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Using the inverse DTFT, we have
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Note that although x(n) is not absolutely summable, by allowing the DTFT to contain impulses, we may consider the DTFT of sequences that contain complex exponentials. As another example, if X(eJ")= r 6 ( w - 9 ) r 8 ( w computing the inverse DTFT, we find x(n) = i
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+ i e - l " w o = cos(nwo)
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2.6 DTFT PROPERTIES
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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 2-2.
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Periodicity
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The discrete-time Fourier transform is periodic in w with a period of 2 n : ~ ( ~ j = x (,jW+zx) w )
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This property follows directly from the definition of the DTFT and the periodicity of the complex exponentials:
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CHAP. 21
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FOURIER ANALYSIS
Table 2-2 Properties of the DTFT
Property Linearity Shift Time-reversal 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
Discrete-Time 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) real-valued. 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 discrete-time 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)
Time-Reversal
DTFT
e - j n o w x (e j w )
Time-reversing 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(w-w)) ! 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 time-shift 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 discrete-time 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)e-jkwy(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 shift-invariant system characterized by the second-order 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,
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