1_signals and systems classifications
TRANSCRIPT
-
7/29/2019 1_Signals and Systems Classifications
1/38
Signals and Systems
Classifications
By Dr. Samer Awad
Assistant professor of biomedical engineering
The Hashemite University, Zarqa, Jordan
Last update: 24 September 2012
mailto:[email protected]:[email protected] -
7/29/2019 1_Signals and Systems Classifications
2/38
Definitions
Signals: Quantitative representation of avariable changing in time &/or space. They
can be functions of 1 or more independent
variables. Examples: force, voltage, stock
market and attendance
-
7/29/2019 1_Signals and Systems Classifications
3/38
Definitions
Systems: An operator acting upon signals toextract useful information
from them
Example: ECG system
Example: Passive LPF
x(t) y(t)System x(t) y(t)
-
7/29/2019 1_Signals and Systems Classifications
4/38
Definitions
Systems: Operator acting upon signals toextract useful information
Example: Ultrasound imaging system
Transducer processor display
summation
amplification
filtering
-
7/29/2019 1_Signals and Systems Classifications
5/38
Definitions
Studying signals & systems involve some sortof mathematical/numerical approach to
model/approximate the behavior of a given
system
Why study signals & systems?
1) Betterinsight better design
2) Compare model with reality3) Save time, money, stress & injuries
-
7/29/2019 1_Signals and Systems Classifications
6/38
Classification of Signals
Different analysis methods
1) Periodic vs non-periodic (a-periodic)
- A signal is periodic if there is a number T s.t.
x(t) = x(t+T)
- The smallest positive T is the period
- The fundamental freq fo=1/T
- Example: ECG (approximately), x(t)=cos(t)- The sum of periodic signals is periodic iff the
ratios of the periods are ratios of integers
-
7/29/2019 1_Signals and Systems Classifications
7/38
Classification of Signals
2) Random vs nonrandom- A signal is random if there is some degree of
uncertainty before the signal actually occurs.
Example: stock market, noise, ECG?
- A signal is non-random if there is no
uncertainty before it occurs
Example: x(t) = e-t
x(t) = cos(t)
- The amplitude for a random variable is
estimated using
n
xx
RMS
2)(
-
7/29/2019 1_Signals and Systems Classifications
8/38
Classification of Signals
3) Power vs Energy signalsIt is useful to estimate the size of the signal
- A signal is an energy signal if x(t) satifies:
Example: x(t) = e-t u(t)
- Power is a time average of energy. "Power
signals" have finite and non-zero power
Example: x(t) = sin(t)
-
7/29/2019 1_Signals and Systems Classifications
9/38
Classification of Signals
3) Power vs Energy signals
- Note that power is the time average of
energy.
- Finite energy zero power
- Finite power infinite energy
- A power signal has infinite energy and
everlasting non-physical
- A signal can be neither energy nor power.
Example: x(t) = r(t)
-
7/29/2019 1_Signals and Systems Classifications
10/38
Classification of Signals
4) Continuous time vs discrete time- x1(t) has values for every t [0,1]
- x2(n) has values at discrete points in time
- t & n are related by sampling intervalt or
sampling fresquency fs = 1/t
- x2(n) = x2(nt )X1(t) continuous
X2(n) discrete
-
7/29/2019 1_Signals and Systems Classifications
11/38
Classification of Signals
5) Analog vs digital- x1(t) has amplitude values [a,b]
- x3(n) has amplitude values finite set of
numbers
X1(t) continuous
X3(n) discrete
-
7/29/2019 1_Signals and Systems Classifications
12/38
Classification of Signals
X1(t): Continuous time & analogX2(n): Discrete time & analogX3(t): Continuous time & digital
X4(n): Discrete time & digital
-
7/29/2019 1_Signals and Systems Classifications
13/38
Classification of Signals
6) Even vs Odd- A signal is even if xe(t) = xe(-t) Ex: cos(t)
- A signal is odd if xo(t) = -xo(-t) Ex: sin(t)
- odd + odd = odd
- even + even = even
- odd + even = neither
- odd x odd = even- even x even = even
- odd x even = odd
-
7/29/2019 1_Signals and Systems Classifications
14/38
Classification of Signals
6) Even vs OddTo find out whether the signal x(t) is odd,
even, or neither:
a) Flip x(t) around the y-axis x(t)
x(t)
-
7/29/2019 1_Signals and Systems Classifications
15/38
Classification of Signals
6) Even vs OddTo find out whether the signal x(t) is odd,
even, or neither:
a) Flip x(t) around the y-axis x(t)
If x(t) = x(t) x(t) is even
x(t)
-
7/29/2019 1_Signals and Systems Classifications
16/38
Classification of Signals
6) Even vs OddTo find out whether the signal x(t) is odd,
even, or neither:
a) Flip x(t) around the y-axis x(t)
If x(t) = x(t) x(t) is even
b) If not, flip x(t) around x-axisx(t)
x(t)
-
7/29/2019 1_Signals and Systems Classifications
17/38
Classification of Signals
6) Even vs OddTo find out whether the signal x(t) is odd,
even, or neither:
a) Flip x(t) around the y-axis x(t)
If x(t) = x(t) x(t) is even
b) If not, flip x(t) around x-axisx(t)
If x(t) = x(t) x(t) is odd
c) If not, x(t) is neither odd, nor even
x(t)
-
7/29/2019 1_Signals and Systems Classifications
18/38
Classification of Signals
6) Even vs OddEven and odd components of a signal:
Any signal can be expressed as the sum of an
even part and odd part
x(t) = xe(t) + xo(t) ----------(1)
x(-t) = xe(-t) + xo(-t)
x(-t) = xe(t) - xo(t) ------(2)
(1) + (2) xe(t) = [ x(t) + x(-t) ]
(1) - (2) xo(t) = [ x(t) - x(-t) ]
Remember:
xe(t) = xe(-t)
xo(t) = -xo(-t)
-
7/29/2019 1_Signals and Systems Classifications
19/38
Classification of Signals
7) Odd Half-wave symmetryOnly for periodic signals
Can pertain to odd or even functions
Odd half wave symmetry (OHWS)
We are left with only ODD harmonics.
Even harmonics disappear.
-
7/29/2019 1_Signals and Systems Classifications
20/38
Even + OHWS
Odd + OHWS
Neither Odd
nor Even
+ OHWS
Classification of Signals
-
7/29/2019 1_Signals and Systems Classifications
21/38
Classification of Systems
1) Causal vs. non-causal- Causal: present o/p doesnt depend on future
values of i/p. It only depends on present &/or
past values
- Non-causal: o/p depends on future values of
i/pnon-physical
x(t)
y(t)
x(t)
y(t)
x(t)
y(t)
-
7/29/2019 1_Signals and Systems Classifications
22/38
Classification of Systems
1) Causal vs. non-causalAn example of a non-causal system. Will be
explained later when convolution is
presented
x(t) y(t)h(t)
zero
-
7/29/2019 1_Signals and Systems Classifications
23/38
Classification of Systems
2) Linear vs. non-linear
The system h(t) is linear if:
a)Additivity rule holds:
b)Homogeneity rule holds:
c)Superposition rule holds:
x2(t) y2(t)h(t)
x1(t) y1(t)h(t)
x1(t)+x2(t) y1(t)+y2(t)h(t)
kx(t) ky(t)h(t)
ax1(t)+bx2(t) ay1(t)+by2(t)h(t)
-
7/29/2019 1_Signals and Systems Classifications
24/38
Classification of Systems
2) Linear vs. non-linearShow that y(t) y(t) + 3y(t) = x(t) is non-linear
Create x1 y1 (eq1), x2 y2 (eq2) & show that:
a(eq1) + b(eq2)
x(t)=ax1+bx2 y(t)=ay1 + by2
y1(t) y1(t) + 3y1(t) = x1(t) -------- eq1
y2(t) y2(t) + 3y2(t) = x2(t) -------- eq2
{ ay1 y1 + by2 y2 }+ 3{ ay1 + by2 } = ax1 + b x2{ay1+ ay2} {ay1+ by2} + 3{ ay1 + by2 } = ax1 + b x2
LHS of the last two equations is not equal
system is nonlinear
-
7/29/2019 1_Signals and Systems Classifications
25/38
Classification of Systems
2) Linear vs. non-linearVin(t) Vout(t)h(t)
Vs - 2V
Vin
Vout
x1 x2
-
7/29/2019 1_Signals and Systems Classifications
26/38
Classification of Systems
3) Time-invariant vs. time varyingc/s doesnt change with time
x(t-) y(t-)h(t)
x(t) y(t)h(t)
x(t)
y(t)
x(t-)
y(t-)
-
7/29/2019 1_Signals and Systems Classifications
27/38
Classification of Systems
HW#1: Classification of systems:
- Linear vs. non-linear- Time-invariant vs. time varying
-
7/29/2019 1_Signals and Systems Classifications
28/38
Classification of Systems
- Most systems that we will deal with are lineartime invariant (LTI) systems continuous and
discrete
4) Continuous time vs. discrete time
- A system is continuous time if i/p & o/p are
continuous time signals
- A system is discrete time if i/p & o/p arediscrete time signals
-
7/29/2019 1_Signals and Systems Classifications
29/38
Classification of Systems
5) Instantaneous vs. dynamic- Instantaneous: depends only on present
values of the i/p memoryless system
Always causal
Example: resistor network
- Dynamic: depends on i/p from a period of
time
Can be causal or non-causalExample: passive LPF
Vin
Vout
-
7/29/2019 1_Signals and Systems Classifications
30/38
System Representations
1) Differential equations:
x(t) y(t)System x(t) y(t)
)()(
)(
])()([1
)(
)()(
txtydt
tdy
RC
tytxR
ti
dt
tdyCti
-
7/29/2019 1_Signals and Systems Classifications
31/38
SystemRepresentations
2) Transfer function H(s) Laplace transform:
x(t) y(t)System x(t) y(t)
-
7/29/2019 1_Signals and Systems Classifications
32/38
System Representations
3) Frequency response H() - Fourier transform:
x(t) y(t)System x(t) y(t)
-
7/29/2019 1_Signals and Systems Classifications
33/38
System Representations
4) Impulse response h(t):
x(t) y(t)System x(t) y(t)
-
7/29/2019 1_Signals and Systems Classifications
34/38
Singularity Functions
Singularity: a notation used to describediscontinuous functions
1)Impulse function:
- VERY IMPROTANT
- Delta, Dirac delta
- Infinite amplitude with zero width
a)
b) (t-to
)=0 every where tto
5
-
7/29/2019 1_Signals and Systems Classifications
35/38
Singularity Functions
1)Impulse function:
c)
d) Sifting/sampling property
Given that f(t) is continuous @ to
)()()( oo tfdttttf
-
7/29/2019 1_Signals and Systems Classifications
36/38
Singularity Functions
1)Impulse function:
e)
a/2
a/2
-
7/29/2019 1_Signals and Systems Classifications
37/38
Singularity Functions
2) Unit step function: u(t)
3) Unit ramp function: r(t)
1
1
-
7/29/2019 1_Signals and Systems Classifications
38/38
Singularity Functions
4) Rect function
rect(t) = u(t+) - u(t-)
-
1