# Signal Processing/Fourier Analysis
## Fourier Analysis

The Fourier Series allows to represent a periodic signal in terms of its frequency components, instead of it's time waveform. The periodic signal can be represented as the sum of sinusoïds of frequencies being all integer multiples of the signal's base frequency, which is the inverse of the signal's period.
The Fourier Transform extends this approach for aperiodic signals. The signal is considered to be the sum of infinitesimal sinusoids. The sinusoïds are no more integer multiples of a base frequency but are found everywhere on the frequency axis.
The Fourier Series and the Fourier Transform both have an inverse transform. The transformation can be carried out from the time domain to the frequency domain as well as from the frequency domain back to the time domain without any loss of information. In other words, the frequency and the temporal representations are equivalent: they are both accurate representations of the signal and convey it's complete information.

## Laplace Analysis

Whilst the Fourier Series and the Fourier Transform are well suited for analyzing the frequency content of a signal, the Laplace transform is the tool of choice for analyzing and developing circuits such as filters.
Indeed, the Laplace transform is used for solving differential and integral equations. With this transform, differentiation and integration respectively become multiplication and division by $s$.

### LTI Systems

Circuits such as filters which implement a set of differential equations on an incoming signal in order to modify it are called Linear Time Invariant (LTI).
In the frequency domain, the input signal's frequency content, $X(s)$, is multiplied by the transfer function of a LTI system, $H(s)$, to determine the output signal's frequency content, $Y(s)$.

### Filters

Time-continuous (analog) filters are LTI systems used to damp unwanted frequency contents out of a signal or to amplify desired frequency bands of the signal.
They are used to:
* damp out noise which has been added to the signal,
* limit the frequency content of a signal in order to mix it with other signals having different frequency contents on a common transmission channel,
* retrieve a single signal which has been mixed with others on a common transmission channel,
* limit the frequency content of a signal before sampling it at a given rate.
From this, the usual shape of a transfer function would be of "brick-wall" type: letting all the signal through in certain frequency bands and letting nothing trough in the others. Alas, this is not possible. It can be shown that brick-wall filters would need an infinite time to provide the very beginning of the output signal. Because of this, filter designers have to fall back on transfer functions close to the brick-wall characteristic but which are physically realizable: Butterworth, Chebychev, …

## Sampled systems analysis

Sampled systems, such as switched-capacitor or digital filters, are not analyzed based on differential equation but on difference equations. At a given time, the output of a sampled filter is a function of the $N$ last values of the filter's input and internal signals.
Solving difference equations is done with the help of the Z-transform. Similarly to the Laplace analysis, the Z-transform of the input signal, $X(z)$, is multiplied by the sampled transfer function, $H(z)$, to determine the output signal's Z-transform, $Y(z)$. The time samples of the output signal, $y(t)$, can be calculated with the help of the inverse Z-transform of $Y(z)$.

### Digital filters

Digital filters can perform the same tasks as their analog counterparts, but on sampled signals. Additionally, they can be used to:
* insert additional samples between the original samples of a signal in order to increase the sampling rate of a signal (interpolation)
* predict future values of a signal (extrapolation)
* adapt their transfer function based on an evaluation of the signal's time-varying characteristics.

### Adaptative filters

The adaptative filtering problem stems from the fact that data is transmitted over noisy channels. At the receiver, the data from a signal needs to be separated from other data streams, inter-symbol interference, and external noise. Notice that both noise and other data streams can be considered to be stochastic (random) processes. These processes can be found to have certain characteristics, such as mean and variance. On top of the problem of separation is the fact that the characteristics of the various components of the received signal, the data itself, the noise, and the other data streams all change over time. For instance, new transmitters could be adding data streams to the channel, or new noise could be added, or the sender of the target data stream could be altered. The mean and variance of the various components could change over time.
As an analogy, consider a room with one person speaking (the speaker), and one person listening (the listener). The room will have certain ambient noises, such as electrical hums from the sound system, or the feint sounds of other speakers in other rooms. In general, if the ratio of the speaker's volume to the volume of the other noises is high, the listener will have no trouble focusing on the speaker. However, if a second speaker sets up a podium and begins speaking, the listener will have trouble focusing, because the received signal has changed.
The filtering problem, in essence, is trying to detect and extract useful data from a noisy signal. The characteristics of the data and the background noise might not be known a priori, and they are likely to change over time.