3.2. The Fourier transform of continuous signals – Digital Filters Design for Signal and Image Processing

3.2. The Fourier transform of continuous signals

3.2.1. Summary of the Fourier series decomposition of continuous signals

3.2.1.1. Decomposition of finite energy signals using an orthonormal base

Let x(t) be a finite energy signal. We consider the scalar product of two functions i(t) and k(t) of finite energy, represented as follows:

where k*(t) denotes the complex conjugate of k(t).

A family {k(t)} of finite energy functions is called orthonormal if it verifies the following relations:

A family {k(t)} is complete if any vector of the space can be approximated as closely as possible by a linear combination of {k(t)}. A family {k(t)} is termed maximal when the sole function x(t) of orthogonal finite energy throughout k(t) is the null function. We can then decompose the signal x(t) on an orthonormal base {k (t)} as follows:

COMMENT 3.1.– when the family is not complete, x(t), k (t) k (t) is an optimum approximation in the least squares sense of the signal x(t).

3.2.1.2. Fourier series development of periodic signals

The Fourier series development of a periodic signal x(t) and of period T0 follows from the decomposition of a signal on an orthonormal base.

To observe this, we look at the family of periodic function {k (t)}k represented as follows:

Here, the scalar product is that of periodic signals of period T0 and of finite power; that is, such as , so:

We then have:

If ik i(t), k (t) = 0; otherwise, k(t), k (t) = 1.

All periodic signals x(t) and of period T0 can be decomposed in Fourier series according to a linear combination of functions . Given equation (3.3), we have:

where ck measures the degree of resemblance between x(t) and exp :

When the signal x(t) is real, we can demonstrate that the Fourier series decomposition of x(t) is written as:

where the real quantities ak and bk verify the following relations:

and

PROOF.− ck is a complex quantity; we can express it as:

When the signal x(t) is real, since the coefficients ck and c−k are complex conjugates |ck| = |c−k|, and . We then have:

comparing relations (3.14) and (3.15) leads to the following identification

The coefficients ck and c−k are then linked to the quantities ak and bk, as follows:

COMMENT 3.2.− Periodic signals do not have finite energy on the interval ] − ∞ + ∞ [. That means that the quantity dt does not have a finite value. We can also say that x(t) is not of a summable square.

COMMENT 3.3.– we also see that, according to Parseval's equality,

If x(t) is real, . The signal's total average power is thus equal to the sum of the average powers of the different harmonics and of the continuous component.

COMMENT 3.4.– we remember that the average power of a periodic signal is given by the relation:

COMMENT 3.5.– if the analyzed signal is even, the complex coefficients ck constitute an even sequence. If the signal is odd, the complex coefficients Ck of the Fourier series decomposition form an odd sequence.

From there, if the analyzed signal is even, the complex coefficients Ck constitute a real even sequence. If the signal is odd and real, the complex coefficients Ck of the Fourier series decomposition form a pure imaginary odd sequence.

COMMENT 3.6.– amplitude and phase spectra. Amplitude spectrum expresses the frequential distribution of the amplitude of the signal. It is given by the module of the complex coefficients Ck according to the frequencies related to the functions .

Figure 3.1. Amplitude spectrum of a periodic signal

According to Figure 3.1, the spectrum of the periodic signal x(t) has a discrete representation. It contains the average value, the fundamental component, and the harmonics of the signal whose frequency is a multiple of the fundamental.

Introducing a delay in the signal x(t) does not modify the amplitude spectrum of the signal, but modifies the phase spectrum, which is expressed by the phase of the complex coefficients Ck according to the frequencies linked to the functions . This phase spectrum is also discrete.

If we let dk be the complex coefficients of the Fourier series development of x(t − τ), we then have:

Now, with equation (3.7), we also have:

According to equations (3.20) and (3.21), we deduce that:

and

EXAMPLE.– let the signal be written as follows:

The signal is periodic and of period . The corresponding amplitude and phase spectra are discrete. That means that there are only certain frequencies in the signal. Here, this corresponds to two Dirac in the frequency domain placed at frequencies f0 and – f0.

3.2.2. Fourier transforms and continuous signals

3.2.2.1. Representations

The Fourier transform of a signal x(t) of total finite energy, with a value in the ensemble of complexes is represented as follows:

The Fourier transform of a signal x(t) being a complex variable, the amplitude and phase spectra respectively represent the module and the phase of X(f) according to the frequency f.

The Fourier transform is then written as:

3.2.2.2. Properties

The Fourier transform is a linear application that verifies certain properties that can be easily proven by using equations (3.24) and (3.25). We will see that this transform goes from the temporal to the frequential domain and that its use facilitates the characterization of continuous signals. Indeed, it helps transform algebraic equations to differential equations and differential equations to algebraic ones:

– when y(t) = x*(t), we have Y(f) = X*(− f);

– when y(t) = x(tt0), we have Y(f) = e−j2πft0 X(f)

– when y(t) = e−j2πft0t x(t), we have Y(f) = X(ff0)

– when y(t) = x(at), we have

– when , we have . We thus have:

– when y(t) = x(t)z(t) where * designates the convolution product, we have Y(f) = X(f)*Z(f).

– if y(t) is real and even, its transform Y(f) is real and even; indeed, since y(t) = y(−t) = y*(t), we have Y(f) = Y(−f) = Y* (− f).

– if y(t) is real and odd, its transform Y(f) is odd and purely imaginary. Since y(t) = −y(−t) = y*(t), we have Y(f) = −Y(− f) = Y* (− f).

– when , we have Y(f) = (j2πf)n X(f);

– when , we have where c designates the mean.

3.2.2.3. The duality theorem

Given the expressions we have seen for the Fourier transform and the inverse Fourier transform, we can now discuss its dual properties. We can easily demonstrate that if x(t) has for a Fourier transform X(f), then X(t) allows for a Fourier transform x(−f). We then have:

As well, if x(t) is real and even, then X(f) is even and real and X(t) allows for the Fourier transform x(f). We will discuss this property again in Chapter 9.

3.2.2.4. The quick method of calculating the Fourier transform

By proceeding with successive derivations, we can easily calculate the Fourier transform of a signal.

EXAMPLE 3.1.– we calculate the Fourier transform of the derivative x′(t) of the rectangular impulse signal of duration θ.

Figure 3.2. Temporal representation of the function gate and its derivative

By deriving the rectangular impulse signal of duration θ′, we can express it according to two Dirac impulses . The Fourier transform of this signal can be easily obtained;

From there, by writing that sinc(x) = , we get:

EXAMPLE 3.21.– here, we look at a signal represented as:

Figure 3.3. Temporal representation of the signal x(t) for a=3 and T0 = 10

By deriving the signal x(t), we also obtain discontinuities such as and .

or we can write:

we get:

The derivation of x1(t) can be expressed according to x(t), as follows:

We end up with the following system:

Using the Fourier transform helps simplify the resolution of this system. We obtain:

Figure 3.4. Temporal representation of the signal x′ (t)

From this:

or:

3.2.2.5. The Wiener-Khintchine theorem

In this section, we look at the Fourier transform of the autocorrelation function Rxx (τ) of a real continuous-time signal x(t):

We then change the variable u = t - τ

Now, since x(t) is real, X(− f) = X* (f). The Fourier transform of the autocorrelation function of the signal x(t) thus satisfies:

where Sxx(f) designates the spectral density of the signal x(t). This relation often represents the Wiener-Khintchine theorem in the case of deterministic signals.

COMMENT 3.7.– another way to obtain this result consists of directly applying the properties of the Fourier transform presented in section 3.2.2.2 to the autocorrelation function that can be seen in the convolution product:

3.2.2.6. The Fourier transform of a Dirac comb

The Dirac comb is a periodic singular distribution of period T0. In order to determine the transform of this signal, we introduce the squared periodic signal x(t) coming from a periodic reproduction of period T0 of the rectangular impulse signal of duration θ and of amplitude .

Figure 3.5. Squared periodic signal

By making θ tend towards 0, the squared periodic signal tends towards the Dirac comb. So we have:

We then calculate the development coefficients by using the Fourier series of the periodic signal:

As well, we have for k ≠ 0:

By then carrying out a limited development of exp when θ tends towards 0, we obtain:

Using equations (3.31) and (3.33), we get:

A Dirac comb thus has a discrete spectrum; each frequency component situated at every and is of amplitude . We say that the Fourier transform of a Dirac comb of period T0 and of unity amplitude is a Dirac comb of period and of amplitude :

COMMENT 3.8.– according to the properties given in section 3.2.2.2., we have:

With equations (3.35) and (3.36), we end up with Poisson's summation formula:

COMMENT 3.9.– by using the properties given in section 3.2.2.2, we can also demonstrate that:

3.2.2.7. Another method of calculating the Fourier series development of a periodic signal

Let x(t) be a signal constructed from the periodization of a pattern m(t) to the period T0. This signal allows for a Fourier series development by satisfying equations (3.7) and (3.8). Calculating the coefficients of this development can be carried out in another way when x(t) can be expressed from a pattern m(t) as follows:

We can then obtain X (f) from equation (3.7) or (3.39):

By identification, it is then possible to express the coefficients of the Fourier series development of the signal x(t) according to M(f), the Fourier transform of the pattern.

We use this result with the signal x(t) shown in Figure 3.6.

Figure 3.6. A periodic signal

Figure 3.7. Pattern linked to periodic signal shown in Figure 3.6

From one of the methods shown in section 3.2.2.3, we find the transform of the pattern described in Figure 3.7:

The coefficients of the Fourier series development of the signal x(t) then equal:

By using equation (3.8), we obtain the same result as in equation (3.43). We then have:

3.2.2.8. The Fourier series development and the Fourier transform

Here we look at the centered rectangular impulse signal of duration θ written as . This signal is called transitory or square summable; that is, its total energy is of finite value:

We reproduce this signal at regular intervals T0 > θ in order to obtain a periodic signal written xp(t). We then develop in Fourier series the signal xp(t) of fundamental period T0.

Figure 3.8. Temporal representation of the centered rectangular impulse signal of duration θ

The signal's spectrum is discrete and equals:

The spectral density of the signal is termed discrete. It is represented by the module of squared complex coefficients ck according to the frequencies linked to the functions exp . As such it equals:

Now, the spectral density of the pattern energy equals:

From this, we deduce that reproducing the signal gate of support θ at period T0 allows us to express the spectral density of the signal xp(t) from that of the pattern x(t):

The spectral density of the pattern sampled at frequency , weighted by a factor of , provides the expression of the spectral density of the periodized signal.

We then observe the evolution of the frequential content of the signal when the period T0 tends towards infinity.

So we propose:

is relative to the complex kth harmonic angular frequency . We then have:

corresponds to a multiplicative constant, with a gap between two successive harmonic angular frequencies.

We have:

From there, equation (3.45) becomes:

Let us look at the limit of xp(t) when the period T0 tends towards infinity. If we assume that XT0k) has a limit written as Xk) when T0 tends towards infinity, we get:

By making the period T0 tend towards infinity, we come to study the frequential behavior of the pattern, which is assumed to be transitory; that is, to the representation of the Fourier transform of the finite energy signal.

APPLICATION.– here, we look again at the above example with θ = 0.02 s and T = 0.05 s. The signal is reconstructed by considering only a limited number of complex coefficients of the Fourier series (11.31, then 61). This is shown in Figure 3.9.

Figure 3.9. Signal obtained by adding a given number of decomposition components using the Fourier series (11, 31, and 61 coefficients) employing the Gibbs phenomenon

Now we will consider the spectrum evolution of the periodized pattern according to the values of the period T0 equal to 0.05 s, 0.1 s, 0.5 s, 1 sand 5 s.

Figure 3.10. Amplitude spectrum of the pattern

Figure 3.11. Evolution of the temporal representation and of the spectrum of the periodized pattern. according to the period value

COMMENT.– it is important to bring together equations (3.7) and (3.8) to represent the Fourier series development of equations (3.24) and (3.25):

Equation (3.8) helps us evaluate the resemblance degree existing between a periodic signal x(t) to be analyzed and exp . Since the signal is periodic and of non-finite energy, integration occurs on a period T0.

Equation (3.7) provides the expression for x(t) according to the family of complex exponentials exp. Only the multiple frequencies of the fundamental frequencies are present in this signal. The spectrum is therefore discrete.

Equation (3.24) allows us to evaluate the resemblance degree existing between a signal x(t) of finite energy and exp(jft). The frequency f is here indeterminate because all can be present in the signal. As well, the integration domain can be since the signal is of finite energy.

Using the example of equation (3.7), equation (3.25) gives the expression of x(t) according to the complex exponentials exp (jft). The discrete sum present in (3.7) becomes an integral on equation (3.25) because all f frequencies can be taken into account.

3.2.2.9. Applying the Fourier transform: Shannon's sampling theorem

In this section, we will look at signal sampling and reconstruction starting from a sampled analog signal written x(t) which we suppose will have bounded support spectrum; that is, the module of the Fourier transform of the signal x(t) is null for throughout the frequency f > fmax. We will later return to this last hypothesis.

Figure 3.12. Bounded spectrum of the analyzed signal

The origin of this spectrum boundary is either a property of the analyzed signal or is due to a low-pass pre-filtering, as we have seen in the acquisition chain and in the process shown in Figure 1.3.

To obtain the sampled signal xs(t), the continuous input signal x(t) is multiplied by a pulse train of period Ts:

The resulting signal xs(t) is then filtered by an ideal passband filter to give the reconstructed signal xr(t). The goal of what follows is to determine if the entire sampling period allows for a reconstruction of the signal after digitization and filtering.

Figure 3.13. Digitization and filtering

The Fourier transform Xs(f) is the Fourier transform of the product between the input signal x(t) and the impulse train. Xs(f) corresponds to the Fourier transform of x(t) convoluted with that of ; it is thus a reproduction of the spectrum X(f) at the frequency .

All sampling frequencies fs do not necessarily guarantee the correct reconstruction of the signal (as shown in Figure 3.13) by low-pass filtering. The spectrum supports X(f) centered at the multiple values of the sampling frequency should not be superimposed.

Figures 3.14 and 3.15 allow us to visualize different situations.

Figure 3.14. Spectrum of sampled signal

Figure 3.15. Spectrum overlap

In order to avoid spectrum distortions of the sampled signal that are due to spectrum overlap, we must take:

In this way, we demonstrate Shannon's sampling theorem, which fixes the sampling choice fs.

If we do not retain the hypothesis of a bounded spectrum signal, folding can occur no matter which sampling frequency we use. The perfect reconstruction of a signal may be impossible if we do not have additional information about the signal. In practice, there is no maximum frequency from which the spectrum can be considered as null. We get around this problem by using a low-pass pre-filtering of the continuous signal before the sampling stage.

is called Shannon's frequency, Nyquist's frequency or folding frequency.