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PROFESSOR: In the last lecture, I introduced and
illustrated the kinds of signals and systems that we'll
be dealing with throughout this course.
In today's lecture I'd like to be a little more specific, and
in particular, talk about some of the basic signals, both
continuous-time and discrete-time that will form
important building blocks as the course progresses.
Let's begin with one signal, the continuous-time sinusoidal
signal, which perhaps you're already
somewhat familiar with.
Mathematically, the continuous-time sinusoidal
signal is expressed as I've indicated here.
There are three parameters, A, omega_0 and phi.
The parameter A is referred to as the amplitude, the
parameter omega 0 as the frequency, and the parameter
phi as the phase.
And graphically, the continuous-time sinusoidal
signal has the form shown here.
Now, the sinusoidal signal has a number of important
properties that we'll find it convenient to exploit as the
course goes along, one of which is the fact that the
sinusoidal signal is what is referred to as periodic.
What I mean by periodic is that under an appropriate time
shift, which I indicate here as T_0, the signal replicates
or repeats itself.
Or said another way, if we shift the time origin by an
appropriate amount T_0, the smallest value T_0 being
referred to as the period, then x(t) is
equal to itself, shifted.
And we can demonstrate it mathematically by simply
substituting into the mathematical expression for
the sinusoidal signal t + T_0, in place of t.
When we carry out the expansion we then have, for
the argument of the sinusoid, omega_0 t + omega_0 T_0 + phi.
Now, one of the things that we know about sinusoidal
functions is that if you change the argument by any
integer multiple of 2 pi, then the function
has the same value.
And so we can exploit that here, in particular with
omega_0 T_0 and integer multiple of 2 pi.
Then the right-hand side of this equation is equal to the
left-hand side of the equation.
So with omega_0 T_0 equal to 2 pi times an integer, or T_0
equal to 2 pi times an integer divided by
omega_0, the signal repeats.
The period is defined as the smallest value of T_0.
And so the period is 2 pi divided by omega_0.
And going back to our sinusoidal
signal, we can see that--
and I've indicated here, then, the period as 2 pi / omega_0.
And that's the value under which the signal repeats.
Now in addition, a useful property of the sinusoidal
signal is the fact that a time shift of a sinusoid is
equivalent to a phase change.
And we can demonstrate that again mathematically, in
particular if we put the sinusoidal signal
under a time shift--
I've indicated the time shift that I'm
talking about by t_0--
and expand this out, then we see that that is equivalent to
a change in phase.
And an important thing to recognize about this statement
is that not only is a time shift generating a phase
change, but, in fact, if we inserted a phase change, there
is always a value of t_0 which would correspond to an
equivalent time shift.
Said another way, if we take omega_0 t_0 and think of that
as our change in phase, for any change in phase, we can
solve this equation for a time shift, or conversely for any
value of time shift, that represents
an appropriate phase.
So a time shift corresponds to a phase change, and a phase
change, likewise, corresponds to time shift.
And so for example, if we look at the general sinusoidal
signal that we saw previously, in effect, changing the phase
corresponds to moving this signal in time
one way or the other.
For example, if we look at the sinusoidal signal with a phase
equal to 0 that corresponds to locating the time
origin at this peak.
And I've indicated that on the following graph.
So here we have illustrated a sinusoid with 0 phase, or a
cosine with 0 phase, corresponding to taking our
general picture and shifting it.
Shifting it appropriately as I've indicated here.
This, of course, still has the property that it's a periodic
function, since we simply displaced it in time.
And by looking at the graph, what we see is that it has
another very important property, a property referred
to as even.
And that's a property that we'll find useful, in general,
to refer to in relation to signals.
A signal is said to be even if, when we reflect it about
the origin, it looks exactly the same.
So it's symmetric about the origin.
And looking at this sinusoid, that, in
fact, has that property.
And mathematically, the statement that it's even is
equivalent to the statement that if we replace the time
argument by its negative, the function
itself doesn't change.
Now this corresponded to a phase shift of 0 in our
original cosine expression.
If instead, we had chosen a phase shift of, let's say,
-pi/2, then instead of a cosinusoidal signal, what we
would regenerate is a sinusoid with the appropriate phase.
Or, said another way, if we take our original cosine and
substitute in for the phase -pi/2, then of course we have
this mathematical expression.
Using just straightforward trigonometric identities, we
can express that alternately as sin(omega_0*t).
The frequency and amplitude, of course, haven't changed.
And that, you can convince yourself, also is equivalent
to shifting the cosine by an amount in time that I've
indicated here, namely a quarter of a period.
So illustrated below is the graph now, when we have a
phase of -pi/2 in our cosine, which is a sinusoidal signal.
Of course, it's still periodic.
It's periodic with a period of 2 pi / omega_0 again, because
all that we've done by introducing a phase change is
introduced the time shift.
Now, when we look at the sinusoid in comparison with
the cosine, namely with this particular choice of phase,
this has a different symmetry, and that symmetry
is referred to odd.
What odd symmetry means, graphically, is that when we
flip the signal about the time origin, we also multiply it by
a minus sign.
So that's, in effect, anti-symmetric.
It's not the mirror image, but it's the mirror
image flipped over.
And we'll find many occasions, not only to refer to signals
more general than sinusoidal signals, as even in some cases
and odd in other cases.
And in general, mathematically, an odd signal
is one which satisfies the algebraic expression, x(t).
When you replace t by its negative, is equal to -x(-t).
So replacing the argument by its negative corresponds to an
algebraic sign reversal.
OK.
So this is the class of continuous-time sinusoids.
We'll have a little more to say about it later.
But I'd now like to turn to discrete-time sinusoids.
What we'll see is that discrete-time sinusoids are
very much like continuous-time ones, but also with some very
important differences.
And we want to focus, not only on the similarities, but also
on the differences.
Well, let's begin with the mathematical expression.
A discrete-time sinusoidal signal, mathematically, is as
I've indicated here, A cos(omega_0 n + phi).
And just as in the continuous-time case, the
parameter A is what we'll refer to as the amplitude,
omega_0 as the frequency, and phi as the phase.
And I've illustrated here several discrete-time
sinusoidal signals.
And they kind of look similar.
In fact, if you track what you might think of as the
envelope, it looks very much like what a continuous-time
sinusoid might look like.
But keep in mind that the independent variable, in this
case, is an integer variable.
And so the sequence only takes on values at integer values of
the argument.
And we'll see that has a very important implication, and
we'll see that shortly.
Now, one of the issues that we addressed in the
continuous-time case was periodicity.
And I want to return to that shortly, because that is one
of the areas where there is an important distinction.
Let's first, though, examine the statement similar to the
one that we examined for continuous time, namely the
relationship between a time shift and a phase change.
Now, in continuous time, of course, we saw that a time
shift corresponds to a phase change, and vice versa.
Let's first look at the relationship between shifting
time and generating a change in phase.
In particular for discrete time, if I implement a time
shift that generates a phase change--
and we can see that easily by simply inserting a
time shift, n + n_0.
And if we expand out this argument, we have omega_0 n +
omega_0 n_0.
And so I've done that on the right-hand side of the
equation here.
And the omega_0 n_0, then, simply corresponds
to a change in phase.
So clearly, a shift in time generates a change in phase.
And for example, if we take a particular sinusoidal signal,
let's say we take the cosine signal at a particular
frequency, and with a phase equal to 0, a sequence that we
might generate is one that I've illustrated here.
So what I'm illustrating here is the cosine
signal with 0 phase.
And it has a particular behavior to it, which will
depend somewhat on the frequency.
If I now take this same sequence and shift it so that
the time origin is shifted a quarter of a period away, then
you can convince yourself--
and it's straightforward to work out-- that that time
shift corresponds to a phase shift of pi/2.
So in that case, with the cosine with a phase of -pi/2,
that will correspond to the expression that I have here.
We could alternately write that, using again a
trigonometric identity, as a sine function.
And that, I've stated, is equivalent to a time shift.
Namely, this shift of pi/2 is equal to a certain time shift,
and the time shift for this particular example is a
quarter of a period.
So here, we have the sinusoid.
Previously we had the cosine.
The cosine was exactly the same sequence, but with the
origin located here.
And in fact, that's exactly the way we drew this graph.
Namely, we just simply took the same values and changed
the time origin.
Now, looking at this sequence, which is the sinusoidal
sequence, the phase of -pi/2, that has a certain symmetry.
And in fact, what we see is that it has an odd symmetry,
just as in the continuous-time case.
Namely, if we take that sequence, flip it about the
axis, and flip it over in sign, that we get the same
sequence back again.
Whereas with 0 phase corresponding to the cosine
that I showed previously, that has an even symmetry.
Namely, if I flip it about the time origin and don't do a
sign reversal, then the sequence is maintained.
So here, we have an odd symmetry, expressed
mathematically as I've indicated.
Namely, replacing the independent variable by its
negative attaches a negative sign to the whole sequence.
Whereas in the previous case, what we have is 0 phase and an
even symmetry.
And that's expressed mathematically as x[n]
= x[-n].
Now, one of the things I've said so far about
discrete-time sinusoids is that a time shift corresponds
to a phase change.
And we can then ask whether the reverse statement is also
true, and we knew that the reverse statement was true in
continuous time.
Specifically, is it true that a phase change always
corresponds to a time shift?
Now, we know that that is true, namely, that this
statement works both ways in continuous time.
Does it in discrete time?
Well, the answer, somewhat interestingly or surprisingly
until you sit down and think about it, is no.
It is not necessarily true in discrete time that any phase
change can be interpreted as a simple time
shift of the sequence.
And let me just indicate what the problem is.
If we look at the relationship between the left side and the
right side of this equation, expanding this out as we did
previously, we have that omega_0 n + omega_0 n_0 must
correspond to omega_0 n + phi.
And so omega_0 n_0 must correspond
to the phase change.
Now, what you can see pretty clearly is that depending on
the relationship between phi and omega_0, n_0 may or may
not come out to be an integer.
Now, in continuous time, the amount of time shift did not
have to be an integer amount.
In discrete time, when we talk about a time shift, the amount
of time shift-- obviously, because of the nature of
discrete time signals--
must be an integer.
So the phase changes related to time shifts must satisfy
this particular relationship.
Namely, that omega_0 n_0, where n_0 is an integer, is
equal to the change in phase.
OK.
Now, that's one distinction between continuous time and
discrete time.
Let's now focus on another one, namely the issue of
periodicity.
And what we'll see is that again, whereas in continuous
time, all continuous-time sinusoids are periodic, in the
discrete-time case that is not necessarily true.
To explore that a little more carefully, let's look at the
expression, again, for a general sinusoidal signal with
an arbitrary amplitude, frequency, and phase.
And for this to be periodic, what we require is that there
be some value, N, under which, when we shift the sequence by
that amount, we get the same sequence back again.
And the smallest-value N is what we've
defined as the period.
Now, when we try that on a sinusoid, we of course
substitute in for n, n + N. And when we expand out the
argument here, we'll get the argument that I have on the
right-hand side.
And in order for this to repeat, in other words, in
order for us to discard this term, omega_0 N, where N is
the period, must be an integer multiple of 2 pi.
And in that case, it's periodic as long as omega_0 N,
N being the period, is 2 pi times an integer.
Just simply dividing this out, we have N, the period, is 2 pi
m / omega_0.
Well, you could say, OK what's the big deal?
Whatever N happens to come out to be when we do that little
bit of algebra, that's the period.
But in fact, N, or 2 pi m / omega_0, may not ever come out
to be an integer.
Or it may not come out to be the one that
you thought it might.
For example, let's look at some
particular sinusoidal signals.
Let's see.
We have the first one here, which is a
sinusoid, as I've shown.
And it has a frequency, what I've referred to as the
frequency, omega_0 = 2 pi / 12.
And what we'd like to look at is 2 pi / omega_0, then find
an integer to multiply that by in order
to get another integer.
Let's just try that here.
If we look at 2 pi / omega_0, 2 pi / omega_0, for this case,
is equal to 12.
Well, that's fine.
12 is an integer.
So what that says is that this sinusoidal signal is periodic.
And in fact, it's periodic with a period of 12.
Let's look at the next one.
The next one, we would have 2 pi / omega_0 again.
And that's equal to 31/4.
So what that says is that the period is 31/4.
But wait a minute.
31/4 isn't an integer.
We have to multiply that by an integer
to get another integer.
Well, we'd multiply that by 4, so (2 pi / omega_0) times 4 is
31, 31 is an integer.
And so what that says is this is periodic, not with a period
of 2 pi / omega_0, but with a period of (2 pi / omega_0)
times 4, namely with a period of 31.
Finally, let's take the example where omega_0 is equal
to 1/6, as I've shown here.
That actually looks, if you track it with your eye, like
it's periodic.
2 pi / omega_0, in that case, is equal to 12 pi.
Well, what integer can I multiply 12 pi by and get
another integer?
The answer is none, because pi is an irrational number.
So in fact, what that says is that if you look at this
sinusoidal signal, it's not periodic at all, even though
you might fool yourself into thinking it is simply because
the envelope looks periodic.
Namely, the continuous-time equivalent of this is
periodic, the discrete-time sequence is not.
OK.
Well, we've seen, then, some important distinctions between
continuous-time sinusoidal signals and discrete-time
sinusoidal signals.
The first one is the fact that in the continuous-time case, a
time shift and phase change are always equivalent.
Whereas in the discrete-time case, in effect, it works one
way but not the other way.
We've also seen that for a continuous-time signal, the
continuous-time signal is always periodic, whereas the
discrete-time signal is not necessarily.
In particular, for the continuous-time case, if we
have a general expression for the sinusoidal signal that
I've indicated here, that's periodic for
any choice of omega_0.
Whereas in the discrete-time case, it's periodic only if 2
pi / omega_0 can be multiplied by an integer
to get another integer.
Now, another important and, as it turns out, useful
distinction between the continuous-time and
discrete-time case is the fact that in the discrete-time
case, as we vary what I've called the frequency omega_0,
we only see distinct signals as omega_0
varies over a 2 pi interval.
And if we let omega_0 vary outside the range of, let's
say, -pi to pi, or 0 to 2 pi, we'll see the same sequences
all over again, even though at first glance, the mathematical
expression might look different.
So in the discrete-time case, this class of signals is
identical for values of omega_0 separated by 2 pi,
whereas in the continuous-time case, that is not true.
In particular, if I consider these sinusoidal
continuous-time signals, as I vary omega_0, what will happen
is that I will always see different sinusoidal signals.
Namely, these won't be equal.
And in effect, we can justify that statement algebraically.
And I won't take the time to do it carefully.
But let's look, first of all, at the discrete-time case.
And the statement that I'm making is that if I have two
discrete-time sinusoidal signals at two different
frequencies, and if these frequencies are separated by
an integer multiple of 2 pi-- namely if omega_2 is equal to
omega_1 + 2 pi times an integer m--
when I substitute this into this expression, because of
the fact that n is also an integer, I'll have m * n as an
integer multiple of 2 pi.
And that term, of course, will disappear because of the
periodicity of the sinusoid, and these two
sequences will be equal.
On the other hand in the continuous-time case, since t
is not restricted to be an integer variable, for
different values of omega_1 and omega_2, these sinusoidal
signals will always be different.
OK.
Now, many of the issues that I've raised so far, in
relation to sinusoidal signals, are elaborated on in
more detail in the text.
And of course, you'll have an opportunity to exercise some
of this as you work through the video course manual.
Let me stress that sinusoidal signals will play an extremely
important role for us as building blocks for general
signals and descriptions of systems, and leads to the
whole concept Fourier analysis, which is very
heavily exploited throughout the course.
What I'd now like to turn to is another class of important
building blocks.
And in fact, we'll see that under certain conditions,
these relate strongly to sinusoidal signals, namely the
class of real and complex exponentials.
Let me begin, first of all, with the real exponential, and
in particular, in the continuous-time case.
A real continuous-time exponential is mathematically
expressed, as I indicate here, x(t) = C e ^ (a t), where for
the real exponential, C and a are real numbers.
And that's what we mean by the real exponential.
Shortly, we'll also consider complex exponentials, where
these numbers can then become complex.
So this is an exponential function.
And for example, if the parameter a is positive, that
means that we have a growing exponential function.
If the parameter a is negative, then that means that
we have a decaying exponential function.
Now, somewhat as an aside, it's kind of interesting to
note that for exponentials, a time shift corresponds to a
scale change, which is somewhat different than what
happens with sinusoids.
In the sinusoidal case, we saw that a time shift corresponded
to a phase change.
With the real exponential, a time shift, as it turns out,
corresponds to simply changing the scale.
There's nothing particularly crucial or
exciting about that.
And in fact, perhaps stressing it is a little misleading.
For general functions, of course, about all that you can
say about what happens when you implement a time shift is
that it implements a time shift.
OK.
So here's the real exponential.
Just C e ^ (a t).
Let's look at the real exponential, now, in the
discrete-time case.
And in the discrete-time case, we have several alternate ways
of expressing it.
We can express the real exponential in the form C e ^
(beta n), or as we'll find more convenient, in part for a
reason at I'll indicate shortly, we can rewrite this
as C alpha ^ n, where of course, alpha = e ^ beta.
More typically in the discrete-time case, we'll
express the exponential as C alpha ^ n.
So for example, this becomes, essentially, a geometric
series or progression as n continues for
certain values of alpha.
Here for example, we have for alpha greater than 0, first of
all on the top, the case where the magnitude of alpha is
greater than 1, so that the sequence is exponentially or
geometrically growing.
On the bottom, again with alpha positive, but now with
its magnitude less than 1, we have a geometric progression
that is exponentially or geometrically decaying.
OK.
So this, in both of these cases, is with alpha
greater than 0.
Now the function that we're talking about is alpha ^ n.
And of course, what you can see is that if alpha is
negative instead of positive, then when n is even, that
minus sign is going to disappear.
When n is odd, there will be a minus sign.
And so for alpha negative, the sequence is going to alternate
positive and negative values.
So for example, here we have alpha negative, with its
magnitude less than 1.
And you can see that, again, its envelope decays
geometrically, and the values alternate in sign.
And here we have the magnitude of alpha greater than 1, with
alpha negative.
Again, they alternate in sign, and of course it's growing
geometrically.
Now, if you think about alpha positive and go back to the
expression that I have at the top, namely C alpha ^ n.
With alpha positive, you can see a straightforward
relationship between alpha and beta.
Namely, beta is the natural logarithm of alpha.
Something to think about is what
happens if alpha is negative?
Which is, of course, a very important and useful class of
real discrete-time exponentials also.
Well, it turns out that with alpha negative, if you try to
express it as C e ^ (beta n), then beta comes out to be an
imaginary number.
And that is one, but not the only reason why, in the
discrete-time case, it's often most convenient to phrase real
exponentials in the form alpha ^ n, rather than e ^ (beta n).
In other words, to express them in this form rather than
in this form.
Those are real exponentials, continuous-time and
discrete-time.
Now let's look at the continuous-time complex
exponential.
And what I mean by a complex exponential, again, is an
exponential of the form C e ^ (a t).
But in this case, we allow the parameters C and a to be
complex numbers.
And let's just track this through algebraically.
If C and a are complex numbers, let's write C in
polar form, so it has a magnitude and an angle.
Let's write a in rectangular form, so it has a real part
and an imaginary part.
And when we substitute these two in here, combine some
things together--
well actually, I haven't combined yet.
I have this for the amplitude factor, and this for the
exponential factor.
I can now pull out of this the term corresponding to e ^ (r
t), and combine the imaginary parts together.
And I come down to the expression that I have here.
So following this further, an exponential of this form, e ^
(j omega) or e ^ (j phi), using Euler's relation, can be
expressed as the sum of a cosine plus j times a sine.
And so that corresponds to this factor.
And then there is this time-varying amplitude factor
on top of it.
Finally putting those together, we end up with the
expression that I show on the bottom.
And what this corresponds to are two sinusoidal signals, 90
degrees out of phase, as indicated by the fact that
there's a cosine and a sine.
So there's a real part and an imaginary part, with
sinusoidal components 90 degrees out of phase, and a
time-varying amplitude factor, which is a real exponential.
So it's a sinusoid multiplied by a real exponential in both
the real part and the imaginary part.
And let's just see what one of those terms might look like.
What I've indicated at the top is a sinusoidal signal with a
time-varying exponential envelope, or an envelope which
is a real exponential, and in particular which is growing,
namely with r greater than 0.
And on the bottom, I've indicated the same thing with
r less than 0.
And this kind of sinusoidal signal, by the way, is
typically referred to as a damped sinusoid.
So with r negative, what we have in the real and imaginary
parts are damped sinusoids.
And the sinusoidal components of that are 90 degrees out of
phase, in the real part and in the imaginary part.
OK.
Now, in the discrete-time case, we have more or less the
same kind of outcome.
In particular we'll make reference to our complex
exponentials in the discrete-time case.
The expression for the complex exponential looks very much
like the expression for the real exponential, except that
now we have complex factors.
So C and alpha are complex numbers.
And again, if we track through the algebra, and get to a
point where we have a real exponential multiplied by a
factor which is a purely imaginary exponential, apply
Euler's relationship to this, we then finally come down to a
sequence, which has a real exponential amplitude
multiplying one sinusoid in the real part.
And in the imaginary part, exactly the same kind of
exponential multiplying a sinusoid that's 90 degrees out
of phase from that.
And so if we look at what one of these factors might look
like, it's what we would expect given the analogy with
the continuous-time case.
Namely, it's a sinusoidal sequence with a real
exponential envelope.
In the case where alpha is positive, then
it's a growing envelope.
In the case where alpha is negative--
I'm sorry--
where the magnitude of alpha is greater than 1, it's a
growing exponential envelope.
Where the magnitude of alpha is less than 1, it's a
decaying exponential envelope.
And so I've illustrated that here.
Here we have the magnitude of alpha greater than 1.
And here we have the magnitude of alpha less than 1.
In both cases, sinusoidal sequences underneath the
envelope, and then an envelope that is dictated by what the
magnitude of alpha is.
OK.
Now, in the discrete-time case, then, we have results
similar to the continuous-time case.
Namely, components in a real and imaginary part that have a
real exponential factor times a sinusoid.
Of course, if the magnitude of alpha is equal to 1, then this
factor disappears, or is equal to 1.
And this factor is equal to 1.
And so we have sinusoids in both the real
and imaginary parts.
Now, one can ask whether, in general, the complex
exponential with the magnitude of alpha equal to 1 is
periodic or not periodic.
And the clue to that can be inferred by examining this
expression.
In particular, in the discrete-time case with the
magnitude of alpha equal to 1, we have pure sinusoids in the
real part and the imaginary part.
And in fact, in a continuous-time case with r
equal to 0, we have sinusoids in the real part and the
imaginary part.
In a continuous-time case when we have a pure complex
exponential, so that the terms aren't exponentially growing
or decaying, those
exponentials are always periodic.
Because, of course, the real and imaginary sinusoidal
components are periodic.
In the discrete-time case, we know that the sinusoids may or
may not be periodic, depending on the value of omega_0.
And so in fact, in the discrete-time case, the
exponential e ^ (j omega_0 n), that I've indicated here, may
or may not be periodic depending on what the value of
omega_0 is.
OK.
Now, to summarize, in this lecture I've introduced and
discussed a number of important basic signals.
In particular, sinusoids and real and complex exponentials.
One of the important outcomes of the discussion, emphasized
further in the text, is that there are some very important
similarities between them.
But there are also some very important differences.
And these differences will surface when we exploit
sinusoids and complex exponentials as basic building
blocks for more general continuous-time and
discrete-time signals.
In the next lecture, what I'll discuss are some other very
important building blocks, namely, what are referred to
as step signals and impulse signals.
And those, together with the sinusoidal signals and
exponentials as we've talked about today, will really form
the cornerstone for, essentially, all of the signal
and system analysis that we'll be dealing with for the
remainder of course.
Thank you.