How to Generate a Basic Sinusoid

Mastering the DSP toolkit starts with a basic sinusoidal signal.

A sinusoid

As a function of time $t$, a real sinusoid is given by:

$$x(t) = A \cos(2\pi f t+\phi)$$

There are three parameters that determine the actual signal:

• Amplitude $A$
• Frequency $f$ in cycles/sec or $\omega=2\pi f$ in radians/sec, and the time period is $T=\frac{1}{f}$ or $\frac{2\pi}{\omega}$
• Phase $\phi$

It’s a convention to use a cosine wave because we know that a sine wave is also a cosine wave with a specific phase.

$$\sin 2\pi f t = \cos \left(2\pi f t - 90^\circ\right)$$

A really basic sinusoid with $A=1$, $f=1$ and $\phi=0$ is shown below: