第 11 讲：离散时间傅里叶变换
本讲内容：非周期信号的表示；离散时间傅里叶变换的性质，包括周期性、线性、对称性、时移与频移、差分与求和、时间与频率尺度变换、频域微分以及帕塞瓦尔关系；卷积与调制、对偶性、极坐标表示；频率响应与单位脉冲响应的计算；总结连续时间和离散时间傅里叶级数、傅里叶变换之间的关系。
授课教师：艾伦·V. 奥本海姆教授
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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.