# Amplitude Frequency Modulation - H Spectrum Analysis 150-1 Amplitude and Frequency Modulation...

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Test & MeasurementApplication Note 150-1

H

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Spectrum Analysis

Amplitude&FrequencyModulation

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HSpectrum Analysis 150-1Amplitude and Frequency Modulation

eAcos(t+ )=Chapter 1 Modulation Methods

Chapter 2 Amplitude Modulation Modulation Degree and Sideband Amplitude Zero Span and Markers The Fast Fourier Transform (FFT) Special Forms of Amplitude ModulationSingle Sideband

Chapter 3 Angular Modulation Definitions Bandwidth of FM Signals FM Measurements with the Spectrum Analyzer AM Plus FM (Incidental FM)

Appendix A Mathematics of Modulation

Appendix B Frequency Modulation in Broadcast Radio

Relevant Products and Training

Contacting Hewlett-Packard Worldwide

ContentsSpectrum Analysis AM and FM

Copyright Hewlett-Packard Company, 1989, 1996. 3000 Hanover Street, Palo Alto California, USA.2

3

HSpectrum Analysis 150-1Amplitude and Frequency Modulation

Modulation is the act of translating some low-frequency or base-band signal (voice, music, data) to a higher frequency. Why do wemodulate signals? There are at least two reasons: to allow the simul-taneous transmission of two or more baseband signals by translatingthem to different frequencies, and to take advantage of the greaterefficiency and smaller size of higher-frequency antennas.

In the modulation process, some characteristic of a high-frequencysinusoidal carrier is changed in direct proportion to the instan-taneous amplitude of the baseband signal. The carrier itself can bedescribed by the equation

e(t) = A cos (t + )where: A = peak amplitude of the carrier,

= angular frequency of the carrier in radians per second,t = time, and = initial phase of the carrier at time t = 0.

In the expression above there are two properties of the carrier thatcan be changed, the amplitude (A) and the angular position(argument of the cosine function). Thus we have amplitude mod-ulation and angular modulation. Angular modulation can be furthercharacterized as either frequency modulation or phase modulation.

1Modulation MethodsMilestones.In 1864 James ClarkMaxwell predicted theexistence of electro-magnetic waves thattravel at the speed oflight. In Germany,Heinrich Hertz provedMaxwell's theory andin 1888 was the first tocreate electromagneticradio waves.

Amplitude modulation of a sine or cosine carrier results in avariation of the carrier amplitude that is proportional to theamplitude of the modulating signal. In the time domain(amplitude versus time), the amplitude modulation of onesinusoidal carrier by another sinusoid resembles Figure 1A.The mathematical expression for this complex wave showsthat it is the sum of three sinusoids of differentfrequencies.

One of these sinusoids has the same frequency andamplitude as the unmodulated carrier. The second sinusoidis at a frequency equal to the sum of the carrier frequencyand the modulation frequency; this component is the uppersideband. The third sinusoid is at a frequency equal to thecarrier frequency minus the modulation frequency; thiscomponent is the lower sideband. The two sidebandcomponents have equal amplitudes, which are proportionalto the amplitude of the modulating signal. Figure 1B showsthe carrier and sideband components of the amplitude-modulated wave of Figure 1A as they appear in thefrequency domain (amplitude versus frequency).

4

HSpectrum Analysis 150-1Amplitude and Frequency Modulation2Amplitude ModulationModulation Degree and Sideband Amplitude

Figure 1A.Time domain displayof an amplitudemodulated carrier.

Figure 1B.Frequency domain(spectrum analyzer)display of an amplitude-modulated carrier.

fcfcfm fc+fm

USBLSB

Am

plit

ud

e(V

olt

s)

t

Am

plit

ud

e(V

olt

s)

5

HSpectrum Analysis 150-1Amplitude and Frequency Modulation

A measure of the amount of modulation is m, the degree ofmodulation. This is usually expressed as a percentage calledthe percent modulation. In the time domain, the degree ofmodulation for sinusoidal modulation is calculated as follows,using variables shown in Figure 2:

Since the modulation is symmetrical,

and

.

From this it is easy to show that

for sinusoidal modulation.

m

E EE E

= +

max min

max min

E EEc

max min+ =2

E E E Ec cmax min =

m

E EE

c

c= max

2Amplitude ModulationModulation Degree and Sideband AmplitudeFigure 2.Calculation of degree of amplitudemodulation from timedomain display.

Emin

EcEmax

6

HSpectrum Analysis 150-1Amplitude and Frequency Modulation

When all three components of the modulation signal are inphase, they add together linearly and form the maximum signal amplitude Emax, as shown in Animation 1.

and, since EUSB = ELSB = ESB, then

For 100% modulation (m=1.0), the amplitude of each sidebandwill be one-half of the carrier amplitude (voltage). Thus, eachsideband will be 6 dB less than the carrier, or one-fourth thepower of the carrier. Since the carrier component does notchange with amplitude modulation, the total power in the 100%modulated wave is 50% higher than in the unmodulated carrier.

m

EE

SB

c= 2

m

E EE

E EE

c

c

USB LSB

c= = +max

E E E Ec USB LSBmax = + +

2Amplitude ModulationModulation Degree and Sideband AmplitudeAnimation 1.Calculation of degree ofamplitude modulationdisplayed in both timeand frequency domain.Click over the chart toactivate the animation.

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Animation 1.

f cf - f c m f + f c m

Ec

E lsb Eusb

m = 0.5

E min

E c

E max

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HSpectrum Analysis 150-1Amplitude and Frequency Modulation

Although it is easy to calculate the modulationpercentage M from a linear presentation in thefrequency or time domain (M = m 100%), thelogarithmic display on a spectrum analyzer offerssome advantages, especially at low modulationpercentages. The wide dynamic range of a spectrumanalyzer (over 70dB) allows measurement ofmodulation percentages as low as 0.06%. This caneasily be seen in Figure 3, where M=2%; that is, wherethe sideband amplitudes are only 1% of the carrieramplitude.

Figure 3A shows a time domain display of anamplitude-modulated carrier with M=2%. It is difficultto measure M on this display. Figure 3B shows thesignal displayed logarithmically in the frequencydomain. The sideband amplitudes can easily bemeasured in dB below the carrier and then convertedinto M. (The vertical scale is 10 dB per division.)

2Amplitude ModulationModulation Degree and Sideband Amplitude Figure 3A.Time and frequencydomain views of lowlevel (2%) AM.

Time Domain

Frequency Domain

Figure 3B.

M%HSpectrum Analysis 150-1Amplitude and Frequency Modulation

The relationship between m and thelogarithmic display can be expressed as

or

Figure 4 shows the modulation percentageM as a function of the difference in dBbetween a carrier and either sideband.

E dB E dB dB mSB c( ) ( ) log + =6 20

E dB E dB

mSB c( ) ( ) log = 20 2

2Amplitude ModulationModulation Degree and Sideband Amplitude

0 -10 -20 -30 -40 -70 0.01

0.10

1.00

10.00

100.00

-50 -60

M[ %

]

Esb Ec [dB]

Figure 4. Modulationpercentage M vs. sideband level(log display).

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HSpectrum Analysis 150-1Amplitude and Frequency Modulation

Figures 5 and 6 show typical displays of a carriermodulated by a sine wave at different modulation levelsin the time and frequency domains.

Figure 5A shows an amplitude-modulated carrier in the time domain. The percent modulation is M=(6-2)/(6+2) = 4/8 = 50%. (Scope calibration is 0.1 msec/division, 50mV/division.)

Figure 5B shows the same waveform measured in thefrequency domain. Since the carrier and sidebands differby 12 dB, M=50%. Frequency span is 10 kHz/division,centered at 60 MHz, and the log reference level is +10 dBm. You can also measure 2nd and 3rd harmonicdistortion on this waveform. Second harmonic sidebandsat are 55 dB down. f fc m 2

2Amplitude ModulationModulation Degree and Sideband AmplitudeFigure 5A.An amplitude-modulated carrier inthe time domain.

Time Domain

Frequency Domain

Figure 5B. The same waveformmeasured in thefrequency domain.

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Figure 6A shows an overmodulated (M>100%)60 MHz signal in the time domain; fm = 2 kHz(0.2 msec/Div, 50 mV/Div). The carrier is cut off at the modulation minima.

Figure 6B is the frequency domain display ofthe signal. Note that the first sideband pair isonly 6 dB lower than the carrier. Also, theoccupied bandwidth is much greater becauseof severe distortion of the modulating signal.(50 kHz span, 10 dB/Div, BW 100 Hz.)

HSpectrum Analysis 150-1Amplitude and Frequency Modulation2Amplitude ModulationModulation Degree and Sideband Amplitude

Figure 6B.The frequency domaindisplay of the signal.

Figure 6A.An over-modulated60MHz signal.

Time Domain

Frequency Domain

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HSpectrum Analysis 150-1Amplitude and Frequency Modulation

AM Signal Animations The effects of varying two of theadjustable parameters in amplitude modulation, thedegree of modulation m and the modulation frequencyfm, are demonstrated in Animations 2 and 3 in thetime domain.

The amplitude of the envelope of the modulatedsignal varies linearly with m. This is demonstrated inAnimation 2.

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