EE313 Linear Systems and Signals
Fall 2010
Transfer Functions
Prof. Brian L. Evans
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
Initial conversion of content to PowerPoint
by Dr. Wade C. Schwartzkopf
Zero-State Response
• Linear constant coefficient differential equation
Input x(t) and output y (t )  y zeroinput (t )  y zero state(t )
Zero-state response: all initial conditions are zero
yt   Y s 
xt   X s 
dr
r


y
t

s
Y s 
r
dt
dk
k


x
t

s
X s 
k
dt
Laplace transform both sides of differential equation with
all initial conditions being zero and solve for Y(s)/X(s)
y ' (t )  y (t )  x(t )
sY ( s)  Y ( s)  X ( s)
y (0  )  0
H ( s) 
Y ( s)
1

X ( s) s  1
19 - 2
Transfer Function
• H(s) is called the transfer function because it
describes how input is transferred to the output
in a transform domain (s-domain in this case)
Y(s) = H(s) X(s)
y(t) = L-1{H(s) X(s)} = h(t) * x(t)  H(s) = L{h(t)}
• Transfer function is Laplace transform of
impulse response
19 - 3
Transfer Function Examples
• Laplace transform

X s     xt  e  s t dt
0
• Assume input x(t) and output y(t) are causal
• Ideal delay of T seconds
Initial conditions (initial voltages in delay buffer) are zero
y t   xt  T 
Y s   X s  e  s T
Y s 
H s  
 es T
X s 
x(t)
T
y(t)
19 - 4
Transfer Function Examples
• Ideal integrator with
y(0-) = 0
y t     x d
t
0
 
1
1
Y s   X s   y 0 
s
s
Y (s) 1
H s  

X (s) s
x(t)
  dt
t
0
y(t)
• Ideal differentiator
with x(0-) = 0
d
y t   x(t )
dt
Y s   s X s   x 0   s X ( s )
Y s 
H s  
s
X s 
 
x(t)
d
y(t)
 
dt
19 - 5
Cascaded Systems
• Assume input x(t) and output y(t) are causal
x(t)
• Integrator first,
then differentiator X(s)
• Differentiator first, x(t)
then integrator
X(s)
 x d
t
1/s
s
0
X(s)/s
d
x(t )
dt
s X(s)
x(t)
s
X(s)
x(t)
1/s
X(s)
• Common transfer functions
A constant (finite impulse response)
A polynomial (finite impulse response)
Ratio of two polynomials (infinite impulse response)
19 - 6
Block Diagrams
X(s)
H(s)
Y(s)
W(s)
X(s)
H1(s)
H2(s)
Y(s)
=
X(s)
H1(s)H2(s)
Y(s)

Y(s)
=
X(s)
H1(s) + H2(s)
Y(s)
Y(s)
=
X(s)
G(s)
1 + G(s)H(s)
Y(s)
H1(s)
X(s)
H2(s)
X(s)
-

E(s)
G(s)
H(s)
19 - 7
Cascade and Parallel Connections
• Cascade
X(s)
W(s) = H1(s) X(s)
Y(s) = H2(s)W(s)
Y(s) = H1(s) H2(s) X(s)  Y(s)/X(s) = H1(s)H2(s)
H1(s)
H2(s)
Y(s)  X(s)
H2(s)
H1(s)
Y(s)
One can switch the order of the cascade of two LTI systems
if both LTI systems compute to exact precision
• Parallel Combination
Y(s) = H1(s)X(s) + H2(s)X(s) Y(s)/X(s) = H1(s) + H2(s)
H1(s)

X(s)
H2(s)
Y(s)
=
X(s)
H1(s) + H2(s)
Y(s)
19 - 8
Feedback Connection
• Governing equations
E s   F s   H s  Y s 
Y s   Gs  E s 
• What happens if H(s) is
a constant K?
Choice of K controls all
poles in transfer function
Common LTI system in
EE362K Introduction to
Automatic Control and
EE445L Microcontroller
Applications/Organization
• Combining equations
Y s   G s   F s   H s  Y s  
Y s   G s  H s  Y s   G s  F s 
G s 
Y s  
F s 
1  G s  H s 
F(s)
-

E(s)
G(s)
H(s)
Y(s)
=
F(s)
G(s)
1 + G(s)H(s)
Y(s)
19 - 9
External Stability Conditions
• Bounded-input bounded-output stability
Zero-state response given by h(t) * x(t)
Two choices: BIBO stable or BIBO unstable
• Remove common factors in transfer function H(s)
• If all poles of H(s) in left-hand plane,
All terms in h(t) are decaying exponentials
h(t) is absolutely integrable and system is BIBO stable
• Example: BIBO stable but asymptotically
unstable
 s  1   1  s  1   1 
H ( s)  




 

s

1
s

1
s

1

 

 s 1 
2
19 - 10
Based on slide by Prof. Adnan Kavak
Internal Stability Conditions
• Stability based on zero-input solution
• Asymptotically stable if and only if
Characteristic roots are in left-hand plane (LHP)
Roots may be repeated or non-repeated
• Unstable if and only if
(i) at least characteristic root in right-hand plane and/or
(ii) repeated characteristic roots are on imaginary axis
• Marginally stable if and only if
There are no characteristic roots in right-hand plane and
Some non-repeated roots are on imaginary axis
19 - 11
Based on slide by Prof. Adnan Kavak
Frequency-Domain Interpretation
est
h(t)
y(t)
• y(t) = H(s) e s t
for a particular value of s
• Recall definition of
frequency response:
ej 2 p f t
h(t)
y(t)
y t   ht   e s t

  h e s t  d

e
st

s 


h

e
d
H s 
y t   ht   e j 2p

ft
  h e j 2p

e
j 2p f t
f t  
d

j 2p f 


h

e
d
Hf

19 - 12
Frequency-Domain Interpretation
• Generalized frequency: s = s + j 2 p f
• We may convert transfer function into
frequency response by if and only if region of
convergence of H(s) includes the imaginary axis
H freq  f   H s 
s  j 2pf
• What about h(t) = u(t)?
1


H s 
s
for Res  0
We cannot convert H(s) to a frequency response
However, this system has a frequency response
• What about h(t) = d(t)?
H s   1 for all s  H freq  f   1
19 - 13
Frequency Selectivity in Filters
• Lowpass filter
|Hfreq(f)|
• Bandpass filter
|Hfreq(f)|
1
1
• Highpass filter
f
|Hfreq(f)|
f
• Bandstop filter
|Hfreq(f)|
f
Linear time-invariant filters are BIBO stable
f
19 - 14