Discrete-time signals are mathematical entities; in particular, they are functions with an independent time variable and a dependent variable that typically represents some kind of real-world quantity of interest. But as interesting as a signal may be on its own, engineers usually want to
do something to it. This kind of action is what discrete-time systems are all about. A
discrete-time system is a mathematical transformation that maps a discrete-time input signal (usually designated $x$) into a discrete-time output signal (usually designated $y$). In other words, it takes an input signal and modifies it to produce an output signal:
System $\mathcal{H}$ takes takes a discrete time signal $x$ as an input and produces an output $y$. There is no end to the possibilities of what a system could do. Systems might be trivially dull (e.g., produce an output of 0 regardless of the input) or incredibly complex (e.g., isolate a single voice speaking in a crowd). Here are a few examples of systems:
A speech recognition system converts acoustic waves of speech into text
A radar system transforms the received radar pulse to estimate the position and velocity of targets
A functional magnetic resonance imaging (fMRI) system transforms measurements of electron spin into voxel-by-voxel estimates of brain activity
A 30 day moving average smooths out the day-to-day variability in a stock price
Signal length and systems
Recall that discrete-time signals can be broadly divided into two classes based upon their length: they are either infinite length or finite length (and recall also that periodic signals, though infinite in length, can be viewed as finite-length signals when we take a single period into account). Likewise, discrete-time systems are also finite or infinite length, depending on the kind of input signals they take. Finite-length systems take in a finite-length input and produce a finite-length output (of the same length), with infinite-length systems doing the same for infinite-length signals.
Examples of discrete-time systems
So a system takes an input signal $x$ and produces an output signal $y$. How does this look, mathematically? Below are several examples of systems and their mathematical expression:
Identity: $y[n] = x[n]$
Scaling: $y[n] = 2\, x[n]$
Offset: $y[n] = x[n]+2$
Square signal: $y[n] = (x[n])^2$
Shift: $y[n] = x[n+m]\quad m\in Z$
\]
Decimate: $y[n] = x[2n]$
Square time: $y[n] = x[n^2]$
Moving average (combines shift, sum, scale): $y[n] = \frac{1}{2}(x[n]+x[n-1])$
Recursive average: $y[n] = x[n]+ \alpha\,y[n-1]$
So systems take input signals and produce output signals. We have seen some examples of systems, and have also introduced a broad categorization of systems as either operating on finite or infinite length signals.
