The only thing that varies with respect to time is the amplitude of the signal, and if you measure this at regular intervals and note the values, then have a recording of the signal.
Once you have a set of figures that represent a waveform you could mathematically manipulate the figures. For example, you could multiply all the values by 2 thereby doubling the signal amplitude. You could add +2 to all the values, which would obviously offset the signal by +2.
Due to a huge number of figures involved, a computer best performs these types of mathematical operations, requiring digital processing.
In electronics, there is a special chip called a DSP or Digital Signal Processor, which performs this type of mathematical manipulation. This chip has dedicated hardware-based circuits to perform mathematical operations on the signal data. This enables the chip to operate very fast on real-time data, without having to store the data first.
Nearly all signal processing requires a multiplication step somewhere in the algorithm, and nearly all will have some form of addition or summation step. Microprocessor manufacturers realised this and incorporated hard-wired circuits to speed-up these types of operations. As a result, many modern processors will have a MAC function that will multiply and accumulate the result in a special register. This function allows programmers to implement filter algorithms such as the Fast Fourier Transform (FFT) much more easily.
History of DSP
Back in the 60s storage devices such as hard disks did not exist for storing all those amplitude values, instead, they hired a pretty typist to enter those values into the computer by hand.
Since computers were slow, mathematicians had to find clever little mathematical steps to process large amounts of data using the least amount of computing resources. They managed to reduce the processing into the fewest number of multiplication steps and compute cycles.
Take a simple set of mathematical steps and convert them into a large head-spinning formula that only you can understand and you are a genius! Hilbert and Fourier did just that with their transforms. The numerical operations required to implement those formulas are so simple that you could do them manually on a simple vintage calculator.
Application: The Eurofighter
This plane has hundreds of transducers and sensors relaying information to the main computer. Unfortunately, the engine strapped at the back, radiates electromagnetic noise.
Since the central computer needs to receive accurate clear information about the engine, as signals contaminated by noise could prove disastrous, DSP chips filter out any noise components.
Of course, the Eurofighter is a fly-by-wire plane and there is no manual override, so it is extremely important that the computers receive clean uncontaminated signals from the engine subsystems.