ANALOG COMMUNICATION Lecture 12

FM Bandwidth and FM Generation

Lesson 12 EEE 352 Analog Communication Systems Mansoor Khan EE Department CIIT Islamabad Campus

Frequency Modulation

Frequency Modulation • The magnitude and direction of the frequency shift (Δf) is proportional to the amplitude of the modulating signal.

• The rate at which the frequency changes are occurring is equal to the frequency of the modulating signal (fm). • (Δf) is called the frequency deviation.

Phase Modulation • The magnitude and direction of the phase shift (Δө) is proportional to the amplitude of the modulating signal.

• The rate at which the phase changes are occurring is equal to the frequency of the modulating signal (fm). • (Δө) is called the phase deviation

Mathematical Analysis For Phase:

e = Ec cos[ωct + KpEicos(ωit)]

For Frequency:

e = Ec cos[ωct + (KfEi/ ωi)sin(ωit)]

Kp = Deviation Sensitivity of the Phase Modulator Kf = Deviation Sensitivity of the Frequency Modulator

Frequency Modulation Waveform Mathematical Analysis In FM, the message signal m(t) controls the frequency fc of the carrier. Consider the carrier

then for FM we may write: FM signal

,where the frequency deviation

will depend on m(t). Given that the carrier frequency will change we may write for an instantaneous carrier signal

where i is the instantaneous angle =

and fi is the instantaneous frequency.

Bessel Function

Examples from the graph  = 0: When  = 0 the carrier is unmodulated and J0(0) = 1, all other Jn(0) = 0, i.e.

 = 2.4: From the graph (approximately)

J0(2.4) = 0, J1(2.4) = 0.5, J2(2.4) = 0.45 and J3(2.4) = 0.2

Significant Sidebands – Spectrum. As may be seen from the table of Bessel functions, for values of n above a certain value, the values of Jn() become progressively smaller. In FM the sidebands are considered to be significant if Jn()  0.01 (1%).

Although the bandwidth of an FM signal is infinite, components with amplitudes VcJn(), for which Jn() < 0.01 are deemed to be insignificant and may be ignored. Example: A message signal with a frequency fm Hz modulates a carrier fc to produce FM with a modulation index  = 1. Sketch the spectrum. n 0 1 2 3 4 5

Jn(1) 0.7652 0.4400 0.1149 0.0196 0.0025 0.0002

Amplitude 0.7652Vc 0.44Vc 0.1149Vc 0.0196Vc Insignificant Insignificant

Frequency fc fc+fm fc - fm fc+2fm fc - 2fm fc+3fm fc -3 fm

Significant Sidebands – Spectrum.

As shown, the bandwidth of the spectrum containing significant components is 6fm, for  = 1.

Significant Sidebands – Spectrum. The table below shows the number of significant sidebands for various modulation indices () and the associated spectral bandwidth.  0.1 0.3 0.5 1.0 2.0 5.0 10.0

No of sidebands  1% of unmodulated carrier 2 4 4 6 8 16 28

Bandwidth 2fm 4fm 4fm 6fm 8fm 16fm 28fm

e.g. for  = 5, 16 sidebands (8 pairs).

Carson’s Rule for FM Bandwidth. An approximation for the bandwidth of an FM signal is given by BW = 2(Maximum frequency deviation + highest modulated frequency)

Bandwidth  2(f c  f m )

Carson’s Rule

Narrowband and Wideband FM Narrowband FM NBFM

From the graph/table of Bessel functions it may be seen that for small , (  0.3) there is only the carrier and 2 significant sidebands, i.e. BW = 2fm. FM with   0.3 is referred to as narrowband FM (NBFM) (Note, the bandwidth is the same as DSBAM).

Wideband FM WBFM For  > 0.3 there are more than 2 significant sidebands. As  increases the number of sidebands increases. This is referred to as wideband FM (WBFM).

For FM Modulator with frequency deviation of 10 kHz, modulating signal frequency of 10 kHz, Carrier amplitude voltage of 50V and Carrier frequency of 500 kHz, determine the following: (a) Minimum Bandwidth using Bessel table (b) Minimum Bandwidth using Carson’s rule (c) Amplitudes of the side frequencies and plot the output frequency spectrum

a)From Bessel function table, m=1 yields three sets of significant sidebands. Thus bandwidth is m

f 10kHz  1 f m 10kHz

B  2(3 10kHz)  60kHz b) Approx. minimum bandwidth is given by Carson’s rule. So

B  2(10kHz  10kHz)  40kHz

c)

J 0  0.77(50)  38.5V J1  0.24(50)  12V J 2  0.11(50)  5.5V J 3  0.02(50)  1V 38.5V 12V

12V 5.5V

5.5V 1V

470

1V

480

490

500

510

520

530

Armstrong Indirect FM Transmitter

Demodulation of FM (cont)

Bandpass Limiter