The Fourier transform of a continuous signal is defined as:
provided is absolutely integrable, i.e.,
Obviously many functions do not satisfy this condition and their Fourier transform do not exist, such as , , and . In fact signals such as , and are not strictly integrable and their Fourier transforms all contain some non-conventional function such as .
To overcome this difficulty, we can multiply the given by an exponential decaying factor so that may be forced to be integrable for certain values of the real parameter . Now the Fourier transform becomes:
The result of this integral is a function of a complex variable , and is defined as the Laplace transform of the given signal , denoted as:
provided the value of is such that the integral converges, i.e., the function exists. Note that is a function defined in a 2-D complex plane, called the s-plane, spanned by for the real axis and for the imaginary axis.
Inverse Laplace Transform
Given the Laplace transform , the original time signal can be obtained by the inverse Laplace transform, which can be derived from the corresponding Fourier transform. We first express the Laplace transform as a Fourier transform:
then can be obtained by the inverse Fourier transform:
Multiplying both sides by , we get:
To represent the inverse transform in terms of (instead of ), we note
and the inverse Laplace transform can be obtained as:
Note that the integral with respect to from to becomes an integral in the complex s-plane along a vertical line from to with fixed.
Now we have the Laplace transform pair:
The forward and inverse Laplace transform pair can also be represented as
In particular, if we let , i.e., , then the Laplace transform becomes the Fourier transform:
This is the reason why sometimes the Fourier spectrum is expressed as a function of .
Different from the Fourier transform which converts a 1-D signal in time domain to a 1-D complex spectrum in frequency domain, the Laplace transform converts the 1D signal to a complex function defined over a 2-D complex plane, called the s-plane, spanned by the two variables (for the horizontal real axis) and (for the vertical imaginary axis).
In particular, if this 2D function is evaluated along the imaginary axis , it becomes a 1D function , the Fourier transform of . Graphically, the spectrum of the signal, can be found as the cross section of the 2D function along the line .
Transfer Function of LTI system
Recall that the output of a continuous LTI system with input can be found by convolution:
where is the impulse response function of the system. In particular if the input is a complex exponential , then the output of the system can be found to be:
This is an eigenequation with the complex exponential being the eigenfunction of any LTI system, corresponding to its eigenvalue defined as:
which is the Laplace transform of its impulse response , called the transfer function of the LTI system. In particular, when , i.e., , the transfer function becomes the frequency response function, the Fourier transform of the impulse response: