Key points you have learned

A system is a transformation from one signal (the input) to another (the response). It can be defined in continuous- or discrete-time.

A particular type of system with nice properties is linear (obeys linear superposition) and time-invariant (nothing inside the system is changing with time).

For mathematical convenience, and only for mathematical convenience, one can define the response of a real-time system to a complex-valued input signal. The real part of the output is the response of the system to the real part of the input, and similarly for the imaginary part.

A sinusoid with arbitrary phase can be represented, for mathematical convenience, as the real part of a complex exponential (in either continuous- or discrete-time).

A linear time- or shift-invariant system has the nice property that a sinusoidal input signal results in an output signal that is also sinusoidal, with the same frequency but possibly different amplitude and phase.

When a complex exponential is input to an LTI system, the output is a complex exponential with the same frequency multiplied by some complex number. That complex number is called the transfer function of the system, and it is a function of frequency.

The magnitude of the transfer function is the factor by which the amplitude of a sinusoidal input frequency is multiplied by the system.

The phase of the transfer function is the amount of phase shift introduced to a sinusoidal input signal by the system at that frequency.

The transfer function is also often called the frequency response. This is because it tells us how the system will respond to different frequencies at its input.

The transfer function can be used to find the response of the system to an input signal consisting of a superposition of sinusoids. We simply find the response to each constituent sinusoid, and then take the superposition of the results.