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zen barcode ssrs WAVEFORMS AND SIGNALS in Software
WAVEFORMS AND SIGNALS QRCode Recognizer In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. QR Code Maker In None Using Barcode maker for Software Control to generate, create QR Code JIS X 0510 image in Software applications. [CHAP. 6
Decoding Quick Response Code In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. Create QR Code ISO/IEC18004 In C# Using Barcode printer for Visual Studio .NET Control to generate, create QR Code 2d barcode image in .NET framework applications. EXAMPLE 6.15 If the switch in Fig. 68(a) is moved to position 2 at t 0 and then moved back to position 1 at t 5 s, express vAB using the step function. vAB V0 u t u t 5 EXAMPLE 6.16 Express v t , graphed in Fig. 69, using the step function. QR Drawer In .NET Using Barcode generator for ASP.NET Control to generate, create QRCode image in ASP.NET applications. Make QR Code 2d Barcode In VS .NET Using Barcode maker for Visual Studio .NET Control to generate, create Denso QR Bar Code image in VS .NET applications. Fig. 69 v t u t u t 2 sin t
Making QR Code JIS X 0510 In VB.NET Using Barcode maker for Visual Studio .NET Control to generate, create QR image in .NET framework applications. Make Code 128A In None Using Barcode encoder for Software Control to generate, create Code 128C image in Software applications. THE UNIT IMPULSE FUNCTION
Code39 Printer In None Using Barcode generation for Software Control to generate, create Code39 image in Software applications. GTIN  128 Encoder In None Using Barcode creator for Software Control to generate, create USS128 image in Software applications. Consider the function sT t of Fig. 610(a), which is zero for t < 0 and increases uniformly from 0 to 1 in T seconds. Its derivative dT t is a pulse of duration T and height 1=T, as seen in Fig. 610(b). 8 for t < 0 <0 30 for 0 < t < T dT t 1=T : 0 for t > T If the transition time T is reduced, the pulse in Fig. 610(b) becomes narrower and taller, but the area under the pulse remains equal to 1. If we let T approach zero, in the limit function sT t becomes a unit step u t and its derivative dT t becomes a unit pulse t with zero width and in nite height. The unit impulse t is shown in Fig. 610(c). The unit impulse or unit delta function is de ned by 1 t 0 for t 6 0 and Barcode Creation In None Using Barcode generator for Software Control to generate, create barcode image in Software applications. Printing EAN / UCC  13 In None Using Barcode drawer for Software Control to generate, create EAN13 Supplement 5 image in Software applications. t dt 1
International Standard Serial Number Creation In None Using Barcode creation for Software Control to generate, create ISSN image in Software applications. Scanning DataMatrix In .NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. 31 UCC.EAN  128 Creator In Visual Studio .NET Using Barcode encoder for .NET Control to generate, create UCC  12 image in Visual Studio .NET applications. Create Code 39 Extended In None Using Barcode printer for Office Word Control to generate, create Code 3/9 image in Microsoft Word applications. An impulse which is the limit of a narrow pulse with an area A is expressed by A t . The magnitude A is sometimes called the strength of the impulse. A unit impulse which occurs at t t0 is expressed by t t0 . Making Barcode In Java Using Barcode generator for Android Control to generate, create barcode image in Android applications. Data Matrix Encoder In .NET Framework Using Barcode creation for ASP.NET Control to generate, create Data Matrix image in ASP.NET applications. EXAMPLE 6.17 The voltage across the terminals of a 100nF capacitor grows linearly, from 0 to 10 V, taking the shape of the function sT t in Fig. 610(a). Find (a) the charge across the capacitor at t T and (b) the current iC t in the capacitor for T 1 s, T 1 ms, and T 1 ms. (a) At t T, vC 10 V. The charge across the capacitor is Q CvC 10 7 10 10 6 . b ic t C dvC dt Encode GTIN  128 In None Using Barcode maker for Font Control to generate, create EAN 128 image in Font applications. ECC200 Decoder In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. CHAP. 6] WAVEFORMS AND SIGNALS
Fig. 610 From Fig. 610, 8 <0 iC t I0 10 6 =T A : 0 for t < 0 for 0 < t < T for t > T 32
For T 1 s, I0 10 6 A; for T 1 ms, I0 10 3 A; and for T 1 ms, I0 1 A. In all the preceding cases, the charge accumulated across the capacitor at the end of the transition period is T Q iC t dt I0 T 10 6 C The amount of charge at t T is independent of T.
It generates a voltage vC 10 V across the capacitor.
EXAMPLE 6.18 Let dT t t0 denote a narrow pulse of width T and height 1=T, which starts at t t0 . Consider a function f t which is continuous between t0 and t0 T as shown in Fig. 611(a). Find the limit of integral I in (33) when T approaches zero. 1 I dT t t0 f t dt 33 dT t t0 Substituting dT in (33) we get I 1 T
1=T 0 t0 T
t0 < t < t0 T elsewhere
f t dt
34a where S is the hatched area under f t between t0 and t0 T in Fig. 6.11(b). Assuming T to be small, the function f t may be approximated by a line connecting A and B. S is the area of the resulting trapezoid. S 1 f t0 f t0 T T 2 I 1 f t0 f t0 T 2 As T ! 0, dT t t0 ! t t0 and f t0 T ! f t0 and from (34c) we get 34b 34c WAVEFORMS AND SIGNALS
[CHAP. 6
Fig. 611 lim I lim 1 f t0 f t0 T 2
34d We assumed f t to be continuous between t0 and t0 T.
T!0 1 Therefore, 34e (34f) (34g) It is also used as another de nition for
lim I f t0 t t0 f t dt
But and so
lim I 1
t t0 f t dt f t0
The identity (34g) is called the sifting property of the impulse function. t .
THE EXPONENTIAL FUNCTION
The function f t est with s a complex constant is called exponential. It decays with time if the real part of s is negative and grows if the real part of s is positive. We will discuss exponentials eat in which the constant a is a real number. The inverse of the constant a has the dimension of time and is called the time constant 1=a. A decaying exponential e t= is plotted versus t as shown in Fig. 612. The function decays from one at t 0 to zero at t 1. After seconds the function e t= is reduced to e 1 0:368. For 1, the function e t is called a normalized exponential which is the same as e t= when plotted versus t=. EXAMPLE 6.19 Show that the tangent to the graph of e t= at t 0 intersects the t axis at t as shown in Fig. 612. The tangent line begins at point A v 1; t 0 with a slope of de t= =dtjt 0 1=. The equation of the line is vtan t t= 1. The line intersects the t axis at point B where t . This observation provides a convenient approximate approach to plotting the exponential function as described in Example 6.20. EXAMPLE 6.20 Draw an approximate plot of v t e t= for t > 0. Identify the initial point A (t 0; v 1 of the curve and the intersection B of its tangent with the t axis at t . Draw the tangent line AB. Two additional points C and D located at t and t 2, with heights of 0.368 and CHAP. 6]

