1.3 The fundamental signal

The sinusoid

The most ubiquitous and important signal in electrical engineering is the sinusoid .

We can also define a discrete-time variant of the sinusoid: $A\cos (2\pi fn+)$ . Here, the independent variable is $n$ and represents the integers. Frequency now has no dimensions, and takes on valuesbetween 0 and 1.

Communicating information with signals

The basic idea of communication engineering is to use a signal's parameters to represent either real numbers or othersignals. The technical term is to modulate the carrier signal's parameters to transmit information from one place to another. To explore the notionof modulation, we can send a real number (today's temperature, for example) by changing a sinusoid's amplitude accordingly.If we wanted to send the daily temperature, we would keep the frequency constant (so the receiver would know what to expect)and change the amplitude at midnight. We could relate temperature to amplitude by the formula $A=A_0(1+kT)$ , where $A_0$ and $k$ are constants that the transmitter and receiver must both know.

If we had two numbers we wanted to send at the same time, we could modulate the sinusoid's frequency as well as itsamplitude. This modulation scheme assumes we can estimate the sinusoid's amplitude and frequency; we shall learn that this isindeed possible.

Now suppose we have a sequence of parameters to send. We have exploited all of the sinusoid's two parameters. What we cando is modulate them for a limited time (say $T$ seconds), and send two parameters every $T$ . This simple notion corresponds to how a modem works. Here, typedcharacters are encoded into eight bits, and the individual bits are encoded into a sinusoid's amplitude and frequency.We'll learn how this is done in subsequent modules, and more importantly, we'll learn what the limits are on such digitalcommunication schemes.