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[MUSIC PLAYING]
PROFESSOR: Last time we began
the development of the
discrete-time Fourier
transform.
And just as with the
continuous-time case, we first
treated the notion of
periodic signals.
This led to the Fourier
series.
And then we generalized that to
the Fourier transform, and
finally incorporated within the
framework of the Fourier
transform both aperiodic
and periodic signals.
In today's lecture, what I'd
like to do is expand on some
of the properties of the
Fourier transform, and
indicate how those properties
are used for
a variety of things.
Well, let's begin by reviewing
the Fourier transform as we
developed it last time.
It, of course, involves a
synthesis equation and an
analysis equation.
The synthesis equation
expressing x of n, the
sequence, in terms of the
Fourier transform, and the
analysis equation telling us
how to obtain the Fourier
transform from the original
sequence.
And I draw your attention again
to the basic point that
the synthesis equation
essentially corresponds to
decomposing the sequence as a
linear combination of complex
exponentials with amplitudes
that are, in effect,
proportional to the
Fourier transform.
Now, the discrete-time Fourier
transform, just as the
continuous-time Fourier
transform, has a number of
important and useful
properties.
Of course, as I stressed last
time, it's a function of a
continuous variable.
And it's also a complex-valued
function, which means that
when we represent it in
general it requires a
representation in terms of its
real part and imaginary part,
or in terms of magnitude
and angle.
Also, as I indicated last time,
the Fourier transform is
a periodic function of
frequency, and the periodicity
is with a period of 2 pi.
And so it says, in effect, that
the Fourier transform, if
we replace the frequency
variable by an integer
multiple of 2 pi, the
function repeats.
And I stress again that the
underlying basis for this
periodicity property is the
fact that it's the set of
complex exponentials that are
inherently periodic in frequency.
And so, of course, any
representation using them
would, in effect, generate
a periodicity with
this period of 2 pi.
Just as in continuous time,
the Fourier transform has
important symmetry properties.
And in particular, if the
sequence x sub n is
real-valued, then the
Fourier transform
is conjugate symmetric.
In other words, if we replace
omega by minus omega, that's
equivalent to applying
complex conjugation
to the Fourier transform.
And as a consequence of this
conjugate symmetry, this
results in a symmetry in the
real part that is an even
symmetry, or the magnitude has
an even symmetry, whereas the
imaginary part or the phase
angle are both odd symmetric.
And these are symmetry
properties, again, that are
identical to the symmetry
properties that we saw in
continuous time.
Well, let's see this in the
context of an example that we
worked last time and that we'll
want to draw attention
to in reference to several
issues as this
lecture goes along.
And that is the Fourier
transform of a real damped
exponential.
So the sequence that we are
talking about is a to the n u
of n, and let's consider
a to be positive.
We saw last time that the
Fourier transform for this
sequence algebraically
is of this form.
And if we look at its magnitude
and angle, the
magnitude I show here.
And the magnitude, as
we see, has the
properties that we indicated.
It is an even function
of frequency.
Of course, it's a function
of a continuous variable.
And it, in addition,
is periodic with a
period of two pi.
On the other hand, if we look
at the phase angle below it,
the phase angle has a symmetry
which is odd symmetric.
And that's indicated clearly
in this picture.
And of course, in addition to
being odd symmetric, it
naturally has to be, again, a
periodic function of frequency
with a period of 2 pi.
OK, so we have some symmetry
properties.
We have this inherent
periodicity in the Fourier
transform, which I'm stressing
very heavily because it forms
the basic difference between
continuous time
and discrete time.
In addition to these properties
of the Fourier
transform, there are a number
of other properties that are
particularly useful in the
manipulation of the Fourier
transform, and, in fact, in
using the Fourier transform
to, for example, analyze systems
represented by linear
constant coefficient difference
equations.
There in the text is a longer
list of properties, but let me
just draw your attention
to several of them.
One is the time shifting
property.
And the time shifting property
tells us that if x of omega is
the Fourier transform of x of n,
then the Fourier transform
of x of n shifted in time is
that same Fourier transform
multiplied by this factor,
which is a
linear phase factor.
So time shifting introduces
a linear phase term.
And, by the way, recall that in
the continuous-time case we
had a similar situation, namely
that a time shift
corresponded to a
linear phase.
There also is a dual to the time
shifting property, which
is referred to as the frequency
shifting property,
which tells us that if we
multiply a time function by a
complex exponential,
that, in effect,
generates a frequency shift.
And we'll see this frequency
shifting property surface in a
slightly different way shortly,
when we talk about
the modulation property in
the discrete-time case.
Another important property that
we'll want to make use of
shortly is linearity, which
follows in a very
straightforward way from the
Fourier transform definition.
And the linearity property says
simply that the Fourier
transform of a sum, or linear
combination, is the same
linear combination of the
Fourier transforms.
Again, that's a property that
we saw in continuous time.
And, also, among other
properties there is a
Parseval's relation for the
discrete-time case that in
effect says something similar
to continuous time,
specifically that the energy
in the sequence is
proportional to the energy in
the Fourier transform, the
energy over one period.
Or, said another way, in fact,
or another way that it can be
said, is that the energy
in the time domain is
proportional to the
power in this
periodic Fourier transform.
OK, so these are some
of the properties.
And, as I indicated, parallel
somewhat properties that we
saw in continuous time.
Two additional properties that
will play important roles in
discrete time just as they did
in continuous time are the
convolution property and the
modulation property.
The convolution property is the
property that tells us how
to relate the Fourier transform
of the convolution
of two sequences to the Fourier
transforms of the
individual sequences.
And, not surprisingly,
what happens--
and this can be demonstrated
algebraically--
not surprisingly, the Fourier
transform of the convolution
is simply the product of
the Fourier transforms.
So, Fourier transform maps
convolution in the time domain
to multiplication in the
frequency domain.
Now convolution, of course,
arises in the context of
linear time-invariant systems.
In particular, if we have
a system with an impulse
response h of n, input x of n,
the output is the convolution.
The convolution property then
tells us that in the frequency
domain, the Fourier transform is
the product of the Fourier
transform of the impulse
response and the Fourier
transform of the input.
Now we also saw and have talked
about a relationship
between the Fourier transform,
the impulse response, and what
we call the frequency response
in the context of the response
of a system to a complex
exponential.
Specifically, complex
exponentials are
eigenfunctions of linear
time-invariant systems.
One of these into the system
gives us, as an output, a
complex exponential with the
same complex frequency
multiplied by what we refer
to as the eigenvalue.
And as you saw in the video
course manual, this
eigenvalue, this constant,
multiplier on the exponential
is, in fact, the Fourier
transform of the impulse
response evaluated at
that frequency.
Now, we saw exactly the same
statement in continuous time.
And, in fact, we used
that statement--
the frequency response
interpretation of the Fourier
transform, the impulse
response--
we use that to motivate an
intuitive interpretation of
the convolution property.
Now, formally the convolution
property can be developed by
taking the convolution sum,
applying the Fourier transform
sum to it, doing the appropriate
substitution of
variables, interchanging order
of summations, et cetera, and
all the algebra works out to
show that it's a product.
But as I stressed when we
discussed this with continuous
time, the interpretation--
the underlying interpretation--
is particularly important
to understand.
So let me review it again in
the discrete-time case, and
it's exactly the same for
discrete time or for
continuous time.
Specifically, the argument was
that the Fourier transform of
a sequence or signal corresponds
to decomposing it
into a linear combination
of complex exponentials.
What's the amplitude of those
complex exponentials?
It's basically proportional
to the Fourier transform.
If we think of pushing through
the system that linear
combination, then each of those
complex exponentials
gets the amplitude modified, or
multiplied, by the Fourier
transform of--
by the frequency response--
which we saw is the
Fourier transform
of the impulse response.
So the amplitudes of the output
complex exponentials is
then the amplitudes of the input
complex exponentials
multiplied by the frequency
response.
And the Fourier transform of the
output, in effect, is an
expression expressing the
summation, or integration, of
the output as a linear
combination of all of these
exponentials with the
appropriate complex
amplitudes.
So, it's important, in thinking
about the convolution
property, to think about it in
terms of nothing more than the
fact that we've decomposed the
input, and we're now modifying
separately through
multiplication, through
scaling, the amplitudes
of each of the complex
exponential components.
Now what we saw in continuous
time is that this
interpretation and the
convolution property led to an
important concept, namely the
concept of filtering.
Kind of the idea that if we
decompose the input as a
linear combination of complex
exponentials, we can
separately attenuate or
amplify each of those
components.
And, in fact, we could exactly
pass some set of frequencies
and totally eliminate other
set of frequencies.
So, again, just as in continuous
time, we can talk
about an ideal filter.
And what I show here is the
frequency response of an ideal
lowpass filter.
The ideal lowpass filter, of
course, passes exactly, with a
gain of 1, frequencies around
0, and eliminates totally
other frequencies.
However, an important
distinction here between
continuous time and discrete
time is the fact that, whereas
in continuous time when we
talked about an ideal filter,
we passed a band of frequencies
and totally
eliminated everything else
out to infinity.
In the discrete time case, the
frequency response is periodic.
So, obviously, the frequency
response must periodically
repeat for the lowpass filter.
And in fact we see that here.
If we look at the lowpass
filter, then we've eliminated
some frequencies.
But then we pass, of course,
frequencies around 2 pi, and
also frequencies around minus
2 pi, and for that matter
around any multiple of 2 pi.
Although it's important to
recognize that because of the
inherent periodicity of the
complex exponentials, these
frequencies are exactly the
same frequencies as these
frequencies.
So it's lowpass filtering
interpreted in terms of
frequencies over a range
from minus pi to pi.
Well, just as we talk about a
lowpass filter, we can also
talk about a highpass filter.
And a highpass filter, of
course, would pass high
frequencies.
In a continuous-time case,
high frequencies meant
frequencies that go
out to infinity.
In the discrete-time case,
of course, the highest
frequencies we can generate
are frequencies up to pi.
And once our complex
exponentials go past pi, then,
in fact, we start seeing the
lower frequencies again.
Let me indicate what I mean.
If we think in the context of
the lowpass filter, these are
low frequencies.
As we move along the frequency
axis, these become high
frequencies.
And as we move further along the
frequency axis, what we'll
see when we get to, for example,
a frequency of 2 pi
are the same low frequencies
that we see around 0.
In particular then, an ideal
highpass filter in the
discrete-time case would be a
filter that eliminates these
frequencies and passes
frequencies around pi.
OK, so we've seen the
convolution property and its
interpretation in terms
of filtering.
More broadly, the convolution
property in combination with a
number of the other properties
that I introduced, in
particular the time shifting and
linearity property, allows
us to generate or analyze
systems that are described by
linear constant coefficient
difference equations.
And this, again, parallels very
strongly the discussion
we carried out in the
continuous-time case.
In particular, let's think of a
discrete-time system that is
described by a linear constant
coefficient difference equation.
And we'll restrict the initial
conditions on the equation
such that it corresponds to a
linear time-invariant system.
And recall that, in fact, in
our discussion of linear
constant coefficient difference
equations, it is
the condition of initial rest
that-- on the equation--
that guarantees for us that
the system will be causal,
linear, and time invariant.
OK, now let's consider a
first-order difference
equation, a system
described by a
first-order difference equation.
And we've talked about
the solution of
this equation before.
Essentially, we run the
solution recursively.
Let's now consider generating
the solution by taking
advantage of the properties
of the Fourier transform.
Well, just as we did in
continuous time, we can
consider applying the Fourier
transform to both sides of
this equation.
And the Fourier transform
of y of n, of
course, is Y of omega.
And then using the shifting
property, the time shifting
property, the Fourier transform
of y of n minus 1 is
Y of omega multiplied by
e to the minus j omega.
And so we have this, using a
linearity property we can
carry down the scale factor, and
add these two together as
they're added here.
And the Fourier transform
of x of n is X of omega.
Well, we can solve this equation
for the Fourier
transform of the output in terms
of the Fourier transform
of the input and an appropriate
complex scale factor.
And simply solving this
for Y of omega yields
what we have here.
Now what we've used in going
from this point to this point
is both the shifting property
and we've also used the
linearity property.
At this point, we can recognize
that here the
Fourier transform of the output
is the product of the
Fourier transform of the input
and some complex function.
And from the convolution
property, then, that complex
function must in fact correspond
to the frequency
response, or equivalently, the
Fourier transform of the
impulse response.
So if we want to determine the
Fourier transform of the--
or the impulse response of the
system, let's say for example,
then it becomes a matter of
having identified the Fourier
transform of the impulse
response, which is the
frequency response.
We now want to inverse
transform to
get the impulse response.
Well, how do we inverse
transform?
Of course, we could do it by
attempting to go through the
synthesis equation for the
Fourier transform.
Or we can do as we did in the
continuous-time case which is
to take advantage
of what we know.
And in particular, we know that
from an example that we
worked before, this is in fact
the Fourier transform of a
sequence which is a to
the n times u of n.
And so, in essence,
by inspection--
very similar to what has gone
on in continuous time--
essentially by inspection, we
can then solve for the impulse
response to the system.
OK, so that procedure follows
very much the kind of
procedure that we've carried
out in continuous time.
And this, of course, is
discussed in more
detail in the text.
Well, let's look at
that example then.
Here we have the impulse
response for that, associated
with the system described
by that
particular difference equation.
And to the right of
that, we have the
associated frequency response.
And one of the things
that we notice--
and this is drawn for a positive
between 0 and 1--
what we notice, in fact, is that
it is an approximation to
a lowpass filter, because it
tends to attenuate the high
frequencies and retain and,
in fact, amplify the low
frequencies.
Now if instead, actually, the
impulse response was such that
we picked a to be negative
between minus 1 and 0, then
the impulse response in the time
domain looks like this.
And the corresponding frequency
response looks like this.
And that becomes an
approximation
to a highpass filter.
So, in fact, a first-order
difference equation, as we
see, has a frequency response,
depending on the value of a,
that either looks approximately
like a lowpass
filter for a positive or
a highpass filter for a
negative, very much like the
first-order differential
equation looked like a
lowpass filter in the
continuous-time case.
And, in fact, what I'd like to
illustrate is the filtering
characteristics--
or an example of filtering--
using a first-order difference
equation.
And the example that I'll
illustrate is a filtering of a
sequence that in fact is
filtered very often for very
practical reasons, namely a
sequence which represents the
Dow Jones Industrial Average
over a fairly long period.
And we'll process the Dow Jones
Industrial Average first
through a first-order difference
equation, where, if
we begin with a equals 0,
then, referring to the
frequency response that we have
here, a equals 0 would
simply be passing
all frequencies.
As a is positive we start to
retain mostly low frequencies,
and the larger a gets, but still
less than 1, the more it
attenuates high frequencies
at the expense of low
frequencies.
So let's watch the filtering,
first with a positive and
we'll see it behave as a lowpass
filter, and then with
a negative and we'll see the
difference equation behaving
as a highpass filter.
What we see here is the Dow
Jones Industrial Average over
roughly a five-year period
from 1927 to 1932.
And, in fact, that big dip in
the middle is the famous stock
market crash of 1929.
And we can see that following
that, in fact, the market
continued a very long
downward trend.
And what we now want to do
is process this through a
difference equation.
Above the Dow Jones average we
show the impulse response of
the difference equation.
Here we've chosen the parameter
a equal to 0.
And the impulse response will
be displayed on an expanded
scale in relation to the scale
of the input and, for that
matter, the scale
of the output.
Now with the impulse response
shown here which is just an
impulse, in fact, the output
shown on the bottom trace is
exactly identical
to the input.
And what we'll want to do now
is increase, first, the
parameter a, and the impulse
response will begin to look
like an exponential with a
duration that's longer and
longer as a moves from 0 to 1.
Correspondingly we'll get more
and more lowpass filtering as
the coefficient a increases
from 0 toward 1.
So now we are increasing
the parameter a.
We see that the bottom trace
in relation to the middle
trace in fact is looking more
and more smoothed or
lowpass-filtered.
And here now we have a fair
amount of smoothing, to the
point where the stock market
crash of 1929 is totally lost.
And in fact I'm sure there are
many people who wish that
through filtering we could, in
fact, have avoided the stock
market crash altogether.
Now, let's decrease a from
1 back towards 0.
And as we do that,
we will be taking
out the lowpass filtering.
And when a finally reaches 0,
the impulse response of the
filter will again be an impulse,
and so the output
will be once again identical
to the input.
And that's where we are now.
All right now we want to
continue to decrease a so that
it becomes negative, moving
from 0 toward minus 1.
And what we will see in that
case is more and more highpass
filtering on the output in
relation to the input.
And this will be particularly
evident in, again, the region
of high frequencies represented
by sharp
transitions which, of course,
the market crash
of 1929 would represent.
So here, now, a is decreasing
toward minus 1.
We see that the high
frequencies, or rapid
variations are emphasized., And
finally, let's move from
minus 1 back towards 0, taking
out the highpass filtering and
ending up with a equal to 0,
corresponding to an impulse
response which is an
impulse, in other
words, an identity system.
And let me stress once again
that the time scale on which
we displayed the impulse
response is an expanded time
scale in relation to the time
scale on which we displayed
the input and the output.
OK, so we see that, in fact,
a first-order difference
equation is a filter.
And, in fact, it's a very
important class of filters,
and it's used very often to
do approximate lowpass and
highpass filtering.
Now, in addition to the
convolution property, another
important property that we had
in continuous time, and that
we have in discrete time, is
the modulation property.
The modulation property tells
us what happens in the
frequency domain when
you multiply
signals in the time domain.
In continuous time, the
modulation property
corresponded to the statement
that if we multiply the time
domain, we convolve the Fourier
transforms in the
frequency domain.
And in discrete time we have
very much the same kind of
relationship.
The only real distinction
between these is that in the
discrete-time case, in carrying
out the convolution,
it's an integration only
over a 2 pi interval.
And what that corresponds to
is what's referred to as a
periodic convolution, as opposed
to the continuous-time
case where what we have is
a convolution that is an
aperiodic convolution.
So, again, we have a convolution
property in
discrete time that is very
much like the convolution
property in continuous time.
The only real difference is
that here we're convolving
periodic functions.
And so it's a periodic
convolution which involves an
integration only over a 2 pi
interval, rather than an
integration from minus infinity
to plus infinity.
Well, let's take a look at an
example of the modulation
property, which will then lead
to one particular application,
and a very useful application,
of the modulation property in
discrete time.
The example that I want to pick
is an example in which we
consider modulating
a signal with--
a signal with another signal,
x of n, or x1 of n as I
indicated here, which
is minus 1 to the n.
Essentially what that says is
that any signal which I
modulate with this in effect
corresponds to taking the
original signal and then going
through that signal
alternating the algebraic
signs.
Now we--
in applying the modulation
property, of course, what we
need to do is develop
the Fourier
transform of this signal.
This signal which I rewrite--
I can write either as minus 1
to the n or rewrite as e to
the j pi n since e to the j
pi is equal to minus 1--
is a periodic signal.
And it's the periodic signal
that I show here.
And recall that to get the
Fourier transform of a
periodic signal, one way to do
it is to generate the Fourier
series coefficients for the
periodic signal, and then
identify the Fourier transform
as an impulse train where the
heights of the impulses in
the impulse train are
proportional, with a
proportionality factor of 2
pi, proportional to the Fourier
series coefficients.
So let's first work out what the
Fourier series is and for
this example, in fact,
it's fairly easy.
Here is the general
synthesis equation
for the Fourier series.
And if we take our particular
example where, if we look back
at the curve above, what we
recognize is that the period
is equal to 2, namely it
repeats after 2 points.
Then capital N is equal to 2,
and so we can just write this
out with the two terms.
And the two terms involved are
x1 of n is a0, the 0-th
coefficient, that's with k
equals 0, and a1, and this is
with k equals 1, and
we substituted in
capital N equal to 2.
All right, well, we can do a
little bit of algebra here,
obviously cross off
the factors of 2.
And what we recognize, if we
compare this expression with
the original signal which is e
to the j pi n, then we can
simply identify the fact that
a0, the 0-th coefficient is 0,
that's the DC term.
And the coefficient
a1 is equal to 1.
So we've done it simply by
essentially inspecting the
Fourier series synthesis
equation.
OK, now, if we want to get the
Fourier transform for this, we
take those coefficients and
essentially generate an
impulse train where we choose
as values for the impulses 2
pi times the Fourier series
coefficients.
So, the Fourier series
coefficients are a0 is equal
to 0 and a1 is equal to 1.
So, notice that in the plot that
I've shown here of the
Fourier transform of x1 of n, we
have the 0-th coefficient,
which happens to be 0, and so
I have it indicated, an
impulse there.
We have the coefficient a1, and
the coefficient a1 occurs
at a frequency which is omega
0, and omega 0 in fact is
equal to pi because the signal
is e to the j pi n.
Well, what's this impulse
over here?
Well, that impulse is a--
corresponds to the
Fourier series
coefficient a sub minus 1.
And, of course, if we drew
this out over a longer
frequency axis, we would see
lots of other impulses because
of the fact that the Fourier
transform periodically repeats
or, equivalently, the Fourier
series coefficients
periodically repeat.
So this is the coefficient a0,
This is the coefficient a1
with a factor of 2 pi,
this is 2 pi times a0
and 2 pi times a1.
And then this is simply an
indication that it's
periodically repeated.
All right.
Now, let's consider what happens
if we take a signal
and multiply it, modulate
it, by minus 1 to the n.
Well in the frequency domain
that corresponds to a
convolution.
Let's consider a signal x2
of n which has a Fourier
transform as I've
indicated here.
Then the Fourier transform of
the product of x1 of n and x2
of n is the convolution
of these two spectra.
And recall that if you could
convolve something with an
impulse train, as this is, that
simply corresponds to
taking the something and placing
it at the positions of
each of the impulses.
So, in fact, the result of the
convolution of this with this
would then be the spectrum that
I indicate here, namely
this spectrum shifted up to pi
and of course to minus pi.
And then of course to not only
pi but 3 pi and 5 pi, et
cetera, et cetera.
And so this spectrum, finally,
corresponds to the Fourier
transform of minus 1 to the n
times x2 of n where x2 of n is
the sequence whose spectrum
was X2 of omega.
OK, now, this is in fact an
important, useful, and
interesting point.
What it says is if I have a
signal with a certain spectrum
and if I modulate--
multiply--
that signal by minus 1 to the
n, meaning that I alternate
the signs, then it takes
the low frequencies--
in effect, it shifts
the spectrum by pi.
So it takes the low frequencies
and moves them up
to high frequencies, and will
incidentally take the high
frequencies and move them
to low frequencies.
So in fact we, in essence,
saw this when we took--
or when I talked about the
example of a sequence which
was a to the n times u of n.
Notice--
let me draw your attention to
the fact that when a is
positive, we have this sequence
and its Fourier
transform is as I show
on the right.
For a negative, the sequence is
identical to a positive but
with alternating sines.
And the Fourier transform of
that you can now see, and
verify also algebraically if
you'd like, is identical to
this spectrum, simply
shifted by pi.
So it says in fact that
multiplying that impulse
response, or if we think of a
positive and a negative, that
is algebraically similar to
multiplying the impulse
response by minus 1 to the n.
And in the frequency domain,
the effect of that,
essentially, is shifting
the spectrum by pi.
And we can interpret that
in the context of
the modulation property.
Now it's interesting that what
that says is that if we have a
system which corresponds to a
lowpass filter, as I indicate
here, with an impulse
response h of n.
And it can be any approximation
to a lowpass
filter and even an ideal
lowpass filter.
If we want to convert that to
a highpass filter, we can do
that by generating a new system
whose impulse response
is minus 1 to the n times the
impulse response of the
lowpass filter.
And this modulation by minus
1 to the n will take the
frequency response of this
system and shift it by pi so
that what's going on here at low
frequencies will now go on
here at high frequencies.
This also says, incidentally,
that if we look at an ideal
lowpass filter and an ideal
highpass filter, and we choose
the cutoff frequencies for
comparison, or the bandwidth
of the filter to be equal.
Since this ideal highpass filter
is this ideal lowpass
filter with the frequency
response shifted by pi, the
modulation property tells us
that in the time domain, what
that corresponds to is an
impulse response multiplied by
minus 1 to the n.
So it says that the impulse
response of the highpass
filter, or equivalently the
inverse Fourier transform of
the highpass filter frequency
response, is minus 1 to the n
times the impulse response
for the lowpass filter.
That all follows from the
modulation property.
Now there's another way, an
interesting and useful way,
that modulation can be used to
implement or convert from
lowpass filtering to
highpass filtering.
The modulation property tells us
about multiplying the time
domain is shifting in the
frequency domain.
And in the example that we
happened to pick said if you
multiply or modulate by minus
1 to the n, that takes low
frequencies and shifts them
to high frequencies.
What that tells us, as a
practical and useful notion,
is the following.
Suppose we have a system that
we know is a lowpass filter,
and it's a good lowpass
filter.
How might we use it as
a highpass filter?
Well, one way to do it, instead
of shifting its
frequency response, is to take
the original signal, shift its
low frequencies to high
frequencies and its high
frequencies to low frequencies
by multiplying the input
signal, the original signal, by
minus 1 to the n, process
that with a lowpass filter where
now what's sitting at
the low frequencies were
the high frequencies.
And then unscramble it all at
the output so that we put the
frequencies back where
they belong.
And I summarize that here.
Let's suppose, for example, that
this system was a lowpass
filter, and so it
lowpass-filters
whatever comes into it.
Down below, I indicate taking
the input and first
interchanging the high and low
frequencies through modulation
with minus 1 to the n.
Doing the lowpass filtering,
which--
and what's sitting at the low
frequencies here were the high
frequencies of this signal.
And then after the lowpass
filtering, moving the
frequencies back where they
belong by again modulating
with minus 1 to the n.
And that, in fact, turns out to
be a very useful notion for
applying a fixed lowpass
filter to do highpass
filtering and vice versa.
OK, now, what we've seen and
what we've talked about are
the Fourier representation for
discrete-time signals, and
prior to that, continuous-time
signals.
And we've seen some very
important similarities and
differences.
And what I'd like to do is
conclude this lecture by
summarizing those various
relationships kind of all in
one package, and in fact drawing
your attention to both
the similarities and differences
and comparisons
between them.
Well, let's begin this summary
by first looking at the
continuous-time Fourier
series.
In the continuous-time Fourier
series, we have a periodic
time function expanded as
a linear combination of
harmonically-related complex
exponentials.
And there are an infinite
number of these that are
required to do the
decomposition.
And we saw an analysis equation
which tells us how to
get these Fourier series
coefficients through an
integration on the original
time function.
And notice in this that what
we have is a continuous
periodic time function.
What we end up with in the
frequency domain is a sequence
of Fourier series coefficients
which in fact is an infinite
sequence, namely, requires all
values of k in general.
We had then generalized that to
the continuous-time Fourier
transform, and, in effect, in
doing that what happened is
that the synthesis equation in
the Fourier series became an
integral relationship in
the Fourier transform.
And we now have a
continuous-time function which
is no longer periodic, this was
for the aperiodic case,
represented as a linear
combination of infinitesimally
close-in-frequency complex
exponentials with complex
amplitudes given by X of omega
d omega divided by 2 pi.
And we had of course the
corresponding analysis
equation that told us how
to get X of omega.
Here we have a continuous-time
function which is aperiodic,
and a continuous function of
frequency which is aperiodic.
The conceptual strategy in the
discrete-time case was very
similar, with some differences
resulting in the relationships
because of some inherent
differences between continuous
time and discrete time.
We began with the discrete-time
Fourier series,
corresponding to representing a
periodic sequence through a
set of complex exponentials,
where now we only required a
finite number of these because
of the fact that, in fact,
there are only a finite number
of harmonically-related
complex exponentials.
That's an inherent property
of discrete-time complex
exponentials.
And so we have a discrete,
periodic time function.
And we ended up with a set of
Fourier series coefficients,
which of course are discrete, as
Fourier series coefficients
are, and which periodically
repeat because of the fact
that the associated complex
exponentials
periodically repeat.
We then used an argument similar
to the continuous-time
case for going from periodic
time functions to aperiodic
time functions.
And we ended up with a
relationship describing a
representation for aperiodic
discrete-time signals in which
now the synthesis equation went
from a summation to an
integration, since the
frequencies are now
infinitesimally close, involving
frequencies only
over a 2 pi interval,
and for which the
amplitude factor X of omega--
well, the amplitude factor
is X of omega d omega
divided by 2 pi.
And this term, X of omega,
which is the Fourier
transform, is given by this
summation, and of course
involves all of the
values of x of n.
And so the important difference
between the
continuous-time and
discrete-time case kind of
arose, in part, out of the fact
that discrete time is
discrete time, continuous time
is continuous time, and the
fact that complex exponentials
are periodic in discrete time.
The harmonically-related ones
periodically repeat whereas
they don't in continuous time.
Now this, among other things,
has an important consequence
for duality.
And let's go back again and look
at this equation, this
pair of equations.
And clearly there is no duality
between these two equations.
This involves a summation, this
involves an integration.
And so, in fact, if we make
reference to duality, there
isn't duality in the
continuous-time Fourier series.
However, for the continuous-time
Fourier
transform, we're talking about
aperiodic time functions and
aperiodic frequency functions.
And, in fact, when we look at
these two equations, we see
very definitely a duality.
In other words, the time
function effectively is the
Fourier transform of the
Fourier transform.
There's a little time reversal
in there, but basically that's
the result.
And, in fact, we had exploited
that duality property when we
talked about the
continuous-time Fourier transform.
With the discrete-time Fourier
series, we have a duality
indicated by the fact that we
have a periodic time function
and a sequence which
is periodic in
the frequency domain.
And in fact, if you look at
these two expressions, you see
the duality very clearly.
And so it's the discrete-time
Fourier
series that has a duality.
And finally the discrete-time
Fourier transform loses the
duality because of the fact,
among other things, that in
the time domain things are
inherently discrete whereas in
the frequency domain they're
inherently continuous.
So, in fact, here there
is no duality.
OK, now that says that there's
a difference in the duality,
continuous time and
discrete time.
And there's one more very
important piece to the duality
relationships.
And we can see that first
algebraically by comparing the
continuous-time Fourier
series and the
discrete-time Fourier transform.
The continuous-time Fourier
series in the time domain is a
periodic continuous function, in
the frequency domain is an
aperiodic sequence.
In the discrete-time case, in
the time domain we have an
aperiodic sequence, and in the
frequency domain we have a
function of a continuous
variable
which we know is periodic.
And so in fact we have,
in the time domain
here, aperiodic sequence.
In the frequency domain
we have a
continuous periodic function.
And in fact, if you look at the
relationship between these
two, then what we see in fact
is a duality between the
continuous-time Fourier
series and the
discrete-time Fourier transform.
One way of thinking of that is
to kind of think, and this is
a little bit of a tongue twister
which you might want
to get straightened out slowly,
but the Fourier
transform in discrete time is a
periodic function of frequency.
That periodic function has a
Fourier series representation.
What is this Fourier series?
What are the Fourier series
coefficients of
that periodic function?
Well in fact, except for an
issue of time reversal, what
it is the original sequence
for which
that's the Fourier transform.
And that is the duality that I'm
trying to emphasize here.
OK, well, so what we see is
that these four sets of
relationships all tie together
in a whole variety of ways.
And we will be exploiting as
the discussion goes on the
inner-connections and
relationships
that I've talked about.
Also, as we've talked about the
Fourier transform, both
continuous time and discrete
time, two important properties
that we focused on, among many
of the properties, are the
convolution property and the
modulation property.
We've also shown that the
convolution property leads to
a very important concept,
namely filtering.
The modulation property leads
to an important concept,
namely modulation.
We've also very briefly
indicated how these properties
and how these concepts have
practical implications.
In the next several lectures,
we'll focus in more
specifically first on filtering,
and then on
modulation.
And as we'll see the filtering
and modulation concepts form
really the cornerstone
of many, many
signal processing ideas.
Thank you.
