Section 4.1. in Software Implement UPC-13 in Software Section 4.1.

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Section 4.1. use software ean-13 supplement 2 printer toproduce ean-13 supplement 5 with software VS 2010 The Lowpass Complex Envelope for Bandpass Signals from the de nition of comple x numbers that A(t) and (t) are related to xd (t) and xq (t) by A(t) = . x(t). = and (t) = arctan xq (t) xd (t) (4.10) x2 (t) + x2 (t) q d (4.9).

Using (4.3) and (4.5), the t Software ean13+2 ime-domain signal x(t) can be written x(t) = Re{[xd (t) + jxq (t)][cos 2 f0 t + j sin 2 f0 t]} which is x(t) = xd (t) cos(2 f0 t) xq (t) sin(2 f0 t) (4.

12) (4.11). Note that (4.12) could have been written by applying the trigonometric identity cos(a + b) = cos(a) cos(b) sin(a) sin(b) (4.13).

to (4.1) and de ning xd (t) GS1-13 for None and xq (t) using (4.7) and (4.

8). Although f0 is typically chosen as the center frequency of the bandpass signal, f0 is arbitrary and can be chosen for convenience. However, as will be illustrated in Example 4.

1, xd (t) and xq (t) are dependent upon the selection of f0 . Example 4.2 illustrates the lowpass representation of an analog FM signal.

Examples 4.3, 4.4 and 4.

5 illustrate the application to digital signals. These last three examples illustrate the development of simulation models for digital modulators. Example 4.

1. Consider the bandpass signal x(t) = A sin (2 fm t) cos (2 fc t + ) (4.14).

where fc is the carrier freq uency and is the carrier phase deviation. We assume that fc fm and desire xd (t) and xq (t) as de ned by (4.3) and (4.

5). The rst step is to choose the frequency f0 de ned in (4.3).

In order not to assume that f0 = fc , let fc = f0 + f . This gives x(t) = A sin (2 fm t) cos (2 f0 t + 2 f t + ) which can be written x(t) = Re {A sin (2 fm t) exp(j2 f t) exp (j ) exp (j2 f0 t)} By inspection, the complex envelope is x(t) = A sin (2 fm t) exp [j(2 f t + )] (4.17) (4.

16) (4.15). i i i i i TranterBook 2003/11/ 18 16:12 page 98 #116. 98 Thus:. Lowpass Simulation Models for Bandpass Signals and Systems 4 . x(t) = A sin (2 fm t) [cos(2 f t + ) + j sin(2 f t + )] from which xd (t) = A sin (2 fm t) cos(2 f t + ) and xq (t) = A sin(2 fm t) sin(2 f t + ). (4.18). (4.19). (4.20). follow directly by equating real and imaginary parts. Note that both xd (t) and xq (t) depend upon the relationship between fc and f0 . The simplest expressions result if f0 = fc (f = 0) but, as previously mentioned, the choice of f0 is arbitrary.

In simulation problems we typically choose f0 so that the computational burden is minimized. Example 4.2.

An analog FM modulator is de ned by the expression. xc (t) = Ac cos 2 fc t + kf m( ) d + (t0 ). (4.21). where Ac and fc represent th e amplitude and frequency of the unmodulated carrier, respectively, m(t) is the message or information carrying signal, kf is the modulation index, t0 is an arbitrary reference time, and (t0 ) is the phase deviation at time t0 . Assuming that the time reference t0 = 0 is selected and that (t0 ) = 0, it follows that xc (t) can be represented. xc (t) = Re Ac exp jkf m( ) d exp (j2 fc t). (4.22). Thus, the complex envelope is x(t) = Ac exp jkf m( ) d (4.23). from which xd (t) = Ac cos kf m( ) d (4.24). xq (t) = Ac sin kf m( ) d (4.25). Thus, in order to represent EAN 13 for None x(t) in a simulation, it is necessary only to generate the two lowpass signals given by (4.24) and (4.25).

. i i i i i TranterBook 2003/11/ Software EAN13 18 16:12 page 99 #117. Section 4.1. The Lowpass Complex Envelope for Bandpass Signals A suite of models for an FM modulator are illustrated in Figure 4.1. Figure 4.

1(a) shows the continuous-time bandpass model. The continuous-time lowpass model, in which the output is the lowpass complex envelope representation of xc (t), is shown in Figure 4.1(b).

The discrete-time equivalent is illustrated in Figure 4.1(c), in which the sampling period is T and n indexes the samples. Note that, in the discrete-time model, the integrator is modeled as an accumulator (summation operator).

This is equivalent to rectangular integration in which the area accumulated in the k th time slot is T m(kT ). Other integrator models can be used, as will be discussed in later chapters. We now consider a few examples involving digital modulation.

In order to do this a simulation model for the modulator is needed. The basic model is illustrated in Figure 4.2.

[Note: Figure 4.2 illustrates a bandpass model for which the output is the bandpass signal xck (t). The lowpass model used for simulation has the outputs xdk (t) and xqk (t), which de ne the lowpass complex envelope of xck (t).

] We assume that ak represents a binary data stream. In addition, M -ary signaling, in which b information bits are grouped together to form a data symbol so that the transmitted signal in the k th signaling interval can carry more than 1 bit of information. Typically M = 2b , but this is not a necessary assumption.

The binary to M -ary symbol mapping shown in Figure 4.2 performs the function of grouping together b bits to form the M -ary symbol. The output of the mapper are the direct and quadrature components of the k-symbol.

These are denoted dk and qk . (The k th symbol itself can be viewed as complex valued, sk = dk + jqk .) The symbols dk and qk can be considered impulse functions having weights determined by the bit-to-symbol mapping.

The impulse response of the pulseshaping lter is denoted p(t) so that the direct and quadrature signals for the k th signaling interval, kT < t < (k + 1)T , are xdk (t) and xqk (t) as shown in Figure 4.2. The transmitter output for the k th signaling interval is xck (t) = xdk (t) cos 2 f0 t xqk (t) sin 2 f0 t The corresponding discrete-time signal model is xck (nT ) = xdk (nT ) cos 2 f0 nT xqk (nT ) sin 2 f0 nT (4.

27) (4.26). which is simply the sampled GS1-13 for None version of the continuous-time signal model. Before leaving this topic, we pause to review a few terms, to brie y discuss scattergrams, and to point out the di erence between scattergrams, as used in the simulation context, and signal constellations, which are familiar to us from our study of digital communication theory. A signal space is de ned as a K-dimensional space generated by K orthonormal basis functions i (t), i = 1, 2, , K and signals are represented as vectors in this space.

For example, assume an M -ary communications system in which one of m signals is transmitted in the k th signaling interval. In terms of the basis functions, the signal transmitted in the k th signaling interval is expressed. xc (t) =. xim i (t),. kT < t < (k + 1)T,.
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