2.1
Signals and Systems
Signals convey information. Systems respond to (or process) information.
Engineers desire mathematical models for signals and systems in order to solve
design problems efficiently and thoroughly.
Signal - encoded information; data; a dynamic (or change) in some quantity that has
meaning. In most cases this is modeled as a function of time or space.
Noise - irrelevant data; variability in a quantity that has no meaning or
significance. In most cases this is modeled as a random variable.
System - a mapping between a set of inputs to a set of outputs; an entity that
processes signals received at its input and produces another set of signals at the
output; a process that responds to actions or events at its input by generating
actions or events at its output. Differential equations or convolution integrals can
approximately model many physical systems.
Kevin D. Donohue: University of Kentucky
2.2
Classifications of Systems
1. Linear vs. Non-linear
For all linear systems superposition holds for input-output relationships.
Denote a general system in the following manner:
y = H[ x]
System H operates on input x to produce output y.
The system H is linear if and only if (iff) for any input-output pair:
and y = H[ x ]
y = H[ x ]
the following statement is also true:
a y + a y = H[ a x + a x ]
where a1 and a2 are constants.
1
1
1
1
2
2
2
1
1
2
2
2
Kevin D. Donohue: University of Kentucky
2.3
Determine whether or not each system described below is linear. Assume inputs
and outputs are functions of time denoted by x and y, and constants are denoted k.
• y = kx
2
d x dx
• y=
+ +x
dt
dt
2
• y = kx + 10
• y= k x+k x
1
2
2
2
d x
• y=x
+x
dt
2
2. Constant-parameter (time-invariant) vs. time-varying systems
Kevin D. Donohue: University of Kentucky
2.4
A system whose model parameters change with time is considered timevarying. Note that the output and input will be varying with time for both constantparameter and time-varying systems.
Systems where outputs differ only by a time shift when the same inputs are applied
at corresponding time shifts is a constant parameter system.
A system is a constant parameter system iff for any input output pair:
y ( t ) = H[ x ( t )]
the following statement is also true:
y ( t − τ ) = H[ x ( t − τ )] for all τ
Determine whether the systems below are time varying or time invariant:
• y ( t ) = kx ( t ) + 10
• y ( t ) = cos( 2πt ) x ( t )
Kevin D. Donohue: University of Kentucky
2.5
3. Instantaneous (memoryless) vs. dynamic (with memory) systems
For an instantaneous system, the present output value depends only on the
present input value. In a dynamic system the present output value depends on the
present and past input values. Dynamic systems usually contain some type of
energy storage elements.
The response of a dynamic system results from two components; the initial
condition and the input. The state of the system refers to the information needed
along with the present input to determine the present output.
Zero-input response - system output due only to system state (or initial condition).
Zero-state response - system output due only to the input of the system.
In general: Total response = Zero-input response + Zero-state response
Determine which systems are instantaneous and which are dynamic.
• y= k x+k x
dx
• y= +x
dt
2
1
2
Kevin D. Donohue: University of Kentucky
2.6
4. Causal vs. Noncausal
Systems where the output depends only on the present and/or past values of the
input are referred to as causal. Note that for a causal system the output cannot
depend on future input values. Systems where the output depends on future input
value is referred to as a noncausal system.
5. Lumped-Parameter vs. Distributed-Parameter Systems
In most real systems the interactions between the signal energy and the system
elements happen continuously over space (i.e. resistance over a wire). In modeling
these systems the interaction can be considered to occur at one point in space. This
is referred to as a lumped-parameter model. This is a reasonable model when the
dimensions of the elements are small with respect to the energy wavelength. When
this is not done (i.e. transmission lines), the model is referred to as a distributed
parameter system.
Kevin D. Donohue: University of Kentucky
2.7
6. Continuous-Time vs. Discrete-Time Systems
The input and output for a discrete-time system is defined only at discrete
points in time:
y ( n ) = H[ x ( n )]
for n ∈ {… -2, -1, 0, 1, 2, …}.
If the inputs and outputs are defined over a continuum of time values then the
system is a continuous-time system:
y ( t ) = H[ x ( t )]
for t ∈ [0,+∞].
7. Analog vs. Digital Systems
A system whose input and output values take on only a set of discrete values is
referred to as a digital system.
If the values of the input and output can take on a continuum of values then the
system is an analog system.
For the following digital signal processing system classify the subsystems at each stage.
Based on the nature of the input and output some subsystems may not fit under a
continuous-time vs. discrete-time or analog vs. digital system classification scheme but
a hybrid combination.
Kevin D. Donohue: University of Kentucky
2.8
Elements of a Digital Signal Processing System
Analog Signal
x a (t )
Discrete-time
Signal
Digital Signal
Coder
Quantizer
xa (nT )
x$ (n)
x$ (nT )
11
10
01
00
Processed
Digital
Signal
x$ (n)
Computing
Hardware
y$ ( n)
Processed
Analog
Signal
Interpolator
and smoothing
Kevin D. Donohue: University of Kentucky
y$ a (t )
2.9
Differential Equation Models for Current and Voltage Systems
Capacitors:
Inductors:
Resistors:
dv (t )
i (t ) = c
dt
1
v (t ) = ∫ i (τ )dτ
c
di ( t )
v (t ) = L
dt
1
i (t ) = ∫ v (τ )dτ
L
v (t ) = Ri (t )
t
i(t)
−∞
v(t)
t
i(t)
−∞
+
v(t)
-
i(t)
+
v(t)
-
Kevin D. Donohue: University of Kentucky
2.10
Example, find the input-output equation relating input is to output vo
iL
L
R1
is
+
vc
-
C
+
v0
-
R2
⎛R
i (t )
1 ⎟⎞
1 ⎜⎛ R1 + R2 ⎞⎟
s
2
⎜
v& +
v
= v&& +
+
Answer:
o
o
o
⎟
⎜
⎜
⎟
CL
L
R
C
CL
R
⎝
⎝
1 ⎠
1 ⎠
or
⎡ R2
⎡ i&L ⎤ ⎢ − L
⎢ v& ⎥ = ⎢ 1
⎣ c⎦ ⎢ −
⎢⎣ C
1 ⎤
0
⎥
L ⎥ ⎡ i L ⎤ + ⎡⎢ 1 ⎤⎥ i
1 ⎥ ⎢v ⎥
s
−
⎣ c ⎦ ⎢⎣ C ⎥⎦
R C ⎥⎦
1
⎡ iL ⎤
v = R
0⎢ ⎥
o
2
v ⎦
⎣
c
Kevin D. Donohue:
University of Kentucky
[
]
2.11
Differential Equation Models for Position and Force Systems
Translational Systems - Consider Position (output) and force (input) in one
direction denoted by y(t) and f(t), respectively:
Mass (M): f ( t ) = My&&( t )
Linear Spring (Stiffness K): f (t ) = Ky (t )
Linear Dashpot (damping coefficient B): f ( t ) = By&( t )
Rotational Systems - Consider an angular position (output) and torque (input)
denoted by θ(t) and T(t), respectively:
Rotational Mass (J): T (t ) = Jθ&&
Torsional Spring (K): T ( t ) = Kθ
Torsional Dashpot (B): T (t ) = Bθ&
Electromechanical Systems - For a DC motor, consider the an angular position
(output), and current (input) denoted by θ(t) and i(t), respectively:
Motor Constant (KT): T (t ) = K i(t )
T
Kevin D. Donohue: University of Kentucky
2.12
Example, consider a torsional spring with stiffness K=2 nt-m/rad fastened to the
rotor of an armature controlled DC motor with motor constant Kr = 5 nt-m/A, and
the rotational mass is J= .5 nt-m/(rad/s2). The friction coefficient for the spinning
rotor is B = .05 nt-m2/(rad/sec). Find the equation that relates the rotor position to
the armature current. Assume the polarity of the motor is such that a positive
current moves the angular position is a positive direction. Describe the motion of
the rotor for a step input going from 0 to .2 A.
Ans: K i = Jθ&& + Bθ& + Kθ ; θ (t ) =.5 − exp( − t / 20)(.5 cos(2t ) − 0.0125 sin(2t ))
In Matlab:
>> t = [0:2*pi/20:100];
>> sig = .5 - exp(-t/20).*(.5*cos(2*t) - .0125*sin(2*t));
>> plot(t,sig)
rotor response
1
>> title('rotor response')
0.9
>>xlabel('seconds')
0.8
0.7
>>ylabel('radians')
r a
radians
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
seconds
Kevin D. Donohue: University of Kentucky