EC--214 Signals and Systems
EC
(Laplace Transform)
Course Instructor
Dr. Deepa Sharma
ECE Department
IIIT Bhopal
Laplace Transform
Convert the time domain function f(t) to the frequency domain function f(jw)
and f(s).
Gives the total solution to the differential equation and corresponding initial
and final value problems.
Widely used for describing continuous time circuits and systems, including
automatic control systems and also for analysing signal flow through causal
linear time invariant systems with non-zero initial conditions.
The laplace transform f(s) of a time function f(t) is
◦ The subscript b: indicate bilateral, or two-sided, laplace transform.
◦ The bilateral laplace transform integral becomes the fourier transform
integral if s is replaced by jw. The laplace transform variable is complex, s =
σ+ jw.
◦ The bilateral laplace transform of a signal f(t) can be interpreted as the
fourier transform of that signal multiplied by an exponential function e-σt .
The unilateral, or single-sided laplace transform, is
The condition for the laplace transform to exist is
Complex s plane
The Region of Convergence:
The range of values of the complex variables s for which the Laplace
transform converges is called the region of convergence (ROC).
Poles and Zeros of X( s):
Usually, X(s) will be a rational function in s,
The coefficients ak, and bk, are real constants, and m and n are positive
integers
The X(s) is called a proper rational function if n > m, and an improper
rational function if n ≤ m.
.
The roots of the numerator polynomial, zk, are called the zeros of X(s) because
X(s) = 0 for those values of s.
Similarly, the roots of the denominator polynomial, pk, are called the poles of
X(s) because X(s) is infinite for those values of s. Therefore, the poles of X(s)
lie outside the ROC since X(s) does not converge at the poles.
The zeros, may lie inside or outside the ROC.
Except for a scale factor a0 /b0, X(s) can be completely specified by its zeros
and poles.
" x " is used to indicate each pole location and an " o" is used to indicate each
zero.
Properties of the ROC:
The ROC of X(s) depends on the nature of x(t). (assume X(s) is a rational
function of s):
Problem1: Consider the signal
Problem2. Consider the signal
Laplace transforms of some common signals
Find the Laplace transform of the unit ramp function f(t) = t
Laplace Transform Pairs for Common Signals:
Properties of the Laplace transform
◦ Linearity:
◦ The resulting ROC is as large as the region in common between the
independent ROCs. However, there may be pole-zero cancellation in the
linear combination, which results in extending the ROC beyond the common
region.
Time Shifting:
◦ As product of X(s) with e-sto will not effect the poles of X(s), ROC remains
unaltered.
Shifting in the s-Domain:
◦ The ROC associated with X(s - so) is that of X(s) shifted by Re(s0).
Time Scaling:
Time Reversal:
Differentiation in the Time Domain:
The associated ROC is unchanged unless there is a pole-zero cancellation at s = 0.
Differentiation in the s-Domain:
Integration in the Time Domain:
The form of R' follows from the possible introduction of an additional pole
at s = 0 by the multiplication by l/s.
Convolution:
Initial value theorems
Final Value Theorem
Tutorial 5
Find the Laplace transform and the associated ROC for each of the
following signals:
Using the various Laplace transform properties, derive the Laplace
transforms of the following signals from the Laplace transform of u(t).
THE INVERSE LAPLACE TRANSFORM
Find the signal x(t) from its laplace transform x(s)
Inversion formula:
◦ Applicable to all classes of transform functions that involves the evaluation
of a line integral in complex s-plane
◦ The real value of c is to be selected such that if the ROC of X(s) is
σ1 < Re(s) <σ2 , then σ1 < c <σ2.
Use of Tables of Laplace Transform Pairs
Partial-Fraction Expansion
◦ If X(s) is a rational function
Tutorial 6
Find the inverse laplace transform
Ans.
The system function
The output y(t) of a continuous-time LTI system equals the convolution of the
input x(t) with the impulse response h(t),
Applying the convolution property
Where Y(s), X(s), and H(s) are the laplace transforms of y(t), x(t), and h(t),
respectively. The Laplace transform H(s) of h(t) is referred to as the system
function (or the transfer function) of the system.
Characterization of LTI Systems:
Causality:
For a causal continuous-time LTI system,
Since h(t) is a right-sided signal, the corresponding requirement on H(s) is that
the ROC of H(s) must be of,
Stability:
a continuous-time LTI system is BIBO stable if and only if
The corresponding requirement on H(s)
is that the ROC of H(s) contains the
jw-axis (that is, s = jw).
Causal and stable systems:
If the system is both causal and stable, then
all the poles of H(s) must lie in the left half of the s-plane; that is, they all
have negative real parts because the ROC is of the form Re(s) >σmax, and
since the jw axis is included in the ROC, we must have σmax < 0.
System Function for LTI Systems Described by Linear Constant-Coefficient
Differential Equations:
A continuous-time LTI system for which input x(t) and output y(t) satisfy the
general linear constant-coefficient differential equation of the form
Applying the Laplace transform and using the differentiation property (3.20) of the
Laplace transform,
Systems Interconnection:
Cascade Connection:
Parallel Connection
Representation of Laplace transform circuit-element models
Tutorial 7
1.
2.
3. Find the transfer function of LTI system described by given differential equation,
Ans.
4. Find a differential equation description of the given system described by transfer
function,
Ans
5.
Ans
Thank You
