Perhaps the most common real-valued signal is the sinusoid. For this signal, A A is its amplitude, f0 f0 its frequency, and φ φ its phase.

The most important signal is complex-valued, the complex exponential. Here, ⅈ denotes -1 -1 (Mathematicians use the symbol ii for -1 -1 ). For electrical engineers, history led to a different symbol. Ampère used the symbol ii to denote current (intensité de current). Euler first used ii for imaginary numbers but that notation did not take hold until roughly Ampère's time. It wasn't until the twentieth century that the importance of complex numbers to circuit theory became evident. By then, using ii for current was entrenched and electrical engineers chose ⅈ for writing complex numbers. Aⅇⅈφ A φ is known as the signal's complex amplitude. Considering the complex amplitude as a complex number in polar form, its magnitude is the amplitude AA and its angle the signal phase. The complex amplitude is also known as a phasor. The complex exponential cannot be further decomposed into more elemental signals, and is the most important signal in electrical engineering! Mathematical manipulations at first appear to be more difficult because complex-valued numbers are introduced. In fact, early in the twentieth century, mathematicians thought engineers would not be sufficiently sophisticated to handle complex exponentials even though they greatly simplified solving circuit problems. Steinmetz introduced complex exponentials to electrical engineering, and demonstrated that "mere" engineers could use them to good effect and even obtain right answers! See Complex Numbers for a review of complex numbers and complex arithmetic.

The complex exponential defines the notion of frequency: it is the only signal that contains only one frequency component. The sinusoid consists of two frequency components: one at the frequency +f0 f0 and the other at -f0 f0 .

Using the complex plane, we can envision the complex exponential's temporal variations as seen in the above figure (Figure 1). The magnitude of the complex exponential is AA, and the initial value of the complex exponential at t=0 t 0 has an angle of φφ. As time increases, the locus of points traced by the complex exponential is a circle (it has constant magnitude of AA). The number of times per second we go around the circle equals the frequency ff. The time taken for the complex exponential to go around the circle once is known as its period TT, and equals 1f 1 f . The projections onto the real and imaginary axes of the rotating vector representing the complex exponential signal are the cosine and sine signal of Euler's relation (Equation 3).

As opposed to complex exponentials which oscillate, real exponentials decay.

The quantity ττ is known as the exponential's time constant, and corresponds to the time required for the exponential to decrease by a factor of 1ⅇ 1 , which approximately equals 0.3680.368. A decaying complex exponential is the product of a real and a complex exponential. In the complex plane, this signal corresponds to an exponential spiral. For such signals, we can define complex frequency as the quantity multiplying tt.

The unit step function is denoted by ut u t , and is defined to be

This kind of signal is used to describe signals that "turn on" suddenly. For example, to mathematically represent turning on an oscillator, we can write it as the product of a sinusoid and a step: st=Asin2πftut s t A 2 f t u t .

The unit pulse describes turning a unit-amplitude signal on for a duration of ΔΔ seconds, then turning it off. We will find that this is the second most important signal in communications.

The square wave sqt sq t is a periodic signal like the sinusoid. It too has an amplitude and a period, which must be specified to characterize the signal. We find subsequently that the sine wave is a simpler signal than the square wave.