signal properties
TRANSCRIPT
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Discrete-Time Signals and Systems
Quote of the Day
Mathematics is the tool specially suited for
dealing with abstract concepts of any kind andthere is no limit to its power in this field.
Paul Dirac
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice HallInc.
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Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 2
Discrete-Time Signals: Sequences
Discrete-time signals are represented by sequence of numbers
The nth number in the sequence is represented with x[n]
Often times sequences are obtained by sampling ofcontinuous-time signals
In this case x[n] is value of the analog signal at xc(nT)
Where T is the sampling period
0 20 40 60 80 100-10
0
10
t (ms)
0 10 20 30 40 50-10
0
10
n (samples)
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Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 3
Basic Sequences and Operations
Delaying (Shifting) a sequence
Unit sample (impulse) sequence
Unit step sequence
Exponential sequences
]nn[x]n[y o!
!
{!H
0n1
0n0]n[
u
!
0n1
0n0]n[u
nA]n[x E!
-10 -5 0 5 100
0.5
1
1.5
-10 -5 0 5 10
0
0.5
1
1.5
-10 -5 0 5 100
0.5
1
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Sinusoidal Sequences
Important class of sequences
An exponential sequence with complex
x[n] is a sum of weighted sinusoids
Different from continuous-time, discrete-time sinusoids
Have ambiguity of 2Tk in frequency
Are not necessary periodic with 2T/[o
? A J[! ncosnx oJ[ !E!E jj eAAande o
? A
? A J[EJ[E!
E!E!E! J[[J
nsinAjncosAnx
eAeeAAnx
o
n
o
n
njnnjnjn oo
J[!JT[ ncosnk2cos oo
integeranisk2
NifonlyNncosncoso
ooo[
T!J[[!J[
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Demo
Rotating Phasors Demo
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Discrete-Time Systems
Discrete-Time Sequence is a mathematical operation thatmaps a given input sequence x[n] into an output sequence
y[n]
Example Discrete-Time Systems Moving (Running) Average
Maximum
Ideal Delay System
]}n[x{T]n[y ! T{.}x[n] y[n]
]3n[x]2n[x]1n[x]n[x]n[y !
_ a]2n[x],1n[x],n[xmax]n[y !
]nn[x]n[y o!
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Memoryless System
Memoryless System
A system is memoryless if the output y[n] at every value of n
depends only on the input x[n] at the same value of n
Example Memoryless Systems
Square
Sign
Counter Example
Ideal Delay System
2]n[x]n[y !
_ a]n[xsign]n[y !
]nn[x]n[y o!
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Linear Systems
Linear System: A system is linear if and only if
Examples Ideal Delay System
_ a _ a
_ a _ a (scaling)]n[xa]n[axand
y)(additi it]n[x]n[x]}n[x]n[x{ 2121
!
!
]nn[x]n[y o!
_ a_ a
_ a ]nn[ax]n[xa]nn[ax]n[ax
]nn[x]nn[x]n[x]}n[x{
]nn[x]nn[x]}n[x]n[x{
o1
o1
o2o112
o2o121
!
!
!
!
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Time-Invariant Systems
Time-Invariant (shift-invariant) Systems
A time shift at the input causes corresponding time-shift at output
Example
Square
Counter Example
Compressor System
_ a]nn[xT]nn[y]n[xT]n[y oo !!
2]n[x]n[y !? A
? A 2oo
2
o1
]nn[xn-nygivesoutputtheDelay
]nn[xnyisoutputtheinputtheDelay
!
!
]Mn[x]n[y !? A
? A ? Aooo1
nnMxn-nygivesoutputtheDelay
]nMn[xnyisoutputtheinputtheDelay
!
!
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Causal System
Causality
A system is causal its output is a function of only the current and
previous samples
Examples
Backward Difference
Counter Example
Forward Difference
]n[x]1n[x]n[y !
]1n[x]n[x]n[y !
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Stable System
Stability (in the sense of bounded-input bounded-output BIBO)
A system is stable if and only if every bounded input produces a
bounded output
Example
Square
Counter Example
Log
gege yx B]n[yB]n[x
2]n[x]n[y !
ge
ge
2x
x
B]n[ybyboundedisoutput
B]n[xbyboundedisinputif
]n[xlog]n[y 10!
? A ? A ? A g!!!ge
nxlog0y0nxforboundednotoutput
B]n[xbyboundedisinputifeven
10
x