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Review of DSP. Signal and Systems:. Signal are represented mathematically as functions of one or more independent variables. Digital signal processing deals with the transformation of signal that are discrete in both amplitude and time. - PowerPoint PPT Presentation
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Review of DSP
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Signal and Systems: Signal are represented mathematically as
functions of one or more independent variables.
Digital signal processing deals with the transformation of signal that are discrete in both amplitude and time.
Discrete time signal are represented mathematically as sequence of numbers.
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Signals and Systems: A discrete time system is defined
mathematically as a transformation or operator. y[n] = T{ x[n] }
T{.}x [n] y [n]
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Linear Systems: The class of linear systems is defined by the
principle of superposition.
And
Where a is the arbitrary constant.
The first property is called the additivity property and the second is called the homogeneity or scaling property.
][][]}[{]}[{]}[][{ 212121 nynynxTnxTnxnxT
][]}[{]}[{ naynxaTnaxT
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Linear Systems: These two property can be combined into
the principle of superposition,
][1 nx
]}[{]}[{]}[][{ 2121 nxbTnxaTnbxnaxT
H
H
Linear SystemH
][][ 21 nbxnax ][2 nx
][][ 21 nbynay ][1 ny
][2 ny
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Time-Invariant Systems: A Time-Invariant system is a system for
which a time shift or delay of the input sequence cause a corresponding shift in the output sequence.
][1 nxH
H][ 01 nnx
][1 ny
][ 01 nny
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LTI Systems: A particular important class of systems consists
of those that are linear and time invariant. LTI systems can be completely characterized by
their impulse response.
From principle of superposition:
Property of TI:
k
knkxTny ][][][
k
knTkxny ][][][
k
knhkxny ][][][
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LTI Systems (Convolution):
Above equation commonly called convolution sum and represented by the notation
k
knhkxny ][][][
][][][ nhnxny
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Convolution properties: Commutativity:
Associativity:
Distributivity:
Time reversal:
][][][][ nxnhnhnx
][][][ nhnxny
])[][(][][])[][( 321321 nhnhnhnhnhnh
])[][(])[][(])[][(][ 2121 nxnhbnxnhanbxnaxnh
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…Convolution properties: If two systems are cascaded,
The overall impulse response of the combined system is the convolution of the individual IR:
The overall IR is independent of the order:
H1 H2
H2 H1
][][][ 21 nhnhnh
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Duration of IR: Infinite-duration impulse-response (IIR).
Finite-duration impulse-response (FIR)
In this case the IR can be read from the right-hand side of:
][...]1[][][ 10 qnxbnxbnxbny q
nbnh ][
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Transforms: Transforms are a powerful tool for
simplifying the analysis of signals and of linear systems.
Interesting transforms for us:Linearity applies:
Convolution is replaced by simpler operation:
][][][ ybTxaTbyaxT
][][][ yTxTyxT
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…Transforms: Most commonly transforms that used in
communications engineering are:Laplace transforms (Continuous in Time & Frequency)
Continuous Fourier transforms (Continuous in Time)
Discrete Fourier transforms (Discrete in Time)
Z transforms (Discrete in Time)
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The Z Transform: Definition Equations:
Direct Z transform
The Region Of Convergence (ROC) plays an essential role.
n
nznxzX ][)(
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The Z Transform (Elementary functions): Elementary functions and their Z-transforms:
Unit impulse:
Delayed unit impulse: ][][ knnx
0:][)(
zROCzzknzXn
kn
][][ nnx
0:1][)(
zROCznzXn
n
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The Z Transform (…Elementary functions):Unit Step:
Exponential: ][][ nuanx n
||||:1
1)(
01
azROCaz
zazXn
nn
otherwise 0,
0n ,1][nu
1||:1
1)(
01
zROCz
zzXn
n
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Important Z Transforms
Z Transform (Cont’d)
][n][ kn
][nu
][nuan
][nx ][zX Region Of Convergence(ROC)
1kz
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1 z
11
1 az
Whole Page
Whole Page
|z| > |a|
|z| > 1
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The Z Transform (Elementary properties): Elementary properties of the Z transforms:
Linearity:
Convolution: if
,Then
)()(][][ zbYzaXnbynax
][][][ nynxnw
)()()( zYzXzW
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The Z Transform (…Elementary properties):Shifting:
Differences: Forward differences of a function,
Backward differences of a function,
)(][ zXzknx k
][]1[][ nxnxnx
]1[][][ nxnxnx
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The Z Transform (…Region Of Convergence for Z transform):Since
the shifting theorem
][]1[][][ nnnxnx
)()1(][ zXznxZ
)()1(][ 1 zXznxZ
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The Z Transform (Region Of Convergence for Z transform): The ROC is a ring or disk in the z-plane
centered at the origin :i.e.,
The Fourier transform of x[n] converges at absolutely if and only if the ROC of the z-transform of x[n] includes the unit circle.
The ROC can not contain any poles.
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The Z Transform (…Region Of Convergence for Z transform): If x[n] is a finite-duration sequence, then
the ROC is the entire z-plane, except possibly or .
If x[n] is a right-sided sequence, the ROC extends outward from the outermost finite pole in to .
The ROC must be a connected region.
0z z
)(zX z
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The Z Transform (…Region Of Convergence for Z transform): A two-sided sequence is an infinite-duration
sequence that is neither right sided nor left sided.
If x[n] is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and not containing any poles.
If x[n] is a left-sided sequence, the ROC extends in ward from the innermost nonzero pole in to
.0z)(zX
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The Z Transform (Application to LTI systems): We have seen that
By the convolution property of the Z transform
Where H(z) is the transfer function of system.
Stability A system is stable if a bounded input
produced a bounded output, and a LTI system
is stable if:
][][][ nhnxny
)()()( zHzXzY
Mnx |][|
k
kh |][|
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Fourier Transform
Fourier Transform
Fourier Series
Discrete Time Continuous Frequency FT
Discrete Time Discrete Frequency FT
Time Frequency Transform Type
Continuous-aperiodic
Discrete-aperiodic
Continuous-aperiodic
Continuous-periodic
Continuous-periodic
Discrete-aperiodic
Discrete-periodic
Discrete-periodic
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Discrete-time Fourier Transform
n
njj enxeX ][)(
The same as Z-transform with z on the unit circle
Continuous in Frequency, periodic with period = 2*pi
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The Discrete Fourier Transform (DFT)
Discrete Fourier transform
It is customary to use theThen the direct form is:
1
0
2
][][N
n
N
knj
enxkX
N
j
N eW2
1
0
][][N
n
nkNWnxkX
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The Discrete Fourier Transform (DFT)With the same notation the inverse DFT is
1
0
][1
][N
k
nkNWkXN
nx
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The DFT (Elementary functions): Elementary functions and their DFT:
Unit impulse:
Shifted unit impulse: ][][ pnnx
kpNWkX ][
][][ nnx
1][ kX
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The DFT (…Elementary functions):Constant:
Complex exponential:njenx ][
2][
NkNkX
1][ nx
][][ kNkX
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The DFT (…Elementary functions):Cosine function:
nfnx 02cos][
][][2
][ 00 NfkNNfkN
kX
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The DFT (Elementary properties): Elementary properties of the DFT:
Symmetry: If
,Then
Linearity: if
and
,Then
][][ kFnf
][][ kXnx
][][][][ kbYkaXnbynax
][][ nNFkf
][][ kYny
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The DFT (…Elementary properties):Shifting: because of the cyclic nature of DFT
domains, shifting becomes a rotation.
if
,ThenTime reversal:
if
,Then
][][ kXnx
])[(])[( NN kXnx
][])[( kXWpnx kpNN
][][ kXnx
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The DFT (…Elementary properties):Cyclic convolution: convolution is a shift,
multiply and add operation. Since all shifts in the DFT are circular, convolution is defined with this circularity included.
1
0
])[(][][][N
pNpnypxnynx
