Ch 1,2 - Intro, Systems, FT
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Transcript of Ch 1,2 - Intro, Systems, FT
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8/3/2019 Ch 1,2 - Intro, Systems, FT
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Digital Signal Processing
Chapter 2: Discrete-time Signals and Systems
2
Digital Signal Processing
Discrete-time signal processing
Sampling, no digitization
Digital Signal Processing Sampling, digitization
Signal:
Digital Signal Processing Any function of one or more
variables which contains useful information
Signals One dimensional
Speech signals
Multi-dimensional
Pictures
3
Digital Signal Processing
Discrete-time signals
May be discrete from the beginning
Could have been the result of sampling a continuous time
signal
Discrete-time
ProcessingD-to-AA-to-D
x(t) y(t)y[n]x[n]
Cont.-time
Signal
4
Discrete-Time Sequences
Discrete-time sequence
Graphical representation of a discrete-time signal.
[ ]{ } ,nxx = ,
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Discrete-Time Sequences
Segment of a continuous-time speech signal.
Sequence of samples obtained with T=125 us
)(][ nTxnx d= ,
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Basic Sequences
Sinusoidal sequences
[ ],),cos(
0nallfornAnx +=
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Periodic Signals
)cos( 0n
11
Periodic Signals
)cos( 0n
12
Basic Sequences
Exponential sequences
[ ] .nAnx =
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Basic Sequences
Exponential sequence where is complex
If
)( 0 += njneA
).sin()cos( 00 +++= nAjnAnn
[ ] njnjn eeAAnx 0 ==
jeAA =0 je=
[ ] )sin()cos( 00)( 0
+++== + nAjnAeAnx nj
1=
[ ] .nAnx =
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Basic Sequences
Complex exponential sequence
[ ]
nj
Aenx
)2( 0 +=
.002 njnjnj AeeAe
==
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Discrete-time Systems
Any operation that maps an input sequence to an output
sequence
[ ] [ ]}{ nxTny =
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Discrete-time Systems
Example: Ideal delay System
Example: Moving average system
[ ] [ ]
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Discrete-time Systems
Moving average
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Discrete-time Systems
Memoryless system
Example:
for each value of n.
[ ] [ ]( ) ,2nxny =
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Linear Systems
If
Then
Or we can combine the two conditions as
[ ] [ ]{ } [ ]{ } [ ]{ } [ ] [ ]
[ ]{ } [ ]{ } [ ],
212221
naynxaTnaxT
nynynxTnxTnxnxT
==
+=+=+
[ ] [ ]
[ ] [ ]nynx
nynx
T
T
22
11
[ ] [ ]{ } [ ]{ } [ ]{ }nxbTnxaTnbxnaxT 2121 +=+
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Linear Systems
Accumulator
Linear?
[ ] [ ]=
=n
k
kxny
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Time Invariance
If
Then
Example:
Time invariant?
[ ] [ ]nynxT
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[ ] [ ]0101 nnynnxT
[ ] [ ]=
=n
k
kxny
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Time Invariance
Example:
Time invariant?
LTI Systems: Linear and time-invariant
[ ] [ ] ,,
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LTI Systems
LTI Systems are completely characterized by their
impulse response.
Given the input signal, the output can be determined.
[ ] [ ]
=
=k
knkxTny ][
[ ] [ ] { } [ ]
=
=
==kk
knhkxknTkxny ][][
[ ] ][*][ nhnxny =
Convolution
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Convolution
1. Flip one sequence (say h[k]) around origin h[-k]
2. Shift the flipped sequence h[n-k]
3. Multiply by the other sequence and add
[ ] [ ] [ ]
=
=
==kk
knxkhknhkxny ][][
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Output of an LTI System
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Output of an LTI System
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Convolution
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Convolution Example
[ ] )3,2,1(
=nh
[ ] )1,1,1(=nx
[ ] ?][*][ == nhnxny
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Convolution Example
In general:
Thus
[ ] [ ] [ ]== Nnununh
.,0
,10,1
otherwise
Nn
[ ] [ ].nuanx n=
.10 Nnfor
[ ] [ ]
=
=
=
=
==
n
k
k
k
k
k
a
knhaknhkxny
0
0
][][
12
1
.1
2
1
21
NNan
Nk
NNk
=
=
+
[ ] .10,1
1 1
=
+
Nna
any
n
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Example
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Example (Cont.)
When n>N-1
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Properties of LTI Systems
These properties can be proved easily using the definition
of convolution operation.
Commutative property
Distributive property
[ ] [ ] [ ] [ ]nxnhnhnx =
[ ] [ ] [ ] [ ] [ ] [ ] [ ]nxnhmnxmhmhmnxnymm
===
=
=
[ ] [ ] [ ] [ ] [ ] [ ] [ ]nhnxnhnxnhnhnx 2121 )( +=+
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Properties of LTI Systems
Serial combination of DT systems
h1[n] h2[n]
h2[n] h1[n]
h1[n]* h2[n]
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Properties of LTI Systems
Parallel combination of DT systems
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Properties of LTI Systems
LTI systems are BIBO stable if and only if the impulse
response is absolutely summable
[ ]
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Example
Causal and stable? FIR or IIR?
Ideal delay system:
Moving average system:
[ ] [ ]dnnxny =
[ ] [ ]dnnnh = nda positive fixed integer
[ ] [ ]= ++=2
11
121
M
Mk
knxMM
ny
[ ] [ ] =++
= =
2
11
1
21
M
MK
knMM
nh
++.,0
,,1
121
21
otherwise
MnMMM
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Example
Causal and stable? FIR or IIR?
Accumulator:
Forward Difference
Backward Difference
[ ] [ ]
=
=n
k
kxny
[ ] [ ]
=
=k
knh
0; a =0.9 (solid curve) and a=0.5 (dashed curve).
Magnitude
Phase
64
Symmetry Property of Fourier Transform
Frequency response for a system with impulse response h[n] = an u[ n].
a > 0; a =0.9 (solid curve) and a=0.5 (dashed curve).
Real part
Imaginary part
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Existence of Fourier Transform
Does the infinite sum converge to a finite value?
If the sequence is absolute summable, its Fourier
transform exists. (Sufficient Condition)
( )