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FREQUENCY SELECTIVITY PARAMETERS ON MULTI-CARRIER WIDEBAND WIRELESS SIGNALS Víctor M Hinostroza, José Mireles and Humberto Ochoa Institute of Engineering and Technology, University of Ciudad Juárez Valle del tigris # 3247,Ciudad Juárez Chihuahua México C. P. 32306 vhinostr@uacj.mx, jmireles@uacj.mx, hochoa@uacj.mx ABSTRACT. This work is a study of the effects of frequency selectivity on multi-carrier wideband signals in three different environments; indoors, outdoor to indoor and outdoors. The investigation was made using measurements carried out with a sounder with a 300 MHz bandwidth. The main part of this work is related to evaluate the contribution of several parameters; frequency selective fading, coherence bandwidth and delay spread on the frequency selectivity of the channel. A description of the sounder parameters and the sounded environments are given. The 300 MHz bandwidth is divided in segments of 60 kHz to perform the evaluation of frequency selective fading. Sub channels of 20 MHz for OFDM systems and 5 MHz for WCDMA were evaluated. Figures are provided for a number of bands, parameters and locations in the three environments. It is also shown the variation of the signal level due to frequency selective fading. The practical assumptions about the coherence bandwidth and delay spread are reviewed and a comparison is made with actual measurements. Statistical analysis was performed over some of the results. Keywords. Coherence bandwidth, frequency correlation, frequency selective fading and multi-carrier modulation . characteristics of the wireless communications I. INTRODUCTION. channel. The dispersive channel characteristics arise from the different propagation paths, i.e. To simulate and evaluate the performance of a multipath, between the receiver and the transmitter. wireless mobile system a good channel model is This dispersion could be measured, if we could needed. Mobile communication systems are using measure the channel impulse response (CIR). As a larger bandwidths and higher frequencies and these general rule the effects of ISI on the transmission characteristics impose new challenges on channel errors is negligible if the delay spread is estimation. The channel models that have been significantly shorter than the duration of the developed for the mobile systems in use may not be transmitted symbol. Due to the expected increase in applicable anymore. To validate that the old demand of higher data rates, wideband multi- models can be used for future systems or to design carrier systems such as; OFDM and WCDMA are new models, it is necessary to answer the question expected to be technologies of choice [1], [12] and about how the same parameters performs at higher [14]. This is because these two technologies can bandwidths? Also, we have to be able to measure provide both; high data rates and an acceptable and validate some parameters and compare them to level of quality of service. However, these systems well known practical assumptions. Measurements need first to address better the problem regarding for analysis of the fading statistics at common channel prediction or estimation, because this frequencies have been performed before, but they condition is the main boundary for higher data have been performed at small bandwidths, it is rates. The study of correlation of the mobile radio necessary to update the models with higher channel in frequency and time domains has helped bandwidths. to understand the problem of channel estimation. One of this work objectives is to evaluate As the data rate (the bandwidth) increases the frequency selective fading (FSF) in several communication limitations come from the Inter environments. This work begins with the results of Symbol Interference (ISI) due to the dispersive measurements made with a sounder that uses the E{H (t1 ; f1 ) * H (t2 ; f 2 )} = chirp technique for sounding. ∞ ∞ ∫ ∫ E{h(t ;τ ) * h(t ;τ )}e − 2πf1τ 1 + j 2πf 2τ 2 1 1 2 2 e dτ 1dτ 2 − ∞− ∞ Multipath fading channels are usually classified into flat fading and frequency selective fading (2) according to their coherence bandwidth relative to the one of the transmitted signal. Coherence By considering the channel to have uncorrelated bandwidth is defined as the range of frequencies scattering (US) and to be wide sense stationary over which two frequency components remain in a (WSS), the subscript for τ is eliminated and f1 and strong amplitude correlation. Physically, it defines f2 can be replaced by f + ∆f and t1 and t2 replaced the range of frequencies over which the channel by t + ∆t, then: ∞ can be considered “flat”. The analytic issue of coherence bandwidth was first studied by Jakes [1] R H (∆t ; ∆f ) = ∫R h (∆t ;τ )e − j 2π∆fτ dτ where by assuming homogeneous scattering, his −∞ work revealed that the coherence bandwidth of a (3) wireless channel is inversely proportional to its In (3) RH and Rh represents the correlation of root-mean-square (rms) delay spread. The same random variations in the channels transfer function issue was subsequently studied by various authors and its impulse response respectively. If there are [4], [8], [9], [10]. Since many practical channel US, then ∆t is 0 then: environments can significantly deviate from the homogeneous assumption, various measurements were conducted to determine multipath delay Rh (0;τ ) = E h (0;τ ) { 2 }= E {h(τ ) } 2 profiles and coherence bandwidths [19], [20], [21], (4) [22], aiming to obtain a more general formula for coherence bandwidth. In this work the variations of substituting into (3) gives: this formula are reviewed and compared with ∫ E {h(τ ) }e ∞ actual results and a comparison is provided. 2 − j 2π∆fτ R H ( ∆f ) = dτ The rest of this document is structured as follow; in −∞ part II the theoretical foundations of the channel (5) impulse response frequency selective fading and coherence bandwidth are reviewed. Also in this part, the characteristics of the three environments where { E h (τ ) 2 } is the average Power Delay sounded are described. In part III, the frequency Profile PDP of the channel. So, under the above selective fading evaluation and analysis are conditions, RH is the Fourier transform of the presented. Plots of the dependency of fading deep average PDP. and frequency separation of two specific points in the response are studied. At part IV, data about the 2.2 Coherence bandwidth. relationship between delay spread and coherence The multipath effect of the channel, the arrival of bandwidth are provided. At the end in part V, different signals in different time delays causes the conclusions and future work are mentioned. statistical properties of two signals of different frequencies to become independent if the frequency II. MATHEMATICAL BACKGROUND separation is large enough. The maximum frequency separation for which the signals are still 2.1 The wideband channel model. strongly correlated is called coherence bandwidth The radio propagation channel is normally (Bc). Besides to contribute to the understanding of represented in terms of a time-varying linear filter, the channel, the coherence bandwidth is useful in with complex low-pass impulse response, h(t, τ). Its evaluating the performance and limitations of time-varying low-pass transfer function is [4] [6] different modulations and diversity models. [8] [10]: ∞ The coherence bandwidth of a fading channel is ∫ h(t;τ )e τ dτ − j 2πf H (t , f ) = probed by sending two sinusoids, separated in −∞ frequency by ∆f = f1- f2 Hz, through the channel. (1) The coherence bandwidth is defined as ∆f, over Where τ represents delay, using (1) the frequency which the cross correlation coefficient between r1 correlation function for the channel can be written and r2 is greater than a preset threshold, say, η0= as: 0.9. Namely: Cov( r1, r 2) ⎛2 λ ⎞ π C r1,r 2 = = η0 (1 + λ ) E ⎜ ⎟ var(r1) var(r 2) ⎜1 + λ ⎟ − 2 ρ ( s, τ ) = ⎝ ⎠ (6) π 2− Then, using (2) 2 J (ω τ ) 2 ∞ ∞ = λ2 = 0 2m 2 R( s, τ ) = r1r 2 = ∫ 1+ s σ 0 ∫ r1r 2 p(r1, r 2)dr dr 0 1 2 (12) (7) It is possible to see in this expression that the Where p(r1,r2) is correlation decreases with frequency separation. This formula has been substituted by several 2π 2π practical expressions some of them are the p( r1, r 2) = ∫ ∫ p(r1, r 2, θ , θ 0 0 1 2 )dθ1dθ 2 following [4], [8], [9], [10]. (8) 1 1 BC =0.9 = (13) BC =0.5 = (14) 50σ rms 5σ rms r1r 2 ⎡ r + r ⎤ ⎛ r1r 2 λ ⎞ 2 2 = exp⎢− I ⎜ 2 ⎥ 0⎜ 1 ⎟ 2 ⎟ 2 1 1 µ (1 − λ ) 2 2 ⎣ 2 µ (1 − λ ) ⎦ ⎝ µ 1 − λ ⎠ BC =0.9 = (15) BC = (16) 8σ mean 2πσ rms Where I0(x) is the modified Bessel function of zero In general order. Then, substituting (8) in (7) and integrating k BC = (17) π 1 1 σ rms R( s,τ ) = b0 F ( − ,− ;1; λ2 ) 2 2 2 (9) It will be shown, comparing with practical measurements that none of these expressions are this may also be expressed as accurate and it is difficult to obtain a comprehensive expression for all environments. ⎛ λ2 ⎞ π R( s, τ ) = b0 ⎜1 + ⎟ (10) 2.3 Sounder systems characteristics and 2 ⎜ ⎝ 4⎟ ⎠ environment description. The sounder system used to make the R ( s, τ ) − r1 r 2 measurements of this work was developed at ρ ( s, τ ) = [r 1 2 − r1 2 ][ r 2 2 − r2 2 ] UMIST in Manchester UK and is described in [2] and [3]. This sounder uses the FMCW or chirp technique. The generated chirp consists of a linearly frequency modulated signal with a ⎛2 λ ⎞ bandwidth of 300 MHz and a carrier frequency of R( s,τ ) = b0 (1 + λ ) E ⎜ ⎟ ⎜1+ λ ⎟ (11) 2.35 GHz. The chirp repetition frequency is 100 ⎝ ⎠ Hertz, which allows having 50-Hertz Doppler range measurements. The receiver has the same architecture than the transmitter. But in the Where E(x) is the complete elliptic integral of the receiver, the generated chirp is not transmitted but second kind. The expansion of the hyper geometric mixed with the incoming signal from the antenna, function gives a good approximation to (9). After which are the multi-path components of the several reductions and considerations, the transmitted chirp. This mixing allows having the correlation coefficient becomes multi-path components at low frequencies, these low frequencies can be sampled, digitized and stored in a computer to perform the required analysis. The three environments where the measurements took place were the following: 1) Indoors, in different floors around a building in an eight stories MHz were calculated. Figure 1 shows a typical building. Each floor form a rectangle with four channel transfer function. In figure 2 are shown the long corridors; two 66 m long and two 86 m long, fading characteristics for a 20 MHz sub channel, in the width of the corridors is 3 m, the total covered this figure there are 15 different lines, each one area was 912 square meters and the ceiling height corresponds to a 20 MHz sub channel in a 300 is 5 meters. The transmitter was static and the MHz bandwidth. To get this figure, the following receiver was moved around the corridors and in has been done; first, the fading information of the different floors, measurements were taken at complete 300 MHz bandwidth was divided in 15 specific distances in each corridor. 2) From a segments of 20 MHz each. Then, each point on that building to different building. These two buildings segment, that represents a 60 kHz sample, was are eight stories high, they have about the same measured and the result was compared to the next high and they are separated by about 200 meters. sample, then the next sample was compared and so The transmitter was located in the top of one of the on up to the complete 20 MHz bandwidth was buildings and the receiver was moved around compared. After that, a second 20 MHz sub specific locations inside the second building in all channel was applied the same procedure and so on eight different floors. 3) An urban environment up to the end of the 300 MHz bandwidth. To form around the city center, a commercial area with figure 3, the same procedure was followed, but in several high rising buildings. Each location in each this case the sub channel bandwidth was of 5 MHz, environment was sampled during one second and then this figure has 60 different lines which come 100 impulse responses were stored for this specific from the 300 MHz total bandwidth. location. When the measurements were taken, every location was sampled with 100 impulse Figure 2 shows that the maximum fading deep responses, an impulse response (IR) was taken on within the 20 MHz sub channel is lower than 14 dB that specific location every 10 milliseconds. in all the bandwidth. On the other hand, in the 5 MHz bandwidth the maximum fading deep was of III. FREQUENCY SELECTIVITY 18 dB. It is possible to see in figure 2, that most of PARAMETERS the lines follows a pattern, which means that the fading in all sub channels is about the same. In To carry out the calculation of frequency selective figure 2, most of the lines stay below 5 dB and only fading, each processed IR was split in samples of a few lines go higher than 6 dB; this could mean 60 kHz, which was the minimum sampled that deep fading in this bandwidth is rare. frequency, each 300 MHz bandwidth was sampled 5000 times every 10 milliseconds, this means that a sample was taken every 2 microseconds or every 60 kHz. Each IR was averaged over the complete second, meaning that every frequency was averaged 100 times for each IR. The level of the frequency response of each 60 kHz segment was calculated and recorded In each environment the channel transfer function for about 50 different locations were calculated and recorded. Each transfer function has 5000 sampled frequencies, i.e. 5000 segments for each location. Every sample represents the signal level on that segment, relative to the maximum over the Figure 1. Typical channel transfer function. complete 300 MHz bandwidth. The outdoors environment has a bandwidth of 120 MHz, the Figure 3 shows the calculations of fading indoors and indoor to outdoor environments has characteristics for the second environment with 15 300 MHz bandwidth. different 20 MHz sub channels, this figure shows that there are more dispersion of the lines, which The fading characteristics in each of the means there are more deep fading in the responses environments were calculated. For each of the of the IR. Since this environment is the locations the fading characteristics that were measurement of the propagation for the penetration calculated were; average delay spread, RMS delay of the signal in different floors in a building, higher spread, coherence bandwidth, channel transfer delay spread, (time dispersion) than the former function and frequency correlation. Using the environment was expected and therefore more channel transfer function for each IR, specific fading. fading characteristics for sub channels of 5 and 20 Another way to look at the statistics of the fading is correlation coefficient, the coherence bandwidth to calculate the CDF of this parameter. To make (Bc) is lower than 10 MHz most of the locations. the calculations of these CDF’s figures, the mean This is corroborated in figure 6, this figure shows of all locations in the involved environment were the average Bc for all locations in the indoors used. Figure 4 shows the CDF of the building-to- environment. Figure 7, shows the RMS delay building environment for both, the 20 and 5 MHz spread for all locations for the same environment. bandwidths, this figure shows that the fading deep Quick calculations comparing figure 11 results and for a 20 MHz sub channel is below 7 dB for 90% expression (13) show that, few calculated values of of the time. On the other hand, for the 5 MHz sub the versions of expression (13) match with the channel, the fades are below 5 dB for 90% of the measured values of figure 7. time. Figure 2 Indoor to outdoor fading Characteristics for a 20 MHz sub channel Figure 4. Fading CDF for indoor to outdoor for a 5 MHz sub channel Figure 3. Outdoor to indoor fading Characteristics For a 5 MHz sub channel Figure 5. Coherence bandwidth for indoors IV. COHERENCE BANDWIDTH EVALUATION. Figure 5 shows the frequency correlation of all locations in the indoors environment. To make this figure the following was done; first the PDP of all locations was calculated. Then a Fourier transform was performed on the PDP, which gave us the frequency correlation for all locations. Then the frequency correlation for each location was plotted in figure 5. On this figure, the thick and dashed line is the line for the maximum coherence bandwidth, when the transmitter and receiver are connected directly. In figure 5, one can see that at 0.9 Figure 6. Average of coherence bandwidth for indoors Figure 8 shows the frequency correlation for the outdoor to indoor environment, this figure shows that in this environment the Bc at frequency correlation of 0.9 is higher than the indoor environment, although the delay spread is not different is both environments. Figure 9, shows the average Bc for the outdoors to indoors environment, one can see in this figure, that the coherence bandwidth is higher than the indoor environment, which was an expected result, but the difference is higher than expected. In indoors the coherence bandwidth is not bigger than 20 MHz in Figure 8. Coherence bandwidth for outdoor to average. In the other hand, in the outdoor to indoor, indoor the average is about 100 MHz, here is relation of 5 to 1. The difference in RMS delay spread is 100 nS versus 200 nS, there is a relation of 2 to 1. Figure 9. Average of coherence bandwidth for indoor to outdoor Figure 7. RMS delay spread for indoors Table 1, shows the comparisons of Bc for the three environments with the different versions of Figure 11 shows the frequency correlation for the expressions 13 -16 and measured results. This outdoors environment. Figure 12, shows the Bc at table shows that the values of the expressions are frequency correlation of 0.9. In this case the Bc can always lower than the measured results, which not be compared to the Bc for the other two induce to conclude that the expressions were environments, since in this environment a lower underestimated, at least in these environments. bandwidth is evaluated, 120 MHz instead of 300 Moreover, it is possible to conclude that these MHz. Despite this difference and observing expressions were deduced with not enough figures 11 and 12, Bc is not significantly lower measured results. Also, table 1 show that the even when we have higher distances and higher relationship between delay spread and coherence delay spread. In outdoors the Bc is not bigger than 2 bandwidth, not necessarily is a single constant. MHz in average. In the other hand, the RMS delay spread is 1.5 µS in average. Figure 11. Coherence bandwidth for outdoors Figure 10. RMS delay spread for outdoors to indoors Figure 12. Average of coherence bandwidth for outdoors V. CONCLUSIONS. In this work the results of analysis of frequency Table 1. Coherence bandwidth calculations selective fading on two indoor and one outdoor environment have been presented. The three Value from Indoors Outdoors Outdoors environments analyzed demonstrate that the fading to indoors is within specific limits, these results could help to (13) 400 kHz 200 kHz 30 kHz the designers of adaptive receivers to estimate the (14) 4 MHz 2 MHz 300 kHz channel more accurately. The division of the (15) 3.3 MHz 2.5 MHz 250 kHz channel impulse bandwidth in segments of 20 and (16) 3.2 MHz 1.6 MHz 212 kHz 5 MHz bandwidths, allow the calculation of fading Measured0.9 5.3 MHz 12 MHz 300 kHz in the bandwidth of interest for OFDM and Measured0.5 19 MHz 72 MHz 5.6 MHz WCDMA transmission. Plots of the frequency selective fading will help for this assessment. The analysis of coherence bandwidth show that the expressions accepted in the literature for its calculation are not accurate and the accepted direct relationship between delay spread and coherence bandwidth is not simple. Also, additional work is require on try to determine how much the combined effect of Doppler spread, time variability and frequency offsets affects the transmission on multi-carrier signals as the ones on OFDM y CDMA Figure 13. RMS delay spread for outdoors References. 1. Jakes W. C., Microwave mobile communications, (Wiley, 1974). 2. 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UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.

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UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.

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