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is possible but at the expense of an increase in the IF bandwidth. This is the large-amplitude signal improvement referred to in Sec. 9.6 and considered further in the following section.
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9.6.3 FM detector noise and processing gain
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At the input to the FM detector, the thermal noise is spread over the IF bandwidth, as shown in Fig. 9.10a. The noise is represented by the system noise temperature Ts, as will be described in Sec. 12.5. At the input to the detector, the quantity of interest is the carrier-to-noise ratio. Since both the carrier and the noise are amplified equally by the receiver gain following the antenna input, this gain may be ignored in the carrierto-noise ratio calculation, and the input to the detector represented by the voltage source shown in Fig. 9.10b. The carrier root-mean-square (rms) voltage is shown as Ec.
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Figure 9.10 (a) The predetector noise bandwidth BN is approximately equal to the IF bandwidth BIF. The LF bandwidth W fixes the equivalent postdetector noise bandwidth at 2W. df is an infinitesimally small noise bandwidth. (b) Receiving system, including antenna represented as a voltage source up to the FM detector.
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2 The available carrier power at the input to the FM detector is Ec /4R, and the available noise power at the FM detector input is kTs BN (as explained in Sec. 12.5), so the input carrier-to-noise ratio, denoted by C/N, is
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E2 c 4RkTs BN
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(9.5)
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When a sinusoidal signal of frequency, fm, frequency modulates a carrier of frequency, fc: The instantaneous frequency is given by f sin 2 fmt , where f is peak frequency deviation. The output fi fc signal power following the FM detector is Ps A f2 (9.6)
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where A is a constant of the detection process. The thermal noise at the output of a bandpass filter, for which fc >> BN has a randomly varying amplitude component and a randomly varying phase component. (It cannot directly frequency modulate the carrier, the frequency of which is determined at the transmitter, which is at a great distance from the receiver and may be crystal controlled). When the carrier amplitude is very much greater than the noise amplitude the noise amplitude component can be ignored for FM, and the carrier angle as a function of time is std 2 fct nstd, where n(t) is the noise phase modulation. Now the instantaneous frequency of a phase modulated wave in general is given by i d (t)/dt and since i 2 fi, the equivalent FM resulting from the noise phase modulation is feq.n fc 1 d nstd 2 dt (9.7)
What this shows is that the output of the FM detector, which responds to equivalent FM, is a function of the time rate of change of the phase change. Now as noted earlier, the available noise power at the input to the detector is kTsBN and the noise spectral density, which is the noise power per unit bandwidth just kTs. A result from Fourier analysis is that the power spectral density of the time derivative of a waveform is (2 f )2 times the spectral density of the input. Thus the output spectral density as a function of frequency is (2 f )2 kTs. The variation of output spectral noise density as a function of frequency is sketched in Fig. 9.11a. Since voltage is proportional to the square root of power, the noise voltage spectral density will be proportional to frequency as sketched in Fig. 9.11b. Figure 9.11a shows that the output power spectrum is not a flat function of frequency. The available noise output power in a very small band 2 df would be given by s2 fd kTs f . The total average noise output power
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