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A special case of this contour integral occurs when C is the unit circle. This contour can be used when the ROC includes the unit circle, which is always guaranteed when X(z){\display style X(z)} is stable, that is, when all the poles are inside the unit circle.

In this case the ROC is the complex plane with a disc of radius 0.5 at the origin “punched out”. ROC shown in blue, the unit circle as a dotted gray circle and the circle | z | = 0.5 is shown as a dashed black circlet x=(0.5)nu{\display style x=-(0.5)^{n}u\} (where u is the Heaviside step function).

Using the infinite geometric series, again, the equality only holds if |0.5 1 z | < 1 which can be rewritten in terms of z as | z | < 0.5. In this case the ROC is a disc centered at the origin and of radius 0.5.

In example 2, the causal system yields an ROC that includes | z | = while the anticausal system in example 3 yields an ROC that includes | z | = 0. The stability of a system can also be determined by knowing the ROC alone.

For stability the ROC must contain the unit circle. If we need both stability and causality, all the poles of the system function must be inside the unit circle.

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To understand this, let X(f){\display style X(f)} be the Fourier transform of any function, x(t){\display style x(t)}, whose samples at some interval, T, equal the x sequence. Comparison of the two series reveals that =2fT{\display style \script style \omega =2\pi ft} is a with units of radians per sample.

And now, with the substitution f=2T,{\display style \script style f={\franc {\omega}{2\pi T}}, } EQ.4 can be expressed in terms of the Fourier transform, X(•) : As parameter T changes, the individual terms of EQ.5 move farther apart or closer together along the f-axis.

In EQ.6 however, the centers remain 2 apart, while their widths expand or contract. When the x(NT){\display style x(NT)} sequence is periodic, its DEFT is divergent at one or more harmonic frequencies, and zero at all other frequencies.

This is often represented by the use of amplitude-variant Dirac delta functions at the harmonic frequencies. Due to periodicity, there are only a finite number of unique amplitudes, which are readily computed by the much simpler discrete Fourier transform (DFT).

The bilinear transform can be used to convert continuous-time filters (represented in the Laplace domain) into discrete-time filters (represented in the Z -domain), and vice versa. }X{*}(s)=X(z){\big |}_{\display style z =ex{St}}} The inverse Laplace transform is a mathematical abstraction known as an impulse-sampled function.

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up=0Nyup=HQ=0MxHQ{\display style \sum _{p=0}^{N}y\alpha _{p}=\sum _{q=0}^{M}x\beta _{q}} Both sides of the above equation can be divided by 0, if it is not zero, normalizing 0 = 1 and the LCD equation can be written This form of the LCD equation is favorable to make it more explicit that the “current” output y is a function of past outputs y, current input x, and previous inputs x.

Data, Mutsuhito, Discrete Time Control Systems 2nd Ed, Prentice-Hall Inc, 1995, 1987. If you’ve studied the Laplace transform, you’re familiar with the concept of transforming a function of time into a function of frequency.

If x(n) is an infinite duration anti-causal sequence, ROC is interior of the circle with radius a. i.e. | z | < a. If x(n) is a finite duration two-sided sequence, then the ROC is entire z -plane except at z = 0 & z = .

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