# Circuit Theory/Fourier Transform

right\|framed\|Joseph Fourier,
after whom the Fourier Transform is named, was a famous mathematician
who worked for Napoleon.

This course started with phasors. We learned how to transform forcing
sinusodial
functions such as voltage supplies into phasors. To handle more complex
forcing functions we switched to complex frequencies. This enabled us to
handle forcing functions of the form:

$$e^{st}\cos(\omega t + \phi)$$ where s is:

$$s=\sigma + j\omega$$ And the convolution integral can do anything.

Along the way \"s\" began to transform the calculus operators back into
algebra. Within the complex domain, \"s\" could be re-attached to the
inductors and capacitors rather than forcing functions. The transfer
function helped us use \"s\" to capture circuit physical
characteristics.

This is all good for designing a circuit to operate at a single
frequency ω. But what about circuits that operate at a variety of
frequencies? A RC car may operate at 27mhz, but when a control is
pressed, the frequency might increase or decrease. Or the amplitude may
increase or decrease. Or the phase may shift. All of these things happen
in a cell phone call or wifi/blue tooth/xbee/AM/FM/over the air tv, etc.

How does a single circuit respond to these changes?

## Fourier analysis

Fourier analysis says we don\'t have to answer all the above questions.
Just one question has to be answered/designed to. Since any function can
be turned into a series of sinusoidals added together, then sweeping the
circuit through a variety of omegas can predict its response to any
particular combination of them.

So to start this we get rid of the exponential term and go back to
phasors.

Set σ to 0:

$$s = j\omega$$

The variable ω is known as the \"radial frequency\" or just frequency.
With this we can design circuits for cell phones that all share the air,
for set-top cable TV boxes that pack multiple channels into one black
cable. Every vocal or pixel change during transmission or reception can
be designed for within this framework. All that is required is to sweep
through all the frequencies that a sinusoidal voltage or current source
can produce.

Analysis stays in the frequency domain. Because everything repeats over
and over again in time, there is no point in going back to the time from
a design point of view.

## Radial Frequency

In the Fourier transform, the value $\omega$ is known as the **Radial
Frequency**, and has units of radians/second (rad/s). People might be
more familiar with the variable f, which is called the \"Frequency\",
and is measured in units called Hertz (Hz). The conversion is done as
such:

$$\omega = 2\pi f$$ For instance, if a given AC source has a frequency
of 60 Hz, the resultant radial frequency is:

$$\omega = 2\pi f = 2\pi(60) = 120\pi$$

## Fourier Domain

The Fourier domain then is broken up into two distinct parts: the
**magnitude graph**, and the **phase graph**. The magnitude graph has jω
as the horizontal axis, and the magnitude of the transform as the
vertical axis. Remember, we can compute the magnitude of a complex value
C as:

$$C = A + jB$$

$$|C| = \sqrt{A^2 + B^2}$$

The Phase graph has jω as the horizontal axis, and the phase value of
the transform as the vertical axis. Remember, we can compute the phase
of a complex value as such:

$$C = A + jB$$

$$\angle C = \tan^{-1}\left(\frac{B}{A}\right)$$

The phase and magnitude values of the Fourier transform can be
considered independent values, although some abstract relationships do
apply. Every fourier transform must include a phase value and a
magnitude value, or it cannot be uniquely transformed back into the time
domain.

The combination of graphs of the magnitude and phase responses of a
circuit, along with some special types of formatting and interpretation
are called Bode Plots.
