In module 2.2 we're going to
study the complex exponential.
We had announced this sequence as being
a fundamental character in our play.
And it is time to meet it more in detail.
It is not a completely simple object, but
it has many features that will be
useful later on in this class.
In particular, there are periodic complex
sequence that we are going to see, but
there is also a notion of frequency
which is a discrete time frequency
that is quite subtle.
In continuous time,
frequency is very natural.
Think of frequency for example in music,
in musical instruments, and so on.
In discrete time, the notion is
more subtle, because there is
an increasing frequency that after a while
looks actually like a smaller frequency.
And this phenomenon, which is well known
if you watch old movies when you have
wagon wheels that turn backwards,
even though the wagon is
actually moving forward.
Is an effect called aliasing
that we need to understand,
because it's very particular to
discrete time signal processing.
Then we will finish the module
2.2 by seeing the relationship
between a discrete time frequency and
an equivalent continuous time frequency,
which is very important, for example,
if you want to synthesize a sound from
the computer into an acoustic system.
Or of course in the reverse,
when you acquire a real-time signal from
the continuous time domain and
you want create a sequence.
The overview is the following.
We are going to introduce a complex
exponential, see periodicity,
then a special effect which is
called the wagonwheel effect, and
the maximum speed of discrete
complex exponentials.
Then we are also going to contrast
digital and real-world frequency notions.
As we have said in the previous
sub-module, oscillations are everywhere.
The cardiac beat,
the whistle on the train, the waves
on the ocean, the violin strings.
The oscillation needs
the following ingredients.
First, we need a frequency.
The unit will be radians.
So, we will always look at the unit circle
and the full circle is in radians 2 pi.
There will be an initial phase, phi.
In the same unit of course, radians.
There'll be an amplitude,
this amplitude will depend how we
measure the physical phenomenon.
And we therefore get the trigonometric
function which is the sequence,
for example, x[n],
which is given by A, the amplitude,
times cos(omega n + phi).
So the phase term phi, the frequency
omega, the time index n, okay?
So this is the basic
ingredient of a trigonometric
sequence here is that thiis going to
be periodic and has initial phase phi.
Now, we always use complex exponentials.
Now, some of you might not like
complex numbers, but complex numbers
are our friends here because they
make trigonometry much simpler.
So a trigonometric function that is
expressed as a complex exponential
of course can be reduced to sines and
cosines using Euler's formulas.
Okay, so we have a sequence x[n],
just like previously.
We have let's say A is the amplitude,
e to the j(omega n + phi).
And by Euler's formulas,
this is the amplitude cos(omega n + phi),
that's the real part.
j times sin (omega n+ phi),
which is the imaginary part, and
together we have a complex exponential.
So, we see that sines and
cosines always come together.
The reason why we prefer
exponentials to sines and
cosines is that
the mathematics are simpler.
Instead of doing trigonometry,
we do algebra.
And, last but not least,
we actually have complex numbers in
all digital systems that we use.
So on your computer,
you have real numbers, but
you have also complex numbers, and
you can do complex arithmetic.
Okay, so let's look at an example of the
advantages of using complex exponentials.
So you want to do the change
of phase of a pure cosine.
So you start with the cos(omega n + phi).
This will be written as
a cos(omega n) + b sin(omega n),
where a and b are given here.
So if you know your trigonometry by heart,
you know this is a very natural thing.
But if you don't remember all
the trigonometric formulas,
you can always look them up on Wikipedia.
But it is a little
a cumbersome way to actually
deal with such simple operations, okay?
So instead, what we do is we look
at the change of phase of a pure
cosine by stating that the cosine is
the real part of a complex exponential,
the change of phase up here is
an addition in the exponent.
And the result is that we simply
have a multiplication here
in the complex exponential.
So the phase that was up there,
of course by the fact that
it's in the exponent it will end up just
being a multiplication here by the phase.
So the phase change is extremely simple
in the complex exponential domain.
At the end, we simply take the real part,
we get the cosine, and
we're all done, okay?
Notation is simpler,
phase shift is a simple multiplication.
So, these two effects are why
we prefer complex exponentials.
Okay, so what is a complex exponential?
It is Euler's formula again,
e to the j alpha
is equal to cos alpha + j sin alpha, okay?
I'm repeating myself, but
it's such a beautiful formula that we can
see it a couple of times, it won't hurt.
Okay, so
let's plot this in the complex plane, so
the complex plane has a real axis,
and an imaginary axis.
So e to the j alpha, given by cos alpha +
j sin alpha lives on a unit circle here.
This is nicely the circle here.
And the complex number e to
the j alpha is, well, cos alpha,
which would be here,
+ j sin alpha, which is here.
And it is a vector of length 1
that reaches the unit circle and
has an angle alpha here
with the real axis, okay?
So if you're not very familiar with
this picture, please look at it,
play with it,
play with Euler's formula, go back and
forth between real numbers and
imaginary numbers, and so on.
Because this is the alpha and omega of
what we will be doing here in this class.
Okay, so if we have a point
on the complex plane, z,
In this location, and
we want to rotate it,
then the argument is that we can
simply multiply it by a complex
number here, e to the j alpha,
that's the previous character, and
it will exactly move z by an angle alpha,
okay?
So rotation amounts to
multiplying with the complex
number which is of unit norm.
So, a complex number of
the form e to the j alpha.
So you can see that a relatively
complicated operation on
the complex plane,
which is moving from here to z',
is actually very simple in this
algebra using complex exponentials.
Let's look at what we will call the
complex exponential generating machine.
x[n] = e to the j omega n.
x[n+1] is simply e to the j times x[n].
And so, recursively we're
going to generate successive
samples of this discrete
time complex exponential.
So let's start at x[0],
it's straight on the real line,
as I will move by omega, that gives x[1[,
by 2 omega, is x[2], etc.
It will keep going x[4], 5, 6, 7, and so
on, and in this particular case,
x[12] happens to be back at the origin,
and therefore, it will be a periodic
discrete complex exponential.
Now, if there is an initial phase,
we don't start from the real line, but
we'll start with an angle theta.
But then the same story goes on,
and as we go,
x[6], 7, etc., at x[12],
we will be back at
the same location as x[0].
Now, one has to be careful,
because not every sinusoid is periodic
in discrete time, and we have an example
here where omega is chosen such that
the result will actually not be periodic.
So x[1], move by omega on the unit circle,
x[2], x[3], 4, 5, 6, 7, 8,
is not back to the origin, and
we can keep going like this.
And it's not hard to see that
this will not be periodic.
What does periodicity mean for
a discrete time exponential?
So e to the j omega n is periodic if and
only if omega is equal to
a rational factor of 2 pi.
So, M over N times 2 pi,
where M and N are integers.
And in this case,
we can actually the fact, of course,
that if we add a multiple of 2 pi to
the angle of a complex exponential,
we are back to e to the j omega.
The quiz is about if a single
e to the jn is periodic.
On the complex plane, a single point
can have many different names, and this
uncertainty, in some sense, is one of
the reasons why complex exponentials and
discrete time are interesting but
sometimes difficult objects to deal with.
If I write the red dot here,
e to the j alpha, I could have also
said that it's actually the same,
except that I had gone alpha plus 2 pi.
Or I could have gone alpha plus 6 pi.
Or any multiple of 2 pi.
I could also have said that the angle
was actually a negative angle of alpha
minus 2 pi, and this will give me
the exact same point, e to the j alpha.
So the question is, how fast can we go?
And here we are going to watch a movie,
and
the movie is a depiction of this famous
wagon wheel effect that you probably have
seen, watching western movies,
old western movies on television.
Let us do an experiment.
Here we have the wheel of a bicycle.
It has four spokes, so it has fourfold
symmetry, and between two frames,
we see it advances by the number of
degrees from the upper left corner.
So we see it goes faster and faster.
Now, because of fourfold symmetry,
everything we have seen before as complex
exponentials will be divided by 4.
So when we approach 45 degrees, we start
to have the feeling the wheel goes
backwards, at 45 it has a funny effect,
and it starts to go backwards.
And it will slow down.
Again, the wheel has a fourfold symmetry.
So we are approaching 90
degrees when it will stop.
80, it gets slower and slower.
Even slower.
And at 90, we are soon there,
it will actually stop.
So the wheel is actually turning.
It turns by a quarter of
the circle every frame.
But of course, because of fourfold
symmetry, it actually stops.
Okay, that's an illustration of aliasing.
To understand this,
we are going to look at frequencies and
we increase the frequency.
So the first frequency omega = 2 pi/12.
So it is a very small frequency.
It's one we have seen before, so
x[0], x[1], etc., turns around and
sure enough,
x[12] is back to the real line.
If we take omega is equal to 2 pi/6, so
that's a much larger frequency here,
but it's still a divisor of 2 pi,
and so we have x[1], 2, 3, 4,
5, and x[6] is equal to x[0].
Again, a periodic signal.
If we go up in frequency, omega is now
equal to 2 pi/5, it is a periodic signal,
sure enough, but you can see that
its frequency now is very fast.
2 pi/4, we simply see the four quadrants
here, x[0], x[1], x[2], and x[3].
2 pi/3 divides nicely the unit
circle into three pieces.
2 pi/2 = pi, that's an interesting signal,
because it simply alternates
between +1 and -1.
And it'll alternate like this,
and so this is the maximum frequency
discrete time complex exponential.
Now, if the frequency is between pi and
2 pi, then
there is a solution that we can easily
think of it as a positive frequency,
slightly larger than pi, or a negative
frequency, slightly smaller than pi.
Now, when omega = 2 pi- alpha,
and alpha is small, then of
course we are going to most likely think
that it's actually a negative frequency.
So here we have an example of x[1],
which is 2 pi- alpha away,
but is also minus alpha
from the real line.
And if we do this, we see now that we
have the sense of a negative frequency.
Let's now think about the difference
between digital and physical frequency.
In discrete time, n has no physical
dimensions, it's just a counter.
And periodicity is how many samples
before the pattern repeats.
In the real world, periodicity is how
many seconds before the pattern repeats,
and the frequency is measured in Hz,
or s-1,
which is 1 over the number of
seconds until the pattern repeats.
Now, if you have a PC that plays sound,
you have to map digital frequency or
a discrete time signal x[n] through
a sound card into a loudspeaker.
And so there has to be a relationship
between the clock frequency on your
computer that maps into a physical
frequency out in the real world.
This is given by the system clock.
So, the sound card has a clock Ts and
every Ts second it will
generate a sample that will be
interpolated and fed to the loudspeaker.
We can set the time Ts between samples.
So periodicity of M samples means a
physical periodicity of M times T seconds.
So the real world frequency
is going to be f = 1/MTs.