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Complex frequency spectra. Amplitude, phase spectrum. Fourier integral theorem. From the Fourier series to the Fourier integral. Fourier transform. Fourier sine and cosine integrals. Convolution theorem. Parseval’s identity.

Complex frequency spectra. For a periodic function *f* (t) with a basic cycle corresponding
to the interval -T/2 < t < T/2 the Fourier series expansion is

where ω_{n} = nω_{0}, ω_{0} = 2π/T and

The complex exponential form of this series is

where

Using the general formula x + iy = re^{iθ} the formulas for C_{n} and C_{-n} can be written as

where

Phase angle. The angle
is called the phase angle of the coefficient C_{n}.

Amplitude spectrum. The amplitude spectrum of a periodic function *f* (t) is a plot of the
magnitude of the complex coefficients C_{n} versus ω_{n} i.e. a plot of |C_{n}| versus ω_{n}.

Phase spectrum. The phase spectrum of a periodic function *f* (t) is a plot of the phase angle
of C_{n} versus ω_{n}

The variable ω is called the angular
frequency. Since ω_{n} = nω_{0} is a
discrete variable, the amplitude and
phase spectra are not continuous
curves but rather present themselves
as a sequence of vertical lines as
shown in Fig. 3. These spectra are
referred to as discrete frequency
spectra or line spectra.

Effect on the frequency
spectra of an increase in the
period T. The frequency spectra
of any periodic function is discrete.
Let us now consider what happens to the frequency spectra of a periodic function when we
increase the length of the period T. Let *f *(t) represent the rectangular pulse train shown in Fig. 1
where a pulse of width d and height A is centered in an interval of length T. We wish to consider
the effect of increasing the period T while keeping the pulse width constant, as shown in Fig. 2,
and asking the question: What happens to the frequency spectra as T → ∞? The amplitude
spectrum for *f *(t) is shown in Fig. 3 for two values of T: T = 1/4 and T = 1/2 — where d = 1/20.
It can be seen from the figure that when the period T doubles from 1/4 to 1/2, the spectral lines
become closer together and the amplitude decreases. We shall now discuss why this is so. The
distance between two spectral lines is given by

6) Δω = ω_{n} - ω_{n-1} = ω_{0} = 2π/T

Thus we see that the distance Δω
between spectral lines is inversely
proportional to the period T. When
the period doubles, the distance
between lines halves. Now let us
consider what happens with the
amplitude. The formulas for the
coefficients C_{n,} as computed by 2)
and 4) above, are

Now the fraction

is of the general form sin θ / θ and for any angle θ,

Thus the fraction 9) is a number that is always less than 1. Formula 8) can be written

and the fraction acts as a scaling factor on C_{0}. We see from 7) that C_{0} is inversely proportional to
the period T. We now make the following general statement: The effect of an increase in the
period T on the frequency spectra is a) a change in the horizontal interval Δω that is inversely
proportional to T, b) a change in the vertical scale that is inversely proportional to T. As we let
T approach infinity the spectral lines merge together and, in the limit, the spectra become
continuous. And the amplitudes decrease, approaching zero in the limit.

Fourier integral theorem. Let a function *f* (x) satisfy the following conditions:

1) *f* (x) satisfies the Dirichlet conditions in every finite interval -L≤ x ≤L

2) The integral

converges, i.e. *f* (x) is absolutely integrable in -∞< x <∞.

Then

where

The result 11) holds if x is a point of continuity of *f* (x). As with a Fourier series, if x is a point
of discontinuity, *f* (x) is given a value equal to its mean value at the discontinuity i.e. if x = a is a
point of discontinuity, *f* (x) is given the value

As with the Fourier series, the Dirchlet conditions are sufficient, but not necessary.

The similarity between 11) and 12) and the formulas for a Fourier series is obvious. The right
side of 11) is sometimes called the Fourier integral expansion of *f* (x). It is also called the
Fourier integral.

From the Fourier series to the Fourier integral. Let us, in a heuristic way, derive the Fourier integral from the Fourier series. We will start with the complex exponential form of the Fourier series

where ω_{n} = nω_{0, }ω_{0} = 2π/T and

Substituting 14) into 13) we obtain

where we have used the dummy integration variable s to avoid confusion with t. Now multiplying both numerator and denominator of the right side by 2π we get

We now replace the factor 2π/T in 16) with Δω (where Δω = ω_{n} - ω_{n-1} = 2π/T) to get

If we now define

equation 17) becomes simply

where ω_{n} is the left-hand end point in the n-th subinterval Δω. Since

and since T → ∞ implies Δω → 0, we make the following statement: If T → ∞, then the
nonperiodic limit of the function *f*(t) is

or

which is the complex, exponential form of the Fourier integral.

Fourier transform. If we define

equation 22) becomes

The functions *f *(t) and *F*(ω) are called a Fourier transform pair. *F*(ω) is the Fourier transform
of *f* (t) and *f* (t) is the inverse Fourier transform of *F*(ω). The symmetries between 23) and 24)
are obvious. If we compare 23) and 24) with 13) and 14) we note the analogies between 24) and
13) and 23) and 14). The relation 22) is known as Fourier’s identity.

Note. The constants preceding the integral signs in 24) and 25) can be any constants whose product is 1/2π. For example, the Fourier transform pair

is known as the symmetric form.

Equivalent forms of the Fourier integral. Let *f* satisfy the Dirichlet conditions on any
finite interval and assume that the integral

exists. Then the following are equivalent forms of the Fourier integral:

or

where

Special cases of the Fourier integral

Fourier cosine integral. If the function *f *(t) is even, Fourier’s integral reduces to

which is called the Fourier cosine integral. It is analogous to the half-range cosine expansion of a periodic function that is even.

Proof. If *f* is an even function, then in 28) above, * f* (s) cos ωs is even (since the product of two
even functions is even), *f* (s) sin ωs is odd, *B*(ω) = 0,

and 27) becomes 29).

Fourier sine integral. If the function *f *(t) is odd, Fourier’s integral reduces to

which is called the Fourier sine integral. It is analogous to the half-range sine expansion of a periodic function that is odd.

Transform pairs for even and odd functions. Transform pairs corresponding to the Fourier cosine and sine integrals are

I Transform pair - cosine integral. *f (*t) even.

II Transform pair - sine integral. *f (*t) odd.

Def. Convolution of two functions. The convolution of two functions *f* (t) and *g*(t),
where -∞< t < ∞, is defined as

and denoted by *f ***g*.

Syn. bilateral convolution

Def. Correlation function. The correlation function for two functions *f* (x) and *g*(x) is
defined as

and represents a measure of correlation (or similarity) between the two functions.

Note. The cross-correlation integral is similar to the convolution integral but has different
properties. For example, in the case of convolution *f*g = g*f*, whereas with cross-correlation this
is not the case.

Functions of type* f(x + h*). The graph of function *f(x + h) *corresponds to that of function
*f(x) *as translated h units to the left or right according to the sign of h. If h is positive the
translation is to the left. When we talk about the function* f(x + h) *we mean that new function
*f _{1}(x)* that we would get if we substituted x + h for x in

Functions of type* f(-x + h*). To understand what the graph of the function *f(-x + h) *looks
like we first note the graph of *f(-x)* is the graph of *f(x) *as reflected about the y axis. The graph of
*f(-x + h) *is the graph of *f(x) *as first reflected about the y axis and then translated h units to the right (assuming positive h).

The Convolution theorem. The Fourier transform of the convolution of *f* and *g* is the
product of the Fourier transforms of *f* and *g* i.e.

*F *(*f* **g*) = *F* (*f* ) *F*(*g*)

Parseval’s identity for Fourier integrals. If the Fourier transform of *f *(t) is *F*(ω), then

This is called Parseval’s identity for Fourier integrals.

References

Wylie. Advanced Engineering Mathematics. Chap. 7

Spiegal. Laplace Transforms. (Schaum) Chap. 6

James & James. Mathematics Dictionary.

Taylor. Advanced Calculus.

Hsu. Fourier Analysis.

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