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barcode lib ssrs Using the inverse DTFT, we have in Software
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Barcode Creation In None Using Barcode encoder for Software Control to generate, create bar code image in Software applications. EAN / UCC  13 Creation In None Using Barcode creator for Software Control to generate, create UCC.EAN  128 image in Software applications. There are a number of properties of the DTFT that may be used to simplify the evaluation of the DTFT and its inverse. Some of these properties are described below. A summary of the DTFT properties appears in Table 22. Draw UPCA Supplement 2 In None Using Barcode maker for Software Control to generate, create UPC Code image in Software applications. Generate EAN13 Supplement 5 In None Using Barcode generation for Software Control to generate, create GTIN  13 image in Software applications. Periodicity
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Paint Code 128 Code Set B In Java Using Barcode generation for Java Control to generate, create Code 128 Code Set A image in Java applications. Code 39 Full ASCII Recognizer In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. FOURIER ANALYSIS
Table 22 Properties of the DTFT
Property Linearity Shift Timereversal Modulation Convolution Conjugation Derivative Multiplication
Note: Given the DTFTs X ( e J W ) and Y ( e J W )of x ( n ) and y ( n ) , this table lists the DTFTs of sequences that are formed from x ( n ) and y ( n ) . Sequence
DiscreteTime Fourier Transform
Symmetry
The DTFT often has some symmetries that may be exploited to simplify the evaluation of the DTFT or the inverse DTFT. These properties are listed in the table below. Real and even Real and odd Imaginary and even Imaginary and odd
Real and even Imaginary and odd Imaginary and even Real and odd
Note that these properties may be combined. For example, if x(n) is conjugate symmetric, its real part is even and its imaginary part is odd. Therefore, it follows that X(eJW) realvalued. Similarly, note that if x(n) is is real, the real part of x(ejw)is even and the imaginary part is odd. Thus, X(ejw)is conjugate symmetric. Linearity
The discretetime Fourier transform is a linear operator. That is to say, if X 1 (ejw)is the DTFT of xl(n), and X2(eJw) the DTFT of x2(n), is Shifting Property
Shifting a sequence in time results in the multiplication of the DTFT by a complex exponential (linear phase term): x(n no) TimeReversal
DTFT
e  j n o w x (e j w ) Timereversing a sequence results in a frequency reversal of the DTFT : Modulation
FOURIER ANALYSIS
[CHAP. 2
Multiplying a sequence by a complex exponential results in a shift in frequency of the DTFT : ejnwx(n) E x(,j(ww)) ! J Thus, modulating a sequence by a cosine of frequency % shifts
the spectrum up and down in frequency by oo: Convolution Theorem
Perhaps the most important result in linear systems theory is that convolution in the time domain is equivalent to multiplication in the frequency domain. Specifically, this theorem says that the DTFT of a sequence that is formed by convolving two sequences, x(n) and h(n), is the product of the DTFTs of x(n) and h(n): Multiplication (Periodic Convolution) Theorem
As with the timeshift and modulation properties, there is a dual to the convolution theorem that states that multiplication in the time domain corresponds to (periodic) convolution in the frequency domain: Parseval's Theorem
A corollary to the multiplication theorem is Parseval's theorem, which is
Parseval's theorem is referred to as the conservation of energy theorem, because it states that the DTFT operator preserves energy when going from the time domain into the frequency domain. APPLICATIONS
In this section, we present some applications of the DTFT in discretetime signal analysis. These include finding the frequency response of an LSI system that is described by a difference equation, performing convolutions, solving difference equations that have zero initial conditions, and designing inverse systems. 2.7.1 LSZ Systems and LCCDEs
An important subclass of LSI systems contains those whose input, x(n), and output, y(n), are related by a linear constant coefficient difference equation (LCCDE): The linearity and shift properties of the DTFT may be used to express this difference equation in the frequency domain as follows: y(eJw)=  a(k)ejkwy(ejw) k= l
b(k)e~*~x(e~~) kO =
CHAP. 21
FOURIER ANALYSIS
Therefore, the frequency response of this system is
EXAMPLE 2.7.1 Consider the linear shiftinvariant system characterized by the secondorder linear constant coefficient difference equation The frequency response may be found by inspection without solving the difference equation for h ( n ) as follows: Note that this problem may also be worked in the reverse direction. For example, given a frequency response function such as a difference equation may be easily found that will implement this system. First, dividing numerator and denominator by 2 and rewriting the frequency response as follows,

