Complex numbers in signal processing

1. How are complex numbers used in signal processing?
2. Why are signals represented as complex numbers?
3. Can a signal take complex value?
4. Why are complex numbers used in waves?

How are complex numbers used in signal processing?

Complex numbers shorten the equations used in DSP, and enable techniques that are difficult or impossible with real numbers alone. For instance, the Fast Fourier Transform is based on complex numbers.

Why are signals represented as complex numbers?

Because complex numbers are a compact way to encode the basic trigonometric functions sin and cos, which are the basic sinusoids. Save this answer.

Can a signal take complex value?

While the domain set determines the arguments of the signal, the range set determines the values of it. Therefore a real valued signal is a function whose range set is the real numbers. Consequently a complex valued signal is the one whose range set is the complex numbers.

Why are complex numbers used in waves?

The simple answer is that a wave has amplitude and phase, which requires two numbers. Complex numbers are one of the ways to represent a vector with magnitude and phase (rectangular coordinates).