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n = 0 N - 1 | x ( n ) | 2 = 1 L k = 0 L - 1 | X ( 2 π k / L ) | 2 = 1 π 0 π | X ( ω ) | 2 d ω .

The second term in [link] says the Riemann sum is equal to its limit in this case.

Examples of dtft

As was true for the DFT, insight and intuition is developed by understanding the properties and a few examples of the DTFT. Severalexamples are given below and more can be found in the literature [link] , [link] , [link] . Remember that while in the case of the DFT signals were defined on the region { 0 n ( N - 1 ) } and values outside that region were periodic extensions, here the signals are defined overall integers and are not periodic unless explicitly stated. The spectrum is periodic with period 2 π .

  • D T F T { δ ( n ) } = 1 for all frequencies.
  • D T F T { 1 } = 2 π δ ( ω )
  • D T F T { e j ω 0 n } = 2 π δ ( ω - ω 0 )
  • D T F T { cos ( ω 0 n ) } = π [ δ ( ω - ω 0 ) + δ ( ω + ω 0 ) ]
  • D T F T { M ( n ) } = sin ( ω M k / 2 ) sin ( ω k / 2 )

The z-transform

The z-transform is an extension of the DTFT in a way that is analogous to the Laplace transform for continuous-time signals being an extension of theFourier transform. It allows the use of complex variable theory and is particularly useful in analyzing and describing systems. The question ofconvergence becomes still more complicated and depends on values of z used in the inverse transform which must be in the “region of convergence" (ROC).

Definition of the z-transform

The z-transform (ZT) is defined as a polynomial in the complex variable z with the discrete-time signal values as its coefficients [link] , [link] , [link] . It is given by

F ( z ) = n = - f ( n ) z - n

and the inverse transform (IZT) is

f ( n ) = 1 2 π j R O C F ( z ) z n - 1 d z .

The inverse transform can be derived by using the residue theorem [link] , [link] from complex variable theory to find f ( 0 ) from z - 1 F ( z ) , f ( 1 ) from F ( z ) , f ( 2 ) from z F ( z ) , and in general, f ( n ) from z n - 1 F ( z ) . Verification by substitution is more difficult than for the DFT or DTFT. Here convergence and the interchange of order of thesum and integral is a serious question that involves values of the complex variable z . The complex contour integral in [link] must be taken in the ROC of the z plane.

A unilateral z-transform is sometimes needed where the definition [link] uses a lower limit on the transform summation of zero. This allow the transformationto converge for some functions where the regular bilateral transform does not, it provides a straightforward way to solve initial conditiondifference equation problems, and it simplifies the question of finding the ROC. The bilateral z-transform is used more for signal analysis andthe unilateral transform is used more for system description and analysis. Unless stated otherwise, we will be using the bilateral z-transform.

Properties

The properties of the ZT are similar to those for the DTFT and DFT and are important in the analysis and interpretation of long signals and in theanalysis and description of discrete-time systems. The main properties are given here using the notation that the ZT of acomplex sequence x ( n ) is Z { x ( n ) } = X ( z ) .

  1. Linear Operator: Z { x + y } = Z { x } + Z { y }
  2. Relationship of ZT to DTFT: Z { x } | z = e j ω = DTFT { x }
  3. Periodic Spectrum: X ( e j ω ) = X ( e j ω + 2 π )
  4. Properties of Even and Odd Parts: x ( n ) = u ( n ) + j v ( n ) and X ( e j ω ) = A ( e j ω ) + j B ( e j ω )
    u v A B e v e n 0 e v e n 0 o d d 0 0 o d d 0 e v e n 0 e v e n 0 o d d o d d 0
  5. Convolution: If discrete non-cyclic convolution is defined by
    y ( n ) = h ( n ) * x ( n ) = m = - h ( n - m ) x ( m ) = k = - h ( k ) x ( n - k )
    then Z { h ( n ) * x ( n ) } = Z { h ( n ) } Z { x ( n ) }
  6. Shift: Z { x ( n + M ) } = z M X ( z )
  7. Shift (unilateral): Z { x ( n + m ) } = z m X ( z ) - z m x ( 0 ) - z m - 1 x ( 1 ) - - z x ( m - 1 )
  8. Shift (unilateral): Z { x ( n - m ) } = z - m X ( z ) - z - m + 1 x ( - 1 ) - - x ( - m )
  9. Modulate: Z { x ( n ) a n } = X ( z / a )
  10. Time mult.: Z { n m x ( n ) } = ( - z ) m d m X ( z ) d z m
  11. Evaluation: The ZT can be evaluated on the unit circle in the z-plane by taking the DTFT of x ( n ) and if the signal is finite in length, this can be evaluated at sample points by the DFT.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
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Damian
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Damian Reply
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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s. Reply
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s. Reply
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Graphene has a hexagonal structure
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Source:  OpenStax, Brief notes on signals and systems. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10565/1.7
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