Other time series are completely man-made, such as in economics.
Here you have the Dow Jones index,
which measures some sort of health state of the economy in certain circles.
And you can see once again that there were some trends, for instance,
the Dow Jones has been pretty low for the best part of last century and
then has grown, again, kind of exponentially.
But what is interesting is probably you're familiar with the crash of 1929,
which is this little dip here,
and compare that, which went down in history as some sort of major disaster,
with the kind of swings that we have today when the Dow is at such high levels.
Okay, so we have seen some examples, now let's try to formalize the concept of
discrete-time signal, for us, this is a sequence of complex numbers.
So it is a one-dimensional sequence, at least for now.
The notation is x[n], where n is in square brackets to indicate that n is an integer.
It's a two-sided sequence, so
n goes from minus infinity to plus infinity, and it's a mapping,
therefore, from Z, the set of integers, to C the set of complex numbers.
n is what we call an a-dimensional time, so we can think of it as time if we want,
but we have to make sure not to associate a physical unit to n.
n is a-dimensional, it just sets an order on the sequence of samples.
Discrete-time signals can be created by an analysis process where we take periodic
measurements of a physical phenomenon,
think of the floods of the Nile if you want.
Or in a synthesis process where we use say a computer program to generate data point
that simulate a physical phenomenon that we want to reproduce,
we will see an example very soon.
Let's now look at some prototypical signals that will appear again and
again in this class.
The simplest non-trivial signal that you can think of is a signal where every
sample is equal to 0 except for n equal to 0 where the samples is equal to 1.
This is called the delta signal, and
it exemplifies a physical phenomenon that has a very, very short duration in time.
To help your memory, you can associate the delta signal to a clapper,
the device that is used in the movie industry,
although perhaps not in the mechanical form that you see here in this picture,
to synchronize the audio and the video tracks.
When you shoot a movie, the video and the audio are recorded on separate devices,
and then you have to synchronize the two tracks together.
So the way this is done, is by filming the clapper and
then having the top part of the clapper slam down on the bottom part.
This will generate a very short instantaneous sound that on the audio
track will look like a delta signal or a combination of positive and
negative delta signals.
When you need to synchronize audio and video, you will look for
this pattern in the audio track.
You will look for the delta, and associate it to the frame,
where the top part of the clapper is hitting the bottom part.
Another useful signal is the unit step.
This is a signal that is 0 for all negative values of the index.
So x[n] = 0 for n less than 0, and
is equal to 1 for n greater than or equal to 0.
This depicts a very simple phenomenon, the flipping of a switch.
So think of a Frankenstein switch when this is pulled up, then the contact
is made, and the signal will go from zero to one and stay at one forever.
Another common signal is the exponential decay.
We take a number a less than 1 in magnitude, and
we take successive powers of the absolute value of a.
Because a is less that 1 in magnitude, successive powers will go down
exponentially to 0, but of course, will never reach 0 unless we go to infinity.
In order to prevent the signal from exploding when n is negative,
we multiply the signal by the unit steps.
So we basically force to 0 all values of the sequence for
negative values of the index.
The exponential decay captures the behavior of a lot of physical systems, for
instance, it shows how your coffee cup gets cold.
Newton's law of cooling says that the rate of change of the temperature of a body
is proportional to the difference in temperature between the environment and
the body itself.
So if you solve this differential equation, you find out that the evolution
of the temperature follows indeed an exponentially decaying trend.
Of course, this is an idealized version of how a coffee gets cold,
because you should have only convection and large conductivity.
But in general, this is a common behavior for a lot of physical systems.
We have seen, for instance, that the rate of discharge of a capacitor in
an RC circuit is also an exponentially decaying curve.
In discrete-time, the exponential decay, a to the power of n,
models this kind of behavior.
And finally, we have sinusoidal signals.
Here we have, for instance, an example using the sin function.
Discrete-time sequence is simply the sine of an angular frequency of omega
0 times the index n + na initial phase theta.
