linear systemskhosrow ghadiri - ee dept. sjsu1 signals and systems linear system theory ee 112 -...
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Linear Systems Khosrow Ghadiri - EE Dept. SJSU 1
Signals and SystemsLinear System Theory
EE 112 - Lecture Eight Signal classification
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 2
Signal A signal is a pattern of variation that carry
information. Signals are represented mathematically as a function
of one or more independent variable A picture is brightness as a function of two spatial
variables, x and y. In this course signals involving a single independent
variable, generally refer to as a time, t are considered. Although it may not represent time in specific application
A signal is a real-valued or scalar-valued function of an independent variable t.
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 3
Example of signals Electrical signals like voltages, current and
EM field in circuit Acoustic signals like audio or speech signals
(analog or digital) Video signals like intensity variation in an
image Biological signal like sequence of bases in
gene Noise which will be treated as unwanted
signal …
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 4
Signal classification
Continuous-time and Discrete-time Energy and Power Real and Complex Periodic and Non-periodic Analog and Digital Even and Odd Deterministic and Random
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 5
A continuous-time signal Continuous-time signal x(t), the independent variable, t is
Continuous-time. The signal itself needs not to be continuous.
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 6
A continuous-time signal
x=0:0.05:5; y=sin(x.^2); plot(x,y);
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 7
A piecewise continuous-time signal
A piecewise continuous-time signal
0; 0 10
1; 10 20( )
1; 20 25
2; 25 30
t
tx t
t
t
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 8
A discrete-time signalA discrete signal is defined only at discrete instances. Thus,
the independent variable has discrete values only. [ ]x n
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 9
Mathlab file
x = 0:0.1:4; y = sin(x.^2).*exp(-x); stem(x,y)
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 10
Sampling A discrete signal can be derived from a continuous-time signal
by sampling it at a uniform rate. If denotes the sampling period and denotes an integer that
may assume positive and negative values, Sampling a continuous-time signal x(t) at time yields a
sample of value For convenience, a discrete-time signal is represented by a
sequence of numbers: We write
Such a sequence of numbers is referred to as a time series.
t n( )x n
[ ] ( )x n x n n
n
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 11
A piecewise discrete-time signal
A piecewise discrete-time signal
0; 0 10
1; 10 20
1; 20 25
2; 25 30
n
nx n
n
n
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 12
Energy and Power Signals X(t) is a continuous power signal if:
X[n] is a discrete power signal if:
X(t) is a continuous energy signal if:
X[n] is a discrete energy signal if:
21
0 lim2
T
TTx t dt
T
2
10 lim
2 1
N
Nn N
x nN
2
0 x t dt
2
0n
x n
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 13
Power and Energy in a Physical System The instantaneous power
The total energy
The average power
21P t v t i t v t
R
2 2 2
1 1 1
21t t t
t t tP t dt v t i t dt v t dt
R
2 2
1 1
2
2 1 2 1
1 1 1t t
t tP t dt v t dt
t t t t R
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 14
Power and Energy By definition, the total energy over the time interval in a
continuous-time signal is:
denote the magnitude of the (possibly complex) number The time average power
By definition, the total energy over the time interval in a discrete-time signal is:
The time average power
1 2t t t x t
2
1
2t
tE x t dt
x t x t
2
1
2
2 1
1 t
tP x t dt
t t
2 1n n n
x n
2
1
2n
n n
E x n
2
1
2
2 1
1
1
n
n n
P x nn n
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 15
Power and Energy Example 1:The signal is given below is energy or power signal. Explain.
This signal is energy signal
12 2
0
11 1 1 9lim lim 3 lim 9 lim 0
02 2 2 2
T
TT T T TP x t dt dt t
T T T T
2 21
0
13 9 9
0
T
TE x t dt dt t
x t x t
3
0 1t
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 16
Power and Energy Example 2:The signal is given below is energy or power signal. Explain.
This signal is energy signal
x n
2
1
32 2
0
1 1 27lim lim 3 lim 0
2 1 2 1 2 1
n
N N Nn n
P x nN N N
2
1
32 2
0
3 27n
n n
E x n
x n
3
0 1n
2
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 17
Real and Complex A value of a complex signal is a complex number
The complex conjugate, of the signal is;
Magnitude or absolute value
Phase or angle
x t
1 2
1 2Where 1 Re Im
x t x t jx t
j x t x t x t x t
x t x t
1 2
1 1Re Im
2 2
x t x t jx t
x t x t x t x t x t x tj
2 21 2
1
2x t x t x t x t x t
12 1tanx t x t x t
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 18
Periodic and Non-periodic A signal or is a periodic signal if
Here, and are fundamental period, which is the smallest positive values when
Example:
x t x n
0
0
; , any integer
; , any integer
x t x t mT t m
x n x n mN n m
oT oN
1m
0( ) cos( ); ; 2ox t A t t f
0 0 0 0
00
( ) cos( [ ] ) cos( )
2; 0,1,2......
x t T A t mT A t
T m
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 19
Analog and DigitalDigital signal is discrete-time signal whose values belong
to a defined set of real numbers
Binary signal is digital signal whose values are 1 or 0
Analog signal is a non-digital signal
1 2{ , ,..., }Na a a
[ ] ( ) 0 1;nx n x t or n
[ ] ( ) ; 1n ix n x t a i N
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 20
Even and Odd
Even Signals The continuous-time signal /discrete-time signal is
an even signal if it satisfies the condition
Even signals are symmetric about the vertical axis Odd Signals The signal is said to be an odd signal if it satisfies the
condition
Odd signals are anti-symmetric (asymmetric) about the time origin
( ) ( );x t x t t
( ) ( ); [ ] [ ];x t x t t x n x n n
( )x t x n
[ ] [ ];x n x n n
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 21
Even and Odd signals:Facts
Product of 2 even or 2 odd signals is an even signal Product of an even and an odd signal is an odd signal Any signal (continuous and discrete) can be expressed
as sum of an even and an odd signal:
( ) ( ) ( ); [ ] [ ] [ ]
( ) ( ) ( ) ( )( ) ; ( )
2 2[ ] [ ] [ ] [ ]
[ ] ; [ ]2 2
e o e o
e o
e o
x t x t x t x n x n x n
x t x t x t x tx t x t
x n x n x n x nx n x n
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 22
Complex-Valued Signal Symmetry For a complex-valued signal
is said to be conjugate symmetric if it satisfies the condition
where
is the real part and is the imaginary part;
is the square root of -1
1 2( ) ( ) ( )x t x t j x t
*( ) ( )x t x t
1 2*( ) ( ) ( )x t x t j x t
1( )x t 2 ( )x t
j
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 23
Deterministic and Random signal A signal is deterministic whose future values can be
predicted accurately. Example: A signal is random whose future values can NOT be
predicted with complete accuracy Random signals whose future values can be statistically
determined based on the past values are correlated signals.
Random signals whose future values can NOT be statistically determined from past values are uncorrelated signals and are more random than correlated signals.
( ) sinx t A t
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 24
Deterministic and Random signal(contd…)
Two ways to describe the randomness of the signal are:
Entropy: This is the natural meaning and mostly
used in system performance measurement.
Correlation: This is useful in signal processing by
directly using correlation functions.
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 25
Taylor series and Euler relation General Taylor series
Expanding sin and cos
Expanding exponential
Euler relation
2
0
...!
nn
n
f a x af x f a f a x a f a x a
n
3 5
sin ...3! 5!
x xx x
2 4
cos 1 ...2! 4!
x xx
2 3 4 5
1 ...2! 3! 4! 5!
x x x x xe x
2 3 4 5
1 ...2! 3! 4! 5!
jx jx jx jx jxe jx
2 3 3 5
1 ... ... cos sin2! 3! 3! 5!
jx x x x xe j x x j x
Signals and Systems Khosrow Ghadiri - EE Dept. SJSU 26
Example The signal and shown below constitute the real and
imaginary parts of a complex-valued signal . What form of symmetry does have?
Answer is conjugate symmetric.
1( )x t 2 ( )x t( )x t
( )x t
( )x t
2
T
2
T0
1( )x t