FREQUENCY MODULATION

A) OBJECTIVES

§  To understand message signal, carrier signal and FM modulated waveform.

§  To use MATLAB to:

- Create FM signal by modulating a sinusoidal and non-sinusoidal message onto a carrier.

- Examine frequency spectrum of frequency modulated signal.

- Evaluate frequency spectrum of the modulated signal when the modulation index is varied.

- Demodulate the FM signal and recover the original message waveform.

§  To illustrate FM blocks using SIMULINK.

B) THEORY OF FREQUENCY MODULATION

FREQUENCY MODULATION (FM)

An alternative system to Amplitude Modulation is Frequency Modulation. In this modulation scheme the carrier frequency increases as the voltage in the information signal increases and decreases in frequency as it reduces. The larger the amplitude of the information signal, the further the frequency of the carrier signal is shifted from its starting point. The frequency of the information signal determines how many times a second this change in frequency occurs. This modulation process does not affect the amplitude of the carrier.

(1)

The amplitude of the modulated carrier is held constant and either the phase or the time derivative of the phase of the carrier is varied linearly with the message signal m(t). Thus the general angle modulated signal is given by:

x(t) = A_c cos (2p f_ct + q (t) )

The quantity 2p f_c + q (t) = y_i(t) is called the instantaneous phase of x(t), while the quantity    q (t)  is called the phase deviation of x(t). The instantaneous angular frequency of x(t), defined as the rate of change of the instantaneous phase and having units of radians per second, is given by:

(2)

The instantaneous frequency f_i (t), having units of Hertz (Hz), of x(t) is accordingly given by:

(3)

The quantity  is called the angular frequency deviation. The two basic types of angle modulation are Phase Modulation (PM) and Frequency Modulation (FM).

 

VARIATION OFq (t) PRODUCES PHASE MODULATION

Phase modulation implies that q (t) is proportional to the modulating signal. Thus q (t)=k_pm(t), where k_p is the deviation constant in radians per unit of m(t). Therefore, the time domain expression for PM is given by:

(4)

x(t) = A_c cos (2p f_ct + k_pm(t))

VARIATION OF   PRODUCES FREQUENCY MODULATION

Frequency modulation implies that  is proportional to the modulating signal. This yield:

(5)

Thus, in FM the instantaneous frequency varies linearly with the message signal and is given by:

	(6)

 

                                                            f_i = f_c  + k_f m(t).

The term k_f , expressed in Hertz per unit of m(t), represents the frequency sensitivity of the FM signal.

The phase angle q (t) of FM signal is given by:

(7)

Therefore, the time domain expression for FM is given by:

(8)

FREQUENCY DEVIATION, MODULATION INDEX AND SPECTRUM OF FM 

Consider a sinusoidal modulating information signal given by:

	(9)

 

m(t) = A_m cos(2p f_m t )

The instantaneous frequency of the resulting FM signal equals:

(10)

f_i(t)  = f_c  + k_f m(t) = f_c  + k_f A_m cos(2p f_m t )

(11)

The maximum change in instantaneous frequency f_i from the carrier frequency f_c, is known as frequency deviation Df, where it is given by:

D f = k_f A_m

Frequency deviation is a useful parameter for determining the bandwidth of FM signals. For example, an information signal of peak-to-peak voltage of 6 volts and a frequency of 10kHz with a frequency deviation of 15 kHz/V would cause a FM carrier to change by a total of 90 kHz (45 kHz above and below the original carrier frequency). The carrier frequency would be swept over this range 10,000 times a second.

Next, the FM modulated signal is given by:

	(12)

 

where  is the modulation index of the modulated signal. In general, for a non-sinusoidal m(t) signal, the modulation index is defined as:

(13)

where, W is the bandwidth of the message signal, m(t).

In case of a sinusoidal message signal, the modulated signal can be represented by:

(14)

where J_n(b) is known as Bessel functions in the order n and argument b. Some of the selected values of J_n(b) is listed in Table 1. In the frequency domain, we have:

(15)

From the equations (14) and (15) and Table 1, we observe that :

1.      The spectrum consists of a carrier-frequency component and an infinite number of sideband components at frequencies f_c ± nf_m (n = 1,2,3,4,5…..).

2.      The relative amplitudes of the spectral lines depend on the value of J_n(b), and the value of J_n(b) becomes very small for large values of n.

3.      The number of significant spectral lines (that is, having appreciable relative amplitude) is a function of the modulation index b. With b << 1, only J₀ and J₁ are significant, so the spectrum will consists of carrier and two sideband lines. But if b >> 1, there will be many sideband lines. The amplitude spectrums of FM signals for several values of b are shown in Figure 2.

n\b	0.1	0.2	0.5	1	2	5
0	0.997	0.990	0.938	0.765	0.224	-0.178
1	0.050	0.100	0.242	0.440	0.577	-0.328
2	0.001	0.005	0.031	0.115	0.353	0.047
3			0.003	0.020	0.129	0.365
4				0.002	0.034	0.391
5					0.007	0.261
6					0.001	0.131
7						0.053
8						0.018
9						0.006
10						0.001

Table 1: Selected value of J_n(b).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2: Amplitude spectrum of sinusoidal modulated FM signals (f_m fixed).

 

 

C) MATLAB AND SIMULINK

MATLAB is an interactive matrix based system for scientific and engineering numeric computation and visualization. Its strength lies in the fact that complex numerical problem can be solved easily and in a fraction of the time required with a programming language such as Fortran or C. It is also powerful in the sense that by using its relatively simple programming capabilities, MATLAB can be easily extended to create new commands and functions.

SIMULINK is a software package in MATLAB used for modeling, simulating and analyzing dynamical systems. It supports linear and nonlinear systems, modeled in continuous time, sampled time or a hybrid of the two. Systems can also be multirate that has different parts that are sampled or updated at different rates. For modeling, SIMULINK provides a graphical user interface (GUI) for building models as block diagrams, using click-and-drag mouse operations. With this interface, you can draw the models just as you would with pencil and paper (or as most textbooks depict them). It also includes a comprehensive block library of sinks, sources, linear and nonlinear components and connectors.

Models are hierarchical, so you can build models using both top-down and bottom-up approaches. You can view the system at a high level, then double-click on blocks to go down through the levels to see increasing levels of model detail. This approach provides insight into how a model is organized and how its parts interact. After you define a model, you can simulate it, using a choice of integration methods, either from the SIMULINK menus or by entering commands in MATLAB's command window. The menus are particularly convenient for interactive work, while the command-line approach is very useful for running a batch of simulations. Using scopes and other display blocks, you can see the simulation results while the simulation is running. In addition, you can change parameters and immediately see what happens, for "what if" exploration. The simulation results can be put in the MATLAB workspace for post-processing and visualization. Model analysis tools include linearization and trimming tools, which can be accessed from the MATLAB command line, plus the many tools in MATLAB and its application toolboxes.

D) EXPERIMENT PROCEDURES – MATLAB

1.      Open and start the MATLAB program by double-clicking the MATLAB icon.

2.      Type the command in the MATLAB COMMAND WINDOW or create a script file in the MATLAB EDITOR.

3.      Given a 100Hz continuous time sinusoidal message signal, m₁(t) = 0.8 sin (200pt) is frequency modulated by a carrier of  s₁(t) = 0.8 cos (2000pt). Using sufficient points and sampling interval of 0.0001 seconds, plot the message and carrier signals for duration of 0.05 seconds. (Use the commands subplot and plot in MATLAB and select an appropriate time interval for each plot).

4.      Write the time domain expression for the modulated signal by using the following equation:

Find the result of  by integrating the message signal, m₁(t). Assuming that the frequency sensitivity, k_f = 625 Hz/V, what is the modulation index for this signal? Plot the modulated signal for the duration of 0.05 seconds.

You can also find the result of  numerically in MATLAB by using cumsum(x)/Fs, function where x and Fs are the message signal and sampling rate used, respectively. Plot the modulated signal using the cumsum function for the duration of 0.05 seconds. Compare the plots. Are they the same?

5.      You can make use of the modulate function in MATLAB to generate your FM signals. Study this function by typing help modulate in the MATLAB command prompt. Repeat your plot for the modulated signal using this function. (Note that the k_f for the modulate function must be set to k_f = 2p(k_f )/Fs, as a normalized value with respect to the sampling rate).

6.      The normalized magnitude spectra (or frequency spectrum) of a signal can be plotted using the built-in fft and fftshift functions in MATLAB. Use the following MATLAB function ft.m in order to plot the magnitude spectra or spectrum for message signal, m₁(t).

% Fourier Transform function, m-input signal, ts-sampling interval

% Save this function as ft.m in MATLAB work directory

function [M,m]=ft(m,ts)            % return the value M and m

df=0.5;                            % frequency resolution

fs=1/ts;                                % sampling frequency

n1=fs/df;                          % refining resolution

n2=length(m);                      % number of point of the signal

n=2^(max(nextpow2(n1),nextpow2(n2)));

M=fft(m,n);                        % fourier transform of the signal

m=[m,zeros(1,n-n2)];

df=fs/n

M=M/fs;                            % scaling

f=[0:df:df*(length(m)-1)]-fs/2;    % frequency vector

plot(f,abs(fftshift(M)))                % plot & shift the DC component

Subsequently, use the above function to plot the magnitude spectra (spectrum) for the carrier, s₁(t) and frequency-modulated signal, x₁(t). You may use the command axis to refine the scales in your plot.

7.      Frequency modulation can be categorized into either narrowband or wideband. When the modulation index, b is very small, it is usually called narrowband FM (NBFM). Generate the spectrum of the above frequency-modulated signal, x₁(t) when the modulation index, b = 0.005, 0.1, 1, 10 and 50. Observe the change of the spectrum shape for various modulation indices. What can you conclude when the modulation index is very large?

8.      MATLAB has a built-in function called demod utilizing the linear frequency-to-voltage transfer characteristic (frequency discriminator) that could be used to demodulate a FM signal. All parameters used in the demod function are as sets in the previous modulate function. Generate the time domain plot for demodulated FM signal when b = 5 by using this function. Did you recover the original message signal?

9.      Now, consider a non-sinusoidal message signal of, m₂(t) as shown below:

with a bandwidth of 20Hz is modulated by a carrier, s₂(t) = cos (400pt). Assuming that the frequency sensitivity, k_f = 50, what is the modulation index for this FM signal?

Plot the time domain and spectrum for the above message m₂(t), carrier s₂(t) and modulated signal x₂(t). Use the functions cumsum and modulate for the modulated signal for comparison purposes. You may look at the command subplot for instruction on plotting several figures in one display window. Limit your time domain plot to a sampling interval of 0.0005 seconds for duration of 0.15 seconds. Demodulate the FM signal and compare to the original message signal.

E) EXPERIMENT PROCEDURES – SIMULINK

1.      Type simulink at the MATLAB COMMAND prompt.

* The Simulink Library Browser window is opened.

2.      Create a new model window by clicking the Create a new model button on the Library Browser toolbar or click File >> New >> Model.

* A new empty workspace window is opened.

3.       Double-click to expand the Simulink folder at the Library Browser window.

4.      Double-click to expand the Sources sub-folder in the Simulink folder.

5.      Drag and drop Signal Generator module into the new empty workspace window.

6.      Go to Communications Blockset -> Modulation -> Analog Passband Modulation sub-folder.

7.      Drag and drop FM Passband and FDM Passband modules into the workspace window.

8.      Go to DSP Blockset -> Filtering -> Filter Designs sub-folder. Drag and drop Analog Filter Design module into the workspace window.

9.      Go to Simulink -> Sinks sub-folder. Drag and drop THREE Scope modules into workspace window.

10.  Go to Simulink Extras -> Additional Sinks sub-folder. Drag and drop FOUR Power Spectral Density modules into workspace window.

11.  Connect all the inserted modules as shown below.

12.  Set the parameters of the different blocks in your workspace as follows:

Block Model	Parameters to be set
Signal Generator	Waveform type: Sine Amplitude: 0.8 Frequency: 100 Hz
FM Passband Modulator & Demodulator	Carrier Frequency: 1 kHz Initial phase: 0 radian Modulation constant: 625 Hz/V Sample time: 10 ms
Analog Filter Design	Filter type: Butterworth Low Pass Filter Filter Order: 5 Passband Edge: 600 rad/s
Power Spectral Density	Length of buffer: 4096 Number of points for fft: 4096 Plot after how many points: 4096 Sample time: 10 ms

13.  Set the simulation parameters (Simulation >> Parameters) as follows:

14.  Run (Simulation >> Start) the simulation and observe the output waveforms in both time and frequency domains of the message, carrier, modulated and demodulated signals. Compare these graphs with that obtained in Section D) EXPERIMENT PROCEDURES – MATLAB.

15.  Change the message to a square wave and run the simulation again. What is the optimum cut-off frequency for the low pass filter at the demodulator?

F) REFERENCES

[1] B. P. Lathi, “Modern Digital and Analog Communication Systems”