low enough to make the transistors saturate in the end of the
restoration (RB < β.RC).
Initial power-up
When the circuit is first powered up, neither transistor will be switched on. However, this
means that at this stage they will both have high base voltages and therefore a tendency to
switch on, and inevitable slight asymmetries will mean that one of the transistors is first to
switch on. This will quickly put the circuit into one of the above states, and oscillation will
ensue. In practice, oscillation always occurs for practical values of R and C.
However, if the circuit is temporarily held with both bases high, for longer than it takes for
both capacitors to charge fully, then the circuit will remain in this stable state, with both bases
at 0.6 V, both collectors at 0 V, and both capacitors charged backwards to −0.6 V. This can
occur at startup without external intervention, if R and C are both very small.
Frequency divider
An astable multivibrator can be synchronized to an external chain of pulses. A single pair of
active devices can be used to divide a reference by a large ratio, however, the stability of the
technique is poor owing to the variability of the power supply and the circuit elements; a
division ratio of 10, for example, is easy to obtain but not dependable. Chains of bistable flipflops provide more predictable division, at the cost of more active elements.[8]
Protective components
While not fundamental to circuit operation, diodes connected in series with the base or
emitter of the transistors are required to prevent the base-emitter junction being driven into
reverse breakdown when the supply voltage is in excess of the Veb breakdown voltage,
typically around 5-10 volts for general purpose silicon transistors. In the monostable
configuration, only one of the transistors requires protection.
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Monostable
For more details on this topic, see Monostable.
Figure 2: Basic BJT monostable multivibrator.
Figure 3: Basic animated interactive BJT bistable multivibrator circuit (suggested values: R1,
R2 = 1 kΩ R3, R4 = 10 kΩ).
In the monostable multivibrator, the one resistive-capacitive network (C2-R3 in figure 1) is
replaced by a resistive network (just a resistor). The circuit can be thought as a 1/2 astable
multivibrator. Q2 collector voltage is the output of the circuit (in contrast to the astable
circuit, it has a perfect square waveform since the output is not loaded by the capacitor).
When triggered by an input pulse, a monostable multivibrator will switch to its unstable
position for a period of time, and then return to its stable state. The time period monostable
multivibrator remains in unstable state is given by t = ln(2)R2C1. If repeated application of the
input pulse maintains the circuit in the unstable state, it is called a retriggerable monostable.
If further trigger pulses do not affect the period, the circuit is a non-retriggerable
multivibrator.
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For the circuit in Figure 2, in the stable state Q1 is turned off and Q2 is turned on. It is
triggered by zero or negative input signal applied to Q2 base (with the same success it can be
triggered by applying a positive input signal through a resistor to Q1 base). As a result, the
circuit goes in State 1 described above. After elapsing the time, it returns to its stable initial
state.
Sawtooth wave
A bandlimited sawtooth wave pictured in the time domain (top) and frequency domain
(bottom). The fundamental is at 220 Hz (A3).
The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named
based on its resemblance to the teeth of a saw.
The convention is that a sawtooth wave ramps upward and then sharply drops. However, in a
"reverse (or inverse) sawtooth wave", the wave ramps downward and then sharply rises. It
can also be considered the extreme case of an asymmetric triangle wave.[1]
The piecewise linear function
based on the floor function of time t is an example of a sawtooth wave with period 1.
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A more general form, in the range −1 to 1, and with period a, is
This sawtooth function has the same phase as the sine function.
Another function in trigonometric terms with period p and amplitude a:
While a square wave is constructed from only odd harmonics, a sawtooth wave's sound is
harsh and clear and its spectrum contains both even and odd harmonics of the fundamental
frequency. Because it contains all the integer harmonics, it is one of the best waveforms to
use for subtractive synthesis of musical sounds, particularly bowed string instruments like
violins and cellos, since the slip-stick behavior of the bow drives the strings with a sawtoothlike motion.
A sawtooth can be constructed using additive synthesis. The infinite Fourier series
converges to a reverse (inverse) sawtooth wave. A conventional sawtooth can be constructed
using
Where A is amplitude.
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In digital synthesis, these series are only summed over k such that the highest harmonic, Nmax,
is less than the Nyquist frequency (half the sampling frequency). This summation can
generally be more efficiently calculated with a fast Fourier transform. If the waveform is
digitally created directly in the time domain using a non-bandlimited form, such as y = x floor(x), infinite harmonics are sampled and the resulting tone contains aliasing distortion.
Animation of the additive synthesis of a sawtooth wave with an increasing number of
harmonics
Analog multiplier
An analog multiplier is a device which takes two analog signals and produces an output
which is their product. Such circuits can be used to implement related functions such as
squares (apply same signal to both inputs), and square roots.
An electronic analog multiplier can be called by several names, depending on the function it
is used to serve (see analog multiplier applications).
Voltage-controlled amplifier versus analog multiplier
If one input of an analog multiplier is held at a steady state voltage, a signal at the second
input will be scaled in proportion to the level on the fixed input. In this case the analog
multiplier may be considered to be a voltage controlled amplifier. Obvious applications
would be for electronic volume control and automatic gain control. Although analog
multipliers are often used for such applications, voltage-controlled amplifiers are not
necessarily true analog multipliers. For example, an integrated circuit designed to be used as
a volume control may have a signal input designed for 1 Vp-p, and a control input designed
for 0-5 V dc; that is, the two inputs are not symmetrical and the control input will have a
limited bandwidth.
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By contrast, in what is generally considered to be a true analog multiplier, the two signal
inputs have identical characteristics. Applications specific to a true analog multiplier are
those where both inputs are signals, for example in a frequency mixer or an analog circuit to
implement a discrete Fourier transform.
A four-quadrant multiplier is one where inputs and outputs may swing positive and negative.
Many multipliers only work in 2 quadrants (one input may only have one polarity), or single
quadrant (inputs and outputs have only one polarity, usually all positive).
Analog multiplier devices
Analog multiplication can be accomplished by using the Hall Effect.
The Gilbert cell is a circuit whose output current is a 4 quadrant multiplication of its two
differential inputs.
Integrated circuits analog multipliers are incorporated into many applications, such as a true
RMS converter, but a number of general purpose analog multiplier building blocks are
available such as the Linear Four Quadrant Multiplier. General-purpose devices will usually
include attenuators or amplifiers on the inputs or outputs in order to allow the signal to be
scaled within the voltage limits of the circuit.
Although analog multiplier circuits are very similar to operational amplifiers, they are far
more susceptible to noise and offset voltage-related problems as these errors may become
multiplied. When dealing with high frequency signals, phase-related problems may be quite
complex. For this reason, manufacturing wide-range general-purpose analog multipliers is far
more difficult than ordinary operational amplifiers, and such devices are typically produced
using specialist technologies and laser trimming, as are those used for high-performance
amplifiers such as instrumentation amplifiers. This means they have a relatively high cost and
so they are generally used only for circuits where they are indispensable.
Some commonly available Analog Multiplier ICs in the market are MPY634 from Texas
Instruments, AD534, AD632 and AD734 from Analog Devices, HA-2556 from Intersil and
much more from other IC manufacturers.
Amplitude modulation
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Amplitude modulation (AM) is a modulation technique used in electronic communication,
most commonly for transmitting information via a radio carrier wave. In amplitude
modulation, the amplitude (signal strength) of the carrier wave is varied in proportion to the
waveform being transmitted. That waveform may, for instance, correspond to the sounds to
be reproduced by a loudspeaker, or the light intensity of television pixels. This technique
contrasts with frequency modulation, in which the frequency of the carrier signal is varied,
and phase modulation, in which its phase is varied.
AM was the earliest modulation method used to transmit voice by radio. It was developed
during the first two decades of the 20th century beginning with Roberto Landell De Moura
and Reginald Fessenden's radiotelephone experiments in 1900.[1] It remains in use today in
many forms of communication; for example it is used in portable two way radios, VHF
aircraft radio and in computer modems.[citation
needed]
"AM" is often used to refer to
mediumwave AM radio broadcasting.
Fig 1: An audio signal (top) may be carried by a carrier frequency using AM or FM methods.
Forms of amplitude modulation
modulation means varying some aspect of a higher frequency continuous wave carrier signal
with an information-bearing modulation waveform, such as an audio signal which represents
sound, or a video signal which represents images, so the carrier will "carry" the information.
When it reaches its destination, the information signal is extracted from the modulated carrier
by demodulation.
In amplitude modulation, the amplitude or "strength" of the carrier oscillations is what is
varied. For example, in AM radio communication, a continuous wave radio-frequency signal
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(a sinusoidal carrier wave) has its amplitude modulated by an audio waveform before
transmission. The audio waveform modifies the amplitude of the carrier wave and determines
the envelope of the waveform. In the frequency domain, amplitude modulation produces a
signal with power concentrated at the carrier frequency and two adjacent sidebands. Each
sideband is equal in bandwidth to that of the modulating signal, and is a mirror image of the
other. Standard AM is thus sometimes called "double-sideband amplitude modulation" (DSBAM) to distinguish it from more sophisticated modulation methods also based on AM.
One disadvantage of all amplitude modulation techniques (not only standard AM) is that the
receiver amplifies and detects noise and electromagnetic interference in equal proportion to
the signal. Increasing the received signal to noise ratio, say, by a factor of 10 (a 10 decibel
improvement), thus would require increasing the transmitter power by a factor of 10. This is
in contrast to frequency modulation (FM) and digital radio where the effect of such noise
following demodulation is strongly reduced so long as the received signal is well above the
threshold for reception. For this reason AM broadcast is not favored for music and high
fidelity broadcasting, but rather for voice communications and broadcasts (sports, news, talk
radio etc.).
Another disadvantage of AM is that it is inefficient in power usage; at least two-thirds of the
power is concentrated in the carrier signal. The carrier signal contains none of the original
information being transmitted (voice, video, data, etc.). However its presence provides a
simple means of demodulation using envelope detection, providing a frequency and phase
reference to extract the modulation from the sidebands. In some modulation systems based on
AM, a lower transmitter power is required through partial or total elimination of the carrier
component, however receivers for these signals are more complex and costly. The receiver
may regenerate a copy of the carrier frequency (usually as shifted to the intermediate
frequency) from a greatly reduced "pilot" carrier (in reduced-carrier transmission or DSBRC) to use in the demodulation process. Even with the carrier totally eliminated in doublesideband suppressed-carrier transmission, carrier regeneration is possible using a Costas
phase-locked loop. This doesn't work however for single-sideband suppressed-carrier
transmission (SSB-SC), leading to the characteristic "Donald Duck" sound from such
receivers when slightly detuned. Single sideband is nevertheless used widely in amateur radio
and other voice communications both due to its power efficiency and bandwidth efficiency
(cutting the RF bandwidth in half compared to standard AM). On the other hand, in medium
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wave and short wave broadcasting, standard AM with the full carrier allows for reception
using inexpensive receivers. The broadcaster absorbs the extra power cost to greatly increase
potential audience.
An additional function provided by the carrier in standard AM, but which is lost in either
single or double-sideband suppressed-carrier transmission, is that it provides an amplitude
reference. In the receiver, the automatic gain control (AGC) responds to the carrier so that the
reproduced audio level stays in a fixed proportion to the original modulation. On the other
hand, with suppressed-carrier transmissions there is no transmitted power during pauses in
the modulation, so the AGC must respond to peaks of the transmitted power during peaks in
the modulation. This typically involves a so-called fast attack, slow decay circuit which holds
the AGC level for a second or more following such peaks, in between syllables or short
pauses in the program. This is very acceptable for communications radios, where
compression of the audio aids intelligibility. However it is absolutely undesired for music or
normal broadcast programming, where a faithful reproduction of the original program,
including its varying modulation levels, is expected.
A trivial form of AM which can be used for transmitting binary data is on-off keying, the
simplest form of amplitude-shift keying, in which ones and zeros are represented by the
presence or absence of a carrier. On-off keying is likewise used by radio amateurs to transmit
Morse code where it is known as continuous wave (CW) operation, even though the
transmission is not strictly "continuous."
Pulse-width modulation
An example of PWM in an idealized inductor driven by a voltage source: the voltage source
(blue) is modulated as a series of pulses that results in a sine-like current/flux (red) in the
inductor. The blue rectangular pulses nonetheless result in a smoother and smoother red sine
wave as the switching frequency increases. Note that the red waveform is the (definite)
integral of the blue waveform.
Pulse-width modulation (PWM), or pulse-duration modulation (PDM), is a technique
used to encode a message into a pulsing signal. It is a type of modulation. Although this
modulation technique can be used to encode information for transmission, its main use is to
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allow the control of the power supplied to electrical devices, especially to inertial loads such
as motors. In addition, PWM is one of the two principal algorithms used in photovoltaic solar
battery chargers,[1] the other being MPPT.
The average value of voltage (and current) fed to the load is controlled by turning the switch
between supply and load on and off at a fast rate. The longer the switch is on compared to the
off periods, the higher the total power supplied to the load.
The PWM switching frequency has to be much higher than what would affect the load (the
device that uses the power), which is to say that the resultant waveform perceived by the load
must be as smooth as possible. Typically switching has to be done several times a minute in
an electric stove, 120 Hz in a lamp dimmer, from few kilohertz (kHz) to tens of kHz for a
motor drive and well into the tens or hundreds of kHz in audio amplifiers and computer
power supplies.
The term duty cycle describes the proportion of 'on' time to the regular interval or 'period' of
time; a low duty cycle corresponds to low power, because the power is off for most of the
time. Duty cycle is expressed in percent, 100% being fully on.
The main advantage of PWM is that power loss in the switching devices is very low. When a
switch is off there is practically no current, and when it is on and power is being transferred
to the load, there is almost no voltage drop across the switch. Power loss, being the product of
voltage and current, is thus in both cases close to zero. PWM also works well with digital
controls, which, because of their on/off nature, can easily set the needed duty cycle.
PWM has also been used in certain communication systems where its duty cycle has been
used to convey information over a communications channel.
Principle
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Fig. 1: a pulse wave, showing the definitions of
,
and D.
Pulse-width modulation uses a rectangular pulse wave whose pulse width is modulated
resulting in the variation of the average value of the waveform. If we consider a pulse
waveform
, with period
, low value
, a high value
and a duty cycle D (see
figure 1), the average value of the waveform is given by:
As
is a pulse wave, its value is
for
and
for
. The above expression then becomes:
This latter expression can be fairly simplified in many cases where
as
. From this, it is obvious that the average value of the signal ( ) is directly
dependent on the duty cycle D.
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Fig. 2: A simple method to generate the PWM pulse train corresponding to a given signal is
the intersective PWM: the signal (here the red sinewave) is compared with a sawtooth
waveform (blue). When the latter is less than the former, the PWM signal (magenta) is in
high state (1). Otherwise it is in the low state (0).
The simplest way to generate a PWM signal is the intersective method, which requires only a
sawtooth or a triangle waveform (easily generated using a simple oscillator) and a
comparator. When the value of the reference signal (the red sine wave in figure 2) is more
than the modulation waveform (blue), the PWM signal (magenta) is in the high state,
otherwise it is in the low state.
Demodulation
Demodulation is the act of extracting the original information-bearing signal from a
modulated carrier wave. A demodulator is an electronic circuit (or computer program in a
software-defined radio) that is used to recover the information content from the modulated
carrier wave.[1] There are many types of modulation so there are many types of demodulators.
The signal output from a demodulator may represent sound (an analog audio signal), images
(an analog video signal) or binary data (a digital signal).
These terms are traditionally used in connection with radio receivers, but many other systems
use many kinds of demodulators. For example, in a modem, which is a contraction of the
terms modulator/demodulator, a demodulator is used to extract a serial digital data stream
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from a carrier signal which is used to carry it through a telephone line, coaxial cable, or
optical fiber.
.
The first type of modulation used to transmit sound over radio waves was amplitude
modulation (AM), invented by Reginald Fessendon around 1900. An AM radio signal can be
demodulated by rectifying it, removing the radio frequency pulses on one side of the carrier,
converting it from alternating current (AC) to a pulsating direct current (DC). The amplitude
of the DC varies with the modulating audio signal, so it can drive an earphone. Fessendon
invented the first AM demodulator in 1904 called the electrolytic detector, consisting of a
short needle dipping into a cup of dilute acid. The same year John Ambrose Fleming invented
the Fleming valve or thermionic diode which could also rectify an AM signal.
Techniques
There are several ways of demodulation depending on how parameters of the base-band
signal such as amplitude, frequency or phase are transmitted in the carrier signal. For
example, for a signal modulated with a linear modulation like AM (amplitude modulation),
we can use a synchronous detector. On the other hand, for a signal modulated with an angular
modulation, we must use an FM (frequency modulation) demodulator or a PM (phase
modulation) demodulator. Different kinds of circuits perform these functions.
Many techniques such as carrier recovery, clock recovery, bit slip, frame synchronization,
rake receiver, pulse compression, Received Signal Strength Indication, error detection and
correction, etc., are only performed by demodulators, although any specific demodulator may
perform only some or none of these techniques.
Many things can act as a demodulator, if they pass the radio waves on nonlinearly. For
example, near a powerful radio station, it has been known for the metal sides of a van to
demodulate the radio signal as sound.
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AM radio
An AM signal encodes the information onto the carrier wave by varying its amplitude in
direct sympathy with the analogue signal to be sent. There are two methods used to
demodulate AM signals:

The envelope detector is a very simple method of demodulation that does not require
a coherent demodulator. It consists of an envelope detector that can be a rectifier
(anything that will pass current in one direction only) or other non-linear that
enhances one half of the received signal over the other and a low-pass filter. The
rectifier may be in the form of a single diode or may be more complex. Many natural
substances exhibit this rectification behaviour, which is why it was the earliest
modulation and demodulation technique used in radio. The filter is usually an RC
low-pass type but the filter function can sometimes be achieved by relying on the
limited frequency response of the circuitry following the rectifier. The crystal set
exploits the simplicity of AM modulation to produce a receiver with very few parts,
using the crystal as the rectifier and the limited frequency response of the headphones
as the filter.

The product detector multiplies the incoming signal by the signal of a local
oscillator with the same frequency and phase as the carrier of the incoming signal.
After filtering, the original audio signal will result.
SSB is a form of AM in which the carrier is reduced or suppressed entirely, which require
coherent demodulation. For further reading, see sideband.
FM radio
Frequency modulation (FM) has numerous advantages over AM such as better fidelity and
noise immunity. However, it is much more complex to both modulate and demodulate a
carrier wave with FM and AM predates it by several decades.
There are several common types of FM demodulators:
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
The quadrature detector, which phase shifts the signal by 90 degrees and multiplies it
with the unshifted version. One of the terms that drops out from this operation is the
original information signal, which is selected and amplified.

The signal is fed into a PLL and the error signal is used as the demodulated signal.

The most common is a Foster-Seeley discriminator. This is composed of an electronic
filter which decreases the amplitude of some frequencies relative to others, followed
by an AM demodulator. If the filter response changes linearly with frequency, the
final analog output will be proportional to the input frequency, as desired.

A variant of the Foster-Seeley discriminator called the ratio detector[2]

Another method uses two AM demodulators, one tuned to the high end of the band
and the other to the low end, and feed the outputs into a difference amplifier.

Using a digital signal processor, as used in software-defined radio.
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POINTS TO REMEMBER:

The sine wave or sinusoid is a mathematical curve that describes a smooth repetitive
oscillation.

The sine wave is important in physics because it retains its wave shape when added to
another sine wave of the same frequency and arbitrary phase and magnitude. It is the
only periodic waveform that has this property.

A phase-shift oscillator is a linear electronic oscillator circuit that produces a sine
wave output.

A multivibrator is an electronic circuit used to implement a variety of simple twostate systems such as oscillators, timers and flip-flops.

Monostable, in which one of the states is stable, but the other state is unstable
(transient).

Bistable, in which the circuit is stable in either state. It can be flipped from one state
to the other by an external trigger pulse. This circuit is also known as a flip flop.

An astable multivibrator consists of two amplifying stages connected in a positive
feedback loop by two capacitive-resistive coupling networks.

In digital synthesis, these series are only summed over k such that the highest
harmonic, Nmax, is less than the Nyquist frequency (half the sampling frequency)
Expected Questions:
1. Explain with neat diagram of Sine wave generation .
2. Illustrate the function of the phase shift oscillators.
3. Explain the function of Analog multipliers.
4. Explain the sawtooth wave generators with a neat diagram.
5. Explain the function of Demodulation.
6. Illustrate the function of the Pulse width modulation.
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