chapter 2 n discrete-time sig signals and...

Discrete-Time Signals and Systems

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Chapter 2



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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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Page 1 0

(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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Page 1 1

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


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Discrete-Time Random Signals

chapter 2 n discrete-time sig signals and...

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