01 FM Principles

FM PRINCIPLES

Frequency Modulation The type of modulation in which the instantaneous frequency of the carrier is varied accoto amplitude of modulating signal is called frequency modulation. Frequency modulatwidely used in VHF communication systems e.g. FM broadcasting, transmission of ssignal in TV, Satellite Communication etc.

Figure 1 Frequency Modulated wave

Frequency modulated wave is shown in Fig.1. The instantaneous frequency varies aboaverage frequency (carrier frequency) at the rate of modulating frequency. The amouwhich the frequency varies away from the average frequency (carrier frequency) is frequency deviation and is proportional to the amplitude of the modulating signal. Analysis of FM waves Equation of a sine wave in the generalised form is e = A sin θ. (1) Where e is instantaneous amplitude, A is peak amplitude and θ is total angular displaceat time t. A frequency modulated wave with sinusoidal modulation has its frequency varied accordthe amplitude of the modulating signal. If ∆f is the maximum deviation of frequencyaverage, then instantaneous frequency is

STI(T) Publication 1 007

1

rding ion is ound

ut the nt by called

ment

ing to from

/2003

FM Transmitter

dtdNow

tcosf2,ortcosffcf

mc

m

θ=ω

ω∆π+ω=ω

ω∆+=

(2)

Integrating both sides

)3()tsinmtsin(Ae

mf

fLet

)tsinf

ft(sinA

)tsinf2t(sinAe

tsinf2t

dt.

mfc

fm

mm

c

mm

c

mm

c

ω+ω=∴

=∆

ω∆

+ω=

ωω∆π

+ω=∴

ωω∆π

+ω=θ

θ=ω∫

Where m is called the Modulation Index of the FM wave. f Thus for a given frequency deviation modulation index varies inversely as the modulating frequency. The frequency components actually contained in the FM wave can be determined by expanding RHS of equation (3), then we get e = AJ0(m ) sin ωf ct +AJ1( fm ) [sin (ωc+ωm)t-sin (ωc-ωm)t] +AJ2( fm ) [sin (ωc+2ωm)t+sin (ωc-2ωm)t] +AJ3( fm ) [sin (ωc+3ωm)t-sin (ωc-3ωm)t] +AJ4( fm ) [sin (ωc+4ωm)t+sin (ωc-4ωm)t]

+AJ5( ) [sin (ωfm c+5ωm)t+sin (ωc-5ωm)t] +……………….. (4)

Where Jn (m ) is the Bessel function of first kind and nth order with argument mf f. Bessel functions Jo to J8 are shown in fig. 2. Equation (4) shows that an FM wave corresponding to sinusoidal modulation is made up of several frequency components spaced apart by the modulating frequency. Thus an FM wave has in addition to the side bands present in an AM wave, higher order sidebands as well. Amplitudes of different frequency components depend upon mf, the modulation index. When the modulation index is less than 0.5 that is when the frequency deviation is less than half the modulation frequency the second and higher order components are relatively small and the frequency band required to accommodate the essential part of the signal is the same as in

STI(T) Publication 2 007/2003

FM Principles

amplitude modulation. This is called Narrowband FM and is used for speech communications. When mf is larger than one (frequency deviation greater than modulating frequency) there are important higher order sideband components contained in the wave and it is called wide band FM.

Fig. 2 Bessel Function Jo to J8 Practical values of modulation index vary considerably with frequency. If fm = 15 kHz and

51575

ffm

kHz75f

mf ==

∆=

=∆

Value of Bessel functions J0, J1 etc. for mf = 5 are plotted in fig. 3. It is clear that the amplitude of sideband pair decreases for pairs of order greater then 5 and becomes less than 1% of the unmodulated carrier amplitude beyond the 8th sideband pair.

J0(5) J3(5) J6(5) J9(5)

+0.5

0

-0.5

ORDER OF SIDEBAND PAIR

AM

PLI

TUD

E O

F C

AR

RIE

R

& S

IDE

BA

ND

PA

IRS

Figure 3 Relative Amplitudes of Carrier and Sideband Pairs for Modulation Index of 5


FM Transmitter

If the amplitude and frequency of a modulating signal are increased in the same ratio, value of mf remains the same and the number of sidebands also remains unchanged. The relative amplitudes of the carrier and sidebands is the same, giving the spectrum pattern but the sideband spacing is greater because of the increased modulation frequency. A typical spectrum pattern for a FM wave for a modulation index of 5 is shown in figure 4. It is seen that modulating frequency does two things: 1. Fixes the separation of sidebands. 2. Determines the rapidity with which the sideband distribution changes.

9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9

0.5

ORDER OF SIDEBAND PAIR

AM

PLI

TUD

E O

F C

AR

RIE

R

& S

IDE

BA

ND

PA

IRS

0.4

0.3

0.2

0.1

0.0

mf = 5 CARRIER

Figure 4 Spectrum of FM Wave for a Modulation Index of 5

Sideband Power In FM signal, the carrier power diminishes during modulation and it is possible for one or more sidebands to contain more power than the carrier. The power withdrawn from the carrier during modulation is distributed among the various sidebands. The louder the modulating signal, the greater will be the energy that is taken away from the carrier. It is therefore, possible for the carrier, during one of these modulation sweeps, to contain no energy at all. This is quite logical because the FM signal does not vary in amplitude. The only way to satisfy this condition during modulation is to transfer part of the energy to the sidebands. The power transfer is a characteristic of frequency modulation. When the intensity of the audio signal is increased the total number of sidebands also increases i.e. the energy of the FM wave is shifted away from the carrier with every sideband and the carrier affected. Thus, energy is taken by some and given up by others. The total energy under all conditions remains constant. The number of significant side bands corresponding to some of the common values of modulation index is given in table below. Thus with an index of 5, there are 8 important sidebands on each side of the carrier with an index of 7, the sidebands increase to 10.


FM Principles

Number of significant sidebands

Modulation Index

Above carrier Below carrier

Bandwidth Required

(fm = frequency of audio signal)

0.1 1 1 2 fm

0.4 1 1 2 fm

0.5 2 2 4 fm

1.0 3 3 6 fm

2.0 4 4 8 fm

3.0 6 6 12 fm

4.0 7 7 14 fm

5.0 8 8 16 fm

6.0 9 9 18 fm

7.0 10 10 20 fm It is interesting to note that when the modulation index is of the order of 0.5 or less, only two sidebands are formed, which is similar to AM operation with one modulating frequency. It is quite confusing to note that although the carrier frequency in the FM transmitter is not shifted beyond the 75 kHz limits, sidebands do appear beyond these limits. As a physical analogy, consider a man moving his finger back and forth at the centre of a small pool of water. Although the man may move his finger only slightly, water ripples will appear far beyond this little area. The greater the distance covered by man’s moving finger, the larger will be the spread of ripples. In FM, the greater the carrier swing, the greater the number of sidebands obtained. In actual practice, it rarely happens that a 15 kHz note will have enough amplitude to spread the carrier to +75 kHz limits. As the frequency of the modulating signal is lowered the number of sidebands that extends beyond the 75 kHz limits also decreases until at 50 Hz a full carrier swing will just produce sidebands up to the 75 kHz limits. Bandwidth in FM In FM, the BW is based on the number of significant sidebands, which depends upon modulation index mf. In practice, the number of significant sidebands is determined by acceptable distortion. These contain about 98% of the radiated power. By way of best approximation, the Carson’s Rule (rule of thumb) gives a simple formula for bandwidth as BW = 2(1+mf)fm = 2(∆f + fm)


FM Transmitter

Guard band

fc-100 kHz fc-90 kHz fc fc+90 kHz fc+100 kHz

Fig. 5 BW of FM signal For modulation index of 5 and maximum modulating frequency of 15 kHz, we have: BW = 180 kHz A guard band of 20 kHz (10 kHz on each side) is provided to prevent adjacent channel interference. Thus the maximum permissible BW in FM broadcasting is 200 kHz. For narrow band FM (mf<0.5), the BW is the same as in AM i.e. 2 fm. When the modulation index is very large (say>20), then the BW becomes 2∆f i.e. 150 kHz. For example, if fm = 100 Hz and ∆f = 75 kHz.

750100

75000==

∆=

mf f

fmthen

In this case the BW will be 150 kHz, but for fm = 15 kHz, BW will be 180 kHz.

Noise Considerations In FM FM offers the advantage of a much better noise performance as compared to AM, the reasons for which are analysed here. The main parameter of interest at the input to the FM detector is the carrier-to-noise ratio (C/N). Since both the carrier and the noise are amplified equally by the various stages of the receiver from antenna input to the detector input, this gain can be ignored and the input to the detector can be represented by the voltage source Es, which is the carrier rms voltage as shown in fig 6(a). Also the thermal noise is spread over the IF bandwidth at the input to the FM detector.

R

(a)

FM Detector

fc-w fc ES

S/N

f fc+w

fn

δf

C/N

(b)

Fig. 6 Noise consideration in FM


FM Principles

At the input to FM detector, carrier power available = R4

E 2s

Available noise power = k TsBN Where Ts = Temp in degrees Kelvin, BN = IF bandwidth and

k = Boltzmann’s constant

Therefore, input carrier-to-noise ratio, ns

2s

BRkT4E

NC

= (1)

The noise voltage, being random, cannot in general be represented by a sinusoid. However, for a very small bandwidth δf, the noise voltage approaches a sinusoidal variation. The phasor diagram for the carrier and noise is shown in Fig.7. Here, it is assumed that the carrier is unmodulated except by the noise. This allows the noise to be shown as a phasor rotating at angular frequency ωn wrt the carrier, where fn = f-fc, and fn =noise frequency as shown in figure 6(b) . It may be seen that the noise produces two types of modulation of the carrier :- a) It changes the resultant amplitude of the signal thereby resulting in AM noise which is

filtered out by the amplitude limiter in FM receiver (before detection). b) It produces phase modulation as the phase of the resultant signal is different from the

phase of the original signal. The instantaneous value of phase modulation is φn(t), its maximum value being (refer to rt. Angle ∆ ABC)

s

n1max E

Esin−=φ (2)

If Es >> En,

s

nmax E

E≅φ (3)

Hence the phase modulation due to noise is given by

tsinEE

)t( ns

nn ω=φ (4)

90°

Es + EnEs - En Es

En

Bφ

90°

A φ

C

Fig. 7 Noise Produces Both Amplitude and Phase Modulation

As we know that phase modulation results in indirect frequency modulation, therefore, the noise indirectly frequency modulates the carrier. The equivalent frequency deviation is given by

)t(cosEE

fn)t(dt

d21f n

s

nnn ω=

φπ

=∆ (5)


FM Transmitter

The peak frequency deviation due to noise is given by

s

nnn E

EfF =∆ (6)

Thus the corresponding noise voltage at the output of the detector will be proportional to f, the amount by which the noise frequency is away from the carrier frequency, fc as shown in fig 8. In other words, we can say that the noise at the detector output

fn

∆Fn

Fig. 8 Noise characteristics at the detector output

increases linearly as the modulating frequency increases. This straight-line relationship plays an important role in the application of pre-emphasis and de-emphasis to the audio signal. Detector Processing Gain The processing gain of the detector is defined as

NCNSKR /

/= (7)

where S/N is signal-to-noise power ratio at the output of the detector and C/N is the carrier to noise power ratio at the detector input. It can be shown that KR = 3(1+mf) mf

2 (8) Where mf is the modulation index for the highest modulating frequency. If mf >> 1, then KR = 3 mf

3. If mf << 1, then KR = 3 mf 2. As shown by equation 7, the output

signal-to-noise (S/N) ratio can be increased by increasing the processing gain, the C/N remaining constant. Equation (8) shows that a high processing gain can be achieved by having a high modulation index. FM is a superior type of modulation system mainly because of its noise suppressing qualities. In an AM system noise can interfere even with the desired signal 100 times stronger and render the reception poor but in FM a noise signal half as strong as the desired signal can be suppressed completely. This effect becomes more and more pronounced as the frequency of the interfering signal approaches that of the desired signal so much so that the weaker noise signal is completely overpowered when their frequencies become equal. This is known as CAPTURE EFFECT. Let us assume that in the worst case, the noise amplitude En is half of the signal amplitude.


FM Principles

i.e. sn E21E =

Then in the right angled triangle ABC of Fig. 7,

radians52.0or30

21sin

EEsin

21

EEsin

o

1

s

n1

s

n

=

==φ

==φ

−−

At the highest modulating frequency of 15 kHz, the frequency deviation due to noise will be ∆fn = 15 x 0.5 = 7.5 kHz But maximum frequency deviation due to signal = 75 kHz. Therefore, the output S/N = 75/7.5 = 10

Thus the 2:1 C/N is transformed into 10:1 S/N at the output of the detector.

40

50

30

20

10

10 20 30 40 50

INPUT CNR, dB

OU

TPU

T SN

R, d

B

4.75dB

AM, ma = 1

THRESHOLD (13dB)

4.5dB

14dB

FM, mf = 1 FM, mf = 5FM, mf = 5

With 13dB Pre-emphasis

Not to scale All levels are in terms

of power ratio

Fig. 9 Detector Processing gain in FM We have seen that noise output of an FM receiver is directly proportional to phase deviation and also modulating frequency (difference in frequency of signal and noise). In case desired frequency and noise frequency are same or in other words fc –f = 0 the resulting indirect frequency deviation and consequently noise output is zero. As already stated this is known as the capture effect. In case two signals (the desired and the interference signal) are at the same frequency or Co-channel, the stronger of the two completely overpowers the weaker, which is not heard at all in the output. As the frequency difference increases the noise output also increases proportionately resulting in what is known as Triangular Noise Response of an FM receiver. Only at a point where difference in frequency becomes equal to direct frequency deviation of the carrier at the transmitter end, S/N at the output of FM receiver becomes equal to


FM Transmitter

S/N at its input. What is heard in the output is the difference in frequencies of the desired and the interfering/noise signals. As audible range of human ears ends at about 15 kHz a noise signal differing in more than 15 kHz in frequency is not heard and hence this is to be neglected. The other important factor is that with the increase in frequency difference, the interfering/noise signal suffers rejection at tuned circuits at the input of the receiver. Taking these factors into consideration noise/interference above 15 kHz has to be neglected. This factor is however applicable equally to AM also. This phenomenon has been depicted pictorially in figure 9. Noise in Narrowband FM Assuming that noise is uniformly spread over the receiver bandwidth, the noise output of an AM receiver remains constant and will be a rectangle. But in FM, the noise output is triangular and increases as we move away from the carrier frequency as shown in Fig.10

FM Noise Triangle

Fc fc+15 fc-75 fc-15 fc fc+15 fc+75

Rectangular AM distribution

a) Narrow band FM (mf=1) b) wideband FM (mf =5)

A

Fig. 10 Comparison of Noise performance of FM over AM It is seen from Fig.10(a) that the average improvement for narrow band FM over AM (point A) will be 2:1 at the average audio frequency of 7.5 kHz at which FM noise appears to be half of the AM noise voltage. But in reality, the picture is more complex and in fact the FM improvement is 3 :1 as a voltage ratio. This gives an increase of 3:1 in power signal-to-noise ratio for narrowband FM as compared to AM. This is equivalent to 4.75 dB improvement, which is quite worthwhile. Noise In Wideband FM In AM, the maximum permissible modulation index m= 1, but in FM there is no such limit. It is the maximum frequency deviation that is limited to 75 kHz in wideband VHF sound broadcasting service. At the highest audio frequency of 15 kHz the modulation index in FM is 5. It will be much higher at lower audio frequencies e.g. if modulating frequency is 1 kHz, the maximum value of modulation index in FM will be 75. It may be seen from figure 11 that as the modulation index is increased from mf =1 to mf = 4, the signal-to-noise voltage ratio will increase proportionately. Thus the S/N power ratio in a FM receiver is proportional to the square of the modulation index. For mf = 5 and modulating frequency of 15 kHz, there will be a 25:1 (14 dB) improvement for FM, as compared to when mf = 1. No such improvement is possible in AM. For an


FM Principles

adequate C/N ratio at the detector input, an overall improvement of 18.75 (4.75 + 14) dB is achieved with wideband FM as compared with AM.

+60fc +15

INAUDIBLEFM NOISEAUDIBLE

FM NOISE

AM NOISE

+45fc +15 +30 fc +15 fc +15

mf = 4 mf = 3 mf = 2 mf = 1

Fig. 11 FM noise increases with reduced modulation index, mf

Threshold Effect The above analysis is not applicable to the case when noise and signal levels are comparable because then the rotating noise phasor En may encircle the origin of Es with the result that the phase modulation is no longer limited to a maximum value as given by equation (3), nor can the noise modulation be assumed to be sinusoidal as in equation (4). In this case there is a sudden decrease in output S/N when the C/N drops below a certain level called the threshold level. For conventional FM detectors, threshold occurs for about 10 to 12 dB as shown in figure 12. Equation 7 can be rewritten as

[ ]dBRdBdB

R101010

KNC

NSor

Klog10NClog10

NSlog10

+

=

+=

This equation applies only for C/N values above threshold can be used to calculate KR for a given value of modulation index. Thus if we want to have the benefits of better noise performance of FM, the input CNR(power) should be greater than about 12 dB. This is called threshold effect. Threshold For practical reasons, the threshold is defined as the value of (C/N) at which the actual curve drops 1 dB below the straight line projection as shown in figure 12.

10 20 30

30dB

20dB

10dB

(C/N) dB

Thresholdmargin

1 dB

Operatingpoint

(S/N)

Not toscale

Fig. 12 Threshold Effect


FM Transmitter

Threshold Margin

The difference in dBs between the operating point (C/N) and the threshold (C/N) is called the threshold margin as shown in figure 12.

Impulse Noise

When a pulse is applied to a tuned circuit, its peak amplitude is proportional to the square root of the bandwidth of the circuit. Similarly, if a noise impulse is applied to the tuned circuit in the IF section of an FM receiver, a large noise pulse will result. When the noise pulses exceed about one-half the carrier size at the amplitude limiter, the limiter fails. When noise pulses exceed carrier amplitude, the limiter limits the signal having been “captured” by noise. Therefore, the maximum deviation and bandwidth cannot be increased indefinitely. As a compromise, the maximum frequency deviation of 75 kHz has been permitted. It can be shown that under ordinary circumstances (En<Es/2), impulse noise in FM is reduced to the same extent as random noise.

Pre-emphasis and De-emphasis

According to noise triangle, the noise output of FM detector increases linearly as the modulating frequency increases. Also we know that in a complex audio signal, the higher audio frequencies are weaker in amplitudes. Thus it is a double tragedy for the high audio frequencies, their amplitudes are small but they have to face higher noise levels as compared to lower audio frequencies. To overcome this problem, the higher audio frequencies are given an artificial boost at the transmitter in accordance with a pre-arranged curve. This process is called pre-emphasis. In the FM receiver, the higher audio frequencies are restored to their normal levels through a reverse process called de-emphasis. The de-emphasis curve is the mirror image of pre-emphasis curve as shown in figure 13.

2kHz kHz 15 kHz

+3dB

0dB

-3dB

Pre-emphasis

3.180 kHz

13 dB

De-emphasis

-13 dB

Frequency

(Not to scale) Fig. 13 Pre-emphasis and De-emphasis


FM Principles

Pre-emphasised AF output

L/R = 50 µ sec.

AF in from FM Detector

Cc L

R

AF in

+V

AF output R

RC = 50 µ sec.

a) Pre-emphasis circuit b) De-emphasis circuit

C

Fig. 14 Pre-emphasis and De-emphasis Circuit

Two types of curves i.e. 50 µ sec and 75 µ sec are in vogue in FM sound broadcasting but AIR has adopted 50 µ sec curve which gives about 13 dB boost at 15 kHz and is 3 dB at the frequency of 3.180 kHz (f = 1/2πRC, where RC = 50 µ sec Figure 15 (a) & (b) illustrate the effect of pre-emphasis on the modulating signal frequency response at the transmitter whereas (c) & (d) show the effect of de-emphasis on the modulating signal and noise at the FM receiver. The de-emphasis cancels out the pre-emphasis on the signal and also attenuates the noise at the receiver. The overall effect is to leave the post detection signal levels unchanged while the high frequency noise is attenuated. Subjective tests with 50 µ sec pre-emphasis and de-emphasis give an improvement of about 4.5 dB. However, one should be cautious that the pre-emphasised signal does not over modulate the carrier by exceeding the 75 kHz deviation. Typical pre-emphasis and d-emphasis circuits are shown in Figs. 14(a) & (b) respectively.

Output before

de-emphasis

Modulating Signal

f

V

a) b)

Modulating Signal after pre-emphasis

f

V

Signal

f

V

Noise

Output after de-

emphasis

Signal

f

V

Noise

c) d)

Fig. 15 Effect of pre-emphasis and De-emphasis on Modulating signal If we compare a 100% AM system with a 100% (mf = D=5) FM system it can be shown that


FM Transmitter

75)D(3)am(N

S)fm(N

S2 ==

Or in decibels S/N improvement in FM over a corresponding AM system is 10 log 75 = 18.75 dB S/N improvement in FM over a corresponding AM system with pre-emphasis/de-emphasis (18.75 + 4.25) = 23.25 dB. (Fig. 16)

2.83 times increase due to pre emphasis at TX & DE at receiver

25 times increase due to wider modulation pass

band (mf = 5)

3 times increase due to phase modulation by noise

Amplitude Modulation S/N

Improvement in S/N Power

Ratio

Improvement (dB)

Fig. 16 S/N Improvement due to FM Transmission

212

75

3

1 0

4.75

18.75

23.25

Other Forms of Interference FM offers not only an improvement in the S/N ratio but also better immunity against other interfering signals.

a) Co-channel Interference

In an FM receiver, the amplitude limiter works on the principle of passing the stronger signal and eliminating the weaker one if the stronger signal is at least twice the amplitude of the weaker signal. In a similar fashion, a relatively weak interfering signal (in the same channel) from another transmitter will also be eliminated. Suppose we are carrying a FM receiver and moving from the coverage area of one transmitter towards that of another co-channel transmitter. The interesting phenomenon of capture will be noted in this case. The second transmitter is virtually inaudible till its signal is less than about half of that from the first. After this, the second transmitter becomes quite audible in the background and eventually dominates, ultimately finishing the first transmitter signal. Thus the moving receiver has been captured by the second transmitter. But in AM, the effect would be totally different, the stronger signal will dominate but the weaker one will also be heard in the background as quite significant interference.


FM Principles


b) Adjacent-channel Interference

We have seen that FM signal with maximum deviation of 75 kHz and 50 µ sec pre-emphasis gives a rejection of 23 to 24 dB better than AM, for noise as well as interfering signals. A guard band of 10 kHz provided on each side of the FM channel also serves to reduce adjacent channel interference. Advantages Of FM over AM

1. Amplitude and hence power of FM wave is constant and independent of depth of

modulation. But in AM, modulation depth determines the transmitted power. Thus additional energy is not required as modulation is raised.

2. FM is more economical than AM due to following reasons :

(a) It is possible to have Low Level Modulation in FM as the intelligence is in the

frequency variations only and the modulated signal can be passed through class C amplifiers. But since the AM signal contains information in amplitude variations, so only high level modulation is possible in an AM transmitter.

(b) All the transmitted power in FM is useful whereas in AM most of it is in the carrier which

contains no useful information.

(c) Antenna gain is possible in FM due to the reason that directive antennas are used in VHF range where the physical dimensions of the antenna are very easy to manage.

3. Better Noise Performance

• Amplitude variations caused by noise are removed by having limiter in FM receiver.

This makes FM reception lot more immune to noise than AM reception. • Noise can further be reduced in FM by increasing the frequency deviation. This is not

possible in AM as modulation cannot exceed 100 % without causing severe distortion . • Less adjacent- channel interference due to better planning as the commercial FM

broadcasts began in 1940s (much later than AM) ------ a guard band has been provided as per CCIR standard frequency allocations.

• FM broadcasts operate in the VHF and UHF ranges in which there happens to be less noise than in the MF and HF ranges occupied by AM bands.

• Due to the use of space wave propagation in which the range of operation is limited to slightly more than line of sight, it is possible to operate several independent transmitters with much less co-channel interference.

4. Stereo transmission is possible with FM due to its wider bandwidth 5. Additional information such as RDS, SCA can be sent along with the stereo signal

Documents

01 FM Principles