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i i i generate, create ean 13 none with software projects .NET Framework 4.0 i i TranterBook 2003/11/ ean13 for None 18 16:12 page 555 #573. Section 14.7. Simulation Methodology p( ). n 2 = T 2 p(nT). Tmax Figure 14.8 Sampled values of the power delay pro le. Source: M. C. Jeruchim, P.

B alaban, and K. S. Shanmugan, Simulation of Communications Systems, 2nd ed.

, New York: Kluwer Academic/Plenum Publishers, 2000.. Sampling An important aspect of the T DL model that deserves additional attention is the sampling rate for simulations. The TDL model shown in Figure 6.8 was derived with continuous time input x(t) and output y (t).

However, in simulation we use sampled values of x(t) and output y (t) which should be sampled at 8 to 32 times the bandwidth, where the bandwidth includes the e ect of spreading due to the time-varying nature of the system as de ned in 13. Note that the Nyquist rate of 2B, B = r/2 was used to derive the TDL model, and the tap spacing of T = 1/r will be much greater than Ts , where Ts is the sampling time for the input and output waveforms. It is of course possible to derive a TDL model with a smaller tap spacing (i.

e., more samples per symbols), but such a model will be computationally ine cient and does not necessarily improve the accuracy of the simulation..

Generation of Tap Gain Processes The tap gain processes are s tationary random processes with Gaussian probability density functions and arbitrary power spectral density functions. The simplest model for the tap gain processes assume them to be uncorrelated, complex, zero mean Gaussian processes with di erent variances but identical power spectral densities. In this case, the tap gain processes can be generated by ltering white Gaussian processes, as shown in Figure 14.

9. The lter transfer function is chosen such that it produces the desired doppler power spectral density. In other words, H(f ) is chosen such that Sgg (f ) = Sd (f ) = Sww (f ) H(f ) .

= H(f ). (14.58). i i i i i TranterBook 2003/11/ 18 16:12 page 556 #574. Modeling and Simulation of Waveform Channels 14 . x(t Tap Input ~ nT ). Input: Unit variance, Complex white ~ Gaussian process w(t) Gain Filter ~ H(f). ~ g(t) ~ gn (t). Tap Output Figure 14.9 Generation of the nth tap gain process. Source: M. C. Jeruchim, P.

B EAN-13 Supplement 2 for None alaban, and K. S. Shanmugan, Simulation of Communications Systems, 2nd ed.

, New York: Kluwer Academic/Plenum Publishers, 2000.. where Sww (f ) is the power spectral density of the input white noise process, which can be set equal to 1, and Sgg (f ) is the speci ed doppler power spectral density of the tap gain processes. The lter gain is chosen such that g(t) has a normalized power of 1. The static gain n in Figure 14.

9 accounts for the di erent power levels or variances for the di erent taps. If the power spectral density of the tap gains are di erent, then di erent lters will be used for di erent taps..

Delay Power Pro les and Doppler Power Spectral Densities As previously mentioned, the BER performance of a communication system is more sensitive to the values of the rms and maximum delay spreads than to the shape of the power delay pro le. Therefore, simple pro les such as uniform or exponential can be used for simulation. The delay pro les are normalized to have unit area (i.

e., total normalized power, or the area under the locally averaged power delay pro le, is set equal to one). Thus.

p( )dt = 1 (14.59). Typical rms delay spreads ar ean13+5 for None e given in Table 14.2. The most commonly used models for doppler power spectral densities for mobile applications assume that there are many multipath components, each having di erent delays, and that all components have the same doppler spectrum.

Each. Table 14.2 Typical rms Delay Spreads Link Type Troposcatter Outdoor Mobile Indoor Link Distance 100 Km 1 Km 10 m rms Delay Spread millisecond EAN 13 for None s (10 3 ) microseconds (10 6 ) nanoseconds (10 9 ). i i i i i TranterBook 2003/11/ EAN 13 for None 18 16:12 page 557 #575. Section 14.7. Simulation Methodology multipath component (ray) is actually made up of a large number of simultaneously arriving unresolvable multipath components, having angle of arrival with a uniform angular distribution at the receive antenna. This channel model was used by Jakes and others at Bell Laboratories to derive the rst comprehensive mobile radio channel model for both doppler e ects and amplitude fading e ects [11]. The classical Jakes doppler spectrum has the form, which was initially simulated in 7 (see Example 7.

11), Sd (f ) = Sgn gn (f ) = K , 1 (f /fd )2 fd f fd (14.60). where fd = v/ is the maximu Software EAN-13 Supplement 2 m doppler shift, v is the vehicle speed in meters per second, and is the wavelength of the carrier. While the doppler spectrum de ned by (14.60) is appropriate for dense scattering environments like urban areas, a Ricean spectrum is recommended for rural environments in which there is one strong direct line-of-sight path and hence Ricean fading.

The Ricean doppler spectrum has the form Sd (f ) = Sgn gn (f ) = 0.41 1 (f /fd )2 + 0.91 (f 0.

7fd), fd f fd (14.61). and is shown in Figure 14.10 . Other spectral shapes used for the doppler power spectral densities include Gaussian and uniform.

Typical doppler bandwidths in mobile applications at 1 GHz will range from 10 to 200 Hz. There are several ways of implementing the doppler spectral shaping lter needed to generate the tap gain processes in the TDL model for the channel when using the model assumed by Jakes. An FIR lter in time domain is the most common implementation, since doppler power spectral densities do not lend themselves easily to implementation in recursive form.

The generation of a Jakes spectrum using FIR ltering techniques was illustrated in 7. A block processing model based on frequency domain techniques is discussed in [13]. In generating the tap gain processes it should be noted that the bandwidth of the tap gain processes for slowly time-varying channels will be very small compared.

Sd ( f ). Jakes Spectrum (continuous).
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