Bandwidth of Angle-Modulated Signals

4.1 PHASE AND FREQUENCY MODULATION DEFINED

4.1.4 Bandwidth of Angle-Modulated Signals

Strictly speaking, the bandwidth of an angle-modulated signal is infinite, since angle modula- tion of a carrier results in the generation of an infinite number of sidebands. However, it can be seen from the series expansion of𝐽𝑛(𝛽)(Appendix F, Table F.3) that for large𝑛

𝐽𝑛(𝛽) ≈ 𝛽𝑛

2𝑛𝑛! (4.40)

Thus, for fixed𝛽,

𝑛→lim∞𝐽𝑛(𝛽) = 0 (4.41)

This behavior can also be seen from the values of𝐽𝑛(𝛽)given in Table 4.1. Since the values of𝐽𝑛(𝛽)become negligible for sufficiently large𝑛, the bandwidth of an angle-modulated signal can be defined by considering only those terms that contain significant power. The power ratio

𝑃𝑟is defined as the ratio of the power contained in the carrier(𝑛 = 0)component and the𝑘 components on each side of the carrier to the total power in𝑥𝑐(𝑡). Thus,

𝑃𝑟=

1 2𝐴2𝑐∑𝑘

𝑛=−𝑘𝐽𝑛2(𝛽)

1

2𝐴2𝑐 =

∑𝑘 𝑛=−𝑘

𝐽𝑛2(𝛽) (4.42)

or simply

𝑃𝑟= 𝐽02(𝛽) + 2∑𝑘

𝑛=1

𝐽𝑛2(𝛽) (4.43)

Bandwidth for a particular application is often determined by defining an acceptable power ratio, solving for the required value of𝑘using a table of Bessel functions, and then recognizing that the resulting bandwidth is

𝐵 = 2𝑘𝑓𝑚 (4.44)

The acceptable value of the power ratio is dictated by the particular application of the system.

Two power ratios are depicted in Table 4.1:𝑃𝑟≥0.7and𝑃𝑟≥0.98. The value of𝑛corre- sponding to𝑘for𝑃𝑟≥0.7is indicated by a single underscore, and the value of𝑛corresponding to𝑘for𝑃𝑟≥0.98is indicated by a double underscore. For𝑃𝑟≥0.98it is noted that𝑛is equal to the integer part of1 + 𝛽, so that

𝐵 ≅ 2(𝛽 + 1)𝑓𝑚 (4.45)

which will take on greater significance when Carson’s rule is discussed in the following paragraph.

The preceding expression assumes sinusoidal modulation, since the modulation index𝛽 is defined only for sinusoidal modulation. For arbitrary𝑚(𝑡), a generally accepted expression for bandwidth results if the deviation ratio𝐷is defined as

𝐷 = peak frequency deviation

bandwidth of𝑚(𝑡) (4.46)

which is

𝐷 = 𝑓𝑑

𝑊(max|𝑚(𝑡)|) (4.47)

The deviation ratio plays the same role for nonsinusoidal modulation as the modulation index plays for sinusoidal systems. Replacing𝛽 by𝐷and replacing𝑓𝑚by𝑊 in (4.45), we obtain

𝐵 = 2(𝐷 + 1)𝑊 (4.48)

This expression for bandwidth is generally referred to asCarson’s rule. If𝐷 ≪ 1, the bandwidth is approximately2𝑊, and the signal is known as anarrowband angle-modulated signal. Conversely, if 𝐷 ≫ 1, the bandwidth is approximately 2𝐷𝑊 = 2𝑓𝑑(max|𝑚(𝑡)|), which is twice the peak frequency deviation. Such a signal is known as awideband angle- modulated signal.

EXAMPLE 4.2

In this example we consider an FM modulator with output

𝑥𝑐(𝑡) = 100 cos[2𝜋(1000)𝑡 + 𝜙(𝑡)] (4.49) The modulator operates with𝑓𝑑= 8and has the input message signal

𝑚(𝑡) = 5 cos 2𝜋(8)𝑡 (4.50)

The modulator is followed by a bandpass filter with a center frequency of 1000 Hz and a bandwidth of 56 Hz, as shown in Figure 4.9(a). Our problem is to determine the power at the filter output.

The peak deviation is 5𝑓𝑑or 40 Hz, and𝑓𝑚= 8 Hz. Thus, the modulation index is 40/5=8. This yields the single-sided amplitude spectrum shown in Figure 4.9(b). Figure 4.9(c) shows the passband of the bandpass filter. The filter passes the component at the carrier frequency and three components on each side of the carrier. Thus, the power ratio is

𝑃𝑟= 𝐽02(5) + 2[𝐽12(5) + 𝐽22(5) + 𝐽32(5)] (4.51)

m(t) = 5 cos 2 (8)tπ xc(t) Output

(a)

(b)

(c)

f, Hz

f, Hz 1 . 3 1 1

. 3 1

1 . 6 2 1

. 6 2

17.8 32.8 32.8

36.5 36.5

4.7 4.7

1 . 9 3 1

. 9 3

952 960 968 976 984 992 1000 1008 1016 1024 1032 1040 1048

1

1000 1028

2 7 9

Amplitude responseAmplitude

FM modulator fc = 1000 Hz fd = 8 Hz

Bandpass f ilter center frequency =

1000 Hz Bandwidth = 56 Hz

Figure 4.9

System and spectra for Example 4.2. (a) FM system. (b) Single-sided spectrum of modulator output. (c) Amplitude response of bandpass filter.

which is

𝑃𝑟= (0.178)2+ 2[

(0.328)2+ (0.047)2+ (0.365)2]

(4.52) This yields

𝑃𝑟= 0.518 (4.53)

The power at the output of the modulator is 𝑥2𝑐= 1

2 𝐴

2 𝑐= 1

2(100)2= 5000W (4.54)

The power at the filter output is the power of the modulator output multiplied by the power ratio. Thus, the power at the filter output is

𝑃𝑟𝑥2𝑐= 2589W (4.55)

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EXAMPLE 4.3

In the development of the spectrum of an angle-modulated signal, it was assumed that the message signal was a single sinusoid. We now consider a somewhat more general problem in which the message signal is the sum of two sinusoids. Let the message signal be

𝑚(𝑡) = 𝐴 cos(2𝜋𝑓1𝑡) + 𝐵 cos(2𝜋𝑓2𝑡) (4.56) For FM modulation the phase deviation is therefore given by

𝜙(𝑡) = 𝛽1sin(2𝜋𝑓1𝑡) + 𝛽2sin(2𝜋𝑓2𝑡) (4.57) where𝛽1= 𝐴𝑓𝑑∕𝑓1> 1and𝛽2= 𝐵𝑓𝑑∕𝑓2. The modulator output for this case becomes

𝑥𝑐(𝑡) = 𝐴𝑐cos[2𝜋𝑓𝑐𝑡 + 𝛽1sin(2𝜋𝑓1𝑡) + 𝛽2sin(2𝜋𝑓2𝑡)] (4.58) which can be expressed as

𝑥𝑐(𝑡) = 𝐴𝑐Re{

𝑒𝑗𝛽1sin(2𝜋𝑓1𝑡)𝑒𝑗𝛽2sin(2𝜋𝑓2𝑡)𝑒𝑗2𝜋𝑓𝑐𝑡}

(4.59) Using the Fourier series

𝑒𝑗𝛽1sin(2𝜋𝑓1𝑡)=

∑∞ 𝑛=−∞

𝐽𝑛(𝛽1)𝑒𝑗2𝜋𝑛𝑓1𝑡 (4.60)

and

𝑒𝑗𝛽2sin(2𝜋𝑓2𝑡)=

∑∞ 𝑚=−∞

𝐽𝑚(𝛽2)𝑒𝑗2𝜋𝑛𝑓2𝑡 (4.61)

X(f)

fc

f Figure 4.10

Amplitude spectrum for(4.63)with𝛽1= 𝛽2and𝑓2= 12𝑓1.

the modulator output can be written 𝑥𝑐(𝑡) = 𝐴𝑐Re

{[ ∞

∑

𝑛=−∞

𝐽𝑛(𝛽1)𝑒𝑗2𝜋𝑓1𝑡

∑∞ 𝑚=−∞

𝐽𝑚(𝛽2)𝑒𝑗2𝜋𝑓2𝑡 ]

𝑒𝑗2𝜋𝑓𝑐𝑡 }

(4.62)

Taking the real part gives 𝑥𝑐(𝑡) = 𝐴𝑐

∑∞ 𝑛=−∞

∑∞ 𝑚=−∞

𝐽𝑛(𝛽1)𝐽𝑚(𝛽2) cos[2𝜋(𝑓𝑐+ 𝑛𝑓1+ 𝑚𝑓2)𝑡] (4.63)

Examination of the signal𝑥𝑐(𝑡)shows that it not only contains frequency components at𝑓𝑐+ 𝑛𝑓1 and 𝑓𝑐+ 𝑚𝑓2, but also contains frequency components at𝑓𝑐+ 𝑛𝑓1+ 𝑚𝑓2for all combinations of𝑛and𝑚.

Therefore, the spectrum of the modulator output due to a message signal consisting of the sum of two sinusoids contains additional components over the spectrum formed by the superposition of the two spectra resulting from the individual message components. This example therefore illustrates the nonlinear nature of angle modulation. The spectrum resulting from a message signal consisting of the sum of two sinusoids is shown in Figure 4.10 for the case in which𝛽1= 𝛽2and𝑓2= 12𝑓1.

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COMPUTER EXAMPLE 4.3

In this computer example we consider a MATLAB program for computing the amplitude spectrum of an FM (or PM) signal having a message signal consisting of a pair of sinusoids. The single-sided amplitude spectrum is calculated. (Note the multiplication by 2 in the definitions ofampspec1and ampspec2in the following computer program.) The single-sided spectrum is determined by using only the positive portion of the spectrum represented by the first𝑁∕2points generated by the FFT program. In the following program𝑁is represented by the variablenpts.

Two plots are generated for the output. Figure 4.11(a) illustrates the spectrum with a single sinusoid for the message signal. The frequency of this sinusoidal component (50 Hz) is evident. Figure 4.11(b) illustrates the amplitude spectrum of the modulator output when a second component, having a frequency of 5 Hz, is added to the message signal. For this exercise the modulation index associated with each component of the message signal was carefully chosen to ensure that the spectra were essentially constrained to lie within the bandwidth defined by the carrier frequency (250 Hz).

0 50 100 150 200 250 300 Frequency-Hz

Frequency-Hz (a)

(b)

350 400 450 500

0 50 100 150 200 250 300 350 400 450 500

0.8 0.6 0.4 0.2

0.5 0.4 0.3 0.2 0.1 0 0

AmplitudeAmplitude

Figure 4.11

Frequency modulation spectra. (a) Single-tone modulating signal. (b) Two-tone modulating signal.

%File: c4ce3.m

fs=1000; %sampling frequency

delt=1/fs; %sampling increment

t=0:delt:1-delt; %time vector npts=length(t); %number of points

fn=(0:(npts/2))*(fs/npts); %frequency vector for plot m1=2*cos(2*pi*50*t); %modulation signal 1

m2=2*cos(2*pi*50*t)+1*cos(2*pi*5*t); %modulation signal 2 xc1=sin(2*pi*250*t+m1); %modulated carrier 1 xc2=sin(2*pi*250*t+m2); %modulated carrier 2 asxc1=(2/npts)*abs(fft(xc1)); %amplitude spectrum 1 asxc2=(2/npts)*abs(fft(xc2)); %amplitude spectrum 2

ampspec1=asxc1(1:((npts/2)+1)); %positive frequency portion 1 ampspec2=asxc2(1:((npts/2)+1)); %positive frequency portion 2 subplot(211)

stem(fn,ampspec1,‘.k’);

xlabel(‘Frequency-Hz’) ylabel(‘Amplitude’) subplot(212)

stem(fn,ampspec2,‘.k’);

xlabel(‘Frequency-Hz’) ylabel(‘Amplitude’) subplot(111)

%End of script file.

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