EE324 Lab Report 6
Written by: Shihao Ni
Introduction
The purpose of this lab is to learn how to derive a discrete-time system as an
approximation of a continuous-time one. We consider the Euler Approximation and
Trapezoidal Approximation for transforming a continuous-time system into discretetime systems. The transformations are performed in the pre-lab section of lab, and the
report shows simulations of the continuous- and discrete-time systems from the two
approximation methods for different sampling times and different inputs.
Pre-Lab
1. The Euler Approximation
1. From the transfer function H(s) (in Laplace domain) of the system, derive the
differential equation that characterizes it.
R1 1k; R2 10k;C 1F
V
H (s)  0 R1 R2 R1R2 SC 11000 10s 1100 s



1000 10s
100 s
Vi
R1 R1R2 SC
The differential equation is as follows:
dy(t)
dt
100 y(t) 1100x(t) 
dx(t)
dt
2. Evaluate the equation at time kT.
d
d
y(kT ) 100 y(kT ) 1100x(kT )  x(kT )
dt
dt
3. Use Euler’s approximation to derive a difference equation, typical of a discrete
system.
y[(k 1)T ] y kT  d
x[(k 1)T ] x  kT 
; x(kT ) 
dt
T
dt
T
y[(k 1)T ] y  kT 
x[(k 1)T ] x kT 

100 y(kT ) 
1100x(kT )
T
T
y[k 1] 100T 1y[k] x[k 1] 1100T 1x[k]
d
y(kT ) 
4. From the difference equation, compute the transfer function HE(z) of the resulting
discrete system (in Z domain).
zY (z) (100T 1)Y (z) zX (z) (1100T 1) X (z)
z 1
1100
H E (z)  Y ( z)  z (1100T 1)  T
z (100T 1)
X (z)
z 1
100
T
(*)
5. What are the relations between the poles and the zeros of the continuous-time system,
and the poles and the zeros of the discrete-time system, respectively? Can the discrete
system be unstable? Why?
For the continuous-time system, the value of zeros makes the rational Laplace transforms
be zeros and the value of poles makes the rational Laplace transforms be infinite. The
poles and zeros in the finite s-plane completely characterize the algebraic expression for
rational Laplace transforms to within a scale factor.
For the discrete-time system, the value of zeros makes the rational z- transforms be zeros
and the value of poles makes the rational z-transforms be infinite. The poles and zeros in
the finite z-plane completely characterize the algebraic expression for rational ztransforms to within a scale factor.
The discrete system can be unstable when the ROC is outside the Unit Circle
6. Verify that HE(z) can be directly obtained from H(s) by taking the transformation
1100 s
z 1
From H (s) 
, substitute s=
100 s
T
z 1
1100
T
H E (z)  z 1
which is the same as equation (*)
100
T
2. The Trapezoidal Approximation
1. Evaluate the differential equation derived previously at times kT and (k + 1)T.
d
y(kT ) 100 y(kT ) 1100x(kT ) 
dt
d
d
x(kT )
dt
y((k 1)T ) 100 y((k 1)T ) 1100x((k 1)T ) 
dt
d
x((k 1)T )
dt
2. Use the trapezoidal approximation to derive a difference equation, typical of a
discrete system.
d y(kT ) 100 y(kT ) 1100x(kT )  d x(kT ) (1)
dt
dt
d y((k 1)T ) 100 y((k 1)T ) 110 0x((k 1)T )  d
x((k 1)T )
dt
dt
using k instead of kT and

(2)
(1) (2)
2
y(k 1) y(k)
x(k 1) x(k )
50 y(k) 50 y(k 1) 
550x(k) 550x(k 1)
2
2
y(k 1) y(k ) y[k 1] y[k ] x(k 1) x(k ) x[k 1] x[k ]

;

2
T
2
T
 y[k 1] y[k] 50Ty[k 1] 50Ty[k] x[k 1] x[k] 550Tx[k 1] 550Tx[k]
(50T 1) y[k 1] (50T 1) y[k] (550T 1)x[k 1] (550T 1)x[k]
3. From the difference equation, compute the transfer function HT (z) of the
resulting discrete system (in Z domain).
(50T 1)zY (z) (50T 1)Y (z) (550T 1)zX (z) (550T 1) X (z)
H T (z) 
Y ( z) (550T 1)z (550T 1)

X (z)
(50T 1)z (50T 1)
(*)
4. What are the relations between the poles and the zeros of the continuous-time system,
and the poles and the zeros of the discrete-time system, respectively? Can the discrete
system be unstable? Why?
For the continuous-time system, the value of zeros makes the rational Laplace transforms
be zeros and the value of poles makes the rational Laplace transforms be infinite. The
poles and zeros in the finite s-plane completely characterize the algebraic expression for
rational Laplace transforms to within a scale factor.
For the discrete-time system, the value of zeros makes the rational z- transforms be zeros
and the value of poles makes the rational z-transforms be infinite. The poles and zeros in
the finite z-plane completely characterize the algebraic expression for rational ztransforms to within a scale factor.
The discrete system can be unstable when the ROC is outside the Unit Circle
5. Verify that HT (z) can be directly obtained from H(s) by taking the transformation
s 
2 z 1
T z 1
1100 s
2
From H (s) 
, substitute s= z 1
100 s
T z 1
2 z 1
1100
T
z
1
( z 1) 550T ( z 1) (550T 1) z (550T 1)
H (z) 
=
=
E
(z 1) 50T (z 1)
(50T 1)z (50T 1)
2 z 1
100
T z 1
So the result is the same as equation (*)
6. Use Simulink to simulate the continuous-time system, as well as the discrete-time
systems from the two approximation methods for different sampling times T and
different inputs, discuss your findings, and suggest a T which provides a satisfactory
approximation for the two methods.
Figure (2)
The result for sinusoidal input with 1rad/s
Figure (3)
The Result for Unit Step
Figure (4)
Figure (5)

Sampling time=0.001s
Sinusoidal input with 1 rad/s
Figure (6)
Unit step input
Figure (7)

Sampling time =0.005s
Sinusoidal input with 1 rad/s
Figure (8)
Unit step input
Figure (9)
Figure (10)

Sampling time=0.001s
Sinusoidal input with 1 rad/s
Figure (11)
Unit step input
Figure (12)

Sampling time =0.005s
Sinusoidal input with 1 rad/s
Figure (13)
Unit step input
Figure (14)
Analysis:
Comparing the simulation result for different input and different sampling frequency of
discrete-time system and the continuous-time system, we can find that the result of
smaller sampling time for both two approximation methods will more approach to the
result of continuous-time system.
So T=0.005s can provides a satisfactory approximation for the two methods.
Conclusion
The purpose of this lab was to learn how to derive a discrete-time system as an
approximation of a continuous-time one. By observing the simulation results, we see that
smaller sampling times for both Euler and Trapezoidal approximation methods are
closer to the result of continuous-time system. For both methods, T=0.005s is a good
approximation.