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# Chapter 4. Angle Modulation

Chapter 4. Angle Modulation. 4.4 Narrow-Band Frequency Modulation. We first consider the simple case of a single-tone modulation that produces a narrow-band FM wave We next consider the more general case also involving a single-tone modulation, but this time the FM wave is wide-band

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## Chapter 4. Angle Modulation

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1. Chapter 4. Angle Modulation

2. 4.4 Narrow-Band Frequency Modulation • We first consider the simple case of a single-tone modulation that produces a narrow-band FM wave • We next consider the more general case also involving a single-tone modulation, but this time the FM wave is wide-band • The two-stage spectral analysis described above provides us with enough insight to propose a useful solution to the problem • An FM signal is The frequency deviation Modulation index The phase deviation

3. The FM wave is • If the modulation index is small compared to one radian, the approximate form of a narrow-band FM wave is • The envelope contains a residual amplitude modulation that varies with time • The angel θi(t) contains harmonic distortion in the form of third- and higher order harmonics of the modulation frequency fm

4. We may expand the modulated wave further into three frequency components • The basic difference between and AM wave and a narrow-band FM wave is that the algebraic sign of the lower side-frequency in the narrow-band FM is reversed • A narrow-band FM wave requires essentially the same transmission bandwidth as the AM wave.

5. Phasor Interpretation • A resultant phasor representing the narrow-band FM wave that is approximately of the same amplitude as the carrier phasor, but out of phase with respect to it. • The resultant phasor representing the AM wave has a different amplitude from that of the carrier phasor, but always in phase with it.

6. 4.5 Wide-Band Frequency Modulation • Assume that the carrier frequency fc is large enough to justify rewriting Eq. 4.15) in the form • The complex envelope is

7. The complex Fourier coefficient

8. In the simplified form of Eq. (4.29)

9. Properties of single-tone FM for arbitrary modulation index β • For different integer values of n, • For small values of the modulation index β • The equality holds exactly for arbitrary β

10. The spectrum of an FM wave contains a carrier component and an infinite set of side frequencies located symmetrically on either side of the carrier at frequency separations of fm,2fm, 3fm…. • The FM wave is effectively composed of a carrier and a single pair of side-frequencies at fc±fm • The amplitude of the carrier component of an FM wave is dependent on the modulation index β The average power of such a signal developed across a 1-ohm resistor is also constant. • The average power of an FM wave may also be determined form

11. 4.6 Transmission Bandwidth of FM waves • Carson’s Rule • The FM wave is effectively limited to a finite number of significant side-frequencies compatible with a specified amount of distortion • Two limiting cases • For large values of the modulation index β, the bandwidth approaches, and is only slightly greater than the total frequency excursion 2∆f, • For small values of the modulation index β, the spectrum of the FM wave is effectively limited to the carrier frequency fc and one pair of side-frequencies at fc±fm, so that the bandwidth approaches 2fm • An approximate rule for the transmission bandwidth of an FM wave

12. Universal Curve for FM Transmission Bandwidth • A definition based on retaining the maximum number of significant side frequencies whose amplitudes are all greater than some selected value. • A convenient choice for this value is one percent of the unmodulated carrier amplitude • The transmission bandwidth of an FM waves • The separation between the two frequencies beyond which none of the side frequencies is greater than one percent of the carrier amplitude obtained when the modulation is removed. • As the modulation index β is increased, the bandwidth occupied by the significant side-frequencies drops toward that value over which the carrier frequency actually deviates.

13. Arbitrary Modulating Wave • The bandwidth required to transmit an FM wave generated by an arbitrary modulating wave is based on a worst-case tone-modulation analysis • The deviation ratio D • The generalized Carson rule is

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