Communication Systems · 96 h · Topic 7 of 11

Noise in analog modulations

Above the threshold, an FM with β = 5 outperforms the best linear modulation by 16 dB. About 5 dB below the threshold, it is already worse than DSB.

Signal-to-noise ratio DSB, SSB and AM FM improvement FM threshold Pre-emphasis

01A common reference

To compare modulations we need a measure that does not depend on them. The reference taken is the transmission of the message in baseband, unmodulated, with the same received power \( S_i \) and white noise of density \( N_0/2 \) over the bandwidth \( W \) of the message:

\[ \gamma = \frac{S_i}{N_0 W} \] γ is the signal-to-noise ratio that would be obtained in baseband. Each modulation is characterized by the ratio at the demodulator output, \( \mathrm{SNR_o} \), compared with γ. Thermal noise and its density were covered in the introduction to the subject.

The noise that reaches the demodulator is bandpass. Written with its complex envelope, it has an in-phase component and a quadrature component, independent and of equal power. A coherent demodulator sees only the in-phase component; an envelope or frequency detector sees both.

02Linear modulations

In DSB-SC with coherent demodulation, the signal occupies \( 2W \) and so does the noise, but the receiver discards the quadrature component of the noise and adds the two sidebands coherently. In SSB the signal occupies \( W \) and half the noise gets in. In both cases the result is that of baseband:

\[ \mathrm{SNR_o^{DSB}} = \mathrm{SNR_o^{SSB}} = \gamma \] \[ \mathrm{SNR_o^{AM}} = \eta\, \gamma = \frac{m^2 \langle s^2 \rangle}{1 + m^2 \langle s^2 \rangle}\, \gamma \] In AM the carrier consumes power and contributes nothing to the message: with a tone and m = 1 the efficiency is one third, and AM ends up 4.8 dB below DSB. The formula holds with coherent detection, or with an envelope detector if the carrier is much larger than the noise.

When the noise approaches the carrier, the envelope detector follows the noise more than the message and the output degrades much faster than γ: this is the threshold effect of AM, which appears at carrier-to-noise ratios of about 10 dB. The coherent detector has no threshold.

03FM: quality in exchange for bandwidth

In FM the message is in the instantaneous frequency, which is the derivative of the phase. Phase noise, when differentiated, has a density that grows with the square of the frequency: the output noise is concentrated at the high frequencies. With the message normalized to \( |s| \le 1 \) and a high carrier-to-noise ratio, the result is:

\[ \mathrm{SNR_o^{FM}} = 3\beta^2 \langle s^2 \rangle\, \gamma \] With a tone, \( \langle s^2 \rangle = 1/2 \) and the improvement over DSB is \( 1.5\,\beta^2 \): 15.7 dB with β = 5. The bandwidth, by Carson's rule, grows only with β, so doubling the bandwidth improves the ratio about four times.

The gain has a limit: the FM threshold. If the carrier-to-noise ratio in the Carson bandwidth falls below about 10 dB, the noise makes the phasor circle around the origin, the phase jumps by \( 2\pi \) and the discriminator delivers impulses, the clicks heard on a badly tuned FM radio. Since the bandwidth grows with β, the threshold is reached at larger γ the wider the FM:

\[ \frac{S_i}{N_0 B_T} = \frac{\gamma}{2(\beta + 1)} \gtrsim 10 \] With β = 5 the threshold is at γ ≈ 20 dB. Above it, FM outperforms any linear modulation; below it, it collapses. Demodulators with a PLL or with frequency feedback lower the threshold by a few dB.

Lab · output SNR and the FM threshold

The output signal-to-noise ratio as a function of γ, with a tone. The gray line is DSB and SSB; the orange one, AM with m = 1; the blue one, FM with the threshold effect, and the dotted one, ideal FM above the threshold. The threshold is modeled with the classic Taub and Schilling approximation for a tone.

Modulation index β =

γ =

04Pre-emphasis and de-emphasis

Since FM noise grows with frequency and audio has little energy at the high frequencies, the transmitter boosts the high frequencies with a pre-emphasis network, and the receiver attenuates them with the inverse network, the de-emphasis. The message stays the same, but the noise at the high frequencies goes down. In commercial FM the time constant is 75 μs in the United States and 50 μs in Europe:

\[ I = \frac{(W/f_1)^3}{3\,\big[W/f_1 - \arctan(W/f_1)\big]} \] \( f_1 = 1/(2\pi\tau) \) is the cutoff frequency of the network. With τ = 75 μs, \( f_1 \) = 2.12 kHz and, for W = 15 kHz, the improvement is 13.2 dB; with 50 μs, 10.2 dB. It is assumed that pre-emphasis does not increase the maximum deviation.

05In the lab

Activity 1 · The FM threshold

Receive an FM signal from a generator, add noise with a white noise generator and lower the signal until clicks appear. Measure the carrier-to-noise ratio at that point with the spectrum analyzer.

Activity 2 · AM versus FM

With the same signal power and the same noise, compare the audio signal-to-noise ratio of an AM receiver and an FM receiver.

Activity 3 · De-emphasis

Measure the noise at the discriminator output with and without the de-emphasis network, as a function of frequency.

06Common mistakes

Believing that DSB improves on SSB because it occupies twice the bandwidth. Twice the noise gets in: the output ratio is the same.
Extrapolating the FM improvement to any signal level. Well below the threshold, FM is worse than a linear modulation.
Comparing modulations with different power. The comparison is made with the same \( S_i \) and the same \( N_0 \).
Forgetting that FM noise is not white at the output. It grows with frequency, which is why pre-emphasis helps.

07Self-assessment

How many dB does AM with m = 1 and a tone lose relative to DSB?

\( 10\log_{10} 3 \approx 4.8 \) dB.

What improvement over DSB does an FM with β = 2 and a tone give?

\( 1.5 \cdot 4 = 6 \): 7.8 dB.

With β = 4, what is the minimum γ for operating above the threshold?

\( 10 \cdot 2 \cdot 5 = 100 \): about 20 dB.

Why are clicks heard on an FM radio with a weak signal?

The noise makes the phase turn \( 2\pi \) and the discriminator delivers an impulse on each turn.

08Further reading

Herbert Taub and Donald L. Schilling. Principles of Communication Systems. 2nd ed., McGraw-Hill, 1986. Noise in AM and FM with the calculation of the threshold and the clicks.
B. P. Lathi and Zhi Ding. Modern Digital and Analog Communication Systems. 5th ed., Oxford University Press, 2019. The performance of analog modulations with noise, and pre-emphasis.
A. Bruce Carlson and Paul B. Crilly. Communication Systems. 5th ed., McGraw-Hill, 2010. A systematic comparison of analog modulations with noise.

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