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.

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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 = nω₀, ω₀ = 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 = nω₀ 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 = ω₀ = 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,

Proof

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 C₀. We see from 7) that C₀ 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,ω₀ = 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

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:

where

Proof

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 f₁(x) that we would get if we substituted x + h for x in f(x). It can be viewed as making the substitution x = u + h in the function f(x) to get the function f₁(u) which is f(x) as referenced to a new u-based coordinate system.

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.