4.1 PHASE AND FREQUENCY MODULATION DEFINED
4.1.2 Spectrum of an Angle-Modulated Signal
The derivation of the spectrum of an angle-modulated signal is typically a very difficult task. However, if the message signal is sinusoidal, the instantaneous phase deviation of the modulated carrier is sinusoidal for both FM and PM, and the spectrum can be obtained with ease. This is the case we will consider. Even though we are restricting our attention to a very special case, the results provide much insight into the frequency-domain behavior of angle modulation. In order to compute the spectrum of an angle-modulated signal with a sinusoidal message signal, we assume that
𝜙(𝑡) = 𝛽 sin(2𝜋𝑓𝑚𝑡) (4.22)
The parameter𝛽is known as themodulation index and is the maximum phase deviation for both FM and PM. The signal
𝑥𝑐(𝑡) = 𝐴𝑐cos[2𝜋𝑓𝑐𝑡 + 𝛽 sin(2𝜋𝑓𝑚𝑡)] (4.23)
can be expressed as
𝑥𝑐(𝑡) = Re[
𝐴𝑐𝑒𝑗𝛽 sin(2𝜋𝑓𝑚𝑡)𝑒𝑗2𝜋𝑓𝑐𝑡]
(4.24) This expression has the form
𝑥𝑐(𝑡) = Re[ ̃𝑥𝑐(𝑡)𝑒𝑗2𝜋𝑓𝑐𝑡] (4.25) where
̃𝑥𝑐(𝑡) = 𝐴𝑐𝑒𝑗𝛽 sin(2𝜋𝑓𝑚𝑡) (4.26) is the complex envelope of the modulated carrier signal. The complex envelope is periodic with frequency𝑓𝑚and can therefore be expanded in a Fourier series. The Fourier coefficients are given by
𝑓𝑚∫
1∕2𝑓𝑚
−1∕2𝑓𝑚𝑒𝑗𝛽 sin(2𝜋𝑓𝑚𝑡)𝑒−𝑗2𝜋𝑛𝑓𝑚𝑡𝑑𝑡 = 1 2𝜋∫
𝜋
−𝜋𝑒−[𝑗𝑛𝑥−𝛽 sin(𝑥)]𝑑𝑥 (4.27) This integral cannot be evaluated in closed form. However, this integral arises in a variety of studies and, therefore, has been well tabulated. The integral is a function of𝑛and𝛽 and is known as theBessel function of the first kind of order𝑛and argument𝛽. It is denoted𝐽𝑛(𝛽) and is tabulated for several values of𝑛and𝛽in Table 4.1. The significance of the underlining of various values in the table will be explained later.
With the aid of Bessel functions, we have
𝑒𝑗𝛽 sin(2𝜋𝑓𝑚𝑡)= 𝐽𝑛(𝛽)𝑒𝑗2𝜋𝑛𝑓𝑚𝑡 (4.28) which allows the modulated carrier to be written as
𝑥𝑐(𝑡) = Re [(
𝐴𝑐
∑∞
𝑛=−∞𝐽𝑛(𝛽)𝑒𝑗2𝜋𝑛𝑓𝑚𝑡 )
𝑒𝑗2𝜋𝑓𝑐𝑡 ]
(4.29) Taking the real part yields
𝑥𝑐(𝑡) = 𝐴𝑐
∑∞ 𝑛=−∞
𝐽𝑛(𝛽) cos[2𝜋(𝑓𝑐+ 𝑛𝑓𝑚)𝑡] (4.30) from which the spectrum of𝑥𝑐(𝑡)can be determined by inspection. The spectrum has com- ponents at the carrier frequency and has an infinite number of sidebands separated from the carrier frequency by integer multiples of the modulation frequency𝑓𝑚. The amplitude of each spectral component can be determined from a table of values of the Bessel function. Such tables typically give𝐽𝑛(𝛽)only for positive values of𝑛. However, from the definition of𝐽𝑛(𝛽) it can be determined that
𝐽−𝑛(𝛽) = 𝐽𝑛(𝛽), 𝑛even (4.31)
and
𝐽−𝑛(𝛽) = −𝐽𝑛(𝛽), 𝑛odd (4.32)
These relationships allow us to plot the spectrum of (4.30), which is shown in Figure 4.5. The single-sided spectrum is shown for convenience.
A useful relationship between values of𝐽𝑛(𝛽)for various values of 𝑛is the recursion formula
𝐽𝑛+1(𝛽) = 2𝑛
𝛽 𝐽𝑛(𝛽) + 𝐽𝑛−1(𝛽) (4.33)
f
f AcJ– 4( )AβcJ–3( )β
AcJ–2( )β AcJ–1( )β
fc–2fm fc+2fm
fc–3fm fc+3fm
fc–4fm fc+4fm
fc– fm fc+ fm fc+2fm fc+3fm fc+4fm
fc+ fm
fc
fc–2fm
fc–3fm
fc–4fm fc– fm fc
AcJ0( )β AcJ1( )β
AcJ J
2( )β Ac 3( )β
AcJ4( )β 0
0
AmplitudePhase, rad
(a)
(b) _π
Figure 4.5
Spectra of an angle-modulated signal. (a) Single-sided amplitude spectrum. (b) Single-sided phase spectrum.
Thus,𝐽𝑛+1(𝛽)can be determined from knowledge of𝐽𝑛(𝛽) and𝐽𝑛−1(𝛽). This enables us to compute a table of values of the Bessel function, as shown in Table 4.1, for any value of𝑛 from𝐽0(𝛽)and𝐽1(𝛽).
Figure 4.6 illustrates the behavior of the Fourier--Bessel coefficients 𝐽𝑛(𝛽), for 𝑛 = 0, 1, 2, 4, and 6 with0≤𝛽 ≤9. Several interesting observations can be made. First, for𝛽 ≪ 1,
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 –0.1 –0.2 –0.3 –0.4
1 2 3 4 5 6 7 8 9
0 β
β J6( ) β
J2( ) β
J1( ) β
J0( )
β J4( )
Figure 4.6
𝐽𝑛(𝛽)as a function of𝛽.
Table 4.2 Values of𝜷for which𝑱𝒏(𝜷) = 𝟎for𝟎≤𝜷≤𝟗
𝒏 𝜷𝒏0 𝜷𝒏1 𝜷𝒏2
0 𝐽0(𝛽) = 0 2.4048 5.5201 8.6537
1 𝐽1(𝛽) = 0 0.0000 3.8317 7.0156
2 𝐽2(𝛽) = 0 0.0000 5.1356 8.4172
4 𝐽4(𝛽) = 0 0.0000 7.5883 --
6 𝐽6(𝛽) = 0 0.0000 -- --
it is clear that𝐽0(𝛽)predominates, giving rise to narrowband angle modulation. It also can be seen that𝐽𝑛(𝛽)oscillates for increasing𝛽 but that the amplitude of oscillation decreases with increasing𝛽. Also of interest is the fact that the maximum value of𝐽𝑛(𝛽)decreases with increasing𝑛.
As Figure 4.6 shows,𝐽𝑛(𝛽)is equal to zero at several values of𝛽. Denoting these values of 𝛽 by 𝛽𝑛𝑘, where𝑘 = 0, 1, 2, we have the results in Table 4.2. As an example, 𝐽0(𝛽) is zero for 𝛽 equal to 2.4048, 5.5201, and 8.6537. Of course, there are an infinite number of points at which𝐽𝑛(𝛽)is zero for any 𝑛, but consistent with Figure 4.6, only the values in the range0≤𝛽 ≤9 are shown in Table 4.2. It follows that since𝐽0(𝛽) is zero at 𝛽 equal to 2.4048, 5.5201, and 8.6537, the spectrum of the modulator output will not contain a component at the carrier frequency for these values of the modulation index. These points are referred to ascarrier nulls. In a similar manner, the components at𝑓 = 𝑓𝑐± 𝑓𝑚 are zero if 𝐽1(𝛽)is zero. The values of the modulation index giving rise to this condition are 0, 3.8317, and 7.0156. It should be obvious why only 𝐽0(𝛽) is nonzero at 𝛽 = 0. If the modulation index is zero, then either 𝑚(𝑡)is zero or the deviation constant 𝑓𝑑 is zero. In either case, the modulator output is the unmodulated carrier, which has frequency components only at the carrier frequency. In computing the spectrum of the modulator output, our starting point was the assumption that
𝜙(𝑡) = 𝛽 sin(2𝜋𝑓𝑚𝑡) (4.34)
Note that in deriving the spectrum of the angle-modulated signal defined by (4.30), the modulator type (FM or PM) was not specified. The assumed𝜙(𝑡), defined by (4.34), could represent either the phase deviation of a PM modulator with𝑚(𝑡) = 𝐴 sin(𝜔𝑚𝑡)and an index 𝛽 = 𝑘𝑝𝐴, or an FM modulator with𝑚(𝑡) = 𝐴 cos(2𝜋𝑓𝑚𝑡)with index
𝛽 =𝑓𝑑𝐴
𝑓𝑚 (4.35)
Equation (4.35) shows that the modulation index for FM is a function of the modulation frequency. This is not the case for PM. The behavior of the spectrum of an FM signal is illustrated in Figure 4.7, as𝑓𝑚is decreased while holding𝐴𝑓𝑑 constant. For large values of 𝑓𝑚, the signal is narrowband FM, since only two sidebands are significant. For small values of𝑓𝑚, many sidebands have significant value. Figure 4.7 is derived in the following computer example.
2 1.5 1
Amplitude 0.5
–5000 –400 –300 –200 –100 0 100 200 300 400 500
ƒ, Hz 2
1.5 1 0.5
Amplitude
0
–500 –400 –300 –200 –100 0 100 200 300 400 500
ƒ, Hz 0.8
0.6 0.4 0.2
Amplitude
–5000 –400 –300 –200 –100 0 100 200 300 400 500
ƒ, Hz Figure 4.7
Amplitude spectrum of a complex envelope signal for increasing𝛽and decreasing𝑓𝑚.
COMPUTER EXAMPLE 4.1
In this computer example we determine the spectrum of the complex envelope signal given by (4.26).
In the next computer example we will determine and plot the two-sided spectrum, which is determined from the complex envelope by writing the real bandpass signal as
𝑥𝑐(𝑡) = 1
2 ̃𝑥(𝑡)𝑒𝑗2𝜋𝑓𝑐𝑡+ 1 2 ̃𝑥
∗(𝑡)𝑒−𝑗2𝜋𝑓𝑐𝑡 (4.36)
Note once more that knowledge of the complex envelope signal and the carrier frequency fully determine the bandpass signal. In this example the spectrum of the complex envelope signal is determined for three different values of the modulation index. The MATLAB program, which uses the FFT for determination of the spectrum, follows.
%file c4ce1.m fs=1000;
delt=1/fs;
t=0:delt:1-delt;
npts=length(t);
fm=[200 100 20];
fd=100;
for k=1:3
beta=fd/fm(k);
cxce=exp(i*beta*sin(2*pi*fm(k)*t));
as=(1/npts)*abs(fft(cxce));
evenf=[as(fs/2:fs)as(1:fs/2-1)];
fn=-fs/2:fs/2-1;
subplot(3,1,k); stem(fn,2*evenf,‘.’) ylabel(‘Amplitude’)
end
%End of script file.
Note that the modulation index is set by varying the frequency of the sinusoidal message signal𝑓𝑚 with the peak deviation held constant at 100 Hz. Since𝑓𝑚takes on the values of 200, 100, and 20, the corresponding values of the modulation index are 0.5, 1, and 5, respectively. The corresponding spectra of the complex envelope signal are illustrated as a function of frequency in Figure 4.7.
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COMPUTER EXAMPLE 4.2
We now consider the calculation of the two-sided amplitude spectrum of an FM (or PM) signal using the FFT algorithm. As can be seen from the MATLAB code, a modulation index of 3 is assumed. Note the manner in which the amplitude spectrum is divided into positive frequency and negative frequency segments (line nine in the following program). The student should verify that the various spectral components fall at the correct frequencies and that the amplitudes are consistent with the Bessel function values given in Table 4.1. The output of the MATLAB program is illustrated in Figure 4.8.
%File: c4ce2.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)-(fs/2); %frequency vector for plot m=3*cos(2*pi*25*t); %modulation
xc=sin(2*pi*200*t+m); %modulated carrier asxc=(1/npts)*abs(fft(xc)); %amplitude spectrum
–400
–500 –300 –200 –100 100 200 300 400 500f, Hz
0 0.25
0.2
0.15
0.1
0.05
0
Amplitude
Figure 4.8
Two-sided amplitude spectrum computed using the FFT algorithm.
evenf=[asxc((npts/2):npts)asxc(1:npts/2)]; %even amplitude spectrum stem(fn,evenf,‘.’);
xlabel(‘Frequency-Hz’) ylabel(‘Amplitude’)
%End of script.file.
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