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Experiment 4: Frequency Modulation and Spectra of FM signals

Frequency Modulation:  A Tutorial

A frequency modulated (FM) wave is most readily described by the carrier signal

The instantaneous angle, (J( t ) , of the above cosine wave is the value in the brackets,

The derivative of q ( t ) is the instantaneous radian frequency w ( t ) of the FM signal. Dividing that by 2p  produces the instantaneous frequency f ( t ), given by

It is now clear that the instantaneous frequency of the FM signal varies around the carrier frequency   f_c by an amount  k _f x ( t ), where   k_f  is the modulation constant. Positive values of x ( t ) produce increases in  f ( t ), whereas negative values of x ( t ) produce decreases in  f ( t ). If x ( t ) is restricted by

then the frequency of the FM wave varies around  f_c  by  ± k_f x_max This is the reason that  k_f x_max is referred to as the maximum frequency deviation of the FM wave

It is nearly impossible to find the spectrum of an FM wave except for special waveforms of x ( t ). The simplest is the sinusoidal case, given by

and we observe that for this sinusoid x_max = A_m, hence the maximum frequency deviation in this case is

( Df )_max = k_f A_{m          (2.7)}

For this modulating waveform, (2.1) becomes

       (2.8)

The notation can be simplified by defining

         (2.9)

so that

x_c( t )  =  A_c cos ( w_ct  +  b sin w_mt )      (2.10) 

It is obvious from the last equation that the parameter b is the peak phase deviation of the signal. It is also referred to as the modulation index. It is noteworthy that the modulation index is inversely related to the modulation frequency  f_c.

We are now in a position to find the spectrum of the signal in (2.10). The easiest way to proceed is to write (2.10) in the exponential form

   (2.11)

which can be recast into

(2.12)

In the above we use a well known mathematical identity in terms of In(I''), the Bessel functions of the first kind,

   (2.13)

to obtain

      (2.14)

or

        (2.15)

The above is identical to

           (2.16)

    We finally have the result that allows us to plot the spectrum of the FM wave for a sinusoidal modulating signal. Some values of J_n(b)  can be found in table 2.1 as well as in figure 2.1. Very thorough listings can be found in E. Jahnke, F. Emde and F. Lösch, Tables of Higher Functions, McGraw-Hill Book Company, 1960. A very affordable (but older) paperback version, by the first two authors, is available from Dover Books.
    The procedure that was used to find the spectrum of the single tone modulated FM wave described in (2.10) can be used to find the spectrum of the multitone modulated signal

x_c( t )  =  A_c cos ( w_ct  +  b₁ sin w₁t   +   b₂ sin w₂t)      (2.17) 

to obtain the result

  (2.18)

    The above procedure can be extended to more than two tones, but it can become rather messy. The above equation tells us that this wave contains four different kinds of frequency components.

Figure 2.1: Curves of Bessel functions.

1. There is the carrier of magnitude A_c J₀ ( b₁)J₀ ( b₂)

2. There are sidebands lines at  f_c  ±  nf₁ due to the first tone.

3. There are sidebands lines at f_c  ±  mf₂ due to the second tone.

4. There are sidebands lines at f_c  ±  nf₁  ±  mf₂   due to both tones. This is  somewhat surprising when compared to linear modulation schemes, such as AM, DSB-SC and SSB, where such lines would not appear at all. But FM is a non-linear modulation scheme, so this is not surprising after all.

In your report address the item below.

1. Derive the result in (2.18) by starting with (2.17) and using (2.13),

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Frequency Modulation - Part 1

The Wavetek generator can be used as an FM modulator by applying a modulating signal to the external input. This signal will change the frequency of the oscillator from the nominal value shown on the dial. To use this feature intelligently we have to know the change in frequency produced by a given external voltage.

Obtain data for a plot of output frequency as a function of input voltage. for DC input voltages in the range of approximately 1 volt. Set the frequency with

zero input voltage to approximately 1 Mhz. Use a frequency counter for accurate determination of frequency. Use a power supply for the DC input voltage. (A 10 ÷ 1 voltage divider across the power supply will make it easier to adjust the voltage. ) Draw a best fit straight line through your data. Since the slope of this straight line is the modulation constant k_f  kHz/volt, you can now determine k_f  when the Wavetek is used as an FM modulator .

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Frequency Modulation - Part 2

The measurement of frequency deviation for a time varying modulating signal is not an easy thing to do in general. However, if the modulating signal is sinusoidal and we have a spectrum analyzer available, there is a nice method available for making accurate deviation measurements. This is based on the fact that at certain values of the modulation index some of the spectral components go to zero. We will use this method to measure the deviation of an FM signal produced by the Wavetek generator for a sinusoidal modulating signal and will compare our results with those of part 1. This will determine if the sensitivity of this modulator is the same for AC signals as it is for DC signals.
    Apply a sinusoidal signal of some convenient frequency ( 10 kHz) to the modulation input (you can use the counter to measure the frequency accurately) and slowly increase the amplitude of the signal, starting with zero, until the carrier component goes to zero, or reaches a minimum. Measure the amplitude of the 10 kHz signal at this point.
    Table 2.2 contains zeros of the Bessel functions. Using this table calculate the frequency deviation. Repeat this procedure using nulls of the first side band and other nulls of the carrier until you reach a deviation in the neighborhood of 100 kHz. Using this data you can plot peak deviation as a function of the peak value of the modulating signal on the same sheet as the plot of part 1. How do the two curves compare ?

Figure 2.2: Method for summing two signals.

Frequency Modulation  - Part 3

Using the values for the sidebands given in table 2.1, adjust the amplitude of the modulating signal to produce modulation indices of 0.2, 1.0 and 5.0. Sketch the spectra for these values. Approximately how many sidebands are needed to represent the signals in each case ?

Frequency Modulation -Part 4

Using the same amplitude square waves as the sine wave amplitudes in part 3, sketch the spectrum for a square wave modulating signal. This will give you an idea of what the spectrum looks like when transmitting data using the FSK (frequency shift keying) method.

Frequency Modulation -Part 5

In this part we will examine the spectrum of an FM signal where the modulating signal is the sum of two sinusoidal signals. This will illustrate the non-linear nature of FM. The mathematical expression for such a signal is given in (1.15) and (1.16). gives insight into the spectrum that can be expected. The sum of two signals can be obtained as shown in figure 2.2.

Choose the two audio frequencies so that the sum   f₁  +   f₂ and the difference  f₁  -   f₂ will not fall on harmonics of f1 or f2. For example choosing   f₁ = 20 kHz   and   f₂ = 10 kHz  would not be a good choice as  f₁  -   f₂  would fall on top of   f₂.

With generator #2 disconnected adjust amplitude of generator #1 to make b₁  =  1. Now do the same backwards to adjust for b₂  =  1. Now connect both generators.

Note the appearance of spectral lines at   f_c  ±  ( f₁  -   f₂ )   and   f_c  ±  ( f₁  +   f₂ )   that were not present when either generator #1 or generator #2 was connected alone. Record the amplitudes and frequencies of terms large enough to be measurable. Identify them as the carrier, lines at   f_c  ±   nf₁   due to the first tone, lines at   f_c  ±   mf₂ due to the second tone and lines at    f_c  ±   nf₁  ±   mf₂   due to both tones.

Compute the theoretical amplitude of these terms using (1.16), and compare measured and calculated values by tabulating the results neatly. Discuss how these results indicate that FM is a non-linear type of modulation.

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