Signals in the time domain

Trigonometry: Trigonometric functions

Signals in the time domain

From now on we use the independent variable $t$ because we usually will interpret it as time. A function $s(t)$ of $t$ is then called a (time) signal.

A signal of the form $s(t)=A \sin(2\pi f t+\varphi)$ is called an alternating signal (also known as sinusoid) with amplitude $A$ , frequency $f$ and phase $\varphi$ .

We will always assume that $A$ and $f$ are positive numbers and call this formula the standard form.
The amplitude determines the size of the oscillations.
The frequency determines the number of oscillations in the time interval [0,1].
The unit of frequency, that is, the number of periods per second, is hertz (Hz).
The phase is related to a horizontal translation
(to the left for positive $\varphi$ , to the right for negative $\varphi$ ).

The number $\displaystyle T=\frac{1}{f}$ is the period of the alternating signal, because $T$ is the smallest positive integer with the property that $s(t+T)=s(t)$ for all $t$ .

In the figure to the right we have draen the graph of the sinusod defined by $s(t)=3\sin\left(\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\right)$ . We denote herein the amplitude ( $3$ ), the period ( $4$ ) en and the horizontal translation ( $-1$ ) of a sine graph that has, in this exampe, a phase of $\tfrac{1}{2}\pi$ .

By making use of the addition formulas any alternating signal can also be written in the form $s(t)=a \sin(2\pi f t)+b \cos(2\pi f t)$ Here $a=A\cos(\varphi)\quad\text{and}\quad b=A\sin(\varphi)$

Conversely, any signal of the form $s(t)=a \sin(2\pi f t)+b \cos(2\pi f t)$ can be written in standard form. You must determine $A$ and $\varphi$ so that at least $A=\sqrt{a^2+b^2}\quad\text{and}\quad \tan(\varphi)=\frac{\sin(\varphi)}{\cos(\varphi)}=\frac{b}{a}$ The appropriate phase $\varphi$ is fixed modulo $2\pi$ .