Which is the better choice: Fourier Transform or Laplace Transform

“There is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.”- Nikolai Lobachevsky (1792-1856)

As we are aware that the concept of Laplace Transformation and Fourier Transformation plays a crucial role in different fields of science and engineering. In Electrical Engineering, Laplace and Fourier Transforms have been used for quite a while as an approach to change the solution of differential equations or to provide alternate representations of signals and data. These strategies, however, overpowered by today's emphasis on digital analysis, still form an invaluable base in the understanding of systems and circuits.

Today, we shall be highlighting the key points in Fourier Transform and Laplace Transform and comparing the similarities as well as the differences between them and concluding which is more reliable and efficient. A firm grasp of the practical aspects of these topics will provide valuable conceptual tools.


Ø Laplace Transform:

Laplace transform is an integral transform method which is particularly useful in solving linear ordinary differential equation. It finds very wide applications in various areas of physics, electrical engineering, control engineering, optics, mathematics, signal processing. Laplace transform can be interpreted as a transformation from time to frequency domain where inputs and outputs are a function of time f(t).

The Laplace Transform of a function of time f (t) is defined as: 

         

This is the bilateral Laplace Transform. In practice, unlike the Fourier Transform which we will consider later, the Laplace Transform is most often used with time-dependent signal functions which are defined for time t = 0' often written as f(t)=u(t)f(t) where u(t) is the unit step function. This allows us to write equation as: 

                                                                   

and this is called the unilateral transform. The inversion formula is:

                                                   

where s= σ +iw, the general complex variable. As is often the case with a terse mathematical formula, as shown above, much is implied.

For a function f(t) to be Laplace transformable, it must satisfy the following Dirichlet conditions:

 1) f(t) must be piecewise continuous; that is, it must be single-valued but can have a finite number of finite isolated discontinuities for t > 0.

2) f(t) must be exponential order; that is, f(t) must remain less than Me^-a₀^t as t approaches ∞, where M is a positive constant and 𝑎₀ is a real positive number. 

Ø  Fourier Transform:

Fourier transform entails representation of a non-periodic function not as a sum but as an integral over a continuous range of frequencies. This is done by converting an infinite Fourier series in terms of series and cosines into a double infinite series involving complex exponentials. On the other hand, Fourier transform involves frequency domain representation of a non-periodic function, in which such representation is valid over the entire time domain and accomplished through Fourier integral.

The Fourier Transform of a function of time, f(t) may be defined as:

                                                         
                                                   

                                         
And its inversion formula is: 

                                                       

           

This is the bilateral Laplace Transform with s = iw, that is, σ =0 , so long as the imaginary axis lies within the region of convergence of the Laplace integral. The function F(w) will, in general, be complex, that is it will have the form:

                                                  

                                            

 where R(w) and X(w) are real functions of the real variable w. This may appear as first to make it less general than the Laplace Transform, but when it comes to the inversion integral the requirement that  possesses a known strip of convergence is more restrictive than that exists for all real w. 

Now as in Laplace Transform, we are faced with the problem of existence of integral. So that the integral in converge, one set of conditions on f(t), known as the Dirichlet conditions: (1) the function is continuous almost everywhere; (2) at any point of discontinuity t_o, that f( t_o +) and f(t_o -) exist; (3) The function f(t) is integrable, that is :

                                                             

                                                            

                                                        

exists. These requirements are, simply stated, that physically realistic signals always possess a Fourier Transform. It must also be pointed out that they are sufficient, not necessary. The function

                                             

is not integrable but has a Fourier Transform. On the other hand, sin(t) and the unit step, u(t) do not, strictly speaking, have Fourier Transforms.

As we have gone through the brief introduction of Laplace and Fourier Transform, now it’s time to focus on their advantages and disadvantages.

Ø Advantages of Laplace Transform Can be listed as :

• Signals which are not convergent in Fourier Transform are Convergent in Laplace Transform. 
• Convolution in the time domain can be obtained by multiplication in the s-domain. 
• Integro-differential equations of a system can be converted into simple algebraic equations, so LTI systems can be analyzed easily. 

Ø Limitations of Laplace Transform Can be listed as :

• Frequency response of the system can’t be drawn or estimated, instead only the pole-zero plot can be drawn. 
• S=jw is used only for sinusoidal steady-state analysis.

         Ø Advantages of Fourier Transform Can be listed as :

• Convolution signals can be easily evaluated using the Fourier transform. 
• Fourier Transform is useful for analyzing signals involved in communication systems because the amplitude and phase characteristics are known. 

Ø Limitations of Fourier Transform Can be listed as :

• In Fourier transform the damping factor is 0 whereas in Laplace transform the damping factor is finite.
• There are many functions for which the Laplace Transformation exists, but Fourier Transform doesn't exist.

But, if you are provided with a signal and asked to obtain its spectrum, and are unable to represent the signal in the frequency domain using Fourier transform, it is advised to check for the Laplace transform of the signal, and if Laplace transform is also not applicable then one can conclude that the signal has no spectrum. Thus, we conclude that Laplace transform is the better choice since it includes both real and imaginary parts while Fourier transform includes only the imaginary part. The signals that can’t be represented using Fourier transform can be represented using the Laplace transform.