Final Paper
“Signals and systems in the complex frequency domain.
The Laplace transform and z-transform”
Student:
Barysenka Dzianis
丹尼斯
Student id:
21806280
Submitted to:
Prof. 邱天爽
2018, Dalian
Introduction
The Laplace transform is an integral transform associating the function F(s) of a complex
variable (image) with the function f(x) of a real variable (original). With its help, the properties of
dynamic systems are investigated and differential and integral equations are solved.
One of the features of the Laplace transform, which predetermined its wide distribution in
scientific and engineering calculations, is that many relationships and operations on the originals
correspond to simpler relations over their images.
1. Direct Laplace transform
The Laplace transform of a function of a real variable f(t) is called a function F(s) of a
complex variable s=σ+iw, such that:
The right side of this expression is called the Laplace integral.
2. Inverse Laplace transform
The inverse Laplace transform of a function of a complex variable F(s) is called a function
f(x) of a real variable, such that:
where is σ1 a real number. The right side of this expression is called the Bromwich integral.
3. Two-way Laplace transform
The two-sided Laplace transform is a generalization to the case of problems in which the
function f(x) includes the values x <0
The bilateral Laplace transform is defined as follows:
4. Discrete Laplace transform
Used in the field of computer control systems. The discrete Laplace transform can be
applied to lattice functions.
There are D- transformation and Z- transformation.
• D –transformation
the lattice function, that is, the values of this function are defined only at discrete points in
time nT, where is n - an integer, and T is the sampling period.
Then applying the Laplace transform we get:
• Z - transformation
If you apply the following variable substitution:
𝑧 = 𝑒 𝑠𝑇
get the z-transform:
5. Properties and theorems
• Absolute convergence
If the Laplace integral converges absolutely with σ = σ0, that is, there is a limit
then it converges absolutely and uniformly for σ ≥σ0 and F(s) is an analytic function as
σ ≥σ0 (σ = Re s is the real part of the complex variable s). The exact lower bound σa of the set of
numbers σ for which this condition is satisfied is called the abscissa of absolute convergence of
the Laplace transform for the function f (x).
• Conditions for the existence of the direct Laplace transform
The Laplace transform L{f(x)} exists in the sense of absolute convergence in the following
cases:
1. Case σ ≥0: the Laplace transform exists if the integral exists
2. The case of σ> σa: the Laplace transform exists if the integral
3. The case of σ> 0 or σ> σa (which of the boundaries is larger): the Laplace transform
exists if there is a Laplace transform for the function f '(x) (derivative to f (x)) for σ> σa.
Note: these are sufficient living conditions.
• Conditions for the existence of the inverse Laplace transform
For the existence of the inverse Laplace transform, the following conditions are sufficient:
1. If the image F (s) is an analytic function for σ ≥σ a and has an order less than −1, then
the inverse transformation for it exists and continuously for all values of the argument,
and L^-1{F(x)}=0 t≤0
3. Let see
then the inverse transform exists and the corresponding direct transform has an absolute
convergence abscissa.
Note: these are sufficient living conditions.
• The convolution theorem
The Laplace transform of the convolution of two originals is the product of the images of
these originals.
• Multiplication of images
The left-hand side of this expression is called the Duhamel integral, which plays an
important role in the theory of dynamical systems.
• Differentiation and integration of the original
The Laplace image of the first derivative of the original in the argument is the product of
the image on the argument of the latter minus the original in zero to the right.
In a more general case (nth order derivative):
The Laplace image of the integral of the original by argument is the image of the original
divided by its argument.
• Differentiation and integration of the image. The inverse Laplace transform of the
derivative of the image with respect to the argument is the product of the original and its
argument, taken with the opposite sign.
The inverse Laplace transform of the image integral over the argument is the original of
this image divided by its argument.
Delay of originals and images. Limit theorems
Image Lag:
The delay of the original:
Note: u (x) is a Heaviside function.
Theorems on the initial and final values (limit theorems):
All poles in the left half-plane. The finite value theorem is very useful, since it describes
the behavior of the original at infinity using a simple relation. This, for example, is used to
analyze the stability of the trajectory of a dynamic system.
• Other properties
Linearity
Multiply by number
6. Forward and inverse Laplace transform of some functions
Below is a Laplace transform table for some functions.
7. Relationship to other transformations.
Fundamental connections
Almost all integral transformations are of a similar nature and can be obtained one from
the other through the expression of conformity. Many of them are special cases of other
transformations. The following are formulas that link Laplace transformations with some other
functional transformations.
Laplace-Carson Transformation
The Laplace-Carson transform is obtained from the Laplace transform by multiplying it by
a complex variable.
Bilateral Laplace transform
The two-sided Laplace Lb transform is connected with the one-sided using the following
formula:
Fourier transform
The continuous Fourier transform is equivalent to a two-sided Laplace transform with a
complex argument s = iω.
The relationship between Fourier and Laplace transforms is often used to determine the
frequency spectrum of a signal or dynamic system.
Mellin's Transformation
The Mellin transform and the inverse Mellin transform are associated with the two-sided
Laplace transform by a simple change of variables. If in the Mellin transform
If we set θ = e - x, then we obtain a two-sided Laplace transform.
Z-transform
The Z-transform is a Laplace transform of the lattice function, produced by changing
variables:
𝑧 = 𝑒 𝑠𝑇
where is T=1/fs the sampling period, and fs is the signal sampling rate. The relationship is
expressed using the following relationship:
Borel Transformation
The integral form of the Borel transform is identical to the Laplace transform; there is also
a generalized Borel transform, by which the use of the Laplace transform extends to a wider
class of functions.
8. Laplace energy conversion
Write the equation
and, without neglecting for simplicity, the dependence of the cross sections Σ (E) and
from E, move from E to new variable.
=E0-E
The solution of this equation can be obtained using the Laplace energy conversion:
It can be considered as a decomposition of the differential flux density in a system of
biorthogonal function exp(p) and exp(-p).
In the second term, it is necessary to change the order of integration and in the integral
over  make the change of variables
=-Q
-transformant Laplace from differential cross section scattering.
The right side of equation (1) is easily transformed, after which we get
If the cross section
rapidly decreasing with increasing Q, the exponent in
can be expanded in a row.
where is β the middle of the energy loss per unit length of the path. Substitute this
decomposition and make the change of variables
k=-iβp
Calculating, the integral with the help of residues and returning from the variable  to the
variable E, we get:
The formula has a simple physical meaning. By definition, Φ (E) = dE is the average path
traveled by a particle during the time when its energy changes from E + dE to E.
9. Laplace transformation by coordinates
Let us write the kinetic equation in the “straight-forward” approximation (i.e., without
taking into account the deviation of particles during scattering) for particles emitted by a
monoenergy source located at the origin:
flux density is 0 and the range of z in equation should be considered a semi-infinite interval (0,).
This circumstance allows us to apply to the solution of equation he Laplace transform along the
coordinates:
where the Laplace transformant F (, E) is expressed in terms of the flux density as
follows:
Multiply both sides of equation by exp(-λz) and integrate over z from 0 to .
Transforming the first term by integrating in parts, taking into account the boundary condition
and using the notation, we get:
After the Laplace transform of the remaining terms of equation , we arrive at the equation
for the flux density transformant:
If the condition is met
then for the transformant of the scattered flux density component we get
We introduce the notation
The function φ (z, a), which is the inverse Laplace transform of s-2exp(a/s), is equal to
where I1 is the modified first order Bessel function. In this way
In particular, for small values of the argument I1 (x)=x/2, therefore
It can be seen that with increasing z, the ratio of scattered radiation to unscattered
increases first linearly (when single scattering plays the main role), then in a more complex way,
and the low-energy part of the spectrum due to multiple scattering grows faster than high-energy.
Conclusions
The Laplace transform is widely used in many areas of mathematics (operational
calculus), physics and technology.
• Solving systems of differential and integral equations using the Laplace transform, it is
easy to move from complex notions of mathematical analysis to simple algebraic relations.
• Calculation of transfer functions of dynamic systems, such as, for example, analog
filters.
• Calculation of the output signals of dynamic systems in control theory and signal
processing - since the output signal of a linear stationary system is equal to the convolution of its
impulse response with the input signal, the Laplace transform allows you to replace this
operation with simple multiplication.
• Calculation of electrical circuits. Produced by solving differential equations describing
the scheme by the operator method.
• Solving non-stationary problems of mathematical physics.
References
1. A. M. Kolchugin, V. V. Uchaikin, "Introduction to the theory of passage of
particles through matter". M., Atomizdat, 1978, 256s.
2. Wn.Rusak "Mathematical physics", Minsk, 1998
3. Gustav Doetsch, "Guide to practical applications of the Laplace and Ztransform."M.: Science, 1971
4. L. G. smyshlyaeva, "the Laplace Transform of functions of many
variables" publishing house of Leningrad state University, 1981