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16.4 Modulators and Demodulators

Back

Sometimes signals are deliberately modified to maintain the accuracy during signal transmission,

conditioning, and processing. In signal modulation, the data signal, known as the modulating signal, is

used to vary a property (such as amplitude or frequency) of a carrier signal. We say that the carrier signal

is modulated by the data signal. After transmitting or conditioning the modulated signal, the data signal

is usually recovered by removing the carrier signal. This is known as demodulation or discrimination.

Many modulation techniques exist, and several other types of signal modification (e.g., digitizing)

could be classified as signal modulation even though they might not be commonly termed as such. Four

types of modulation are illustrated in Figure 16.12. In amplitude modulation (AM), the amplitude of a

periodic carrier signal is varied according to the amplitude of the data signal (modulating signal),

frequency of the carrier signal (carrier frequency) being kept constant. Suppose that the transient signal

Box 16.2

FILTERS

Active Filters (Need External Power)

Advantages:

* Smaller loading errors (have high input impedance and low output impedance, and hence

do not affect the input circuit conditions and output signals)

* Lower cost

* Better accuracy

Passive Filters (No External Power, Use Passive Elements)

Advantages:

* Useable at very high frequencies (e.g., radio frequency)

* No need for a power supply

Filter Types

* Low pass: Allows frequency components up to cutoff and rejects the higher frequency

components

* High pass: Rejects frequency components up to cutoff and allows the higher frequency

components

* Band pass: Allows frequency components within an interval and rejects the rest

* Notch (or band reject): Rejects frequency components within an interval (usually narrow)

and allows the rest

Definitions

* Filter order: Number of poles in the filter circuit or transfer function

* Antialiasing filter: Low-pass filter with cutoff at less than half the sampling rate (i.e., Nyquist

frequency), for digital processing

* Butterworth filter: A high-order filter with a very flat pass band

* Chebyshev filter: An optimal filter with uniform ripples in the pass band

* Sallen-Key filter: An active filter whose output is in phase with input

Signal Conditioning and Modification 16-29

(a)

(b)

(c)

Time t

(d) t

(e)

FIGURE 16.12 (a) Modulating signal (data signal); (b) amplitude-modulated (AM) signal; (c) frequencymodulated

(FM) signal; (d) pulse-width-modulated (PWM) signal; (e) pulse-frequency-modulated (PFM) signal.

16-30 Vibration and Shock Handbook

shown in Figure 16.12(a) is used as the modulating signal. A high-frequency sinusoidal signal is used as

the carrier signal. The resulting amplitude-modulated signal is shown in Figure 16.12(b). Amplitude

modulation is used in telecommunications, radio and TV signal transmission, instrumentation, and

signal conditioning. The underlying principle is useful in other applications such as fault detection and

diagnosis in rotating machinery.

In frequency modulation (FM), the frequency of the carrier signal is varied in proportion to the

amplitude of the data signal (modulating signal), while the amplitude of the carrier signal is kept

constant. If the data signal shown in Figure 16.12(a) is used to frequency modulate a sinusoidal carrier

signal, then the result will appear as shown in Figure 16.12(c). Since information is carried as frequency

rather than amplitude, any noise that might alter the signal amplitude will have virtually no effect on the

transmitted data. Hence, FM is less susceptible to noise than AM. Furthermore, since the carrier

amplitude is kept constant in FM, signal weakening and noise effects that are unavoidable in longdistance

data communication will have less effect than in the case of AM, particularly if the data signal

level is low in the beginning. However, more sophisticated techniques and hardware are needed for signal

recovery (demodulation) in FM transmission, because FM demodulation involves frequency

discrimination rather than amplitude detection. Frequency modulation is also widely used in radio

transmission and in data recording and replay.

In pulse-width modulation (PWM), the carrier signal is a pulse sequence. The pulse width is changed in

proportion to the amplitude of the data signal, while keeping the pulse spacing constant. This is

illustrated in Figure 16.12(d). Pulse-width modulated signals are extensively used in controlling electric

motors and other mechanical devices such as valves (hydraulic, pneumatic) and machine tools. Note

that, in a given short time interval, the average value of the pulse-width modulated signal is an estimate of

the average value of the data signal in that period. Hence, PWM signals can be used directly in controlling

a process without one having to demodulate it. Advantages of PWM include better energy efficiency (less

dissipation) and better performance with nonlinear devices. For example, a device may stick at low

speeds due to Coulomb friction. This can be avoided by using a PWM signal that provides the signal

amplitude that is necessary to overcome friction while maintaining the required average control signal,

which might be very small.

In pulse-frequency modulation (PFM) as well, the carrier signal is a pulse sequence. In this method,

the frequency of the pulses is changed in proportion to the data signal level, while the pulse width is

kept constant. PFM has the advantage of ordinary frequency modulation. Additional

advantages result due to the fact that electronic circuits (digital circuits, in particular) can handle

pulses very efficiently. Furthermore, pulse detection is not susceptible to noise because it involves

distinguishing between the presence and absence of a pulse rather than accurate determination of

the pulse amplitude (or width). PFM may be used in place of PWM in most applications with

better results.

Another type of modulation is phase modulation (PM). In this method, the phase angle of the carrier

signal is varied in proportion to the amplitude of the data signal.

Conversion of discrete (sampled) data into the digital (binary) form is also considered to be

modulation. In fact, this is termed pulse-code modulation (PCM). In this case, each discrete data sample is

represented by a binary number containing a fixed number of binary digits (bits). Since each digit in the

binary number can take only two values, 0 or 1, it can be represented by the absence or presence of a

voltage pulse. Hence, each data sample can be transmitted using a set of pulses. This is known as

encoding. At the receiver, the pulses have to be interpreted (or decoded) in order to determine the data

value. As with any other pulse technique, PCM is quite immune to noise because decoding involves

detection of the presence or absence of a pulse rather than determination of the exact magnitude of the

pulse signal level. Also, since pulse amplitude is constant, long-distance signal transmission (of this

digital data) can be accomplished without the danger of signal weakening and associated distortion. Of

course, there will be some error introduced by the digitization process itself, which is governed by the

finite word size (or dynamic range) of the binary data element. This is known as quantization error and is

unavoidable in signal digitization.

Signal Conditioning and Modification 16-31

In any type of signal modulation, it is essential to preserve the algebraic sign of the modulating signal

(data). Different types of modulators handle this in different ways. For example, in PCM an extra sign bit

is added to represent the sign of the transmitted data sample. In AM and FM, a phase-sensitive

demodulator is used to extract the original (modulating) signal with the correct algebraic sign. Note that,

in these two modulation techniques, a sign change in the modulating signal can be represented by a 1808

phase change in the modulated signal. This is not noticeable in Figure 16.12(b) and (c). In PWM and

PFM, a sign change in the modulating signal can be represented by changing the sign of the pulses, as

shown in Figure 16.12(d) and (e). In PM, a positive range of phase angles (say 0 to p) can be assigned for

the positive values of the data signal and a negative range of phase angles (say 2p to 0) can be assigned

for the negative values of the signal.

16.4.1 Amplitude Modulation

Amplitude modulation can naturally enter into many physical phenomena. More important, perhaps, is

the deliberate (artificial) use of AM to facilitate data transmission and signal conditioning. Let us first

examine the related mathematics.

Amplitude modulation is achieved by multiplying the data signal (modulating signal), xðtÞ; by a high

frequency (periodic) carrier signal, xcðtÞ: Hence, amplitude-modulated signal, xaðtÞ; is given by

xaðtÞ ¼ xðtÞxcðtÞ ð16:57Þ

Note that the carrier could be any periodic signal such as one which is harmonic (sinusoidal), square

wave, or triangular. The main requirement is that the fundamental frequency of the carrier signal (carrier

frequency), fc; be significantly larger (say, by a factor of five or ten) than the highest frequency of interest

(bandwidth) of the data signal. Analysis can be simplified by assuming a sinusoidal carrier frequency;

thus

xcðtÞ ¼ ac cos 2p fct ð16:58Þ

16.4.1.1 Modulation Theorem

Modulation theorem is also known as the frequency-shifting theorem, and it relates the fact that if a signal

is multiplied by a sinusoidal signal, the Fourier spectrum of the product signal is simply the Fourier

spectrum of the original signal shifted through the frequency of the sinusoidal signal. In other words, the

Fourier spectrum, Xaðf Þ; of the amplitude-modulated signal, xaðtÞ; can be obtained from the Fourier

spectrum, Xðf Þ; of the data signal, xðtÞ; simply by shifting through the carrier frequency, fc:

To mathematically explain the modulation theorem, we use the definition of the Fourier integral

transform to obtain

Xaðf Þ ¼ ac

ð1

xðtÞ cos 2p fct expð2j2p ftÞdt

However, since

cos 2p fct ¼

2 ½expð j2p fctÞ þ expð2j2p fctÞ􀀉

we have

Xað f Þ ¼

ð1

xðtÞ exp½2j2pðf 2 fcÞt􀀉dt þ

ð1

xðtÞ exp½2j2p ð f þ fcÞt􀀉dt

Xað f Þ ¼

ac½Xð f 2 fcÞ þ Xð f þ fcÞ􀀉 ð16:59Þ

Equation 16.59 is the mathematical statement of the modulation theorem. It is illustrated by an example

in Figure 16.13. Consider a transient signal, xðtÞ; with a (continuous) Fourier spectrum, Xðf Þ; whose

magnitude, lXðf Þl; is as shown in Figure 16.13(a). If this signal is used to modulate the AM of a

high-frequency sinusoidal signal, the resulting modulated signal, xaðtÞ; and the magnitude of its

16-32 Vibration and Shock Handbook

Fourier spectrum are as shown in Figure 16.13(b). It should be kept in mind that the magnitude has been

multiplied by ac=2: Note that the data signal is assumed to be band limited, with bandwidth fb: Of course,

the theorem is not limited to band-limited signals but, for practical reasons, we need to have some upper

limit on the useful frequency of the data signal. Also for practical reasons (not for the theorem itself), the

carrier frequency, fc; should be several times larger than fb so that there is a reasonably wide frequency

band from 0 to ðfc 2 fbÞ, within which the magnitude of the modulated signal is virtually zero. The

significance of this should be clear when we discuss applications of amplitude modulation.

Figure 16.13 shows only the magnitude of the frequency spectra. It should be remembered, however,

that every Fourier spectrum has a phase angle spectrum as well. This is not shown for conciseness, but

clearly the phase-angle spectrum is also similarly affected (frequency shifted) by AM.

16.4.1.2 Side Frequencies and Side Bands

The modulation theorem, as described above, assumed transient data signals with associated continuous

Fourier spectra. The same ideas are applicable to periodic signals (with discrete spectra) as well. The case

(a)

(b)

(c)

(d)

Time t

x(t)

Frequency f

X( f )

Xa( f )

−fb fb

xa(t) = x(t)ac cos 2π fc t

−fc−fb fc

Mac

−fc −fc+fb fc−fb fc+fb

Xa( f )

x(t) = a cos 2πfot

f f −fo 0 o

xa(t) = aac cos 2πfot cos 2πfct

f f −fc 0 c

aac

−fc−fo −fc+fo fc−fo fc+fo

FIGURE 16.13 Illustration of the modulation theorem: (a) a transient data signal and its Fourier spectrum

magnitude; (b) amplitude-modulated signal and its Fourier spectrum magnitude; (c) a sinusoidal data signal;

(d) amplitude modulation by a sinusoidal signal.

Signal Conditioning and Modification 16-33

of periodic signals is merely a special case of what was discussed above. This case can be analyzed by using

Fourier integral transform itself, from the beginning. If this method is chosen, however, we will have to

cope with impulsive spectral lines. Alternatively, Fourier series expansion could be employed to avoid the

introduction of impulsive discrete spectra into the analysis. However, as shown in Figure 16.13(c) and

(d), no analysis is actually needed for the periodic signal case because a final answer can be deduced from

the transient signal results. Specifically, each frequency component, fo; that has amplitude a/2 in the

Fourier series expansion of the data signal will be shifted by ^fc to the two new frequency locations

fc þ fo and 2fc þ fo with an associated amplitude aac=4: The negative frequency component 2fo should

also be considered in the same way, as illustrated in Figure 16.13(d). Note that the modulated signal does

not have a spectral component at carrier frequency, fc; but rather on each side of it, at fc ^ fo: Hence,

these spectral components are termed side frequencies. When a band of side frequencies is present, we

have a side band. Side frequencies are very useful in fault detection and diagnosis of rotating machinery.

16.4.2 Application of Amplitude Modulation

The main hardware component of an amplitude modulator is an analog multiplier. They are

commercially available in the monolithic IC form, or one can be assembled using IC opamps and

other discrete circuit elements. A schematic representation of an amplitude modulator is shown in

Figure 16.14. In practice, to achieve satisfactory modulation, other components such as signal

preamplifiers and filters are needed.

There are many applications of AM. In some applications, modulation is performed intentionally. In

others, modulation occurs naturally as a consequence of the physical process, and the resulting signal is

used to meet a practical objective. Typical applications of AM include the following:

1. Conditioning of general signals (including DC, transient, and low-frequency) by exploiting the

advantages of AC signal conditioning hardware

2. Improvement of the immunity of low-frequency signals to low-frequency noise

3. Transmission of general signals (DC, low-frequency, etc.) by exploiting the advantages of AC

signals

4. Transmission of low-level signals under noisy conditions

5. Transmission of several signals simultaneously through the same medium (e.g., same telephone

line, same transmission antenna, etc.)

6. Fault detection and diagnosis of rotating machinery

The role of AM in many of these applications should be obvious if one understands the frequencyshifting

property of AM. Several other types of application are also feasible due to the fact that the power

of the carrier signal can be increased somewhat arbitrarily, irrespective of the power level of the data

(modulating) signal. Let us discuss, one by one, the six categories of application mentioned above.

AC signal conditioning devices such as AC amplifiers are known to be more “stable” than their DC

counterparts. In particular, drift problems are not as severe and nonlinearity effects are lower in AC signal

conditioning devices. Hence, instead of conditioning a DC signal using DC hardware, we can first use the

Modulated

Signal

Carrier

Signal

Modulating

Input

(Data)

Multiplier

Out

FIGURE 16.14 Representation of an amplitude modulator.

16-34 Vibration and Shock Handbook

signal to modulate a high-frequency carrier signal. Then, the resulting high-frequency modulated signal

may be conditioned more effectively using AC hardware.

The frequency-shifting property of AM can be exploited in making low-frequency signals immune to

low-frequency noise. Note from Figure 16.13 that, by AM, low-frequency spectrum of the modulating

signal can be shifted into a very high-frequency region by choosing a sufficiently large carrier frequency, fc:

Then, any low-frequency noise (within the band 0 to fc 2 fb) will not distort the spectrum of the modulated

signal. Hence, this noise can be removed by a high-pass filter (with cutoff at fc 2 fb) without affecting the

data. Finally, the original data signal can be recovered by demodulation. Note that the frequency of a noise

component can be within the bandwidth, fb; of the data signal and, hence, if AM is not employed, noise can

directly distort the data signal.

Transmission of AC signals is more efficient than that of DC signals. Advantages of AC transmission

include lower problems with energy dissipation. Hence, a modulated signal can be transmitted over long

distances more effectively than could the original data signal alone. Furthermore, transmission of lowfrequency

(large wave-length) signals requires large antennas. Hence, when AM is employed (with an

associated reduction in signal wave length), the size of broadcast antenna can be effectively reduced.

Transmission of weak signals over long distances is not desirable because signal weakening and

corruption by noise could produce disastrous results. By increasing the power of the carrier signal to a

sufficiently high level, the strength of the modulated signal can be elevated to an adequate level for longdistance

transmission.

It is impossible to transmit two or more signals in the same frequency range simultaneously using a

single telephone line. This problem can be resolved by using carrier signals with significantly different

carrier frequencies to modulate the amplitude of the data signals. By choosing carrier frequencies that are

sufficiently farther apart, the spectra of the modulated signals can be made nonoverlapping, thereby

making simultaneous transmission possible. Similarly, with AM, simultaneous broadcasting by several

radio (AM) broadcast stations in the same broadcast area has become possible.

16.4.2.1 Fault Detection and Diagnosis

A use of the AM principle that is particularly important in the practice of mechanical vibration is in the

fault detection and diagnosis of rotating machinery. In this method, modulation is not deliberately

introduced, but rather results from the dynamics of the machine. Flaws and faults in a rotating machine

are known to produce periodic forcing signals at frequencies higher than, and typically at an integer

multiple of, the rotating speed of the machine. For example, backlash in a gear pair will generate forces at

the tooth-meshing frequency (equal to the number of teeth £ gear rotating speed). Flaws in roller

bearings can generate forcing signals at frequencies proportional to the rotating speed times the number

of rollers in the bearing race. Similarly, blade passing in turbines and compressors and eccentricity and

unbalance in rotors can produce forcing components at frequencies that are integer multiples of the

rotating speed. The resulting vibration response will be an amplitude-modulated signal, where the

rotating response of the machine modulates the high-frequency forcing response. This can be confirmed

experimentally by Fourier analysis (fast Fourier transform or FFT) of the resulting vibration signals. For a

gear box, for example, it will be noticed that, instead of obtaining a spectral peak at the gear toothmeshing

frequency, two side bands are produced around that frequency. Faults can be detected by

monitoring the evolution of these side bands. Furthermore, since side bands are the result of modulation

of a specific forcing phenomenon (e.g., gear-tooth meshing, bearing-roller hammer, turbine-blade

passing, imbalance, eccentricity, misalignment, etc.), one can trace the source of a particular fault (i.e.,

diagnose the fault) by studying the Fourier spectrum of the measured vibrations.

Amplitude modulation is an integral part of many types of sensors. In these sensors, a high-frequency

carrier signal (typically the AC excitation in a primary winding) is modulated by the motion. Actual

motion can be detected by demodulation of the output. Examples of sensors that generate modulated

outputs are differential transformers (LVDT, RVDT), magnetic-induction proximity sensors, eddycurrent

proximity sensors, AC tachometers, and strain-gage devices that use AC bridge circuits.

Signal Conditioning and Modification 16-35

Signal conditioning and transmission is facilitated by AM in these cases. However, the signal has to be

demodulated at the end for most practical purposes such as analysis and recording.

16.4.3 Demodulation

Demodulation, or discrimination or detection, is the process of extracting the original data signal from a

modulated signal. In general, demodulation must be phase sensitive in the sense that the algebraic sign of

the data signal should be preserved and determined by the demodulation process. In full-wave

demodulation, an output is generated continuously. In half-wave demodulation, no output is generated

for every alternate half-period of the carrier signal.

A simple and straightforward method of demodulation is by the detection of the envelope of the

modulated signal. For this method to be feasible, the carrier signal must be quite powerful (i.e., the signal

level has to be high) and the carrier frequency also should be very high. An alternative method of

demodulation that generally provides more reliable results involves the further step of modulation

performed on the already-modulated signal, followed by low-pass filtering. This method will be

explained by referring to Figure 16.13.

Consider the amplitude-modulated signal, xaðtÞ; shown in Figure 16.13(b). If this signal is multiplied

by the sinusoidal carrier signal, 2=ac cos 2pfct; we obtain

x~ðtÞ ¼

xaðtÞcos 2p fct ð16:60Þ

Now, by applying the modulation theorem (Equation 16.59) to Equation 16.60, we obtain the Fourier

spectrum of x~ðtÞ as

X~ ð f Þ ¼

ac{Xðf 2 2fcÞ þ Xð f Þ} þ

ac{Xð f Þ þ Xð f þ 2fcÞ}

􀀒 􀀓

X~ ðf Þ ¼ Xð f Þ þ

Xð f 2 2fcÞ þ

Xð f þ 2fcÞ ð16:61Þ

The magnitude of this spectrum is shown in Figure 16.15(a). Note that we have recovered the spectrum,

Xð f Þ; of the original data signal, except for the two side bands that are present at locations far removed

(centered at ^2fc) from the bandwidth of the original signal. Hence, we can easily use a low-pass filter on

this signal, x~ðtÞ; using a filter with cutoff at fb to recover the original data signal. A schematic

representation of this method of amplitude demodulation is shown in Figure 16.15(b).

Carrier

cos 2pfc t

(a)

(b)

−2 fc −fb 0 fb 2 fc Frequency f

Multiplier

Out

Modulated Signal

Low-Pass

Filter

Cutoff = fb

Original Signal

xa(t) x(t) x(t)

X (f)

FIGURE 16.15 Amplitude demodulation: (a) spectrum of the signal after the second modulation; (b) demodulation

schematic diagram (modulation þ filtering).

16-36 Vibration and Shock Handbook

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16.4 Modulators and Demodulators

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