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Frequency Modulation(Simulation experiment)
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The classic definition of FM is that the instantaneous output frequency of a transmitter is varied in accordance with the modulating signal. Recall that we can write an equation for a sine wave as follows: e(t) = Ep sin(ωt + φ)

While amplitude modulation is achieved by varying Ep, frequency modulation is realized by varying ω in accordance with the modulating signal or message. Notice that one can also vary φ to obtain another form of angle modulation known as phase modulation (PM).

An important concept in the understanding of FM is that of frequency deviation. The amount of frequency deviation a signal experiences is a measure of the change in transmitter output frequency from the rest frequency of the transmitter. The rest frequency of a transmitter is defined as the output frequency with no modulating signal applied. For a transmitter with linear modulation characteristics, the frequency deviation of the carrier is directly proportional to the amplitude of the applied modulating signal. Thus an FM transmitter is said to have a modulation sensitivity,represented by a constant, kf, of so many kHz/V, kf = frequency deviation/V = kf kHz/V.

FM analysis


From the definition of frequency deviation, an equation can be written for the signal frequency of an FM wave as a function of time  :  

V(t) = A sin((fc +m(sinfmt))t+ Θ)

The constant is the modulation index for FM. It is defined as follows: m= δ/fm

The Greek letter δ represents the frequency deviation and fm represents the modulating frequency that causes the deviation. As in the case of AM, this time domain representation of the FM signal can be converted to an equivalent frequency-domain expression that includes the carrier and sidebands.  Because the mathematics required for this conversion are quite complex, we will only consider the result:

The Jn(x) functions are known as Bessel Functions of First kind. Graphs of Jn(x) look like slowly decreasing sine and cosine functions. The Jn(x) functions are a closely related family of functions in the same way that sin(nx) and cos(nx) for a family of similar functions.

The zeroth order Bessel Functions, J0(m) determines the amplitude of the carrier. The nth Bessel function Jn(m) determines the amplitude of the nth pair of sidebands. There are two important concepts contained in the expression shown above.

The amplitude of the carrier depends on m. the modulation index. This is quite different from AM, where the amplitude of the carrier was independent of the value of m.There are an infinite number of sidebands. Thus the theoretical bandwidth of FM is infinite.


An infinite bandwidth signal would be very difficult to transmit. Fortunately, the higher order sidebands in FM have extremely low amplitude and may be ignored.  For example: if the modulation index is 5, only the first 7 sidebands are significant in value. 

There is a rule of thumb, known as Carson’s Rule, that predicts the bandwidth occupied by the significant sidebands of an FM signal, based on the maximum modulation frequency and its corresponding modulation index:

B = 2 fm(m+1) = 2( δmax+ fm

Parameter fm is the frequency of the modulating signal, δmax is the maximum deviation, and m is the corresponding modulation index. If a range of frequencies is used to modulate the carrier, the maximum modulating frequency and its corresponding modulation index are used.

Commercial FM broadcasting uses a maximum deviation of 75 KHz and a maximum modulating frequency of 15 KHz. Substituting these values into Carson’s Rule gives:

B = 2*(75+15) = 180 KHz.

Applications 


FM applications are divided into two broad categories:

       Wideband FM (WFM)

       Narrowband FM (NBFM)

The primary difference between the two types of FM is the number of sidebands in the modulated signal. Wideband FM has a large number  (theoretically infinite) number of sidebands. Narrowband FM has only a single pair of significant sidebands. 

It is possible to determine if a particular FM signal will be wide or narrow band by looking at a quantity called the Deviation Ratio (DR). It is defined as the ratio of the maximum deviation of the FM signal to the maximum modulating frequency:

DR= δ/fmax

The DR is also the modulation index of the highest modulating frequency. If the DR ≥ 1.0 it is called wideband FM (WFM) and DR< 1.0 the modulation is narrow band FM (NBFM). 

One of the drawbacks of wideband FM is the large bandwidth required.Commercial FM broadcasting requires 150 KHz of bandwidth  to transmit a15 KHz audio signal, 5 times the bandwidth required for an AM signal.

Commercial WBFM broadcasts occur in the VHF range, between 88 and 108 MHz.  The carrier frequencies start at 88.1 MHz and are separated by 200 KHz intervals. The maximum audio bandwidth allowed is 15 KHz and the deviation is limited to +/- 75 KHz. Limiting the deviation to this value leaves a 25 KHz guard band at each end of the channel that limits inter-channel interference. The DR for commercial FM broadcasting is 75/15 = 5.0. This is clearly a wideband FM signal.

NBFM is widely used in business and public service communications.The DR for NBFM is restricted to values between 0.5 and 1.0. By holding the DR to such small values, only the carrier and the first sideband are of significant amplitude. When only one sideband and the carrier are transmitted, the NBFM signal occupies the same bandwidth as an AM signal. This overcomes one of the drawbacks of wideband FM, the large bandwidth required. The FCC permits the bandwidth of NBFM signals to be from 10 to 30 KHz, depending on the assigned carrier frequency and the type of operation authorized. 

The figure below compares the spectra of a WFM signal (DR = 5) and a NBFM signal (DR = 0.5). The separation between sidebands is equal to the modulating frequency. Thus the bandwidth for NBFM is 2*fm , which is the same as for AM. However, for WFM, the bandwidth is approximately 2N* fm , where N = the number of sidebands.

Cite this Simulator:

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