Syllabus Overview
UCLA ECE102 - Kambiz Shoarinejad
why signals and systems
Speech 1 audio
processing
Speech Recognition 1 Synthesis
Digital audio
Equalization
Speech Encoding compression etc
Image 1 Video processing CZ D signals
Enhancements
Image
Pattern Recognition
Image 1 video Encoding compression
Digital TV
Military Space signalprocessing
Radar
processing
Estimation Navigation Guidance
Sonar processing
Biomedical signal
processing
Scanners
ECG Analysis
EGG Brain mappers
X ray analysis
Consumer Electronics
Circuit design and analysis
Wireless Communication
Digital cameras
Control systems
Electromechanical systems Robotics
Automotive
process control
Aviation
What is a signal anyway
A function of one or more independent
variables
Temporal Variable Time
e.g speech stock price a
Spatial variable coordinates
e.g Image
Both e.g video
In 102mostly deal w functions of
we
a
single variable generally referred to as
Time t C continuous time or n discrete
time index
Continuous time Discrete time signals
us
Nct depends on real valued continuous
time variable t
NE n depends on integer valued discrete
time index n
Alt anted
step signal
f jo n
Analog vs Digital signals
Analogs continuous timer signal
act which may take values
any
Digital Discrete time signal
which is a k
3ed and
N u
can take values from a limited
only
Set of discrete numbers
so what is a
system
A process which inputs stimuli
by or
or excitations are transformed into
outputs or
responses
Discrete time
Mn 1 f YIM
system
YIN T Trans
t ue
Nlt
System yet
T sects
yet
Example Simple series RC
Tmr yet
seat
1 T
1
put Nlt Rice sect Rcdyct
It
Red yet Sect
y
t Y
Let a
plz diff a
y a rat
Ordinary Differential Equation
ODE
This is an example of a continuous time Dynamic
system modeled by an ODE
Example Accumulator
an Hn
f
Epigeal
YENI FE ft n so yen yen D seen
Difference Equation
This is
example of a discrete time
an
Dynamic system modeled by a DifferenceEg
echaracteristems
o static Cmemoryless Dynamic vs
statics system output at anytime only
depends on the input at that
same time
Ideal resistor
yet Vct R Ict Rat
YEN a Ksn b KKn
has i e
Dynamic System memory
its output at time it depends on its
input not just at't but at other times
too
EI Rc circuit
E Discrete time Accumulator
EI YInJ x n7 3 acn
Dg Finite
YENI IIon n
og memory
YCnJ Hn KI Infinite
memory
Linear VS Non Linear Systems
A system TC is linear if and
only if it satisfies both
scaling and superposition conditions
Scaling
T fret a yet T
yet aka
superposition
y Ctl TFR Ct Zz Ct T Exacts
y Lt
YzCt T Kilt 12247
so in general
T1 fairies Edit kilts
in discrete time
Similarly
A Nin a
T
t x say
aan FI Ha
Corollary Zero input will always yield to
Zero output
n
EE ya sett d YE
Tfa Nitti Az kilt a R Ct Azeez LED
online
ATat at Africa 2aiazacet Ralt
f A T FK Cti t.az fKzCt7J apefCts azaCt
EI Accumulator
Tfn k YIN IT Mn
This is an example of a discrete time system
described
by a Linear Constant Coefficient
Difference Equation LLCCDE
Now is this system always linear let's
investigate
Let's assumesystem has the initial condition
the
ye D and any new input is applied at
time n 0 n
din ENT yin 7 YE D Foa Rick
n
again YIN Dt
Yf
EzakiKI
aiken tavern
Yin YC Mt area eaten
E yen YIN
will satisfy linearity only if ya so
i e only if the system is at rest or relaxed
But are most systems linear or nonlinear
why care about linear systems anyway
0 Time Invariant Vs Time
varying systems
recti 4 T facts
1 yet
act e
yet 2 T Ncte
Ft e fact
i e
Any time shift in the
input will yield
the same time shift at the output
Similarly for discrete time systems
T Mon Hn K T Cafu KI
yay
Time In variant is sometimes called
Shift In variant for discrete time systems
sif
EI.ME
yct Sincxfti
let Rz Ct Kitt to y Lt Ey Lt to
Sin kits Sin CR Ct to
Yett L Lt to
Time Invariant
EI yl n n Rf n
n Rf n KI Yfn k n K Hn k
Time
varying
0 Stable us Unstable systems
A
system is Bounded Input Bounded
stable if and if
Output BIBO only
bounded input bounded
any yields a
output ice
if F Ma sat sect I Made for all t
Then IM a sit I yet I I E My L x for
y afl
EI Accumulator
I Dt Nnl
Y n
y n
Esack Yf Dt I omen
Let run Nn n
Step signal or
sequence
and
YE 17
assume 0
I I I
yf I
n
n Is
clearly as n x
y Cn a
i e a bounded input leads to an unbounded
output Unstable
The continuous time analogue of accumulator
is an integrator dyCt Rct YG o
which is likewise Unstable
0 Causal us Non Causal systems
A
system is called Causal if its output
depends on its input at present and past
times and future times
only on
Jlt T fact
yet
is a function of sect for t Eto Fto
physical systems are causal
All
Any real time processing system will
always be causal
We can implement non causal processing
systems if our
input data is pre recorded
E RC circuit like other circuit
any
is causal
E A non causal Moving Average Filter
Nn D In Mn til
YE n Iz
0 Single Input Single Output CSIS O vs
Mutti Input Multi Output MIMO systems
Y Ct
n'I
F fE
a
Knut inHult
Unit Impulse and Unit Step Functions
0 Discrete time
FO
UnitImpulse Sen I
I n
ascent
0
H n
unitstp sun
runs
I 3
Notice 8 n u Cnj Ucu e
RENT 8Th Mo S n
NCKT SC n KJ seen SC n k
Kz Kz
Eat k3S n K e ERT n SEN K
k K Kak
IIISfm kg Mn if k En Eka
Rcn a
0 otherwise
andingeneraty
NE n
k
E NE KI S Cn 14
Siftingproperty of Unit Impulse
Continuous time
anti
unit
1
uu
i.ir 3
Discontinuous at time 7 0
Unit Impulse Dirac Delta
Let's first define the pulse function
It DssCt
Saltidt I FA I
consider o b
sect 8g Ct dt Itn
L sect Seco as D o
Dirac Delta cont time unit impulse is
then defined as i
see hi
A o
Sae If
Ltd t 1 and Kit'S Lt Idt Nco
Seti is not function in strict sense rather a
a
generalized function or distribution and a linear
functional that maps every function to its value
at zero And as we shall see an extremely useful
tool for us
Unit and Unit Impulse properties
step
sifting property Mct
L ace Sct e de
8 Ct SC ti
f Seo do Ult
tutti
area
duct Sct
Tt
RaIpfmotion t
ITU co do tact Unit
UL t I Uct
System Impulse Response
Sit T Sct G http
htt e
Response of the system TC at
time t to a unit impulse
inputattimer
Now if the system is time invariant then i
het e he t e o E h Ct e
If the system is causal
h Ct e o for T L Z
And if it is both causal and time invariant then
htt 7 0 for tco
Response of a Linear system to
arbitrary inputs
Mn Tle
4 f yen
Thanks to the sifting property of unit impulse
we can write
A
Un E NE KI S n K
ke a
Taking advantage of the superposition principle
for Linear systems we have
yEnJ T men T
EINENSEN El
L NE KI T ES En KI
D 9
A
E KE KI h En k superposition Sum
T
YE n
k a
where hcn KI E T SC n KI
Impulse Response at time n to impulse
input at time k
And if the system is Linear Time Invariant
LTI then
yen I I awh En Es
k 2
This is called linear Convolutison of
signals x and h and is denoted a h s
hen
Y EnJ men I
be h n nEk7h In 14
seen DT LTI YEN Nen ten
4 h
This is a very significant result It shows
that the input output relationship for
any
Linear Time Invariant CTI system can
be described
by the linear convolution
of the input with the system impulse
response
Equivalently it shows that any LTI system
be
can
uniquely characterized by its
impulse response
