6.11 Signal-to-noise ratio of an amplitude-modulated signal

When we consider the much more realistic situation when we have a channel that introduces attenuation and noise, we can make useof the just-described receiver's linear nature to directly derive the receiver's output. The attenuation affects the outputin the same way as the transmitted signal: It scales the output signal by the same amount. The white noise, on the other hand,should be filtered from the received signal before demodulation. We must thus insert a bandpass filter havingbandwidth $2W$ and center frequency $f_c$ : This filter has no effect on the received signal-related component, but does remove out-of-band noisepower. As shown in the triangular-shaped signal spectrum , we apply coherent receiver to this filtered signal, with the result thatthe demodulated output contains noise that cannot be removed: It lies in the same spectral band as the signal.

As we derive the signal-to-noise ratio in the demodulated signal, let's also calculate the signal-to-noise ratio of thebandpass filter's output $\tilde{r}(t)$ . The signal component of $\tilde{r}(t)$ equals $\alpha A_{c}m(t)\cos (2\pi f_{c}t)$ . This signal's Fourier transform equals

Thus, the total signal-related power in $\tilde{r}(t)$ is $\frac{\alpha ^2{A}_c^2}{2}\mathrm{power}(m)$ . The noise power equals the integral of the noise power spectrum;because the power spectrum is constant over the transmission band, this integral equals the noise amplitude $N_0$ times the filter's bandwidth $2W$ . The so-called received signal-to-noise ratio —the signal-to-noise ratio after the de rigeur front-end bandpass filter and before demodulation—equals

The demodulated signal $\widehat{m}(t)=\frac{\alpha A_{c}m(t)}{2}+n_{\text{out}}(t)$ . Clearly, the signal power equals $\frac{\alpha ^2{A}_c^2\mathrm{power}(m)}{4}$ . To determine the noise power, we must understand how the coherent demodulator affects the bandpass noise found in $\tilde{r}(t)$ . Because we are concerned with noise, we must deal with the power spectrum since we don't have the Fouriertransform available to us. Letting $P_{ñ}(f)$ denote the power spectrum of $\tilde{r}(t)$ 's noise component, the power spectrum after multiplication by the carrier has the form

Let's break down the components of this signal-to-noise ratio to better appreciate how the channel and the transmitter parametersaffect communications performance. Better performance, as measured by the SNR , occurs as it increases.

• More transmitter power—increasing $A_c$ —increases the signal-to-noise ratio proportionally.
• The carrier frequency $f_c$ has no effect on SNR, but we have assumed that $\gg (f_c, W)$ .
• The signal bandwidth $W$ enters the signal-to-noise expression in two places: implicitly throughthe signal power and explicitly in the expression's denominator. If the signal spectrum had a constant amplitude as we increased the bandwidth, signal power would increase proportionally. On the otherhand, our transmitter enforced the criterion that signal amplitude was constant . Signal amplitude essentially equals the integral of the magnitude of the signal's spectrum.
  This result isn't exact, but we do know that $m(0)=\int_{()} \,d f$∞ ∞ M f .
  Enforcing the signal amplitude specification means that as the signal's bandwidth increases we must decrease thespectral amplitude, with the result that the signal power remains constant. Thus, increasing signal bandwidth doesindeed decrease the signal-to-noise ratio of the receiver's output.
• Increasing channel attenuation—moving the receiver farther from the transmitter—decreases the signal-to-noise ratio as the square. Thus, signal-to-noiseratio decreases as distance-squared between transmitter and receiver.
• Noise added by the channel adversely affects the signal-to-noise ratio.

In summary, amplitude modulation provides an effective means for sending a bandlimited signal from one place to another. Forwireline channels, using baseband or amplitude modulation makes little difference in terms of signal-to-noise ratio. Forwireless channels, amplitude modulation is the only alternative. The one AM parameter that does not affectsignal-to-noise ratio is the carrier frequency $f_c$ : We can choose any value we want so long as the transmitter and receiver use the same value. However, supposesomeone else wants to use AM and chooses the same carrier frequency. The two resulting transmissions will add, and both receivers will produce the sum of the two signals. What we clearly need to do is talk to the otherparty, and agree to use separate carrier frequencies. As more and more users wish to use radio, we need a forum for agreeingon carrier frequencies and on signal bandwidth. On earth, this forum is the government. In the United States, the FederalCommunications Commission (FCC) strictly controls the use of the electromagnetic spectrum for communications. Separate frequencybands are allocated for commercial AM, FM, cellular telephone (the analog version of which is AM), short wave (also AM), andsatellite communications.