Signal and System, CT Signal DT Signal, Signal Processing(amplitude and time scaling)

Signal and System(definitions)
Continuous-Time Signal
Discrete-Time Signal
Signal Processing
Basic Elements of Signal Processing
Classification of Signals
Basic Signal Operations(amplitude and time scaling)

1. TOPICS
•Signal and System(definitions)
•Continuous-Time Signal
•Discrete-Time Signal
•Signal Processing
•Basic Elements of Signal Processing
•Classification of Signals
•Basic Signal Operations(amplitude and time scaling)
1

2. 2
•Signal:
Asignalisdefinedasafunctionofoneormorevariables
whichconveysinformationonthenatureofaphysical
phenomenon.Thevalueofthefunctioncanbeareal
valuedscalarquantity,acomplexvaluedquantity,or
perhapsavector.
•System:
Asystemisdefinedasanentitythatmanipulatesoneor
moresignalstoaccomplishafunction,therebyyielding
newsignals.

3. 3
•Continuos-TimeSignal:
Asignalx(t)issaidtobeacontinuoustimesignalifitis
definedforalltimet.
•Discrete-TimeSignal:
Adiscretetimesignalx[nT]hasvaluesspecifiedonlyat
discretepointsintime.
•SignalProcessing:
Asystemcharacterizedbythetypeofoperationthatit
performsonthesignal.Forexample,iftheoperationis
linear,thesystemiscalledlinear.Iftheoperationisnon-
linear,thesystemissaidtobenon-linear,andsoforth.
Suchoperationsareusuallyreferredtoas“Signal
Processing”.

4. 4
Basic Elements of a Signal Processing
System
Analog
Signal Processor
Analog input
signal
Analog output
signal
Analog Signal Processing
Digital
Signal Processor
A/D
converter
D/A
converter
Digital Signal Processing
Analog
input
signal
Analog
output
signal

5. 5
Classification of Signals
•DeterministicSignals
Adeterministicsignalbehavesinafixedknownwaywith
respecttotime.Thus,itcanbemodeledbyaknown
functionoftimetforcontinuoustimesignals,oraknown
functionofasamplernumbern,andsamplingspacingT
fordiscretetimesignals.
•RandomorStochasticSignals:
Inmanypracticalsituations,therearesignalsthateither
cannotbedescribedtoanyreasonabledegreeofaccuracy
byexplicitmathematicalformulas,orsuchadescriptionis
toocomplicatedtobeofanypracticaluse.Thelackof
sucharelationshipimpliesthatsuchsignalsevolveintime
inanunpredictablemanner.Werefertothesesignalsas
random.

6. 6
Even and Odd Signals
Acontinuoustimesignalx(t)issaidtoanevensignalifit
satisfiesthecondition
x(-t)=x(t)forallt
Thesignalx(t)issaidtobeanoddsignalifitsatisfiesthe
condition
x(-t)=-x(t)
Inotherwords,evensignalsaresymmetricaboutthe
verticalaxisortimeorigin,whereasoddsignalsare
antisymmetricaboutthetimeorigin.Similarremarks
applytodiscrete-timesignals.
Example:
even
odd odd

7. 7
Periodic Signals
Acontinuoussignalx(t)isperiodicifandonlyifthere
existsaT>0suchthat
x(t+T)=x(t)
whereTistheperiodofthesignalinunitsoftime.
f=1/TisthefrequencyofthesignalinHz.W=2/Tisthe
angularfrequencyinradianspersecond.
Thediscretetimesignalx[nT]isperiodicifandonlyif
thereexistsanN>0suchthat
x[nT+N]=x[nT]
whereNistheperiodofthesignalinnumberofsample
spacings.
Example:
0 0.2 0.4
Frequency=5Hzor10rad/s

8. 8
Continuous Time Sinusoidal Signals
Asimpleharmonicoscillationismathematically
describedas
x(t)=Acos(wt+)
Thissignaliscompletelycharacterizedbythree
parameters:
A=amplitude,w=2f=frequencyinrad/s,and=
phaseinradians.
A T=1/f

9. 9
Discrete Time Sinusoidal Signals
Adiscretetimesinusoidalsignalmaybeexpressedas
x[n]=Acos(wn+) -<n<
Properties:
•Adiscretetimesinusoidisperiodiconlyifitsfrequencyisarational
number.
•Discretetimesinusoidswhosefrequenciesareseparatedby
anintegermultipleof2areidentical.
•Thehighestrateofoscillationinadiscretetimesinusoidis
attainedwhenw=(orw=-),orequivalentlyf=1/2(orf=-
1/2).
0 2 4 6 8 10
-1
0
1

10. 10
Energy and Power Signals
•Asignalisreferredtoasanenergysignal,ifandonlyif
thetotalenergyofthesignalsatisfiesthecondition
0<E<
•Ontheotherhand,itisreferredtoasapowersignal,if
andonlyiftheaveragepowerofthesignalsatisfiesthe
condition
0<P<
•An energy signal has zero average power, whereas a power
signal has infinite energy.
•Periodic signals and random signals are usually viewed as
power signals, whereas signals that are both deterministic and
non-periodic are energy signals.

11. 11
Basic Operations on Signals
(a)Operationsperformedondependent
variables
1.AmplitudeScaling:
letx(t)denoteacontinuoustimesignal.Thesignaly(t)
resultingfromamplitudescalingappliedtox(t)is
definedby
y(t)=cx(t)
wherecisthescalefactor.
Inasimilarmannertotheaboveequation,fordiscrete
timesignalswewrite
y[nT]=cx[nT]
x(t)
2x(t)

12. 12
2.Addition:
Let x
1[n] and x
2[n] denote a pair of discrete time signals.
The signal y[n] obtained by the addition of x
1[n] + x
2[n]
is defined as
y[n] = x
1[n] + x
2[n]
Example: audio mixer
3.Multiplication:
Letx
1[n]andx
2[n]denoteapairofdiscrete-timesignals.
Thesignaly[n]resultingfromthemultiplicationofthe
x
1[n]andx
2[n]isdefinedby
y[n]=x
1[n].x
2[n]
Example:AMRadioSignal

13. 13
(b)Operationsperformedonindependent
variable
•TimeScaling:
Lety(t)isacompressedversionofx(t).Thesignaly(t)
obtainedbyscalingtheindependentvariable,timet,by
afactorkisdefinedby
y(t)=x(kt)
–ifk>1,thesignaly(t)isacompressedversionof
x(t).
–If,ontheotherhand,0<k<1,thesignaly(t)isan
expanded(stretched)versionofx(t).

14. 14
Exampleoftimescaling
0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
exp(-2t)
exp(-t)
exp(-0.5t)
Expansion and compression of the signal
e
-t
.

15. 15
-3 -2 -1 0 1 2 3
0
5
10
x[n]
-1.5-1 -0.5 0 0.5 1 1.5
0
5
10
x[0.5n]
-6 -4 -2 0 2 4 6
0
5
x[2n]
n
Time scaling of discrete time systems

16. 16
TimeReversal
•This operation reflects the signal about t = 0
and thus reverses the signal on the time scale.
0 1 2 3 4 5
0
5
x[n]
n
0 1 2 3 4 5
-5
0
x[
-
n]
n

17. 17
TimeShift
Asignalmaybeshiftedintimebyreplacingthe
independentvariablenbyn-k,wherekisan
integer.Ifkisapositiveinteger,thetimeshift
resultsinadelayofthesignalbykunitsoftime.If
kisanegativeinteger,thetimeshiftresultsinan
advanceofthesignalby|k|unitsintime.
x[n
]
x[n+3]
x[n
-
3]
n