chapter 2 n discrete-time sig signals and...
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Discrete-Time Signals and Systems
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Chapter 2
Discrete-Time Signals and Systems
Discrete-Time Signals and Systems
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Outline
2 .0 Introduction2 .1 Discrete-Time Signals : Sequences2 .2 Discrete-Time Systems2 .3 Linear Time-Invariant Systems2 .4 Properties of Linear Time-Invariant Systems2 .5 Linear Constant-Coefficient Difference Equations
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2 .6 Frequency-Domain Representation of Discrete-Time Signals and Systems2 .7 Representation of Sequences by Fourier Transforms2 .8 Symmetry Properties of the Fourier Transforms2 .9 Fourier Transforms Theorems
Outline (cont.)
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In this chapter we consider the fundamental concepts of discrete-time signals and signal processing systems for one-dimensional signals(one independent variable).
2 .0 Introduction
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A signal can be defined as a function that conveys information, generally about the state or behavior of a physical system. Signals are represented mathematically as functions of one or more independent variables.
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1 Continuous - time signals(CTS)2 Discrete - time signals(DTS)
Concept of the signals :
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x = {x[n]} -∞<n<∞ (2 .1 )
Where n is integer , T is sampling period.
x[n] = {xa[nT]} -∞<n<∞ (2 .2 )
For analog signal :
A sequence of number x :
2 .1 Discrete-Time Signals : Sequences
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Graphical representation of a discrete-time signal (from [1 ] p.1 0)
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(a) Segment of a continuous-time speech signal. (b) Sequence of samples obtained from (a) with T=1 2 5 μs. (from [1 ] p.1 0)
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Basic Sequences and Sequence Operation
1 Unit sample sequence
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Shift or delay sequence :
y[n] = x[n-n0], n0 is an integer
Periodic sequence :
x[n] = x[n+N], for all n, N is period
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p[n] = a-3δ[n+ ]+3 a1δ[n- ]+1 a2δ[n- ]2 +a7δ[n- ]7
x [n]= ∑k=−∞
∞
x [k ][n−k ]
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Unit step sequence2
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Exponential sequences3
x[n] =An
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Sinusoidal sequences4
x[n] =Acos(0n+) for all n
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Complexe exponential sequence5
x[n] =An
=∣∣e j0 A=∣A∣e j
When
x [n]=∣A∣∣∣n e j 0n
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We consider (ω0+2π) :
x [n]=Ae j 02n=Ae j0n
x [n]=Acos [02 r n]
=Acos 0n
A periodic sequence :x [n]=x [nN ]
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x t =cos0t
A continuous-time sinusoidal signal :
As Ω0 increases, x(t) oscillates more and more rapidly.
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x [n]=Acos 0n
The discrete-time sinusoidal signal :
0≤0≤ x[n] oscillates more and more rapidly.
≤0≤2 the oscillation become slower.
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. Discrete-time Systems2 2
A discrete-time system is defined mathematically as a transformation or operator that maps an input sequence with values x[n] into an output sequence with values y[n].
y[n] = T{x[n]}
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Representation of a discrete system(from [ ] p. )1 17
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Memoryless System
A system is referred to as memoryless if the output y[n] at every value of ndepends only on the input x[n] at the same value of n.
y[n]=(x[n])2Example
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Linear System
The class of linear systems is define by the principle of superposition. If y1[n] and y2[n] are the response of a system when x1[n] and x2[n] are the respective input, then the system is linear if and only if
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T{ax1[n]+bx2[n]}=aT{x1[n]}+bT{x2[n]} =ay1[n]+by2[n]
Linear systems have two properties 1 The additivity property2 The homogeneity or scaling property
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Time-Invariant Systems
A time-invariant system is one for which a time shift or delay of the input sequence causes a corresponding shift in the output sequence. Specifically, A system transforms the input sequence with value x[n] into the output sequence y[n].
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A system is said to be time-invariant if for all n0 the input sequence x1[n]=x[n-n0] produce the output for sequence y1[n]=y[n-n0].
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Example
The system defined by the relation :
y [n]=x [Mn]
y1[n]=x1[Mn]=x [Mn−n0]
y [n−n0]=x [M n−n0]≠ y1[n]
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Causality
A system is causal if for every choice of n0 the output sequence value at index n=n0 depends only on the input sequence values n≤n0. This implies that if x1[n]=x2[n] for n≤n0, then y1[n]=y2[n] for n≤n0That is, the system is nonanticipative.
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Stability
A system is stable in the bouned-input bounded-output (BIBO) sense if and only if every bounded input sequence produces a bounded output sequence. For every bouned input Bx there exist a fixed positive finite value output By.
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. Linear Time-Invariant Systems2 3
Linear Time-Invariant Systems is a particularly important class of systems consists of linear and time-invariant.
A linear system can be characterized by its impulse response. Let hk[n] be the response of the system to [n-k].
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Representation of sequences x[n] and h[n]
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Example
Suppose h[k] is the sequence :
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First consider h[-k] plotted against k; h[-k] is simply h[k] reflected or flipped about k= . 0
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Replacing k by k-n where n is a fix integer, leads to a shift of the origin of the sequence h[-k] to k=n, for n=4.
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. Properties of Linear Time-2 4Invariant Systems
Since all linear time-invariant systems are described by the convolution sum, the properties of these class of systems are defined by the properties of discrete time convolution. Therefore the impulse response is a complete characterization of the properties of a specific linear time-invariant system.
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Commutative
x [n]∗h [n]=h [n]∗x [n]
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Distributes over addition
x [n]∗h1[n]h2[n]=x [n]∗h1[n]x [n]∗h2[n]
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Stability and Causality
if x1[n]=x2[n] for n≤n0, then y1[n]=y2[n] for n≤n0
Stability
Causality
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Ideal Delay
h[n]=[n−nd ]
Moving Average
h[n]= 1M 1M 21
∑k=−M 1
M 2
[n−k ]
={ 1M 1M 21
, −M 1≤n≤M 2
,0 otherwise
Example of Impulse Response
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h[n]= ∑k=−∞
n
[k ]
Accumulator
={ ,1 n≥0,0 n0=u [n]
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h[n]=[n1]−[n]
h[n]=[n]−[n−1]
Forward Difference
Backward Difference
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Inverse System
If a linear time-invariant system has impulse response h[n], then its inverse system, if it exists, has impulse responsehi[n] defined by the relation .
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. Linear Constant-Coefficient 2 5Difference Equations
An important subclass of linear time-invariant systems consists of those systems for which the input x[n] and the output y[n] satisfy an Nth-order linear constant coefficient difference equation of the form.
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Block diagram of a recursive difference equation representing an accumulator
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. Frequency-Domain Representation 2 6of Discrete-Time Signals and Systems
Discrete-time signals may be represented in the terms of sinusoids or complex exponentials.
Complex exponential sequences are eigenfunctions of linear time-invariant systems.
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To demonstrate the eigenfunction property of complex exponentials for discrete-time systems, consider an input sequence x[n] = ejn for (-∞< n<∞) , i.e., a complex exponential of radian frequency, the corresponding output of a linear time-invariant system with impulse response h[n] is
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y [n]= ∑k=−∞
∞
h[k ]e jn−k
=e jn∑k=−∞
∞
h[k ]e− j kH e j= ∑
k=−∞
∞
h[k ]e− j k
y [n]=H e je jn
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Consequently,ejn is an eigenfunction of the system, and the associated eigenvalue is H(ejn). H(ejn) describes the change in complex amplitude of a complex exponential as a function of the frequency . The eigenvalue H(ejn) is called the frequency responseof the system.
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In general, is complex and can be expressed in terms of its real and imaginary parts as
or in terms of magnitude and phase as
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Ideal lowpass filter (p. [ ])43 1
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. Representation of Sequences by 2 7Fourier Transforms
Many sequence can be represented by a Fourier integral of the form
where
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In general, the Fourier transform is a complex-value function of . We sometimes express X(ej) in rectangular form as
or in polar form as
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The quantities |X(ej)| and ∢X(ej)are called the magnitude and phase (or magnitude spectrum and phase spectrum), respectively, of the Fourier transform.
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Sinc function
sinc =sin
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Example
X j={ ,1 ∣∣W,0 ∣∣W
x t = 12∫−w
w
e j t d =sin Wt t
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Example Fine impulse response of a DT ideal low-pass filter.
x t = 12∫−
H e je jn d
=sin c n
n
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. Symmetry Properties of the 2 8Fourier Transforms
Symmetry properties of the Fourier transform are often very useful. Before presenting the properties, we begin with some definitions.
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Conjugate-symmetric sequence 1
A conjugate-symmetric sequence xe[n] is defined as a sequence for which
xe[n] = xe*[-n]
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A conjugate-antisymmetric sequence xo[n] is defined as a sequence for which
xo[n] = -xo*[-n]
Conjugate-antisymmetric sequence 2
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Any sequences x[n] can be expressed as a sum of conjugate-symmetric and conjugate-antisymmetric sequence.
where
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A conjugate-symmetric real sequence is also called an even sequence, and a conjugate-antisymmetric real sequence is called an odd sequence.
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A Fourier transform can be decomposed into a sum of conjugate-symmetric and conjugate-antisymmetric functions as
where
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Clearly, Xe(ej) is conjugate-symmetric function, and Xo(ej) is conjugate-antisymmetric function.
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Symmetry Properties of the Fourier Transform
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. Fourier Transforms Theorems2 9
Linearity of the Fourier Transform1
Time Shifting and Frequency shifting 2
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Time Reversal 3
If x[n] is real, then
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Differentiation in Frequency 4
Parseval’s Theorem5
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. Convolution Theorem6
y [n]= ∑k=−∞
∞
x [k ]h[n−k ]=x [n]∗h[n]
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The Modulation or Windowing Theorem7