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**COMMUNICATION SYSTEM EECB353 Chapter 3(II) ANGLE MODULATION**

Dept of Electrical Engineering Universiti Tenaga Nasional

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**Frequency Analysis of Angle-Modulated Waves**

The output of angle-modulated signal of can be expressed as a series of sinusoids by using Bessel function of the first kind. The table and graph of Bessel function represents normalized voltages for the various freq components of an FM and PM signals. (i.e the number in the table represents actual voltages if the unmodulated carrier has an amplitude of 1V).

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**Consider the result of expression m(t) above:**

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**Frequency Analysis of Angle-Modulated Waves**

Note that amplitude values with –ve sign represent phase shifts of 180 and SB freqs are considered insignificant if their amplitude values less than 0.01 of the unmodulated carrier amplitude.

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**Frequency Analysis of Angle-Modulated Waves**

Figure 1: FM in Frequency Domain

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**Frequency Analysis of Angle-Modulated Waves**

Generally, J0(m) determines the amplitude of the carrier. The nth Bessel function Jn(m) determines the amplitude of the nth pair of sidebands All the Bessel terms should be multiplied Vc. to find the actual SB amplitude. If m = 0 zero SB, if m , SB pairs Example of the meaning Jn(m): J1(2.5)- represent the amplitude of the 1st pair of sidebands for the modulated signal m(t) with m = 2.5. J1(2.5)- from the table = 0.5, means that 1st SB pairs amplitude are 50% of the unmodulated carrier amplitude (i.e = 0.5Vc).

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Jn(m) versus m Figure shows the curve for the relative amplitudes of the carrier and several sets of side frequencies for values of m up to 10. It can be seen that the amplitudes of both the carrier and the side frequencies vary at a periodic rate that resembles a damped sine wave. The negative values of J(m) indicate the relative phase of that side frequency. 1st carrier null

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**Example 1 Consider an angle-modulation system where the message signal**

m(t)=2 sin(5000t) , frequency deviation sensitivity, Kf = 3000Hz/v and phase deviation sensitivity, Kp = 1.2rad/v. Given the carrier frequency is 1MHz and amplitude of 10V. Determine, a) The peak frequency and phase deviation. b) The modulation index for both modulators. c) Number of sets of significant side frequencies d) Their amplitudes, then e) Draw the frequency spectrum showing their relative amplitudes.

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**Bandwidth Requirement**

The actual BW required to pass all the significant SBs for an angle-modulated wave is equal to two times the product of the highest modulating-signal frequency, fm and the number of SBs, n found from the Bessel function. Carson’s rule approximate the BW necessary to transmit an angle-modulated wave as twice the sum of peak freq deviation, f and the highest modulating-signal freq.fm.

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Example 2 a) For an FM modulator with a peak frequency deviation f = 10kHz, a modulating-signal frequency fm = 10kHz, Vc = 10V, and a 500kHz carrier, determine Actual minimum bandwidth from the Bessel function table Approximate minimum bandwidth using Carson’s rule, then Plot the output frequency spectrum for the Bessel approximation. b) Compare a percentage saving of a bandwidth in an FM system using Carson’s rule and the one obtained using Bessel Function table. Given a modulating signal frequency is 3 kHz and maximum deviation of 12 kHz

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Deviation Ratio (DR) Definition – is defined as the worst-case modulation index and is equal to the maximum peak frequency deviation divided by the maximum modulating-signal frequency. DR = deviation ratio (unitless) f(max) = maximum peak freq deviation (Hz) f(max) = maximum modulating-signal freq (Hz) The worst-case modulation index produces the widest output freq spectrum. Example, for the maximum frequency deviation of 50kHz and the maximum modulating signal frequency of 15kHz, the DR is 3.33. This does not mean that whenever a modulation index of 3.33 occurs, the widest BW also occurs at the same time. It means that whenever a modulation index of 3.33 occurs for a maximum modulating-signal freq, the widest BW occurs.

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**Frequency Analysis of Angle-Modulated Waves**

Example 3 Determine the deviation ratio and bandwidth for the worse-case (widest-bandwidth) modulation index for an FM broadcast-band transmitter with a maximum frequency deviation of 75kHz and a maximum modulating-signal frequency of 15kHz. Determine the deviation ratio and maximum bandwidth for an equal modulation index with only half the peak frequency deviation and modulating-signal frequency.

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**Frequency Analysis of Angle-Modulated Waves**

Example 4 Given FM and PM modulators with the following parameters: A) Determine the modulation index and sketch the output spectrums for both modulators. B) Change the modulating signal amplitude for both modulators to 4Vp and repeat step (a) C) Change the modulating signal frequency for both modulators to 1kHz and repeat step (a)

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**Average Power of an Angle-Modulated Wave**

With angle modulation, the power that was originally in the unmodulated carrier is redistributed among the carrier and its sidebands. It’s independent of modulating signal, modulation index and frequency deviation. The average power in the unmodulated carrier is : where Vc = peak unmodulated carrier voltage (volts) R = load resistance (ohm)

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**Average Power of an Angle-Modulated Wave**

The modulated carrier power is the sum of the power of the carrier and side frequency components. Therefore, total modulated wave power is :

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EXAMPLE 5 : Determine the unmodulated carrier power for the FM modulator with a modulation index, m =1, a modulating signal vm(t)=Vmsin(21000t), and an unmodulated carrier vc(t)=10sin(2500kt), assuming load resistance RL = 50. Determine the total power in the angle-modulated wave.

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EXAMPLE 6 : Consider an FM Modulator with output Xc(t)=100cos[21000t ), The modulator operates with Kf=8 and has the input message signal vm(t)=5 cos(28t), The modulator is followed by a BPF with a centre frequency of 1000 Hz and a bandwidth 56Hz. Determine : Peak frequency deviation Modulation index Number of sets of significant side frequencies Their amplitude Plot the output frequency spectrum for the Bessel approximation –show all the amplitude and frequency Power of the unmodulated carrier Power at the filter output Power at modulator output

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