qr code vb.net library Analog Signal in Sampled Form. Unit Impulse Notation in .NET framework

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Analog Signal in Sampled Form. Unit Impulse Notation
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Let v t denote the instantaneous continuous values of an analog signal. Then one way of GRAPHICALLY representing the SAMPLED FORM of v t is shown in Fig. 331, where T is the uniform amount of time between samples.
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Fig. 331
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In the gure, each distance line, drawn from the horizontal axis, represents the value of v t at that particular sampling instant. Thus v 2T value of v t at t 2T v T value of v t at t T v 0 value of v t at t 0 v T value of v t at t T v 2T value of v t at t 2T and so on, thus, v nT value of v t at any nth sampling instant Now let s assume the analog signal starts at some time t 0, that is, v t 0 for t < 0. For this condition, all the sample values to the left of the origin in the above gure are, of course, equal to zero. Then the equation for vs t , the sampled form of v t , can be written in terms of time-delayed unit impulses;* thus vs t v 0  t v T  t T v 2T  t 2T v nT  t nT 571
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where n is a positive integer including zero, n 0; 1; 2; 3; . . . . Because of the unit impulse factors, vs t 0 at all times EXCEPT at the instants t 0; t T; t 2T, and so on, to t nT. At each such instant one of the terms in eq. (571) will not be equal to zero; for example, at t 2T, all the terms in the equation are equal to zero except the one term v 2T  t 2T . The graphical representation of eq. (571) at t 2T is shown below.
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Immediately following the instant at t 2T, all terms in eq. (571) become and remain equal to zero until the time t 3T, at which time only the term v 3T  t 3T is not equal to zero; the situation at t 3T is shown graphically in the following gure.
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* See note 31, then note 32, in Appendix.
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Let us note, now, that eq. (571) can also be written using the convenient sigma or summation notation, thus vs t
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n 1 X n 0
v nT  t nT
572
P in which the symbol is the capital Greek letter sigma. The sigma notation is read as the SUM of all such terms from n 0 to n equals in nity, where here in nity means that n, the number of terms, must be allowed to become in nitely great.
The z-Transform
Equation (572) is called a time series, because the independent variable is the real quantity time, t. It has been discovered, however, that the ALGEBRAIC WORK associated with the manipulation of sampled analog signals is much simpli ed if the situation in Fig. 331 is mathematically de ned in terms of a COMPLEX VARIABLE z instead of the real variable time. This is done as follows. Let us begin by arbitrarily writing down the following in nite series, in which v 0 ; v T ; v 2T and so on are the actual sampled values of an analog signal, and where z is the complex variable referred to above, F z v 0 v T z 1 v 2T z 2 v nT z n where z A j!T in which A is a positive real constant, with the restriction that A be greater than 1 A > 1 . Note that now the independent variable is sinusoidal FREQUENCY, ! 2f ; thus we are now said to be working in the frequency domain instead of the time domain. The quantity F z above is called the z-transform of the sequence of samples. As always, T is the constant time between samples. Note now that the above can be neatly summarized by making use of the sigma notation, thus F z where z A j!T and where A > 1.*
* Basically the value of A, in the summation of a given v nT series, must be large enough so that, as n ! 1, A n decreases faster than the sum of the v nT series increases; this allows us to get a de nite answer for F z , instead of the indeterminate answer in nitely great.
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