barcode lib ssrs 1J.6 Signal Manipulations in Software

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1J.6 Signal Manipulations
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In our study of discrete-time signals and systems we will be concerned with the manipulation of signals. These manipulations are generally compositions of a few basic signal transformations. These transformations may be classified either as those that are transformations of the independent variable n or those that are transformations of the amplitude of x ( n ) (i.e., the dependent variable). In the following two subsections we will look briefly at these two classes of transformations and list those that are most commonly found in applications.
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sequence that is conjugate symmetric is sometimes said to be hermitian.
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CHAP. 11
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SIGNALS AND SYSTEMS
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Transformations of the Independent Variable
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Sequences are often altered and manipulated by modifying the index n as follows:
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where f (n) is some function of n. If, for some value of n, f (n) is not an integer, y(n) = x( f (n)) is undefined. Determining the effect of modifying the index n may always be accomplished using a simple tabular approach of listing, for each value of n, the value of f (n) and then setting y(n) = x( f (n)). However, for many index transformations this is not necessary, and the sequence may be determined or plotted directly. The most common transformations include shifting, reversal, and scaling, which are defined below.
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Shifting This is the transformation defined by f (n) = n - no. If y(n) = x(n - no), x(n) is shifted to the right by no samples if no is positive (this is referred to as a delay), and it is shifted to the left by no
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samples if no is negative (referred to as an advance).
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Reversal This transformation is given by f (n) = - n and simply involves "flipping" the signal x(n)
with respect to the index n.
Time Scaling This transformation is defined by f (n) = Mn or f (n) = n/ N where M and N are positive integers. In the case of f (n) = Mn, the sequence x(Mn) is formed by taking every Mth sample of x(n) (this operation is known as down-sampling). With f (n) = n / N the sequence y(n) = x ( f (n)) is defined as follows:
otherwise
(this operation is known as up-sampling). Examples of shifting, reversing, and time scaling a signal are illustrated in Fig. 1-2.
(a) A discrete-time signal.
( h ) A delay by no = 2.
(c)Time reversal.
-2 -1
;;;,
-2 -1
;(;/2; 1
2 3 4
(d) Down-sampling by a factor of 2.
(e) Up-sampling by a factor of 2.
Fig. 1-2. Illustration of the operations of shifting, reversal, and scaling of the independent variable n.
5 6 7 8 9
SlGNALS AND SYSTEMS
[CHAP. 1
Shifting, reversal, and time-scaling operations are order-dependent. Therefore, one needs to be careful in evaluating compositions of these operations. For example, Fig. 1-3 shows two systems, one that consists of a delay followed by a reversal and one that is a reversal followed by a delay. As indicated. the outputs of these two systems are not the same.
x(n)
Trio
x ( n - no)
x ( - n - no)
( a )A delay Tn,followed by a time-reversal Tr .
x(n)
x(-n)
x(-n
+ no)
(b)A time-reversal Tr followed by a delay T",,
Fig. 1-3. Example illustrating that the operations of delay and reversal do
not commute.
Addition, Multiplication, and Scaling
The most common types of amplitude transformations are addition, multiplication, and scaling. Performing these operations is straightforward and involves only pointwise operations on the signal.
Addition The sum of two signals
is formed by the pointwise addition of the signal values.
Multiplication The multiplication of two signals
is formed by the pointwise product of the signal values.
Scaling Amplitude scaling of a signal x ( n ) by a constant c is accomplished by multiplying every signal value by c:
y(n)=cx(n) -oo<n<oo
This operation may also be considered to be the product of two signals, x ( n ) and f ( n ) = c.
1.2.7 Signal Decomposition
The unit sample may be used to decompose an arbitrary signal x ( n ) into a sum of weighted and shifted unit samples as follows:
This decomposition may be written concisely as
where each term in the sum, x(k)S(n - k ) , is a signal that has an amplitude of x ( k ) at time n = k and a value of zero for all other values of n . This decomposition is the discrete version of the svting property for continuous-time signals and is used in the derivation of the convolution sum.
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