Review of DSP 1 Signal and Systems: Signal are represented - - PowerPoint PPT Presentation
Review of DSP 1 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
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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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A discrete time system is defined
mathematically as a transformation or
y[n] = T{ x[n] }
T{.}
x [n] y [n]
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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.
] [ ] [ ]} [ { ]} [ { ]} [ ] [ {
2 1 2 1 2 1
n y n y n x T n x T n x n x T
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These two property can be combined into
the principle of superposition,
] [
1 n
x
]} [ { ]} [ { ]} [ ] [ {
2 1 2 1
n x bT n x aT n bx n ax T
H H Linear System H
] [ ] [
2 1
n bx n ax
] [
2 n
x
] [ ] [
2 1
n by n ay
] [
1 n
y
] [
2 n
y
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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 n
x
H H
] [
1
n n x
] [
1 n
y
] [
1
n n y
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A particular important class of systems consists
LTI systems can be completely characterized by
their impulse response.
From principle of superposition: Property of TI:
k
k n k x T n y ] [ ] [ ] [
k
k n T k x n y ] [ ] [ ] [
k
k n h k x n y ] [ ] [ ] [
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Above equation commonly called convolution
sum and represented by the notation
k
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Commutativity: Associativity: Distributivity: Time reversal:
] [ ] [ ] [ n h n x n y
3 2 1 3 2 1
]) [ ] [ ( ]) [ ] [ ( ]) [ ] [ ( ] [
2 1 2 1
n x n h b n x n h a n bx n ax n h
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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
2 1
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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 [ ] [ ] [
1
q n x b n x b n x b n y
q
n
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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:
] [ ] [ ] [ y bT x aT by ax T
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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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Definition Equations:
Direct Z transform The Region Of Convergence (ROC) plays an
essential role.
n n
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Elementary functions and their Z-transforms: Unit impulse: Delayed unit impulse:
] [ ] [ k n n x
n k n
] [ ] [ n n x
n n
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Unit Step: Exponential:
] [ ] [ n u a n x
n
1
n n n
0, n , 1 ] [n u
1
n n
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Important Z Transforms
Region Of Convergence (ROC) Whole Page z≠0
|z| > |a| |z| > 1
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Elementary properties of the Z transforms:
Linearity: Convolution: if
,Then
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Shifting: Differences:
Forward differences of a function, Backward differences of a function,
k
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Since
the shifting theorem
] [ ] 1 [ ] [ ] [ n n n x n x
1
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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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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.
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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 .
) (z X
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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:
M n x | ] [ |
k
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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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n n j j
The same as Z-transform with z on the unit circle Continuous in Frequency, periodic with period = 2*pi
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Discrete Fourier transform
It is customary to use the Then the direct form is:
1 2
N n N kn j
N j N
e W
2
1
N n nk N
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With the same notation the inverse DFT is
1
N k nk N
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Elementary functions and their DFT: Unit impulse: Shifted unit impulse:
] [ ] [ p n n x
kp N
] [ ] [ n n x
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Constant: Complex exponential:
n j
e n x
] [
1 ] [ n x
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Cosine function:
n f n x 2 cos ] [
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Elementary properties of the DFT:
Symmetry: If
,Then
Linearity: if
and ,Then
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Shifting: because of the cyclic nature of DFT
domains, shifting becomes a rotation. if ,Then
Time reversal:
if ,Then
N N
kp N N
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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
N p N