LAPLACE TRANSFORM

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Biomedical Engineering
Signal theory and Medical biocomputation
Lab 09: Laplace Transform
Lab Instructor: Diego Almeida
Developed by: Diego Almeida
Group: Marilyn Figueroa, Adriana Estrella, Gabriela Gonzalez
1. Introduction
The Laplace transform is a mathematical tool that allows us to go from the temporal to
the frequency domain.
The Laplace transform models a signal as a sum of exponentially decaying sinusoids
having various frequencies and rates of decay. It is especially used for calculating the
time response of a linear system to an arbitrary input. In most practical cases it is
usually straight forward to determine the frequency response of a filter if the Laplace
transform of its impulse response is known.
The Laplace transform is used in process control circuits. For example, monitoring to
control the temperature and humidity of houses and buildings.
In biomedical engineering to control the variables of a glucose prediction model.
Another application in biomedical engineering is digital filters to eliminate noise in
signals from the human body.
Figure 1. Applications in Biomedical Engineering
1. 1 Laplace transform
The Laplace transform provides an s-domain or frequency domain version of a time
signal. Consider the common Laplace transform pair:
where is the unit step function. The time function and the frequency function𝑢(𝑡) ℎ(𝑡)
are alternate ways of describing the same signal. In the time domain, is𝐻(𝑠) ℎ(𝑡)
exponential. In the frequency domain, is rational. The choice of the letter h for the𝐻(𝑠)
above signal is commonly used for filter functions. In the time domain, is theℎ(𝑡)
impulse response function of the filter.
1. 2 The Z- transform
Z transform is used to convert discrete time domain signal into discrete frequency
domain signal. It has wide range of applications in mathematics and digital signal
processing, it is an equivalent to Laplace transform but in discrete time domain.
Figure 2. the relationship between the s- plane in Laplace domain and the z- plain in z-
transform domain
Relationship of the sampled system in Laplace domain and its digital system in
z-transform domain:
Substituting into relationship of sampled system in Laplace domain, it𝑠 =− α±𝑗ω
follows that . In the polar form we have:𝑍 = 𝑒−α𝑇±𝑗ω𝑇
From the Laplace transfer function, we can achieve the analog filter steady – state
frequency. If we set s = jω, then we get the complex frequency response function H(jω).
1.3 Circuit Analysis with Laplace Transforms
One application of the Laplace transform is its use in circuit analysis. In this section we
will derive the voltage-current relationships for basic passive circuit devices, i.e.,
resistors, inductors, and capacitors in the complex frequency domain.
a. Resistor
The time and complex frequency domains for purely resistive circuits are shown in
figure.
b. Capacitor
The time and complex frequency domains for purely capacitive circuits in figure 4.
Note: In the complex frequency domain, the terms sL and 1/sC are called complex
inductive impedance, and complex capacitive impedance respectively. Likewise, the
terms and sC and 1/sL are called complex capacitive admittance and complex inductive
admittance respectively.
1.4. Transfer Functions
In a s-circuit, the ratio of the output voltage to the input voltage under𝑉
𝑜𝑢𝑡
𝑠( ) 𝑉
𝑖𝑛
𝑠( ) 
zero state conditions, is of great interest in network analysis. This ratio is referred to as
the voltage transfer function and it is denoted as , that is,𝐺
𝑉
𝑠( )
1. 5 Objects and functions of MATLAB
Freqz function
Given a transfer function, the MATLAB function freqz () can be used to determine the
frequency response. The syntax is given by.
ℎ 𝑤[ ] = 𝑓𝑟𝑒𝑞𝑧 (𝐵, 𝐴, 𝑁)
where all the parameters are defined as
h=an output vector containing frequency response.
w =an output vector containing normalized frequency values distributed in the range
from 0 to π radians.
B = an input vector for numerator coefficients.
A= an input vector for denominator coefficients.
N = the number of normalized frequency points used for calculating the frequency
response.
Unwrap function
Unwraps: the radian phase angles in a vector P. Whenever the jump between
consecutive angles is greater than or equal to π radians, unwrap shifts the angles by
adding multiples of ±2π until the jump is less than π.
tf function
Use tf to create real-valued or complex-valued transfer function models, or to convert
dynamic system models to transfer function form.
sys = tf(numerator,denominator)
Pzmap, zero and pole functions
pzmap (H) graph the poles and zeros in a simple plane that MATLAB will show us the
command, where p is pole, z comes from zero.
Filter function
y = filter (b, a, x) : filters the input data x using a rational transfer function defined by the
coefficients of the numerator and the denominator b and a respectively.
2. Objectives:
● The goal in this lab is to get acquainted with these tools and develop some
familiarity with some MATLAB commands that are useful when treating Laplace
transforms and their applications.
3. Materials and Methods
3. 1 Materials
● MATLAB
● CALCULATOR
3.2 Methodology
Part 1:
Consider the linear system:
ℎ 𝑡( ) = 𝑒−2𝑡 − 𝑒−3𝑡 + δ(𝑡)
Calculate the Laplace transform, representing by the transfer function.
Compare your answer with the answer by applying the laplace function in
MATLAB.
𝐻 𝑠( ) = 𝑠 + 1𝑠 + 2 −
1
𝑠 + 3 =
𝑠+1
𝑠2+5𝑠+6
Using the command subplot of MATLAB, find and plot h(t), system impulse response,
system step response and system zero-state response due to f(t).
a) The h(t) signal.
b) The system impulse response.
c) The system step response.
d) The system zero-state response due to the input signal .𝑓 𝑡( ) = sin 𝑠𝑖𝑛 2𝑡( ) 𝑢(𝑡)
● Xlabel: Time
● Ylabel: Name of response: Impulse response, step response, Sinusoidal signal.
Part 2:
Given each of following digital transfer functions,
a) 𝐻 𝑧( ) = 𝑧𝑧−0.5 
b) 𝐻 𝑧( ) = 1 − 0. 5 𝑧−1
c) 𝐻 𝑧( ) = 0.5𝑧
2−0.4
𝑧2−0.5𝑧+0.3 
d) 𝐻 𝑧( ) = 1− 0.8𝑧
−1+0.91𝑧−2
1−0.5𝑧−1−0.7 𝑧−2
1. Plot the poles and zeros on the z- plane and based on their, explain why are
they stable?
● Use MATLAB functions tf, pole, zero and pzmap to obtain poles and zeros on the
z- plane.
2. Plot the magnitude frequency response and phase response vs frequency
response (radians) for each transfer function.
● Use MATLAB function freqz() to obtain the transfer function in a vector.
● Use MATLAB function 'unwrap' will correct the value to be higher than pi and
so forth.
3. Identify the corresponding filter type such as lowpass, highpass, bandpass or
bandstop. Explain your answer.
● Include MATLAB figure
Lowpass filters: Allow passing only low frequency details, attenuates the high
frequency details. Example: Smoothening filters.
Highpass filters: Allows passing only high frequency details, attenuates the low
frequency details. Example: Sharpening mask filters.
Bandpass filters: Only allows signals within a certain band to pass, attenuates the
frequencies below a threshold and above another threshold to pass.
Bandstop filters: Attenuate signal in range of a certain frequency. Allows frequency
below a certain threshold and above another threshold to pass.
Part 3:
Find the transfer functions for the networks (a) and (b). Using MATLAB, plot
versus frequency in Hertz, on a semilog scale. The values of R ( ) and C (𝐺 (𝑠)| | 𝐾Ω
) are defined by the student.µ𝐹
Calculations:
a. Network (a)
;𝐶 = 1𝐶∙𝑠 𝑠 = 𝑗𝑤
𝐺 𝑠( ) = 𝑉𝑜𝑢𝑡(𝑠)𝑉𝑖𝑛(𝑠) =
𝑍
𝑜𝑢𝑡
𝑍
𝑖𝑛
=
1
𝐶∙𝑠
𝑅+ 1𝐶∙𝑠
= 1𝑅𝐶𝑠+1
b. Network (b)
;𝐶 = 1𝐶∙𝑠 𝑠 = 𝑗𝑤
𝐺 𝑠( ) = 𝑉𝑜𝑢𝑡(𝑠)𝑉𝑖𝑛(𝑠) =
𝑍
𝑜𝑢𝑡
𝑍
𝑖𝑛
= 𝑅
𝑅+ 1𝐶∙𝑠
= 𝑅𝐶𝑠𝑅𝐶𝑠+1
● Include MATLAB figure and code.
Part 4:
Biomedical Application
Create a Lowpass filter with the transfer function from part 2. a), apply this filter
with an ECG signal 𝐻 𝑧( ) = 𝑧𝑧−0.5 
a. Load the file noisyecg.mat
b. Usedthe noisyECG_withTrend as data, ECG+noise
c. Used the function filter to create your filter
d. With the subplot command: Plot ECG with noise and another graph with ECG filter.
e. Compare these two last graphs.
Conclusions:
The use of the Laplace transform does not allow changing the time domain to the
frequency domain easily. It is especially used to compute the time response of a linear
system to an arbitrary input and in process control circuits. In this laboratory he
becomes familiar with the use of the Laplace transform, using MATLAB with specified
code, obtaining the expected results without difficulty.
References:
- Karris, S. (2003). Signals and Systems with MATLAB Applications. 2nd edition.
Orchard Publications
- Semmlow, J (2005). Circuits, signals and Systems for Bioengineers. Elsevier
Academic.
- Tan, L., & Jiang, J. (2019). Digital Signal Processing Systems, Basic Filtering Types,
and Digital Filter Realizations. Digital Signal Processing,
173–228. doi:10.1016/b978-0-12-815071-9.00006-3