Linear Systems

Definition 13973

A discrete real time series is a map from ${\fam11\tenbbb Z}^+$ or ${\fam11\tenbbb Z}$ into ${\fam11\tenbbb R}$; we think of the integers as consecutive ticks of a clock, and the real values u_n or u(n) as being the value of some physical variable at the n^th clock tick.

For example, the clock ticks might be once a day and the values might be the Dow-Jones averages on those days, or the clock ticks might be once every twentieth of a millisecond, and the values the voltages output by a microphone as you talk into it. Some notations have u(n) for the value and hence u as the time series, which is evidently a function; others write the sequence $\{u_1,u_2,\cdots,u_n,\cdots\}$ which is clumsier but more familiar to the older generation. I shall use the function language because it is neater and cuts down on the number of different kinds of things you have to think about.

Definition 13980

A continuous real time series is a continuous map from ${\fam11\tenbbb R}$ or ${\fam11\tenbbb R}^+$ into ${\fam11\tenbbb R}$. We think of the input as being continuous time and the output as being the value of some changing measurement at that time.

Since we are going to be dealing with real life, where it is impractical to take a measurment at every conceivable instant of time, we have no immediate interest in such things except that we may like to believe for philosophical reasons that a discrete time series came from sampling a continuous one. Of course, you have to be pretty committed to metaphysics to believe that using ${\fam11\tenbbb R}$ for time or space is anything more than a linguistic convenience. Maybe space-time comes in very, very small lumps.

It is common to look at stock exchange figures and to feel that there is altogether too much wild variation in consecutive values. A standard way of smoothing the stock exchange time series is to replace the central value on a day by the average of the three days consisting of the original day, the preceding day, and the succeeding day. This cannot be done in real time, you get to be a day late with your figures, but predictions on smoothed data might be more useful, so it could be worth it. This is an example of a moving average filter; it takes in one time series and outputs another. In fact this is the definition of a filter.

Definition 13982

A filter is a map from a space of time series to a space of time series.

The usual application is to filter out random noise, as in the moving average example. In the old days, when life was hard and real engineers had to do three impossible things before breakfast, the time series was likely to be continuous, was almost invariably a voltage, and a filter was made out of capacitors, inductances, resistors and similar knobbly things you could pick up and lose. The signal went in, and emerged purified, like water having the dirt filtered out by pouring it through well, through filters. The mental image of a dirty, insanitary time series coming in, and a somewhat improved one leaving, still pervades the language. In these latter days, the time series is more likely to be a sequence of numbers input to a computer by some data acquisition system or output from it to some mechanism, while the filter is likely to be a software construct. In this case they become known as digital filters. I shall consider only digital filters here, with a brief aside now and again to indicate the directions for the continuous case.