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Noise power spectral density
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Noise voltage spectral density
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Figure 9.11 (a) Output noise power spectral density for FM. (b) The corresponding noise voltage spectral density.
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would be the sum of all such increments, which is twice the area under the curve of Fig. 9.11a, twice because of the noise contributions from both sides of the carrier. The detailed integration required to evaluate the noise will not be carried out here, but the end result giving the signal power to noise ratio is S N Ps Pn 1.5
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2 C BN f N W3
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(9.8)
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The processing gain of the detector is the ratio of signal-to-noise ratio to carrier-to-noise ratio. Denoting this by GP gives GP S/N C/N 1.5 BN f 2 W3 Using Carson s rule for the IF bandwidth, BIF 2( f W), and assuming BN BIF, the processing gain for sinusoidal modulation becomes after some simplification GP 3s 1d
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(9.9)
(9.10)
Here, f/W is the modulation index for a sinusoidal modulation frequency at the highest value W. Equation (9.10) shows that a high modulation index results in a high processing gain, which means that the signal-to-noise ratio can be increased even though the carrier-to-noise ratio is constant.
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9.6.4 Signal-to-noise ratio
The term signal-to-noise ratio introduced in Sec. 9.6.3 is used to refer to the ratio of signal power to noise power at the receiver output. This ratio is sometimes referred to as the postdetector or destination signalto-noise ratio. In general, it differs from the carrier-to-noise ratio at the detector input (the words detector and demodulator may be used interchangeably), the two ratios being related through the receiver processing gain as shown by Eq. (9.9). Equation (9.9) may be written in decibel form as 10 log10 S N 10 log10 C N 10 log10GP (9.11)
As indicated in App. G, it is useful to use brackets to denote decibel quantities where these occur frequently. Equation (9.11) therefore may be written as c S d N c C d N [GP] (9.12)
This shows that the signal-to-noise in decibels is proportional to the carrier-to-noise in decibels. However, these equations were developed for the condition that the noise voltage should be much less than the carrier voltage. At low carrier-to-noise ratios this assumption no longer holds, and the detector exhibits a threshold effect. This is a threshold level in the carrier-to-noise ratio below which the signal-to-noise ratio degrades very rapidly. The threshold level is shown in Fig. 9.12 and is defined as the carrier-to-noise ratio at which the signal-to-noise ratio is 1 dB below the straight-line plot of Eq. (9.12). For conventional FM detectors (such as the Foster Seeley detector), the threshold level may be taken as 10 dB. Threshold extension detector circuits are available which can provide a reduction in the threshold level of between 3 and 7 dB (Fthenakis, 1984). In normal operation, the operating point will always be above threshold, the difference between the operating carrier-to-noise ratio and the threshold level being referred to as the threshold margin. This is also illustrated in Fig. 9.12.
Example 9.4 A 1-kHz test tone is used to produce a peak deviation of 5 kHz in an
FM system. Given that the received [C/N] is 30 dB, calculate the receiver processing gain and the postdetector [S/N].
Solution Since the [C/N] is above threshold, Eq. (9.12) may be used. The modulation index is
5 kHz/1 kHz
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Figure 9.12 Output signal-to-noise ratio S/N versus input carrierto-noise ratio C/N for a modulating index of 5. The straight-line section is a plot of Eq. (9.12).
Hence GP 3 52 s5 1d 450
and [GP] From Eq. (9.12) [S/N] 30 26.5 56.5 dB 26.5 dB
9.6.5 Preemphasis and deemphasis
As shown in Fig. 9.11b, the noise voltage spectral density increases in direct proportion to the demodulated noise frequency. As a result, the signal-to-noise ratio is worse at the high-frequency end of the baseband,
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a fact which is not apparent from the equation for signal-to-noise ratio, which uses average values of signal and noise power. For example, if a test tone is used to measure the signal-to-noise ratio in a TV baseband channel, the result will depend on the position of the test tone within the baseband, a better result being obtained at lower test tone frequencies. For FDM/FM telephony, the telephone channels at the low end of the FDM baseband would have better signal-to-noise ratios than those at the high end. To equalize the performance over the baseband, a deemphasis network is introduced after the demodulator to attenuate the high-frequency components of noise. Over most of the baseband, the attenuation-frequency curve of the deemphasis network is the inverse of the rising noisefrequency characteristic shown in Fig. 9.11b (for practical reasons it is not feasible to have exact compensation over the complete frequency range). Thus, after deemphasis, the noise-frequency characteristic is flat, as shown in Fig. 9.13d. Of course, the deemphasis network also will attenuate the signal, and to correct for this, a complementary preemphasis characteristic is introduced prior to the modulator at the transmitter. The overall effect is to leave the postdetection signal levels unchanged while the high-frequency noise is attenuated. The preemphasis, deemphasis sequence is illustrated in Fig. 9.13. The resulting improvement in the signal-to-noise ratio is referred to variously as preemphasis improvement, deemphasis improvement, or simply as emphasis improvement. It is usually denoted by P, or [P] decibels, and gives the reduction in the total postdetection noise power. Preemphasis curves for FDM/FM telephony are given in CCIR Recommendation 275-2 (1978) and for TV/FM in CCIR Recommendation 405-1 (1982). CCIR values for [P] are 4 dB for the top channel in multichannel telephony, 13.1 dB for 525-line TV, and 13.0 dB for 625-line TV. Taking into account the emphasis improvement, Eq. (9.12) becomes c S d N c C d N [GP] [P] (9.13)
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