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

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

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

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

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

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

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.

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.

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

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.

Properties of the Convolution

Properties of the Laplace transform

Heavyside stepfunction

Laplace Transformation of a step function

Laplace Transformation of a step function

Examples for Laplace Transforms (from wikipedia)

Examples for Laplace Transforms II (from wikipedia)

Examples for Laplace Transforms III (from wikipedia)

Laplace Transform: Easy Example I

Laplace Transform: Easy Example I

Laplace Transform: Easy Example II

Laplace Transform: Easy Example II

Laplace Transform: More Complex Example

Laplace Transform: More Complex Example

Properties of the Laplace transform II

Properties of the Laplace transform III

z-Transform

Laplace periodicity of a period signal

Transform a continuous into a discrete signal

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

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:

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.

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

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

Properties of the z-Transform

Properties of the z-Transform

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

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):

Feed back system + ∑ + a unit delay

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):

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:

Zero-pole transfer function II Transform into product form:

Zero-pole transfer function III Transform into product form:

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

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.

BIBO stability BIBO=Bounded input – Bounded output system

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.

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

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.