What methodology to build an electronic system?
That's why I'm wondering if there are methods that can be used to build an entire electronic systems just from a set of requirements.


You've got the fundamentals right. Start with a list of requirements and specifications.
Next, break the system down into functional blocks. For example, an audio mixer would include

Power supply.
Input module

Preamplifier / gain control
Tone control
Pan (panorama) control


Output module
Headphone amplifier
VU (volume unit) indicators






simulate this circuit &ndash; Schematic created using CircuitLab
Figure 1. Audio mixer block diagram.
Each of these blocks would have their own sub-specification and can be designed and built from reference designs with tweaks to meet the requirements of your project. It should be clear that each block can be designed in sequence and tested before moving on to the next section.
A final part of the exercise might be to see if certain sections can be merged to reduce the component count, power consumption, PCB area, etc. In this example an obvious one would be to combine the preamp and tone controls.