傅里叶变换和拉普拉斯变换的比较

1. Overview

The Fourier and Laplace transforms are commonly used in many fields of science and engineering. In this tutorial, we’ll compare their definitions and then their applications.

2. The Fourier Transform

In this section, we’ll discuss the Fourier transform.

2.1. Definition

It’s possible to express a continuous-time periodic function as a linear combination of harmonically related complex exponentials using the Fourier series. The Fourier transform is the generalization of the Fourier series for aperiodic functions:

$X(jw)=\int_{-\infty}^{+\infty}x(t)e^{-jwt}dt$

The inverse Fourier transform represents the aperiodic x(t) function as a summation of complex exponentials at a continuum of frequencies:

$x(t))=\dfrac{1}{2\pi}\int_{-\infty}^{+\infty}X(jw)e^{jwt}dw$

The Fourier transform of x(t) converges as long as x(t) satisfies the Dirichlet conditions:

The integral of x(t) must be absolutely integrable, i.e., $\int_{-\infty}^{+\infty}\lvert x(t) \rvert dt < {\infty}$ .
x(t) must have a finite number of maxima and minima within any finite interval.
x(t) must have a finite number of discontinuities within any finite interval.

The discrete-time counterpart of the Fourier transform is the discrete-time Fourier transform (DTFT), which is a continuous function of frequency. The discrete Fourier transform (DFT) is the sampled version of DTFT. The fast Fourier transform (FFT) is an efficient implementation of DFT.

2.2. Applications of the Fourier Transform

The Fourier transform plays an important role in the analysis of signals and systems.

The Fourier transform maps a signal in the time domain to a signal in the frequency domain, i.e., it shows the frequencies in the signal and their amplitudes. Frequency domain analysis is the basis of signal and system analysis.

If x(t) is the input of a linear time-invariant (LTI) system, then the output of the system, y(t), is calculated using the convolution integral in the time domain:

$y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)dt$

h(t) is the impulse response of the LTI system.

The Fourier transform has many useful properties. However, one of its most important properties is the convolution property. Taking the Fourier transform of the convolution integral yields:

Y(jw)=X(jw)H(jw)

H(jw) and X(jw) are the Fourier transforms of h(t) and x(t), respectively. Similarly, Y(jw) is the Fourier transform of y(t). H(jw) is the frequency response of the system.

As a result, convolution in the time domain corresponds to multiplication in the frequency domain. This property makes the analysis of LTI systems in the frequency domain much easier since multiplication is simpler than convolution.

3. The Laplace Transform

In this section, we’ll discuss the Laplace transform.

3.1. The Bilateral Laplace Transform

The Laplace transform of a function x(t) is defined as:

$X(s)=\int_{-\infty}^{\infty}x(t)e^{-st}dt$

s is a complex frequency of the form s = $\sigma$ + jw, where $\sigma$ is the real part and w is the imaginary part. Hence, the Laplace transform maps a function to the complex plane.

This definition is the bilateral Laplace transform. The integration is from $-\infty$ to $+\infty$ .

The Fourier transform is equivalent to the Laplace transform when s is purely imaginary, i.e., s = jw. Therefore, the Fourier transform is a special case of the Laplace transform.

To show the relationship between the two transforms, let’s rewrite the Laplace transform as:

$X(\sigma+jw)=\int_{-\infty}^{\infty}[x(t)e^{-\sigma t}]e^{-jwt}dt$

This integral is the Fourier transform of $x(t)e^{-\sigma t}$ . Therefore, the Laplace transform of x(t) is the Fourier transform of x(t) multiplied by a real exponential function. Since the real exponential function decays for positive $\sigma$ , the Laplace transform converges for a broader number of functions than the Fourier Transform.

The complex s values for which the Laplace transform converges is a region in the complex plane. This region’s name is the region of convergence (ROC). Two different functions’ Laplace transforms may be the same but with different ROCs. So, while specifying the Laplace transform of a function, we also specify its ROC.

The Laplace transform’s properties are similar to the Fourier transform. For example, the Laplace transform has the convolution property as well. If X(s) is the Laplace transform of x(t) with ROC Rx, and H(s) is the Laplace transform of h(t) with ROC Rh, then the Laplace transform of their convolution is:

Y(s) = H(s)X(s) with ROC containing $R_h \cap R_x$

H(s) is known as the system or transfer function.

The Laplace transform’s discrete-time counterpart is the z-transform.

3.2. The Unilateral Laplace Transform

There’s also another form of the Laplace transform:

$X(s)=\int_0^{\infty}x(t)e^{-st}dt$

This form is the unilateral Laplace transform. The difference is in the lower limit on the integral. The lower limit is 0 for the unilateral transform.

3.3. Applications of the Laplace Transform

The Laplace transform provides additional tools for the analysis of signals and systems. Besides, we can use it in different contexts in which the Fourier transfer can’t be used since the Laplace transform exists for a broader set of functions. Therefore, it’s possible to apply it to many unstable systems.

An example is the investigation of the stability of systems in control system theory. For instance, the Routh-Hurwitz criterion tests the absolute stability of an LTI system, which has a characteristic equation with constant coefficients. We obtain the characteristic function of an LTI system by setting the denominator polynomial of its transfer function to 0. The criterion checks whether any root of the characteristic function lies in the right-half complex plane.

Initial-value problems involving a linear differential equation can be more easily solved since the Laplace transform maps the problem to an algebraic problem. This is due to its differentiation property in the time domain. Once the algebraic problem is solved in the complex domain, we can transform the solution to the time domain as the transform is reversible.

Solving linear differential equations in the s domain is particularly useful in the analysis of LTI circuits under general excitation, i.e., when the circuit isn’t in the sinusoidal steady state. An RLC circuit consisting of resistors, inductors, and capacitors is a typical example of an LTI circuit.

4. Conclusion

In this article, we compared the Fourier and Laplace transforms. We reviewed the definitions of both transforms and saw that the Laplace transform is a generalization of the Fourier transform. Then, we learned that both transforms have an important role in the analysis of signals and systems. Finally, we saw that the Laplace transform is widely used in control and circuit theories, and in solving linear differential equations.

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