1. Convolution sum

3. DT System Impulse Response, h[n]
A very important way to analyse a system is to study the response signal, h[n], when the input is an impulse d[n]
Loosely speaking, this corresponds to giving the system a unit kick at n=0, and then seeing what happens
The specific notation, h[n], is used to denote the impulse response signal, rather than the more general y[n].
For a DT LTI system, the impulse response signal, h[n], has all the information that exists in the difference equation
d[n]
System
h[n]
1) Having broken down the DT (input) signal into a number of simple bits (time shifted impulses), we’re interested in evaluating how the system responds to simple impulse signal – this is known as the impulse response and is denoted by h[n], rather than the more general y[n] which denotes the system output to any input signal.
2) The DT impulse response signal h[n] corresponds to the unforced dynamics of the DT system, i.e. if the system is disturbed/has initial conditions at n=0, how does the system evolve? The DT impulse response signal uniquely characterizes the system when the system is LTI, i.e. you don’t need to know the difference equation that relates x -> y, equivalent information is contained in h[n].
3) Using this information we can see how the system responds to a general input signal, because we have broken up the complex input signal and we can see how it responds to each individual bit.

4. Superposition & DT Convolution
For any LTI discrete time system, the response to an input signal x[n] is given as follows.
Using the sifting property:
h[n] is the system impulse response to d[n].
Because the system is time invariant,
Using the superposition property (linear):
The system response of any LTI system can be calculated as a linear combination of impulse responses & is sometimes denoted by:
We’re now in a position to state/derive the convolution expression. It actually follows immediately from the sifting property for the DT input signal x[n] and superposition property for the DT LTI system.
The sifting property says that we can represent any DT signal as:
x[n] = Sk x[k]d[n-k]
2) We also know that the system response to d[n] is h[n] and if the system is time invariant
d[n-k] is h[n-k]
3) If the system is also linear, the superposition property applies so if the input given as a linear combination of little bits and we know the response to each little bit, the output is the same linear combination of little outputs, i.e.
y[n] = Sk x[k]h[n-k]
4) It is the combination of two signals (input signal and impulse response signal) to produce a third signal (response signal) and is sometimes denoted by y[n] = x[n]*h[n], where multiplying two signals means taking the convolution