Modulators and Demodulators

1

• Chapter 8
• Modulators and Demodulators

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• Modulation is the modification of a
  high-frequency carrier signal to include the
  information present in a relatively low frequency
  signal. This is necessary because radio wave
  propagation is more efficient at higher
  frequencies and smaller antennas can be used. A
  larger bandwidth can be obtained at higher
  frequencies, enabling many information-containing
  signals to be multiplexed onto one carrier and
  sent simultaneously.
• Frequency conversion, modulation and detection
  are common tasks performed in a communication
  circuit.
• Frequency Mixers
• The most commonly used device for frequency
  modification is the mixer. It is basically a
  multiplier
• The output consists of the sum and difference of
  the two input frequencies, one of which is the
  desired component. The other will be filtered
  out. This combination of a mixer and filter to
  remove an output frequency is known as
  single-sideband mixer.
• There are 2 main classes of mixers -- nonlinear
  or switching-type.

3
Switching-Type Mixers

• One or more switches, realized by diodes or
  transistors, will function as the time-varying
  circuit elements.

4

• For ideal center-tapped transformer, the voltages
  will be indicated as below
• The local oscillator VL is a constant- amplitude
  signal. VL gtgt Vi so that D1 is on when VL is
  positive and D2 is on when VL is negative. Thus

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• The output consists of the local oscillator plus
  Vi switched by 180? at the frequency of the local
  oscillator. If the switched form of Vi is
  represented by Vi then
• The Fourier series for P(t) and Vi are
• If Vi is a sine wave then
• Since the mixer output
  consists of the local oscillator signal plus an
  infinite number of additional frequencies created
  in the mixer. The output frequencies in addition
  to the upper and lower sidebands are called
  spurious. The desired component is obtained by
  filtering.

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• The preceding analysis assumed that the local
  oscillator signal was much larger than the input
  signal and sufficiently large to turn on the
  diodes instantly. Deviations from these
  assumptions will result in distortions in the
  desired frequency component.
• A disadvantage of the circuit above is that VL
  appears in the output. If the oscillator
  frequency is much larger than the input
  frequency, then the desired mixing product
  may be close to the oscillator freq. and
  will be difficult to separate by filtering. In
  the new circuit below

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• The local oscillator signal does not appear in
  the output. For ideal transformer the voltages
  are shown below

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• If VL is positive and much larger than Vi than
  both diodes are conducting . The local
  oscillator current balance out in the output
  transformer VoVi. If VL is negative, the diodes
  will be open and the output signal will be zero.
  Thus
• If Vi is a sine wave ViVsin?it, the output is
• The output of this mixer differs from the
  previous one in that it does not contain the
  local oscillator signal but it does contain a
  signal at the same frequency as the input signal.
• Four Diode Switching Type Mixer
• The construction of this type of mixer is shown
  below

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• Neither the local oscillator signal and the input
  signal appears at the output. If the local
  oscillator, VL, is positive, then diodes D2 and
  D3 will conduct and the equivalent circuit is
  shown below
• rd is the diode on resistance. The loop
  equations are
• Thus
• If the local oscillator signal is negative,
  diodes D1 and D4 conduct and the equivalent
  circuit is

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• In this mixer the output voltage is proportional
  to the input voltage is switched at the local
  oscillator frequency. Therefore
  if
• then
• A double-balanced mixer with perfectly matched
  diodes and ideal transformer coupling will
  generate the upper and lower sidebands plus an
  infinite number of spurious frequencies centered
  on odd harmonics of the local oscillator
  frequency. Their excellent performance is due in
  part to modern fabrication techniques to
  construct closely matched diodes. High frequency
  Schottky barrier diodes are often used today.

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Conversion Loss

• Mixer conversion loss is defined as the ratio of
  output power in one sideband to signal input
  power. It is a most important mixer parameter,
  particularly for the receiver.
• From Fig. 12.13, and the load impedance seen by
  Vi is
• Normally RLgtgtrd so the input will be matched for
  maximum power transfer if RLRs. Under this
  condition ViVs/2 and
• The output voltage in on sideband, for RLgtgtrd, is
  
• the output power is
• So the conversion gain of the double-balanced
  mixer is
• The conversion loss is
• For an ideal double-balanced mixer matched to the
  source impedance, and ignoring the power lost in
  the transformer and switching diodes,
  approximately 40 of the signal input power will
  be transferred to the output.

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• For the single-balanced mixer, the output voltage
  of one sideband is
• If the port is matched for maximum power transfer
• The power gain is .
  The conversion loss is 4 times (6 dB) larger than
  double-balanced mixer
• Distortion
• As the mixer input signal power increases, it
  will reach the level at which it is larger than
  the local oscillator.
• The input signal then assumes the switching role,
  and the output power becomes proportional to the
  local oscillator power. Since the local
  oscillator is constant the output power will be
  constant.

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• Intermodulation Distortion
• Consider a diode-ring mixer with a resistance R
  in series with each diode as shown below
• The purpose of the additional resistors will
  become clear once the IMD is determined.
• If the local oscillator power is sufficiently
  large, the circuit during either half-cycle is as
  shown below

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• The diode current then consists of a constant
  component I, due to the local oscillator, and a
  small component i, due to the input signal. The
  diode current is described by
  where Vd is the voltage drop across the
  diode and VTkT/q
• The input signal Vi causes a signal current 2i to
  flow through the load. Because of the circuit
  symmetry one-half flows through each diode. That
  is
• The currents are shown in Fig. 12.18. The
  voltage equations are
• adding the two equations we get
• and since the
  relation between input voltage and diode current
  is

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• This can be expanded for iltltI to
• The even order terms cancel out so
• Since the first term of the power series is not
  zero, the series can be inverted

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• Square-law Mixers
• The square-law characteristic is approximated by
  several electronic devices which square the sum
  of two sine waves
• An ideal square-law device will provide the upper
  and lower sidebands, together with a dc component
  and the second harmonic of both input waveforms.
  The circuit is frequently used at microwave
  frequencies for down conversion to the lower
  side-band, which is at a lower frequency than
  either of the input signals. A simple square law
  mixer is shown below

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• Schottky barrier diodes are typically used for
  high speed applications.
• At lower frequencies this form of the diode
  mixing is normally not used because of the large
  conversion loss. Transistors mixers are
  preferred because they can provide conversion
  gain. Transistors are often used to approximate
  the square-law characteristic. The input and
  local oscillator signal voltages are applied to
  the transistor so that they effectively add to
  the dc bias voltage to produce the total
  gate-source of base-emitter voltage. The
  composite signal is then passed through the
  device nonlinearity to create the sum and
  difference frequencies.
• BJT Mixers
• This is illustrated
• in the figure

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• The base to emitter voltage is
  where VDC is the base-to-emitter
  bias voltage. The collector current in a bipolar
  transistor is described by (Vbe gt 0)
• If
  then the current can be expanded in a
  series of modified Bessel functions as
• where
  and In is the nth-order modified Bessel function.
• The collector current consists of a dc component
  IC, components at both the input and oscillator
  frequencies, components at the frequencies
  , and an infinite number of
  high-frequency components. The amplitude of
  either the upper or lower-sideband component is

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• The local oscillator voltage amplitude is
  constant and V2gtgtV1, then the collector direct
  current will not vary with changes in the
  amplitude of the input signal since
  .
• The mixer should have a linear response to
  changes in the amplitude of the input amplitude.
  The ratio is given as
  .
• So if the input amplitude is sufficiently small
  the mixer upper- and lower-sideband outputs will
  be a linear function of the input signal. For
  ylt0.4 (V1lt10.5 mV) the response will be within 2
  percent of a linear response. The amplitude of
  the sideband current is
  

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FET Mixers

• If an FET is operated in its constant-current
  region, the idealized FET current transfer
  characteristics is the square-law relation
• where Vgs is
  the gate-to-source voltage and Vp is
• the transistor
  pinch-off voltage. Because of the square-law
  characteristic, the FET will not generate any
  harmonics higher than second-order
  intermodulation distortion. However, in reality,
  the transfer characteristic deviates from the
  idealized version, version and some
  intermodulation distortion will be produced.
  Still, a properly biased and operated FET mixer
  will produce much smaller high-order mixing
  products than a bipolar transistor. This is one
  reason why an FET is usually preferred to a
  bipolar transistor mixer.
• The FET also provides at least 10 times as great
  an input voltage range as the BJT. The following
  figure illustrates an FET mixer circuit. The
  drain current is

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• where VDC is the gate-to-source bias voltage (or
  VGS-VT for a MOSFET). If the applied signals are
  sine waves
• 
  then the output current is
• The amplitude of the sum and difference
  frequencies is
• where K3/Vi is referred to as the conversion
  transconductance gc. In general the device with
  the lowest pinch-off voltage has the highest
  gain, and the conversion transconductance is
  directly proportional to the amplitude of the
  local oscillator signal.

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• It would also appear that FETs with high IDSS are
  preferred, but IDSS and Vp are related. It is
  usually the case that devices selected for high
  IDSS also have a high Vp and a lower conversion
  transconductance that low- IDSS devices. Since
  the device is to be operated in the
  constant-current region, VL must be less than the
  magnitude of the pinch off volgate. If
  then K3Vi IDSS/2Vp and the sideband
  current is
• 
  
• Since for a JFET the transconductance is
• The conversion transconductance is one-fourth the
  small-signal tansconductance evaluated at Vgs0
  (provided VLVp/2). For a MOSFET it can be shown
  that the conversion conductance cannot exceed 1/2
  of the transconductance of the device when it is
  used as a small-signal amplifier.
• Although the conversion transconductance is
  smaller than the small-signal transconductance,
  it is large enough that the circuit can be
  operated as a mixer with power and voltage gains.
  This is an important difference from the
  diode-switching mixer.

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• An FET mixer is capable of producing lower
  intermodulation and harmonic products than a
  comparable bipolar or diode mixer. Also, an FET
  mixer operating a high level has a larger dynamic
  range and greater signal-handling capacity than a
  diode mixer operated at the same local oscillator
  level. However, the noise figure of FET mixers
  is currently higher than that of diode mixers.
  The best intermodulation and cross-modulation
  performance is obtained with the FET operated in
  the common-gate configuration, where the input
  impedance is much lower than that for the
  common-source configuration.
• The figure below illustrates double balanced
  mixer in which the FET transistors are operated
  in the common-gate configuration. The push pull
  output cancels the even-order output harmonics.

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• The dual-gate MOSFETs is often used as a mixer.
  A typical dual-gate MOSFET mixer circuit is shown
  below
• If the input signals are sinusoidal, the output
  will contain frequency components at both the sum
  and difference frequencies. Several other
  frequency components are also present in the
  output. The magnitude of either the sum or
  difference frequency is proportional to

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• so the conversion gain is proportional to the
  magnitude of the local oscillator voltage. For
  maximum conversion gain, the local oscillator
  amplitude should be selected so that it drives
  the gate just to the point of transistor
  saturation.
• The input signal is normally connected to the
  lower (closest to the ground) input gate terminal
  and the local oscillator signal to the upper
  gate. If the input is connected to the upper
  terminal, then the drain resistance of the lower
  transistor section appears as a source resistance
  to the input signal. The source resistance will
  reduce the voltage gain at the collector. Also,
  the connection has a larger drain-to-gate
  capacitance with a lower bandwidth than is
  attainable when the input signal is connected to
  the lower gate. The device is usually biased so
  that both transistors are operating in their
  nonsaturated region.
• The small-signal drain current is
• The drain current can be written as
• Since the drain current contains the product of
  the 2 signals, the dual-gate MOSFET can be used
  as a mixer when both transistors are operated in
  the linear region.

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• Amplitude and Phase Modulation and Demodulation
• Amplitude modulation (AM) is the process of
  varying the amplitude of a constant frequency
  signal with a modulating signal. An
  amplitude-modulated wave can be mathematically
  expressed as
• where g(t) is the modulating signal and ?c is
  the carrier frequency. Normally the modulating
  signal varies slowly compared with the carrier
  signal frequency. Conventional AM is in the form
  of
• 
  where m is the modulation factor and is normally
  less than 1. Consider a simple modulating
  signal
• The frequency spectrum of the modulated signal is
  shown

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• The equation above shows that for mlt1 the
  amplitude of the carrier is at least twice as
  large as the amplitude of either sideband
  component, so at least 2/3 of the signal power
  will be in the carrier and at most 1/3 in the 2
  sidebands. Because the carrier does not contain
  any information, it is often removed or
  suppressed in the signal
• which is referred to as a double-sideband (DSB)
  suppressed-carrier signal. The carrier component
  is not present in the DSB signal. However, as
  the waveform gets more efficient in terms of
  power-to-information content, the detection
  method gets more complex. Some means of
  recovering the carrier component is needed for
  the detector to recover the amplitude and
  frequency of the modulating signal. The DSB
  signal, although more efficient in terms of
  transmitted power, still occupies the same
  bandwidth as a normal AM signal. Since both
  sidebands contain the same information, one
  sideband can be removed, resulting in a
  single-sideband-signal (SSB).

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Amplitude Modulators

• Full-carrier double-sideband amplitude modulation
  is achieved either modulating the oscillator
  signal at a relatively low power level and
  amplifying the modulated signal with a cascade of
  amplifiers or by using the modulating signal to
  control the supply voltage o fthe power
  amplifier. Both methods are illustrated below

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• The power requirements of the modulator and
  modulating signal can be estimated by considering
  the power in an amplitude-modulated waveform
  . The peak
  power is
• so if the maximum modulation index is unity,
• The modulator must be designed to handle 4 times
  the average carrier power with 100 modulation
  the output power will be 4 times the carrier
  power.
• The diode mixer can be used to realize low-level
  modulation. If VL is a sine wave
  and if a low-pass filter is added to the
  output with a bandwidth of
  then the output will be
• .
  Since the low-pass filter removes the
• higher-frequency component, the modulation index
  of the resulting AM waveform is
  . This particular amplitude modulator
  functions well only for low indices of
  modulation.
• Both FET and BJT mixers can function as amplitude
  modulators with a relatively high modulation
  index. The final amplifier will need to be
  linear. The output will then be linearly related
  to the input provided

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• the amplifier output circuit is not
  current-limited.
• The most frequently used method of amplitude
  modulation at high power levels is to modulate
  the supply voltage to the power amplifier, as
  shown in Fig. 12-27b. In the figure below
• the modulating signal is applied in series with
  the dc supply voltage, so the total low-frequency
  supply for the transistor is

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• For Class C power amplifiers the amplitude of the
  output signal under saturation-limited conditions
  equals the power supply voltage. Therefor
  changing the transistor supply voltage modulates
  the output signal amplitude proportionally, and
  the output voltage becomes
• 
  . For 100 modulation the peak value of the
  voltage Vm(t) must equal VCC. The total output
  power is
• . The unmodulated
  carrier power is supplied by the power supply.
  The remaining power must be furnished by the
  modulator. One reason that output modulation has
  been the most frequently used method is that
  collector modulation results in less
  intermodulation distortion.
• All the information in an AM wave is contained in
  one sideband. It is possible to eliminate the
  other sideband without loss of information thus
  the required transmitter power is reduced to
  one-third of that previously required.
• The simplest method of SSB generation is to
  generate the DSB signal using a double-balanced
  modulator and then remove one of the

32

• sidebands with a filter. A block diagram of
  this form of SSB is shown below
• Another technique know as phasing method is shown
  below

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• Here both the modulating signal and the carrier
  signal are processed through phase splitters,
  which each generate two signals 90? out of phase
  with each other. The summing network output
• is the desired SSBsignal. The phasing method
  has the advantage of not requiring the sharp
  cutoff filters of the filtering method of
  SSBgeneration, but it is difficult to realize a
  broadband phase-shifting network for the lower
  frequency modulating signal.
• Demodulators
• AM detection can be divided into synchronous and
  asynchronous detection. Synchronous detection
  employs a time-varying or nonlinear element
  synchronized with the incoming carrier
  frequency. Otherwise the detection is
  asynchronous. The simplest asynchronous
  detector, the average envelope detector, is
  described below

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• Average Envelope Detectors
• A block diagram of the average envelope detector
  is shown in the fig.
• The rectifier output
• can be written as
• If S(t) is periodic with a frequency ?c, since
• If S(t) is the AM wave described by

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• If the low-pass filter bandwidth is chosen to
  filter out the component at ?c and all higher
  harmonics, the output will be
• which is a dc term plus the modulating
  information.
• Two additional points will be made to further
  describe the operation of the envelope detector.
  First, consider the case where
• The
• The output will contain a term at the frequency
  , which must also be removed by
  the low-pass filter. This is not possible if ?m
  is close to ?c. To ensure this distortion does
  not occur the max modulating frequency should be
  and the corresponding
• low-pass filter bandwidth B must be selected so
  that
• This is only possible if m is not greater than 1,
  and the carrier term is present. Average
  envelope detection will only work for normal AM
  with a modulation index less than 1. However, if
  a large carrier component Acos?ct is added to the
  SSB signal, the resultant signal can also be
  detected with an envelope detector.

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• A simple diode envelope detector circuit is shown
  in the figure below
• It is assumed here that the input signal
  amplitude is large enough that the diode can be
  considered either on or off, depending upon the
  input signal polarity. The diode can then be
  replaced by a open circuit when it is
  reverse-biased and by a constant resistance when
  it is forward-biased. The series capacitor Cc is
  included to remove the dc component. The purpose
  of the load capacitor C in the circuit is to
  eliminate the high-frequency component from the
  output and to increase the average value of the
  output voltage. The effect of the load capacitor
  can be seen from the figure below

37

• which illustrates the input and the output signal
  waveforms of a diode detector. As the input
  signal is applied, the capacitor charges up until
  the input waveform begins to decrease. At this
  time the diode becomes open-circuited and the
  capacitor discharges through the load resistance
  RL as where
  Vp is the peak value of the input signal, and the
  diode opens at time t0. The larger the value of
  capacitance used, the smaller will be the output
  ripple. However, C cannot be too large or it
  will not be able to follow the changes in the
  modulated signal. The time constant is often
  selected as

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Angle Modulation

• Information can also be transmitted by modulating
  the phase frequency. Angle modulation occupies a
  wider bandwidth, but it can provide better
  discrimination against noise and other
  interfering signals. An angle-modulated waveform
  can be written as
• 
  where ?(t) representing the angle modulation.
  Angle modulation can be further subdivided into
  phase and frequency modulation, depending on
  whether it is the phase or the derivative of
  phase that is modulated. Frequency modulation
  and phase modulation are not distinct, since
  changing the frequency will result in a change in
  phase and modulating the phase also modulates the
  frequency.
• Angle Modulators
• Frequency modulation can be achieved directly by
  modulating a VCO (direct FM) or indirectly by
  phase-modulating the RF waveform by the
  integrated audio input signal (indirect FM).
  Another method of FM is to use a
  phase-locked-loop as shown below

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• The output in response to the modulating signal
  Vm is
• where Kd is the phase-detector gain constant and
  Ko is the VCO sensitivity (Hertz per volt). In
  the steady state, the output phase will be
  proportional to the modulating voltage. So the
  PLL can serve either as a phase modulator or, if
  VM is the integral of the modulating signal of
  interest, as a frequency modulator.

40

• FM Demodulators
• The same type of circuitry is used for detecting
  both types of angle modulation, and we will refer
  to either process as FM detection. FM detectors
  are often referred to as frequency
  discriminators.
• The ideal FM detector produces an output voltage
  that changes linearly with changes in the input
  frequency as shown
• The output voltage is usually 0 at the carrier
  frequency. Any deviation from the linear
  characteristic distorts the detected waveform.
  Amplitude modulation caused by noise can also
  cause distortion in the recovered signal.
  Limiting circuitry is usually included in FM
  detector to reduce the amount of amplitude
  modulation. The transfer characteristic of an
  ideal

41

• limiter is shown below

The limiter output is restricted to the values
that depend only on the sign of the input. A
single stage differential-pair limiter is shown
42

• The circuit gives a close approximation to the
  ideal limiter characteristics. If the input
  signal is too small, several differential-pair
  stages may be cascaded in order for the output to
  be saturated. Integrated-circuit limiters
  frequently contain 3 cascaded stages.
• An analytical basis of FM detection is obtained
  by considering the derivative of the FM signal
• The derivative of an angle-modulated signal is an
  amplitude-modulated FM waveform. All the
  modulating information is contained in the
  amplitude of the differentiated waveform.
  Normally if so the
  amplitude modulation can be removed with an
  envelope detector. The output of the envelope
  detector will be proportional to
• , which is
  for a frequency-modulated waveform. If
  the output is then high-pass filtered to remove
  the constant term , the remainder will be
  proportional to the modulating signal. This
  technique has the disadvantage that any dc
  components in the modulating signal is lost.

43

• The most often used circuit for realizing the
  differentiator is the single-tuned circuit. The
  frequency response of an ideal differentiator
• has a 90 phase shift, and the magnitude
  increases with increasing frequency at 6 db per
  octave. The frequency response of a simple tuned
  circuit will approximate this response at
  frequencies sufficiently below the circuits
  resonant frequency.
• The frequency response magnitude of the parallel
  tuned circuit is
• Values for Q and ?0 for a parallel tuned circuit
  are, in which Rp, C and L and parallel to each
  other

44

• The magnitude of the frequency response of the
  parallel resonant circuit is shown below
• At frequency
• provided ?c is close enough to ?0 so that

45

• Also if
• The output consists of a constant term
  corresponding to ?c plus a component proportional
  to the frequency deviation ??. Balanced
  discriminators are often used to eliminate the
  constant term. A simplified balanced
  discriminator is illustrated below
• output is then
• which is proportional to the frequency deviation
  from the carrier frequency.

The upper resonant cirucit is tuned to the
frequency ?0- ?c, and the output is proportional
to ?c- ??. The differential