In
this lecture, we introduce ourselves to the concept of the Fourier integral and the
Fourier transform. Let us recall our motivation in discussing the Fourier series.
Recapitulate, that the characteristic property of a linear time and variant system was,
that if it is driven by sinusoidal excitation. The responsible also be sinusoidal in
character. This particular property is valid only for sinusoidal. No, other periodic
function will have the such a characteristic, will provide such a characteristic.
And once, this particular property is satisfied. We brought in additional concept, like
frequency response function, impedance and impedance concepts, which will simplify
the analysis of linear systems, driven with sinusoidal excitation functions. Now, this
naturally let us to enquire, how we can analyze the linear system. When, it is driven
by a periodic non sinusoidal excitation function. And the Fourier series came in very
handy in the contest. In the sense that, very non sinusoidal periodic
function can be thought of as composed of several sinusoidal of different frequencies.
And by virtue of linearity, principle, we can find out the response to each one of the
excitation functions. And superpose the responses to get the total response, under
steady state conditions. When, the linear system is driven by a periodic, but non sinusoidal
excitation function. Naturally, you like to see, whether this analysis
can be carried forward to a case. When, the excitation function is no longer
periodic. We will call such excitation functions, a periodic function. So, when a
function is periodic, is it possible to for us
to consider, that a periodic function has composed of different sinusoidal functions
of different frequencies.
Surprisingly, the answer is yes. And this constitutes the discussion of the subject
matter of our discussion under the Fourier integral, Fourier transform methods. I use .the words surprisingly not without justification.
Because, when Fourier proposed that any a periodic function can also be a thought
of as composed of several sinusoids. You did not gain immediate acceptance from
all the mathematicians. And actually, the fact that the periodic function
can be composed of several trigonometric functions of different frequencies.
Appears have to be known, even before Fourier, even though, the earlier work
is did not carried out this analysis to a logical limit logical extent. But, it is the
singular and unique contribution of Fourier to
extended this kind of analysis to a periodic functions as well.
What he really said was, that if you have the periodic function, it can be thought of,
as the sum of several sinusoidal with infinite
number of frequencies. Infinite number of such small components, each with vanishingly
the small amplitude. And if you put all of them together, you will get the given the
periodic function. So, let us see, how we go about splitting up a given a periodic f
of t into various sinusoidal components of the description, which we gave just now.
The key to the whole idea is, that if a particular periodic function is given. We can
assume that the period is infinity. We are extending from minus infinity to plus
infinity. And what happens, beyond plus infinity on one side and what happens,
beyond minus infinity, we will not bother about it. So, any periodic function can be
a thought of as a periodic function with period
extending from minus infinity to plus infinity it occurs just ones.
So, that means, the period of this is going to infinity. What is the consequence? The
consequence is, that the frequency of this is 0. And so, if the frequency of this 0 is
0, then how are we going to think of various
harmonic components. Because, in the Fourier series, various frequency components,
that are presented integrally related, integral multiple of the fundamental frequency.
Here, the fundamental frequency is 0. And in the Fourier series the only term, which
has the fundamental frequency, whose frequency is 0 is the dc term and after all, the
dc term can split, any periodic function. So, must look for a different approach all
together. So, let us see, how we develop this. So, basically the key idea as I mentioned
is, that we think of the period of this periodic phenomenon to be infinite extending
from minus infinity to plus infinity. .So, what we will do is, will start with the
Fourier series of a periodic function. And then see, what happens, when the period becomes
larger and larger, till it goes out to infinity. So, the limiting process, we will
find out, how the various frequency components arise and what their amplitude
will be and so on and so forth. . So, let me start with a function, which is
periodic, period T naught. Just give a concrete example; I will take a specific wave form.
But, all over discussion will be in completely general terms. So, we have periodic pulse
strain of amplitude A and pulse duration d and the periodicity being the T naught. .. We know the spectrum for this, would be something like this. Where, this is C n and
we know that, C of n is A d upon T naught sin n omega naught d upon 2 by n omega
naught d upon 2. This is something which we have talked about several times. So, let
us look at the spectrum. And since, this is going to be a real number, because the
function is even. So, it is going to be a real number.
I am not plotting the phase and magnitude separately; I am plotting the entire C n
itself directly. So, it is either way. So, it is always either positive or negative.
The height of this is A d upon T naught and the
first 0 occurs, when n omega naught d by 2, happens to be pi. Therefore, the frequency
corresponding to this 2 pi upon d. So, this is the nature of the spectrum, for a
periodic pulse strain like this. .. And if we wanted to find out, f of t from
this, we certainly write this as n from minus infinity to plus infinity of C n e to the
power of j n omega naught t. That is what gets.
This is the general that we are getting. This particular C n; that we are getting apply
to this specific wave form, this is the general
formulation. The general formulation for C n; in general case would be 1 over T naught
minus T naught upon 2 to plus T naught upon 2 of f of t, e to the power of minus
j n omega naught t, d t as we already know. So, these are the two general expressions
for C n and this is the f of t, which is the Fourier series for this wave form. So, these
are two general expressions. In the particular case of a periodic pulse strain,
this would be the particular value of C n. That would derive for our special case that
we have considered. .. Now, let us see, what happens, when we have
a periodic pulse strain, a periodic function again. But, the period become very
large, large period. I will simply say T naught of L will indicate as the large period.
We have the same wave form, but the period becomes very large. So, I can put in
this fashion. I have the same pulse of amplitude A, duration d. But, the period is
very large. That means, the next pulse occurs after long period, very large period.
So, consequently, I will the angular frequency corresponding to this. Angular
frequency here is fundamental period angular frequency is omega naught. So, we will
say, fundamental of this is delta omega naught, I am saying. So, it include emphasize,
there is the small angular frequency delta omega naught. So, T naught of l is 2 pi over
delta omega naught. So, what would be the consequence as far the
spectrum is concerned. Now, look at this, this is the formula for C n. Now, what
happens, if T naught becomes larger and larger, A d upon T naught becomes smaller.
That means, the initial height here, the amplitude will become smaller. And if you
have very large T naught, this becomes almost negligible smaller and smaller.
I am sorry, what we should say is, C n is naught continuous thing. C n will be, this
is the envelope, what we have drawn is the envelope.
The actual C n will be at discrete point on this. Because, the spacing between
two lines being delta omega naught. The .fundamental frequency, this is what we had
in the first place. What I have to drawn here is, the envelope of C n.
So, the actual C n will be presented only at discrete points. And this is the line
spectrum and the spacing between two adjacent components
is omega naught. Now, when we make the period larger and larger,
keeping the pulse with same d; what you have is, A d upon T naught is smaller, very,
very small. So, the result is, that after all
2 pi by d remains invariant. Because, the 0 occurs here, when n omega naught t by 2
is pi and that corresponds to 2 pi upon d. . And therefore, what you are having here therefore
is, that you will have, if you plot C n, the envelope will be like this. And what
happens to the lines, previously the separation between two lines is omega naught.
Now, since the fundamental frequency is delta omega naught, which is smaller, the
lines comes closer and closer. The lines become closer and closer, the whole amplitude
comes down. Now, what we really want to know is, what
happens when T naught close to infinity Because, we want to treat a periodic function
as function of infinite period. So, if T naught becomes infinitely large. Then, obviously,
the whole thing squashes down to the horizontal line, because the amplitude
becomes 0. So, that will not give us any useful information. So, must see some other
way of handling this information. .So, what we should like to see is, you have
some kind of new quantity, which we should plot, which will not become 0, which
will be meaningful. And which will give us an idea of the relative changes of the
various wave harmonic components. So, to do this, what we will think of is, instead of
C n, after all the whole problem has come because, C n has 1 over T naught upon this.
. So, C n is the Fourier coefficient. It represents
to us the amplitude in a sort of fashion of e to the power of j n omega naught t term.
Instead of that, if I plot C n time T naught, so
in our case, C n stands T naught of l. This I can write it as, C n times delta f naught.
Because, the delta omega naught is the fundamental angular frequency, delta f not is the
corresponding frequency. So, I can write this as C n upon delta omega naught upon 2
pi. So, you plot this quantity, we are going to
multiply this C n by T naught of l. That means, as far this particular wave form is
concerned. Here, multiplying C n with T naught, then you are going to plot A d sin
omega naught t by two upon n omega naught d upon 2. Therefore, the spectrum will now
be restored to it is original side. .. So, what you are plotting now will be having
this envelop, which is exactly the same as this. But, the lines will become closer and
closer. And what you are plotted here is, C n
time T naught of l large T naught. And the amplitude will be, since A d is multiplied
by T naught. This will be A d. And the spacing
between two particular lines becomes delta omega naught. And this frequency corresponds
to 2 pi upon d, because that is not changing, d is the same.
So, we now see, that if we do not want to deal with C n, but C n times T naught of l.
Then, we can think of a meaningful way of representation the various harmonic
components and what is the relative size. .. Since, this is C n. The coefficient per unit
frequency, this is called coefficient density. Whereas, this is Fourier coefficient, this
is called Fourier coefficient density. This is
called Fourier coefficient density and once, you have the concept of coefficient density.
Then, f of t can be written as further. I will write here, f of t can be written as,
you recall f of t is the Fourier series expansion
is this, n minus infinity to plus infinity C n, e
to the power of j n omega naught t. But, instead of C n, I would like to talk
in terms of the coefficient density. So, I will
write this as, C n upon delta omega naught over 2 pi; e to the power of j n omega
naught t, sum from n minus infinity to plus infinity. And since, I have divided by delta
omega naught by 2 pi. I have to multiply by delta omega naught and then, I have to
write here 1 over 2 pi. So, that would be the form of the Fourier series now, in terms
of coefficient density rather than the coefficients. .. Now, let us see what happens, when you have
a periodic function. In the periodic function, we still take this same function
of same pulse of amplitude A, duration d. But,
there is no other pulse involved, the period is infinity. So, infinite period, so imagine
that we have this situation here, where now delta omega naught tends to 0. So, what
this consequence of this T naught goes to infinity, delta omega naught become smaller
and smaller. So, what happens the spectrum now, A d is
independent of your T naught. So, that remains the same. This 2 pi by d remains the
same. All that happens now is, as you increase the period, delta omega naught become
smaller and smaller. These lines come closer and closer will be more lines. And
the limit, you will find a particular line at
every point on the x axis. So, that in other words, you will get a frequency
component at every point on the x axis rather than the discrete point along the x
axis. Because, delta omega naught become smaller and smaller. So, the line gets crowding
almost next each other at every point on the curve. And therefore, we can say n delta
omega naught, which n delta omega naught what is significant n delta omega n
delta omega is the frequency at which a line exists in this spectrum now a line theoretically
exists at every point on the spectrum. So, n delta omega can be replaced by continuous
running variable omega. So, that is how the limiting process occurs as the fundamental
frequency become smaller and .smaller the harmonic exists at every point
along the omega axis. So, in the limit you will have a frequency component at every point
along the omega axis. So, the spectrum now would be just like this. But, a continuous
spectrum, because you have at every point, there is a meaning for this amplitude.
So, this is the continuous spectrum and the dc value here it is A d. And x axis in terms
of omega as before, every where is the omega, this is n omega naught of course here.
This is n delta omega naught and here, the running variable is the omega. So, what we
have here is, when arrangement where you have every frequency presents. But, the
coefficient density corresponding to each frequency is different. That is dictated that
is given by the spectrum.
. So, let us see what happens, the coefficient
density is now, C n times T naught,
wherever we are having is called now F of j omega. That is F of j omega. How do we
get now F of j omega, here you observe, that in the C n is given by this formula, the
coefficient density is obtained by multiplying C n by T naught. So, once, you multiply
C n by T naught, what you are having is, the integral relation minus T naught 2 plus T
naught upon 2, f of t, e to the power of minus j n omega naught t, d t.
So, in our case it happens, because T naught is multiplied by this. So, what you here is,
it is this formula, applicable to the periodic case. And what is the period T naught by
minus T naught upon 2 and plus T naught upon 2 are infinity. So, you are integrating F .of t multiplied by something over a complete
period, which is extended from minus infinity to plus infinity, f of t and e to
the power of minus j n. N omega naught now, in the first case, it
is replaced by n delta omega naught here. And
when you go to a periodic case, n delta omega naught is a continuous running variable
omega naught. Therefore, this will be the f, e to the power of minus j omega naught
d t. So, the coefficient density in the A periodic
case will be f j omega, which is minus infinity to plus infinity of f of t, e to
the power of minus j omega naught t d t. And this
is called the Fourier transform of f of t.
Now, what is the corresponding result for the Fourier series, that means, Fourier
transform is an equivalent of what we had for c of n. The formula to getting the Fourier
coefficient, we had some formula. The corresponding formula here is, the Fourier
transform, instead of C n, we deal with F of j omega. Why I will explain for a moment
later, about the notation. Now, what is the result we have corresponding
to this, f of t in the periodic case is expressed as the sum of different exponential
functions. Now, what is that we have here, f of t as you recall in the case of
large period is this. So, now, we have to take the
limit of this as delta omega naught becomes goes to 0. So, limit of f of t, limit of this
series as delta omega naught goes to 0, means we have to take the integration. After all,
C n omega naught by 2 pi, f of j omega, this is F of j omega.
. .Therefore, we have to write here as the f
of t, write it here. Limit as delta not omega naught goes to 0 of 1 over 2 pi, the summation
of C n over delta omega naught over 2 pi, we call that f of j omega. So, f of j
omega, e to the power of j omega t, because n
omega naught becomes, n delta omega naught becomes. Here, also delta omega naught
in the large case, n delta omega naught becomes omega and then, this is delta omega.
And the frequency omega, because n from minus infinity to plus infinity. That means,
omega is minus infinity to plus infinity. So, thus really becomes 1 over 2 pi. So, that
summation in the limit becomes integration from minus infinity to plus infinity and
omega f of t e to the power of j omega t d omega, this is f of t. So, this is f of omega,
this is must be F of j omega the same. So, instead of Fourier series for f of t,
instead of f of t is being thought of the summation of several exponential functions
like this. What we are really having is, f of
t is consisting of similar exponential functions, e to the power of j omega naught t,
where omega can take any value from minus infinity to plus infinity. Each of these is
multiplied by a coefficient, which is F j omega d omega by 2 pi.
. And you are integrating that over the complete
omega axis from minus infinity to plus infinity, this is f of t. And this is called
Fourier integral, this is called Fourier integral. So, instead of series, you have an integral
and instead of a formula like this for .calculating the Fourier coefficient. You
are calculating the Fourier coefficient density, which is referred to as a Fourier transform.
Now, a few words about this, why we get this 2 pi here. This 2 pi arises, because we are
calculating the Fourier density, coefficient density in terms of frequency rather than
omega naught. This is, how it is conventionally defined, therefore, because we are
talking about the coefficient density. In terms of having the denominator frequency
rather than in hertz rather than radius per second, we get this 2 pi term.
And secondly, we call this F of j omega instead of f of omega. Because, later on, we
like to relate the Laplace transform, where this f of j omega can be a thought of as a
special case f of s, we deal with Laplace transform domain. Therefore, it would be
convenient for us to put this f as j omega, because after all, when you talk about the
exponential terms here. Every j is accompanied by every frequency
is accompanied by j. So, j omega would be all right. You can think of f of omega as
well, but f of j omega would be more convenient for us to use.
. Now, let me quickly summaries, what we have
done up to this stage. We started with a periodic function. The formula for finding
the Fourier coefficients for the periodic function is this, we know general function.
And once, we have the various coefficients, .we can construct the f of t by combining
all such exponential terms from minus infinity to plus infinity. This is the Fourier series
and this is the integral; that we use to calculating the Fourier coefficient.
Taking a particular case of a periodic pulse strain, we showed that it has the spectrum
like this and the characteristic of the spectrum is, that is the line spectrum. You have
coefficients only at integral multiples of the basic frequency. Therefore, this spacing
between two adjutants lines is the fundamental frequency omega naught in radians per
second. Now, then we enquired as to what happens,
if we keeps this particular pulse constant at
the center around the origin and allow this T naught to become larger and larger. So,
that is the transition, before we go make T naught equal to infinity to see the trends.
So, as T naught becomes larger and larger, we
observe that the amplitude of this various C
n coefficients become smaller and smaller. So, the spectrum, line spectrum we had becomes
shallower and shallower. It will be of this order. It will be like this and the height
will diminish as T naught increases for them. But, the another fact, that we noticed
was, the spacing between the lines now. Instead of omega naught, it was the fundamental
frequency, earlier becomes smaller and smaller.
Because, as T naught increases omega naught becomes smaller and smaller. But, this is
not meaningful to us in the limit, when T naught becomes infinity, because when we
are moving towards the direction in which the period is considered to be infinitely
large. So, if the period is going to infinity large, this whole thing become 0. And this
C n will become 0. And therefore, we will now,
no way of knowing, what is the relative sizes of two different frequency components.
So, to beat this problem, what we said was, instead talking about the Fourier coefficient
C n, let us talk about Fourier coefficient density, which is C n divided by delta f not.
So, if you request this new quantity, which is C n times T naught the period. Then,
immediately the trouble that arose, because T naught being the denominator will
vanish. And then, we had C n T naught will be minus T naught upon 2, T naught upon
2, f of t, e to the power of j omega naught t d t. And immediately this spectrum would
be like this, corresponding to this periodic pulse strain. .So, you have depending upon the pulse width
and amplitude, you have certain value A d. And this no longer, as period becomes larger
and larger. This no longer has to become smaller, A d will remain the same,
the line will become closer. And the spacing between two adjutant sides, delta omega naught,
which is the new fundamental frequency, which is quite small.
And we can put f of t as 1 over 2 pi times, this coefficient density times e to the power
of j n delta omega naught t. I am writing this as times delta omega naught. So, the
Fourier series now will be in this form. But, instead of C n, we are talking about, we are
talking in terms of coefficient density. Now, at this stage, we take the limit as T naught
becomes infinitely large. So, if T naught becomes infinitely large,
we observed that delta omega naught become smaller and smaller. That means, the line,
the spacing between the lines becomes 0 in the limit; you have a frequency component
at every point on the omega axis. So, what we started as a line spectrum now becomes
continuous spectrum. That means, almost every single frequency is presented.
Of course, almost everywhere, because at this point, we can say this is 0. But, some
other, nearly everywhere, you have the frequency component present. And what we are
plotting now is not any more the Fourier coefficient C n, but the coefficient density.
And we give the new name for the coefficient density, we call that of j omega the
Fourier transform, which is given by the formula minus infinity to plus infinity of f of t,
e to the power of minus j omega t d t. That is F of j omega.
To recover the original function f of t from the f of j omega. Instead of the Fourier
series, we will have now, take the limit as the summation as delta omega tends to 0 of
1 over 2 pi times the Fourier coefficient density,
times e to the power of j omega t delta omega. And that leads us to the formula like
this, which is called the Fourier integral. So, basically, then we have when we are dealing
with Fourier transform for a periodic function.
We have two basic integral to deal with, from the given f of t, we can generate this
Fourier transform, which is really Fourier transform coefficient density as explained
here. And once, we have the Fourier transform, we can recover the Fourier function of .time through this integral. These are the
two basic integrals for characteristic of the
Fourier integral theory. Now, we will look into the meaning of the
coefficient density and the similarities. And
the contrast between the periodic function and a periodic function in a greater detail
at this point of time. After introduced ourselves
to the concept of Fourier integral and Fourier transform. Now, let us look at the
points of comparison of the spectra of a periodic signal and a periodic signal.
. So, if you had a periodic function of time,
say period T naught. Then, you have an amplitude spectrum
magnitude spectrum C n. And a phase spectrum, as you recall the
magnitude is the human function of n and the phase spectrum is not function of n. So,
this x axis can be graduated in terms of frequency n omega naught or in terms of n.
Now, in particular C n here and the corresponding phase together these two components
together will give me. That means, this specifies the magnitude,
this specifies the phase. So, it is C n, e to the
power of j phi n, e to the power of j n omega naught t. So, the significance of these two
terms, one specifies the magnitude. The other is specifying the phase is C n, e to the
power of j phi n is the coefficient of this exponential term, e to the power of j n omega
naught t. .Likewise, for the negative value of for this
n, will have these two terms together will give me the conjugate of this. Because, when
C n and c minus n are related by conjugate relationship. So, this will become
C n same magnitude e to the power of j Phi n, e to the power of minus j omega naught
t. So, all these four put together. That means, the magnitude and phase together will give
me 2 C n, cos n omega naught t plus phi n. So, the meaning of this negative frequency
minus n omega naught is only, it means that, this is the exponential. This is the
coefficient of this j t terms in this exponential representation. Even though, the frequency
is negative. These two terms together combine to provide you with a real frequency
of n omega naught t, n omega naught. Now, this is the line spectrum and there will
be frequency component present only at discrete value, along the frequency axis.
And they are all integral multiples of fundamental frequency and as I said the phasing
between two lines is omega naught. Whereas, when we have got the, this is the
periodic case. . In the A periodic case, we have some phenomena,
which lose, only one it does not repeat itself. The period T naught tends to
infinity and we said this will lead to a spectrum, which is continuous. And we call
that coefficient density F of j omega naught, magnitude. And we also have associate
with after all F of j omega naught F of j omega in general is a complex number. It is
complex function; therefore, we have magnitude and sign. So, also an angle associated
with that it is a continuous function. .This is argument of f of t omega, which we
may call phi j omega. Now, we said this coefficient density, what does it really mean.
If you take any small interval delta omega and the corresponding here, this lies on the
spectrum of the two spectra, the magnitude spectrum and the phase spectrum together.
These two will imply a constituent of our f
of t. What is that function, F of j omega magnitude, this is the coefficient density.
That means, over this interval of frequency. The density at the centre is F of j omega.
Therefore, the total magnitude is F of j omega magnitude time delta omega. It has the
phase e to the power of j t omega. And then, the exponential representation of function
of time is e to the power of j omega t. So, this magnitude and this phase together will
give rise function of time. Naturally, on the other side, corresponding
to, this is omega. This is minus omega and again, if you take two sections of the spectrum
of width delta omega. These two together will give me, F of j omega magnitude,
times delta omega times e to the power of minus j Phi omega, e to the power of minus
j omega t. So, this portion of the spectrum combined with portion of the spectrum.
These two together, if you combine these two, you will get 2 times F of j omega
delta omega cos omega t. So, even though you have components, corresponding
to negative frequency, what it really means. For every negative frequency
here, you have positive frequency and all together will lead a real function of a cosine
term. This is magnitude of course. And you have frequency at every point. In other
words, what we are having here is not and at any given, if you have A periodic case
at any given point of time, at any given point of time at point of frequency, if you want
to know, what is the amplitude, the amplitude is 0, vanishingly small.
But, if you take a small along the omega axis and if you consider this entire slice to be
representative of one single frequency term at the centre. Then, the amplitude of that
particular exponential term will be given by this. So, this is called coefficient density.
So, in other words, f of t is in volts. If f of t is in volts the dimension of F of j
omega is in volts per cycle per second or dimensions
will be volt seconds. Whereas, in the periodic case, if f of t is
in volts, C n will be in volts. It gives the amplitude of each individual component, whereas,
here it gives not the amplitude, but .the magnitude density. It is something similar
to, what we you know in your applied mechanics courses and so on.
. You have beam here, we have concentrated loads
at point. This is one way of looking at it. And then, sometimes you have distributed
loads. In case of distributed loads what you say is, so much kilogram per unit length.
That means, at given point, there is no load. But, if you take a small interval and
the total load in the interval will be the density times the small elemental distances.
So, it similarly here, you have situation where, each individual frequency component
is almost negligible amplitude. But, you have
over a small interval of along the omega axis. You can think of that the sections of
the spectrum to represent a function of time, which has the amplitude, which is F j omega
times delta omega. This is the density, this is the total amount.
So, what we have done is, to demonstrate by taking the limiting case of the Fourier
series as the frequency becomes smaller and smaller and as the period becomes larger
and larger. That even in A periodic case can be a thought of, as composed of number of
different frequency components. In contrast to the Fourier series, which concerns itself,
with the periodic case. .Here, we have a frequency component at every
conceivable frequency all along the omega axis from minus infinity to plus infinity
in the exponential representation. But, their amplitudes are vanishingly small. So,
the amplitude at any given frequency is 0, but over a small width, you can think of an
amplitude. So, what we are really talking about is, the amplitude density of the magnitude
density, otherwise called coefficient density.
And that coefficient density is, we term it is F of j omega, because function of omega
and you like associate j with the term omega. Therefore, it is F of j omega. So, we have
basically these two relations, which are important, which will use to our further work.
. The coefficient density F of j omega is obtained
from your given f of t as the integral from minus infinity to plus infinity of f
of t, e to the power of minus j omega t d t. This
is the coefficient density, which we call the Fourier transform. And once, we have the
Fourier transform. We can recover the time function f of t by writing the integral
relation from minus infinity to plus infinity of F of j omega, e to the power of j omega
t d.
Both integrals from minus infinity to plus infinity, in one case, we get 1 over 2 pi
here. The other case, we do not. And as I said,
we get this 2 pi, because we are defining the
coefficient density as the amplitude per cycle per second, f not instead of omega .naught. And this is called the Fourier integral.
That means, you always look at this as f of t is being composed of an exponential term
the function of time like this. And watch its amplitude, F of j omega this
is the coefficient density delta omega over 2
pi. So, this is the amplitude, F of j omega is the coefficient density. So, at this
frequency, if you want to look this, you take a small interval along the omega axis. So,
delta omega over 2 pi this is the base, this is the amplitude density. Therefore, this
is the amplitude. And take the limits as delta
omega goes to 0 overall possible frequencies and summation overall possible frequencies
and that leads to this integral. So, what we have done in this lecture is to
say that, the any periodic function can be thought of as an infinite summation of elementary
frequency components. Each with vanishingly small amplitude and with all frequencies
present along the omega axis. We continue next class, next lecture discussion
of the properties of the Fourier transforms. .
