 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
CHAP. 11 in VS .NET
CHAP. 11 Recognize QR Code In VS .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Draw QR Code ISO/IEC18004 In .NET Using Barcode printer for .NET framework Control to generate, create QR image in .NET framework applications. SIGNALS AND SYSTEMS
Scanning QR Code JIS X 0510 In .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Bar Code Drawer In Visual Studio .NET Using Barcode drawer for .NET Control to generate, create bar code image in .NET framework applications. Then the unit step function u(t) can be expressed as
Scanning Barcode In .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in VS .NET applications. Generate QR Code 2d Barcode In C#.NET Using Barcode creator for .NET framework Control to generate, create QR Code image in .NET framework applications. ( t )= Quick Response Code Printer In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create Quick Response Code image in ASP.NET applications. Paint QR Code ISO/IEC18004 In VB.NET Using Barcode creation for Visual Studio .NET Control to generate, create Denso QR Bar Code image in .NET applications. S(r)di
Make Linear 1D Barcode In VS .NET Using Barcode creation for Visual Studio .NET Control to generate, create Linear image in VS .NET applications. GS1 DataBar Expanded Printer In Visual Studio .NET Using Barcode encoder for VS .NET Control to generate, create GS1 DataBar Truncated image in VS .NET applications. (1.31) Make Bar Code In .NET Using Barcode maker for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. Painting EAN8 Supplement 5 AddOn In .NET Using Barcode generator for .NET framework Control to generate, create EAN / UCC  8 image in .NET framework applications. Note that the unit step function u(t) is discontinuous at t = 0; therefore, the derivative of u(t) as shown in Eq. (1.30)is not the derivative of a function in the ordinary sense and should be considered a generalized derivative in the sense of a generalized function. From Eq. (1.31) see that u(t) is undefined at t = 0 and we Barcode Encoder In VS .NET Using Barcode drawer for ASP.NET Control to generate, create barcode image in ASP.NET applications. GTIN  128 Creator In .NET Using Barcode encoder for Reporting Service Control to generate, create GTIN  128 image in Reporting Service applications. of by Eq. (1.21)with $(t) = 1. This result is consistent with the definition (1.18) u(t). Drawing ECC200 In .NET Framework Using Barcode generation for ASP.NET Control to generate, create ECC200 image in ASP.NET applications. Encode Code 128 Code Set C In ObjectiveC Using Barcode drawer for iPad Control to generate, create Code 128 Code Set C image in iPad applications. C. Complex Exponential Signals: The complex exponential signal
EAN / UCC  13 Creation In Java Using Barcode printer for Java Control to generate, create European Article Number 13 image in Java applications. Encode EAN 13 In Java Using Barcode drawer for Java Control to generate, create EAN 13 image in Java applications. Fig. 17 ( a ) Exponentially increasing sinusoidal signal; ( b )exponentially decreasing sinusoidal signal. Create Code 128A In None Using Barcode generation for Software Control to generate, create Code128 image in Software applications. Printing Code 39 In Java Using Barcode printer for Android Control to generate, create USS Code 39 image in Android applications. SIGNALS AND SYSTEMS
[CHAP. 1
is an important example of a complex signal. Using Euler's formula, this signal can be defined as ~ ( t = eiUo'= cos o,t ) +jsin w0t
(1.33) Thus, x ( t ) is a complex signal whose real part is cos mot and imaginary part is sin o o t . An important property of the complex exponential signal x ( t ) in Eq. (1.32) is that it is periodic. The fundamental period To of x ( t ) is given by (Prob. 1.9) Note that x ( t ) is periodic for any value of o,. General Complex Exponential Signals: Let s = a + jw be a complex number. We define x ( t ) as ~ ( t = eS' = e("+~")'= e"'(cos o t ) +j sin wt ) ( 135) Then signal x ( t ) in Eq. (1.35) is known as a general complex exponential signal whose real part eu'cos o t and imaginary part eu'sin wt are exponentially increasing (a > 0) or decreasing ( a < 0) sinusoidal signals (Fig. 17). Real Exponential Signals: Note that if s = a (a real number), then Eq. (1.35) reduces to a real exponential signal x(t) = em' (1.36) Fig. 18 Continuoustime real exponential signals. ( a ) a > 0; ( b )a < 0.
CHAP. 1 1
SIGNALS AND SYSTEMS
As illustrated in Fig. 18, if a > 0, then x(f ) is a growing exponential; and if a < 0, then x ( t ) is a decaying exponential. Sinusoidal Signals: A continuoustime sinusoidal signal can be expressed as
where A is the amplitude (real), w , is the radian frequency in radians per second, and 8 is the phase angle in radians. The sinusoidal signal x ( t ) is shown in Fig. 19, and it is periodic with fundamental period The reciprocal of the fundamental period To is called the fundamental frequency fo: fo=70 .
h ertz (Hz) From Eqs. (1.38) and (1.39) we have
which is called the fundamental angular frequency. Using Euler's formula, the sinusoidal signal in Eq. (1.37) can be expressed as where "Re" denotes "real part of." We also use the notation "Im" to denote "imaginary part of." Then
Fig. 19 Continuoustime sinusoidal signal.
SIGNALS AND SYSTEMS
[CHAP. 1
1.4 BASIC DISCRETETIME SIGNALS A. The Unit Step Sequence: T h e unit step sequence u[n] is defined as which is shown in Fig. 110(a). Note that the value of u[n] at n = 0 is defined [unlike the continuoustime step function u(f) at t = 01 and equals unity. Similarly, the shifted unit step sequence ii[n  k ] is defined as which is shown in Fig. 1lO(b). Fig. 110 ( a ) Unit step sequence; (b) shifted unit step sequence.
B. The Unit Impulse Sequence: T h e unit impulse (or unit sample) sequence 6[n] is defined as
which is shown in Fig. 1  l l ( a ) . Similarly, the shifted unit impulse (or sample) sequence 6[n  k ] is defined as which is shown in Fig. 11l(b). Fig. 111 ( a ) Unit impulse (sample) sequence; (6) shifted unit impulse sequence.
CHAP. 11
SIGNALS AND SYSTEMS
Unlike the continuoustime unit impulse function S(f), S [ n ] is defined without mathematical complication or difficulty. From definitions (1.45) and (1.46) it is readily seen that which are the discretetime counterparts of Eqs. (1.25) and (1.26), respectively. From definitions (1.43) to (1.46), 6 [ n ] and u [ n ] are related by (1.49) ( 1SO) which are the discretetime counterparts of Eqs. (1.30) and (1.31), respectively. Using definition (1.46), any sequence x [ n ] can be expressed as which corresponds to Eq. (1.27) in the continuoustime signal case. C. Complex Exponential Sequences: The complex exponential sequence is of the form x[n]= e ~ n ~ " Again, using Euler's formula, x [ n ] can be expressed as

