1. Research Methods in Acoustics Lecture 9: Laplace Transform and z-Transform
Jonas Braasch

2. Review: Fourier transform and convolution
Now we easily separated
s and h
Note that we made use of the exponential function’s
property:
This property is not restricted to

3. Let us generalize the Fourier transform
Instead of ejwt we allow a complex exponent est
The transformation still works!

4. Definition Laplace Transform
The unilateral Laplace transform is defined as:
with: s=s+jw
The bilateral Laplace transform is defined as:
The Laplace transform has many advantages when analysing a dynamical linear system:
differentiation and integration become multiplication and division (similar to logarithms)
integral and differential equations become polynomial equations which are easier to solve

5. Causality
A causal system is a system that does not respond to an input before it actually occurs. For a causal continuous-time LTI, we find:
For the convolution of a causal LTI system we find
Alternatively, we can write

6. Causality II
We find for a causal continuous LTI system and a causal input signal

7. Inverse Laplace transform
The inverse Laplace transform is defined as
The variable g is a real number to avoid integrating outside the region of convergence. g must be greater than the real part of every singularity. If the real part of all singularities is negative, g can be set to zero and the inverse Laplace transform becomes identical to the inverse Fourier transform.

8. Fourier Transform vs. Laplace Transform
The Fourier Transform is a special case of the Laplace Transform with s=jw.
The relationship between the Fourier transform and the Laplace transform is often used to determine the frequency response of a system.

9. Eigenfunctions
We would like to find a system T with zn as eigenfunction:
The following transformation has the eigenfunction zn:

10. Eigenfunction and Eigenvector
An eigenfunction fE of a transformation T is a function that is not changed by this transformation except for its magnitude. The factor by which the magnitude is scaled is called the eigenvalue l.
An eigenvector x of a matrix transformation T is a non-null vector (e.g., x=(0 0)) that is not changed by this transformation except for its magnitude. The factor by which the magnitude is scaled is called the eigenvalue l.
eigenvector
not an eigenvector
The term Eigen=(German) own was first used be Hilbert in 1904, although the concept was derived ealier.

11. Properties of the Convolution

12. Properties of the Laplace transform

13. Heavyside stepfunction

14. Laplace Transformation of a step function

15. Laplace Transformation of a step function

17. Examples for Laplace Transforms (from wikipedia)

18. Examples for Laplace Transforms II (from wikipedia)

19. Examples for Laplace Transforms III (from wikipedia)

20. Laplace Transform: Easy Example I

21. Laplace Transform: Easy Example I

22. Laplace Transform: Easy Example II

23. Laplace Transform: Easy Example II

24. Laplace Transform: More Complex Example

25. Laplace Transform: More Complex Example

26. Properties of the Laplace transform II

27. Properties of the Laplace transform III

28. z-Transform

29. Laplace periodicity of a period signal

30. Transform a continuous into a discrete signal

31. Definition of the z-Transform
The unilateral z-Transform is used for causal
Signals x[n].

32. Inverse z-Transform The inverse z-Transform is defined as:
The inverse discrete-time Fourier Transform is a special case of the z-Transform with C the unit circle:
The integration path C is counterclockwise and encircles the origin and the region of convergence (ROC). It must encircle all poles of X(z):
The region of convergence (ROC) includes all transformed signals
with finite sums:

33. Laplace vs. z-Transform
The bilateral z-Transform is the two-sided Laplace transform of the ideal sampled
function:
where x[n] = x(nT) is the sampled continuous-time function x(t), and the sampling
period T. The relationship between z and s is defined as:
The same relationship exists between the unilateral Laplace and z-Transforms.

34. Relationship between z-Plane and s-Plane
s-Plane z-Plane
left complex plane inner unit circle
imaginary axis unit circle
right complex plane outer unit circle
origin (s=0) z=1
fA/2 (s=jp/T) z=−1

35. Bilinear Transform
The Bilinear transform is an approximation to convert continuous time filters
of the Laplace space into discrete time filters of the z-space. We can use the following substitutions in H(s) or H(z):
Laplace to z
z to Laplace

36. Properties of the z-Transform

37. Properties of the z-Transform

38. Definition of the z-Transform
To avoid periodicity problems we can define the
z-Transform:

39. Transfer Function We can arrange the results into:
Using the fundamental theorem of algebra we can transform the transfer function into the following form:
The numerator has M roots – called zeros – while the denominator has N roots (poles):

40. Feed back system + ∑ + a
unit
delay

41. Systems described by difference equations
The Nth-order difference equation is the discrete version of the general differential equation:
The coefficients ak and bk are real constants. N is the largest delay of the equation. The instantaneous value of y[k] consist of linear combinations of the input b[k] and old values of y[k] (feed back):

42. Zero-pole transfer function
Let us take the difference equation:
Due to the time invariance we find:
We can now use the time-shift property:
We now find the zero-pole transfer function:

43. Zero-pole transfer function II
Transform into product form:

44. Zero-pole transfer function III
Transform into product form:

45. Convergence The mathematical term Convergence describes the limiting
behavior of an infinite sequence or series toward a limit. This limit has to exist, but can be unknown.
This function converges with the limit 0
This function does not converge

46. Region of convergence
The Laplace transform typically has so-called singularities. A singularity is an undefined area – going infinite – usually caused by a division through zero.
The region in which the Laplace transform is defined is called the region of convergence (ROC). In general, the one-sided Laplace transform F(s) exist for all complex numbers with Re{s}>a, with a a real number which is determined on the growth behavior of f(t).
The bilateral Laplace transform ROC is often a convergence strip with a<Re{s}>b.

47. BIBO stability
BIBO=Bounded input – Bounded output
system

48. BIBO system
A BIBO system is a system which always supplies a bounded output (e.g., |g(t)|≠∞) if the input is bounded (e.g., |g(t)|≠∞). A system is a BiBo system if its integrated impulse response is bounded:
The BIBO criteria determines whether a system is stable.

49. Proof Let us start with a bounded input signal x(t):
Then we find for the system output y(t):
Consequently, the output y(t) is bounded if

50. BIBO Examples 1.) y(t)=a·x(t)
Since h(t)=a is bounded this Example is BIBO stable.
2.) y(t)=a/x(t)
This example does not show BIBO stability, y(t) becomes infinite if x(t)=0.