Signals and Noise
Fall 2018

ECE 110

Course Notes

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In lab you encounter electrical noise as random fluctuations in voltage or current measurements. One source of this noise is the random motion of electrons in conductors. This noise is called thermal noise since the fluctuations increase as the temperature of the conductor increases. Another source of noise in your circuits is the nonideal output of your power supply.

Figure 1

Analog TV static
Fig. 1: Analog TV static. In the days of over-the-air analog TV, noise is what you would see when you tuned to a channel with no TV signal. Each pixel randomly fluctuates to different levels of brightness creating a "snowy" effect. The main sources of this noise are electromagnetic interference from other radio transmissions and atmospheric noise such as lightning. If you tuned to a strong TV signal, the noise would still be present but much less noticeable. Image source.

As you know from lab, noise limits your ability to collect precise measurements. Similarly, it limits the accuracy and/or bandwidth of electronic communication. For these reasons, substantial effort is sometimes put towards suppressing sources of noise and then processing signals to filter out as much of the remaining noise as possible. The simplest signal processing technique to reduce noise is averaging a set of measurements.

A varying voltage signal that is overlaid with additive noise can be represented as $W(t)=S(t)+N(t)$, where $S(t)$ is the signal and $N(t)$ is the noise. The signal-to-noise ratio (SNR) is defined as ratio of the signal power to the noise power. As an equation,
\end{aligned} \label{SAN-PRX}
where $P_{\text{avg},S}$ and $P_{\text{avg},N}$ are the average powers of the signal $S(t)$ and the noise $N(t)$, respectively. In this way, a "snowy" picture on an anolog TV display corresponds to a low SNR and a clear picture to a high SNR.

Since $P_{\text{avg}}=(V_{\text{rms}})^2/R$,
\text{SNR}= \frac{(V_{\text{rms},S})^2}{R} \frac{R}{(V_{\text{rms},N})^2} =\left( \frac{V_{\text{rms},S}}{V_{\text{rms},N}} \right)^2
\end{aligned} \label{SAN-VRX}
where $V_{\text{rms},S}$ and $V_{\text{rms},N}$ are the RMS voltages of the signal $S(t)$ and the noise $N(t)$, respectively.

Let us now consider only those signals that could possibly be generated as voltage signals in practice. For example, a square wave can be the output of a function generator found in lab.

Figure 2

Square wave
Fig. 2: Square wave. Let $V(t)$ be this square wave with amplitude $A=1 \text{ V}$ and period $T=1\text{ s}$.

Even though $V(t)$ is not smooth, it turns out that it can be represented as an infinite sum of sine and cosine functions of different frequencies.
V(t) &= \frac{1}{2}\cos(2\pi 0t)+\frac{2}{\pi}\sin(2\pi 1t)+\frac{2}{3\pi}\sin(2\pi 3t)+\frac{2}{5\pi}\sin(2\pi 5t)+\cdots\\
&= \frac{1}{2}+\frac{2}{\pi}\sin(2\pi t)+\frac{2}{3\pi}\sin(6\pi t)+\frac{2}{5\pi}\sin(10\pi t)+\cdots
\end{aligned} \label{SAN-SFS}
This infinite sum is called a Fourier series. Since sinusoids $\sin(2\pi ft)$ and $\cos(2\pi ft)$ have frequency $f\text{ Hz}$, we say that $V(t)$ has components at $0\text{ Hz}$, $1\text{ Hz}$, $3\text{ Hz}$, $5\text{ Hz}$, and so on.

In fact, the theory of Fourier series says that any periodic signal (that can be realized as a voltage signal) can be represented as a possibly infinite sum of sinusoids. In ECE 210: Analog Signal Processing, you will learn how to decompose periodic signals into their Fourier series components and you will also generalize this result to aperiodic signals.

It is common that a signal has a Fourier representation that is restricted to a certain frequency band. If the Fourier representation of the noise has components with frequencies outside the band, those components can be filtered out using frequency-selective filters. Only the signal itself and the noise components within the frequency band would remain. In this way, the average signal power would stay the same, but the average noise power would reduce, causing a boost in the SNR.

Figure 3

Fig. 3: WILL-TV. The TV station of the University of Illinois at Urbana-Champaign, WILL-TV, used to transmit analog broadcasts on VHF channel 12 from 1955 to 2009. By Federal Communications Commission (FCC) regulations, signals on channel 12 were restricted between frequencies of 204 MHz and 210 MHz. An analog TV tuned to WILL-TV would effectively filter out all the noise and interference outside this range. Image source.

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