In an SDR receiver complex numbers let us use algebra to solve differential equations representing signals. This stuff is imaginative magic. 

Early scientists created equations to explain nature. Occasionally, these equations involved √-1 which has no real solution. But humans are imaginative. When facing the impossibility of solving for √-1, we created the imaginary unit j as a tool to find real solutions after all the math was done. It worked. While j does not exist in the real world, imagining that it does lets us find all the roots of polynomial equations.

Complex numbers contain both real and imaginary terms. Mathematicians built on Descartes and invented the complex plane to visualize these numbers. For example, in the complex number a + jb, a and b are real numbers. “a” lies on the real axis and “b” lies on the imaginary axis, as shown above.

While ordinary numbers have only one value, complex numbers allow us to work with two different values as part of one number. For example, a complex number lets us define a single algebraic object that contains both voltage and current, or amplitude and phase. So, solving equations in electronics became a lot easier.

Lastly, some of the scientific giants like Euler found equivalence between algebra, trigonometry and exponential forms. Complex numbers let us deal with trigonometric angles and rotations using simple algebra. Differential equations suddenly have linear solutions. Perhaps most important for SDR, complex math lets us easily conquer and move between time and frequency domains.

So, to summarize, complex numbers do indeed contain imaginary components. Using these is an interim step or placeholder that lets us do the math faster and better. When all is said and done, the math ends up with real solutions.

SDR Receiver Complex Numbers and Signals

Nature is full of signals. All signals are sinusoidal. Sines and cosines are trigonometric functions of time that have magnitude, frequency and phase. Since frequency is just the first derivative of phase, you can represent magnitude and phase in a complex number which can be interpreted as a vector or phasor in the complex plane, as shown above.

The problem with sinusoids is that they are transcendental functions. This simply means that they cannot be solved with algebra, they “transcend” algebra. You cannot use familiar operations like addition, multiplication and root extraction on sinusoids.

But you can treat sinusoids as a vector rotating about a circle in the complex plane.  One full rotation period of a sine wave can be viewed vector moving in a circle. At any instant, the vector has a magnitude and a phase angle, just like the signal it represents.

Differential equations involving sinusoids are tough to solve. However, really smart guy named Euler discovered that sinusoids could be viewed as rotating complex exponentials. Since the derivatives of complex exponentials are just more complex exponentials, the math became much easier. Euler discovered that although they are also transcendental functions, you can actually do some basic algebra with exponentials. Simply put, it is simpler to perform calculus using the natural logarithm than equivalent transcendental functions.

If you want a deeper dive into complex numbers and signals, watch the AC Analysis videos at the wonderful Khan Academy.

Negative Frequencies

Negative frequencies do not exist in the real world, but they do exist in the complex plane. This derives from the fact that complex numbers may contain either a “+j” or “-j” term. Positive frequency vectors rotate counter clockwise, as shown above. Negative frequency vectors rotate clockwise. More on how complex math helps us get rid of negative frequencies in the next article.