Estimating frequency correlation functions from propagation ...

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 6, AUGUST 2002

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Estimating Frequency Correlation Functions From Propagation Measurements on Fading Radio Channels: A Critical Review Robert J. C. Bultitude, Senior Member, IEEE

Abstract—Recently a significant number of propagation and channel modeling papers have reported channel parameters like rms delay spread, correlation bandwidth, and the Rician factor, derived by various new methods from instantaneous, or snapshot measurements. This conflicts with the original definitions of these parameters, which formally should be derived from time averages, under an assumption of ergodicity and applied for the assessment of time-averaged link performance. It appears that the origins of fading channel characterization parameters, and the conditions under which they can be estimated and applied are now often ignored, leaving interpretation of new results subject to skepticism. This paper, therefore, provides a critical review of the estimation of frequency correlation and one of these parameters, correlation bandwidth, using channel impulse response estimates derived from propagation measurements. The practical application for knowledge of frequency correlation is summarized. Then, the derivation of the equation relating a channel’s average power delay profile to its frequency correlation function via a Fourier transform is reviewed, with emphasis on conditions needed for validity. An alternate method, free from most of these conditions is also reported. Examples are given and comparisons are made of results from analyzes using the two methods to estimate frequency correlation on Rayleigh and Rician mobile radio channels, which are shown to have significantly different frequency correlation characteristics. Finally, a measure of frequency correlation that is free from ambiguity concerning the value (e.g., 0.5, 0.75, or 1/ ) at the correlation band edges is recommended. Index Terms—Correlation bandwidth, fading channels, frequency correlation, frequency-selective fading, multipath channels, radio channel modeling, radio propagation measurements, Rayleigh, Rician.

I. INTRODUCTION

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FTEN THE quality of radio channels must be assessed from propagation measurements well in advance of digital radio system design. If the channels of interest do not vary with time, received signal constellations can be computed directly by multiplication of a measured channel transfer function with the spectrum of proposed transmission signals. Error performance can then be predicted by superposition of probability density functions (pdfs) for noise on computed constellation points: there is no need for knowledge of channel correlations. However, if the channels are randomly time

Manuscript received March 14, 2001; revised September 30, 2001 and February 26, 2002. The author is with the Communications Research Centre, Ottawa, ON K2H 8S2, Canada (e-mail: [email protected]). Publisher Item Identifier 10.1109/JSAC.2002.801212.

varying (fading), statistical moments of their variations must be estimated if analytical methods are to be used for predicting performance. A key function involved in expressions for such moments on any fading channel is its spaced frequency correlation function (FCF) [1]. This portrays information on the correlation during fading among different spectral regions over the proposed transmission bandwidth. If there is significant degradation in this correlation, time-varying signal distortion and concomitant time-varying intersymbol interference (ISI) will occur. This could result in an irreducible error rate floor below which error performance cannot be improved by simple increases in signal-to-noise ratio (SNR). Bello and Nelin, in a classic analytical treatment [2] with an assumed Gaussian-shaped FCF, showed that with binary transmission, ISI induced irreducible error rate floors occur on a , is small compared fading channel unless the symbol rate, with the channel’s correlation bandwidth, . Thus, knowledge of , which they defined as the bandwidth at which the magnitude of the channel’s FCF decreases to 1/ , or 0.37, can be used to predict whether error floors will be encountered. Such knowledge is, therefore, of interest in the specification of ISI-mitigating technology, such as equalizers, for deciding upon OFDM subchannel bandwidths in order to avoid ISI and in other applications, such as deciding upon frequency spacings for frequency hopping and diversity systems. A channel’s FCF can sometimes be related through a Fourier transform to its average power delay profile (APDP), the average of the power represented in its impulse response at different multipath delays. However, very specific, wide sense stationary uncorrelated scattering (WSSUS) channel conditions are required for this relationship. Assuming directly from an these hold, to facilitate speed in estimating APDP, numerous publications [2]–[5] have reported relationand a power-weighted standard deviation ( ) ships between of multipath delays in the APDP. The parameter is commonly referred to as the rms delay spread of the channel. The product and has been shown [5] to have a lower bound, but can of itself vary considerably depending on the value of FCF magniis defined, and the shape of the APDP. Because tude at which of this, and the fact that the Fourier transform relationship is not always valid, it is clearly of interest to study alternate methods for estimating FCFs in order to determine . The subject of this paper is the FCF estimation problem. A derivation of the above-mentioned Fourier transform relationship is reviewed. Conditions for its validity are emphasized, including the fact that averages, rather than just measurement snapshots are required. Focus is then placed on applying the

0733-8716/02$17.00 © 2002 IEEE

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Fourier transform in the analysis of measured data. An alternate method [6], which is more robust to various channel characteristics is also presented. Both methods are then applied in the estimation of FCFs on Rayleigh and Rician fading mobile radio channels. Results are compared, and differences for the two channel types are discussed. The alternate method is recommended as preferable in comparison with the well-known Fourier transform method, because of its robustness. II. DERIVATION OF FCFS FROM CHANNEL IMPULSE RESPONSE FUNCTIONS Let a linear radio channel, defined as a composite of the influence all linear wave interactions that occur between the transmit and receive antennas of a single radio link, be represented by a time-varying linear filter, with complex low-pass . Its time-varying complex low-pass impulse response, transfer function is then given by (1) where represents delay. Equation (1) can be used [7] to write the time frequency correlation function for the channel as

(2) are unWhen random variations of the multipath at delay correlated with variations at delay , a condition referred to as uncorrelated scattering (US), and the complex components of and have means that are constant in , (2) can be rewritten as

(3) has been rewhere a subscript for is no longer required, . If a second placed by , and has been replaced by is wide sense stationary condition is established that (WSS) in , then can be replaced by , and can be replaced , to give by

or (4) and represent the correlation of random variaIn (4), tions in the channel’s transfer function and its impulse response, respectively. In [7], it is assumed that the complex components and have zero means and, therefore, there is of no subtraction of means in the equation that corresponds to (2) above. For a channel on which these components have nonzeromeans (e.g., a Rician channel), subtraction of them in the computation of such a correlation would allow calculation of the

similarity at different frequencies of random variations superimposed on the means. However, the purpose of the work described herein is to assess how important such random variations are in determining transmission characteristics on the channel. If there are nonzero means, the influence of these must also be considered, since they could be dominant, rendering the randomness insignificant. If the channel is slowly time-varying (i.e., symbol durations are small compared with its correlation time [2]), the distortion it imparts to each received symbol is nearly constant over a symbol interval. This distortion depends only upon amplitude and phase relationships at different frequencies across as it exists during the time that symbol is being received. An adverse combination of this distortion and noise causes decision errors in a receiver. Changes in distortion depend on whether or change not the amplitude and phase relationships in significantly as time evolves, which is reflected in the value of the channel’s FCF. In evaluating distortion at an instant of time, the whole transmission bandwidth must be considered as a single deterministic entity. Frequency selectivity within at that instant cannot be classified as a random process. Randomness is brought about when there is fading, the characteristics of which can either be the same at all frequencies (as in frequency-flat fading) or different for different frequencies (as in frequency-selective fading). In real-world applications, except in frequency-hopping systems, the only randomness that must take place along the axis, since can occur in transmit frequencies and bandwidths are fixed. The expectation associated with the FCF in (4) must, therefore, be taken along the axis. Its calculation is affected by multiplying the comat a given plex amplitude at each different frequency in ] by the complex time instant [i.e., in a given snapshot of amplitude at every other frequency at the same time instant. Results are then averaged along the axis. Since the products must from the be calculated by multiplying components of in (4) must be set equal to zero, giving same time instant, (5) The quantity in (5) is referred to as the channel’s spaced frequency correlation function. equal to zero is mistakenly inSometimes the setting of terpreted to imply a correlation calculated by averaging along the axis in a single snapshot of products formed with comhaving the same frequency spacings. Such ponents of a correlation is not relevant since signal distortion at any instant is determined by the specific amplitude and phase rela, rather than tionships at different frequencies across merely the correlation among them. Additionally, if the condition of US does not hold, frequency correlation characteristics change as a function of frequency, rather than just frequency spacings ( ). Therefore, if FCFs are computed by averaging along the axis outside of proposed transmission bandwidths, results could be incorrectly weighted by channel characteristics in spectral regions that would never be occupied by transmitted signals. Finally, averaging along in a snapshot makes it impossible for changes in multipath conditions as a function of time to be reflected in the resulting FCF. This mistake is one that is

BULTITUDE: ESTIMATING FREQUENCY CORRELATION FUNCTIONS

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frequently made in current publications on mobile radio propagation measurements. The result is that correlation bandwidths are reported on the basis of instantaneous measurements. These, so far as the author knows, have never been shown to be related to bit-error performance, yet they are often interpreted as having such a relationship. It is only when spectral phases and amplitudes over transmit bandwidths that would be occupied in proposed systems have the possibility of varying differently in time at different frequencies that correlation becomes important. The aforementioned misinterpretation often arises because of as an autocorrelation. The nomenclature is correference to is a two-dimensional rect, but it must be recalled that process, and its “autocorrelation” can, therefore, be calculated along any axis representing lags (i.e., spacings) in or or their Fourier transform equivalents that represent dimensions within is also sometimes taken to imply the process. The zeroing of removal of the expectation. It is important to note that the expectation must be retained. As a final step in the derivation of the Fourier transform relationship [7], if there is US, one can substitute (6) into (5), to give (7) is the channel’s APDP. For validity of this rewhere lationship, there is no requirement for Gaussian amplitude variand , or ations of the complex components of for them to have a zero-mean. However, as explained in the following section, these conditions can help justify the assumption of ergodicity, needed to replace expectations with time averages in practical measurement scenarios. This is one reason for the popularity of Bello’s Gaussian wide sense stationary uncorrelated channel (GWSSUS) model, which incorporates all the required conditions for the validity of (7). Problems arise, however, in the interpretation of results from thje analysis measured data when (7) is merely applied without verifying that the GWSSUS model is reasonable for the measured channels. III. ESTIMATION OF FCFS FROM MEASURED DATA Determination of FCFs as described in Section II requires an ensemble of channel impulse response functions. If ergodicity can be substantiated, ensemble averages can be replaced by time averages from a single sample record. Such a record can be derived using pseudonoise (PN) channel sounding measurements [3], [8], [9]. As part of the sounding process, time series of complex baseband channel impulse response estimates (IREs) can be formed from recordings of received PN sequences after their passage over a channel and complex down conversion to baseband. To derive actual time series, complete with information on Doppler phenomena, mobile channel data collection systems must be triggered to sample sequentially in time. Unless a constant velocity is maintained, the influence of moving objects in the environment could be obscured by triggering as a function of distance.

If (7) is to be applied, data recorded during experiments must be preanalyzed to substantiate assumptions of ergodicity, US, and a constant transfer function mean along the axis. Results must also be adjusted to compensate for nonuniformity in the sounding spectrum across its bandwidth [10], [11]. These practicalities are detailed in the following sections. A. Substantiating Assumptions of Ergodicity In [12], it is reported that a random process is ergodic if it is WSS and Gaussian. It is recognized that these criteria may not be unique. However, they afford the only method known by the author to demonstrate it is reasonable to assume an ergodic channel model. Unless this is demonstrated, the expectation in (7) can’t be replaced by time averages, making it necessary to conduct measurements over an ensemble. The latter, in most cases, would be prohibitively expensive. Sample records for which an assumption of ergodicity is not reasonable should be discarded, or at least it should be recognized that statistics from the analysis of them may not be representative of the channel. They represent only characteristics of the record under study. In such cases, for many engineering applications, heuristic or practical interpretation of experimental results on the basis of sample averages or sample distributions from multiple sequential measurements can lead to reasonable conclusions. However, care must be taken in using moment and distribution estimates from such samples as a basis for extended formal mathematical analyses. The WSS criterion can be verified by studying variations in the series of values taken on by spectral lines in a series of experas time evolved during imentally derived estimates of the experiment in which they were measured. Such values can be computed from IREs in accordance with (1). It should be ensured that the running means of the complex parts of a selected line or a few lines are constant. If the means are time varying, either the data interval to be analyzed must be shortened until a constant mean is realized, the time-varying mean must be removed, or another analysis approach that initially assumes a nonstationary process must be taken. Once a constant mean is verified, compliance with the remaining characteristic of a WSS process, that its autocorrelation is invariant to shifts of its index parameter, must be assessed. Either the evaluation of structure functions [13] or the RUN test can be used for this. Application of the RUN test for this purpose is clearly described in [12] and is the method used in the examples of Section IV. It involves counting the number of variance estimates for sequential subintervals of the data record that occur in runs on either side of their mean or median. The Gaussian criterion for ergodicity must be verified by methods other than time series analysis, since this would inherently assume ergodicity to begin with. One way is to make a central limit theorem argument based on knowledge that the sum of multiple [14], [15] multipath components would have a distribution that approaches Gaussian. It is shown in [14] that the pdf for the sum of independent components with identical uniform distributions differs from a Gaussian pdf by only 1% at its mean when the number of components is greater than 15. The equations in [14] were used to verify that this holds within three standard deviations ( ) of the mean, and slowly increases

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thereafter. The number of components required for a good comparison with a Gaussian pdf at 5 is 30. It is considered reasonable to expect that this number would have to be greater in cases where the components are not independent with identical distributions. Regardless, the occurrence of fewer than 15 multipath components is considered a good indicator of when Gaussian conditions cannot be assumed. The existence of considerably more components is considered to be an indication that Gaussian conditions might apply when information on distributions or independence is not available. In cases where WSS properties are demonstrated and there are sufficient multipath components that it is considered Gaussianity is a good assumption, ergodicity can also be assumed. Agreement between pdfs estimated from time-series measurements and a Gaussian model can then be taken as additional support for the assumption of Gaussianity. It is noteworthy that the GWSSUS model, and the modeling considerations discussed in the foregoing sections are based on consideration of many multipath contributors. They, therefore, , apply to random variations within spectral regions of rather than random variations at specific excess delays in . It is only within that the sum of all multipath components influences the channel characteristics, and this influence is exhibited across the entire channel bandwidth. B. Verification of US and a Constant Mean in Equation (3) implies that if there is US and the variation in complex amplitudes as a function of frequency across for any channel has a constant mean, its FCF should be symmetrical in with respect to any reference frequency. Therefore, if the FCF can be estimated by a method other than by application of (7), symmetry can be taken as a demonstration of both conditions. Such an alternate method is to directly cross-correlate time variations in the complex amplitudes of different spectral against amplilines within experimental estimates of , using tude variations of the line at a reference frequency, the equation (8) where

orders the frequencies across the measurement system bandwidth, and orders sequential estimates in the experimental time series. The result from applying (8) for all spectral lines is herein referred to as a frequency correlation estimate (FCE). If an FCE is not symmetrical with respect to the reference frequency, US and a constant mean for variations of the complex along the axis cannot be assumed. The components of double integral in (2) must, therefore, be retained. This corresponds to applying (8) with all frequencies across the channel bandwidth used, in turn, as the reference frequency to give a three-dimensional (3-D) function. Such 3-D correlation functions can be used, as in the moment equations of [16], for the

rigorous estimation of error performance, but it is noteworthy that results from this are inherently frequency-dependent. C. Compensation for Measurement System Transfer Function Nonuniformity If a radio channel is modeled as a linear filter, the impulse response of the filter model could, in theory, be determined as the cross correlation of a white noise input with the resulting filter output [9]. The infinite bandwidth of white noise makes it impossible to make a measurement of this, but an estimate could be measured by applying band-limited noise and performing the required cross correlation. In practice, however, because the time variability of noise makes it difficult to obtain repeatable results, PN signals are used instead. For sounding radio channels, PN sequences are transmitted on an appropriate carrier and resulting received signals are taken as the channel model output. These are then cross correlated with a replica of the originally transmitted sequence, regarded as the model input, to form the desired impulse response estimate. In addition to having finite bandwidth, PN sequences have a power spectrum that has a shape, rather than having uniform power across their bandwidths. These nonideal characteristics must be compensated for when deriving FCFs from PN measurements. The compensation method is simple [10], [11] and involves division of the results from (7) by the Fourier transform of the expectation of the squared magnitude of the impulse response of the measurement system. Estimates of FCFs derived by applying (7) to IREs from measurements are hereinafter referred to as frequency correlation profiles (FCPs). After compensation, they are referred to as adjusted FCPs, and are given by (9) where

denotes a Fourier transform, orders the IREs in the is the impulse response time series from one to , and of the measurement system with the transmitter connected back-to-back with the receiver via a transmission line. For a stable measurement system, the expectation in the denominator of (9) can either be ignored, or applied as a time average for noise reduction. It is recognized that the back-to-back test omits the antenna characteristics from this adjustment. However, even if an anechoic chamber were used for full system measurements, it would be difficult to eliminate all unwanted multipath components in order to measure only the system characteristics. For calibrating the antennas, it is sufficient to ensure their input impedance is properly matched to that of their feed lines and has a phase characteristic that is linear across the bandwidth of operation. D. Phase Synchronization on Mobile Radio Channels If propagation measurements are made using a moving receiver in areas where a consistent specular component is received, its phase will rotate deterministically at a rate that is


a function of the receiver velocity (direction and speed) with respect to the location of the specular source. This corresponds to a situation in which the received signal envelope would be Rician. However, unless the deterministic phase rotation is eliminated, the complex components in estimates of at any frequency exhibit mean values of zero, which is typical of Rayleigh, rather than Rician fading. The phase rotation corresponds to a sinusoidal variation of the nonzero means over distance, which has itself a mean of zero. If the specular component has the greatest of multipath powers, this rotation can be eliminated in IRE time series by subtracting its phase in each IRE snapshot from the phase of all samples in the snapshot. In cases where the specular signal is not so dominant, a linear fit to observed deterministic trends in the mean of the unwrapped phase of the vector sum of multipath components can be subtracted. These procedures are referred to herein as phase synchronization. On Rayleigh channels, the absence of a consistent phase variation will result in an inability to stop the phase rotation, and both complex signal components will retain an almost zero time average even after synchronization. On Rician channels, phase rotation will be reduced to a total radians, and variation within a range that is less than depends on the ratio of specular to random power ( ) in the received signal. The mean value of the real component will, therefore, be greater than zero. It will be shown in the following sections that phase synchronization has little effect on frequency correlation, but it is important in choosing the most appropriate model for the channel. This is particularly true when there are only a limited number (say, 100) of independent samples in the data under analysis, resulting in large error margins around experimentally-estimated envelope cumulative distribution functions (CDFs). In 10 is less than unity, tests such such cases, when as the Kolmogorov–Smirnov (KS) test [17] indicate that both the Rayleigh and the Rician models are acceptable. The presence or absence of a mean value in the real component after synchronization adds additional information with which to make a choice. This is important in some studies [16], since the selection of a Rician model reflects the presence of a coherent component in the received signal even though the envelope dis1. tributions for the two models are very similar when This influences directions taken in subsequent analyses in such studies. IV. DATA ANALYSIS EXAMPLES The following sections demonstrate the analysis of data measured on fading mobile radio channels. Two examples are given. On the channel represented by measured data in the first example, the envelope values at single spectral lines exhibit fading that nearly conforms to a Rayleigh model. In the second example, a Rician model is most appropriate. This was determined by substituting the means and variances of time series values for the envelope of the first spectral line above midband in experiinto the equations for Rayleigh and mental estimates of Rician CDFs. The KS test was then applied to verify goodness of fit between CDFs estimated from the data and the resulting CDF model.

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Fig. 1. Time series of IREs from measurements on a Rayleigh fading channel.

IREs used in the examples were determined from PN measurements on urban microcellular-type radio channels in Ottawa. A microcellular configuration was simulated by transmitting from a biconical (omnidirectional) antenna mounted at the top of a 6 m mast extended from the roof of a closed-in utility trailer parked near the curb on one of the streets in the measurement area. The trailer simulated a base station and housed the transmitter, which fed the antenna with a 40 dBm, 511 chip, 5 Mchps PN sequence modulated on a 1.98-GHz carrier using binary phase-shift keying (BPSK). The receiver was housed in a minivan equipped with a quarter-wavelength monopole mounted in the center of its roof, resulting in an in-situ receive antenna pattern that was measured to be omnidirectional within 3 dB. Received signals were sampled at a rate of 10 Msamples/s with windowing such that four sequence lengths (yielding four sequential IREs) could be recorded every 6.66 ms. The second IRE in each group of four was used in subsequent analyzes. This 150-Hz effective IRE sample rate is just under twice the previously-measured [16] Doppler spread at a very low relative power of 40 dB in the power spectrum of continuous wave (CW) measurements recorded when the minivan moved in traffic at typical urban speeds. The Nyquist criterion is, therefore, considered to have been satisfied. The transmit sequence ran continuously during the measurements, and all clocks and sampling circuits in both the transmitter and receiver systems were slaved to calibrated, coherent rubidium frequency standards. The measurements, therefore, preserved the channel’s amplitude and phase fluctuations from one IRE to another, allowing accurate representation of both spatial and time variations on the channel. A. Results for a Rayleigh Fading Channel ] in a Fig. 1 shows the magnitudes [i.e., estimates of series of 500 IREs that were studied for this example. Two preprocessing steps were taken to derive this time series from the recorded data. In the first step, every fourth recorded sequence was cross correlated with the sequence from a recording made with the measurement system connected back-to-back to give an IRE. In the second step, the IREs were processed to differentiate between noise spikes and legitimate multipath components

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Fig. 2. Time series of the midband sections of channel transfer function estimates for a Rayleigh fading channel.

using the method outlined in [18]. Parameters used to set the threshold for this were based on estimates of the noise variance in each IRE. They were such that only one noise spike would be mistakenly interpreted to be a multipath component in one out of every 100 IREs. For this example the noise exclusion threshold varied from IRE to IRE as the received signal faded, but had a mean 16 dB below the power of the strongest peak in the IRE, and a standard deviation of 2 dB. The extent along the delay axis of the IREs in Fig. 1, where the sample indexes represent 100-ns delay increments, indicates that there was considerable time delay spread on the channel. It can also be seen that the amplitude of most peaks varied from snapshot-to-snapshot. These variations could have been caused either by obstruction shadowing, or by the time-varying vector summation of more than one multipath component received within the 200-ns resolution interval of the measurement system. Although it is difficult to differentiate between the two phenomena, it is most likely that over the 3.3-s interval taken to record the data under study the rapid variations that are evident were the result of the latter multipath fading phenomenon. Fig. 2 is the time series of values for the middle 800-kHz wide sections of the channel transfer function estimates derived through a Fourier transform of each IRE in Fig. 1. The figure shows the large dynamic range of fading for each spectral line as time progressed. In addition, it can be seen that there was considerable nonuniformity of fading across the bandwidth shown, indicative of a quickly decaying FCF. In accordance with Section III-A, further processing of the data series was preceded by a demonstration that ergodicity is a reasonable assumption. The first step was to show that there were many received multipath components, forming the basis for an argument that the underlying channel process is Gaussian. Several analyzes were undertaken to demonstrate this. The first was to separate each IRE into multipath groups in 200-ns resolution intervals and count the intervals in which there was significant received power. Fig. 3(a) is a bar chart of the results, where it can be seen that there were 25 intervals with average relative powers greater than 25 dB below that of the strongest multipath group. Next, based on the knowledge

Fig. 3. Multipath group model for a Rayleigh fading channel. (a) Relative powers in each group. (b) ratio corresponding to the best fit Rician model for each group. (c) Correlation coefficient for complex amplitude variations of each group with respect to those of the first group. (d) Rough estimate of the fewest multipath components in each group.

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that at least two multipath components are required if a group is to exhibit fading, the time series of each was studied to ensure there was fading. It was also noted whether or not the nulls in the fading pattern remained above the noise floor of the measurement system. It was found that all groups faded and most of the fading remained above noise in the first 11 groups, but some excursion into the noise, as detected by the probability of false alarm algorithm, occurred in the other groups. The envelope fading distributions for all groups were found using the KS test to be acceptable fits to either the Rician or Rayleigh distribution. For the purpose of graphical representation, fading of those groups for which both Rayleigh and Rician models would be acceptable was identified only by the best fit Rician ratio. Thus, even though theoretically for Rayleigh value (e.g., 2 dB) in the results plotted in fading, a low Fig. 3(b) can be taken to indicate near Rayleigh fading for practical purposes. It is clear that many groups did exhibit values and, thus, can be considered to be the vector low sum of several components of comparable strength, plus noise. Of interest also were the cross correlations of fading in the different groups. High correlations would have indicated the possibility that adjacent groups overlapped the resolution interval boundaries and that counting components in each group would involve some duplication. However, correlations were fairly low, as exemplified by the cross correlation of fading in each group with that in group 1, plotted in Fig. 3(c). Finally, an eigenanalysis [19] and subsequent estimation of the number of signals in each multipath group was conducted by


estimating the minimum description length (MDL) associated with its time series. Spatial smoothing [20] was used to eliminate correlations in multipath signals. The ratio of the number of data in each of the subintervals used in this process to the total length of the series was set equal to 5/7, for reasons reported in [21]. After estimating the covariance matrix for the spatially smoothed data, its eigenvalues were plotted. In the data and there noiseless case, if each subinterval contained multipath components, there would be distinct were equal smaller noise larger signal eigenvalues and eigenvalues [19]. In practice, noise destroys the equality of the smaller eigenvalues, making it difficult to estimate the number of signals. However, the eigenvalues tend to cluster, with those having the larger values representing the signals. These can either be counted, requiring some subjectivity, or the MDL for the series can be calculated and used to estimate the number of signals as outlined in [19]. In the Rayleigh fading case under discussion here, the method in [19] gave estimates in the range between 30 and 40 for the number of components in each multipath group. It is not clear whether these are reasonable values, given the low resolution of the measurement system, or if the low SNR resulting from NLOS operation to realize Rayleigh fading resulted in failure of the method. However, it was possible to observe a number of eigenvalue clusters, enabling a rough estimate of the fewest signals believed to be present in each multipath group. These estimates are shown in Fig. 3(d). Results in Fig. 3 show that there were 25 multipath signal groups in adjacent resolution intervals above the noise floor. They also show that each group contains at least two signal components, since all groups fade in time. Finally, they show that rough estimates of the fewest multipath components in a group ranged between 6 and 15. These results all support the conclusion that over 100 multipath components were received on the channel under study. Whether or not they were all independent and identically distributed (i.i.d.), it is considered reasonable to conclude that the underlying channel ensemble would be well modeled as a Gaussian random process. This is because 100 is much greater the value of 15 cited earlier for a pdf within 1% of Gaussian in the i.i.d. case. The RUN test was applied to the spectral line at 19.57 kHz above midband (hereinafter referred to as the “19-kHz line”) to substantiate an assumption that the measured sample record is from a WSS process. The center frequency was not used for this because suppressed carrier operation, coupled with the possible presence of dc offsets in the equipment, could have obscured the true channel characteristics at this frequency. Fig. 4 is a plot depicting intermediate RUN test results. Fig. 4(a) shows the time series corresponding to the real part of the 19-kHz line before phase synchronization (hereinafter just written as “synchronization”). The constant mean is clear. Subplot (b) shows the 20 subinterval variances estimated during the test. There were seven runs on either side of the median shown by the straight line. From tables in [12] the value of seven is within the range of runs for 20 subintervals that would occur at the 95% confidence level for independent subinterval variances. This can be interpreted to indicate the channel process from which the sample record was taken can reasonably be as-

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Fig. 4. Intermediate results from the test for WSS characteristics on a Rayleigh fading channel. (a) Real part of the 19-kHz line before synchronization. (b) Subinterval variances estimated from the data represented in subplot (a). (c) Real part of the 19-kHz line after synchronization. (d) Subinterval variances estimated from the data represented in subplot (c).

sumed to be WSS. Subplot (c) shows the real part of the 19-kHz line after synchronization. Small changes can be seen, including a small positive shift of the mean. The six runs in Fig. 4(d) also indicate WSS properties. Since it has been demonstrated that the process under study would probably be Gaussian, and tests on the available sample record indicate it is probably also WSS, it is considered reasonable to assume it is ergodic. This allows a time series analysis , and in the estimation of an FCF. Fig. 5 shows the FCPs, , as well as the FCE, , for the time series under study. The 19-kHz line was taken as the reference for estimation of , using (8). Since the symmetry in the FCE shown by the curve through the circles in Fig. 5 indicates US with a constant mean in , and WSS temporal properties of the time series have already been demonstrated, the Fourier transform relationship in (7) can be , is shown by considered applicable. The unadjusted FCP, the the dashed curve. The solid curve without markers is FCP after its adjustment to compensate for the nonideal spectrum of the sounding signal. It can be seen that changes made by this adjustment are not very significant, particularly for small values of frequency shift with respect to midband ( ). The figure shows also that the FCE (circles), given by (8) is very similar to the FCPs, with the differences being negligible for small , and increasing as increases, but never really becoming significant. It is believed that the FCP is the less accurate of the two estimates because of the number of conditions required for validity of the method by which it was computed, and because of the necessity for making adjustments derived through

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Fig. 5.


FCF estimates for a Rayleigh fading channel.

Fig. 6. Time series of IREs from measurements on a Rician fading channel.

division by an unavoidably-noisy back-to-back measurement. The crosses indicate the post-synchronization FCE, and show that the synchronization had little effect. It can be observed in Fig. 5 that correlation decreases rapidly with frequency separation from midband on this Rayleigh channel. This makes it possible to define a bandwidth at which correlation drops to some value such as 75%, 50%, or 1/ , all used in previously published papers to define the correlation band edges. On Rician fading channels, however, correlation rarely drops below 50%, making it often impossible to define correlation bandwidths in terms of frequently used definitions. B. Results for a Rician Fading Channel Fig. 6 shows a series of 500 IREs from measurements on a line of sight channel. Equivalent CW fading (i.e., envelope fading at the 19-kHz line) was in accordance with a Rician equal to 8 dB. This ratio could have been estimodel for mated using either envelope, or complex data in accordance with the methods of [22] and [23], respectively. Herein, envelope data were analyzed as in [22] and the corresponding Rician model was then verified to be a good fit using the KS test at the 90% confidence level. Envelope fading at other spectral positions was

Fig. 7. Time series of the midband sections of channel transfer function estimates for a Rician fading channel.

also analyzed and it was found that varied by 2 dB across the center 2 MHz of the measurement bandwidth. Greater variations occurred outside that range because of the decreasing SNR resulting from the probing spectrum roll off. It can be seen from Fig. 6 that the IREs had a more limited extent along the delay axis than those shown in Fig. 1. In addition, the strongest peak is consistent and more dominant. This led to success in phase synchronization, which stopped phase rotation of the vector sum of the samples in the IREs and resulted in a strong positive mean for the real component of each spectral line. The middle 800-kHz wide sections of the series of transfer functions corresponding to Fig. 6 are shown in Fig. 7. The uniformity in fading across the 800-kHz bandwidth as well as its shallowness is notable in comparison with the series in Fig. 2, for the Rayleigh channel. The same processing to determine if a Gaussian model is appropriate as that reported in Section IV-A was applied. In preprocessing the same false alarm rate was allowed, but because of the greater SNR in this line of sight case, the average noise threshold that had to be applied to eliminate false multipath echoes was calculated to be 27 dB with respect to peak IRE power. Its standard deviation was 1.2 dB. Fig. 8(a) shows the average power of the multipath group in each resolution interval where the signal power was greater than 25 dB with respect to that of the strongest signal group. There are six of these spanning an excess delay range of 1.2 s. Fig. 8(b) shows the ratio for the best fit Rician distribution for each group. It can be seen that four of these were greater than zero and clearly conformed to a Rician, rather than a Rayleigh fading model. Correlations with fading of the first group are shown in Fig. 8(c) to be greater than in the Rayleigh case, but less than perfect. Application of the eigenanalysis for estimating the number of multipath components resulted in sharp MDL minima for between 7 and 11 components for the six groups shown. In total there are estimated to be 43 multipath components. It is, therefore, considered reasonable to assume a Gaussian model since 43 is considerably greater than 15. Fig. 9 shows intermediate results that were obtained during application of the RUN test. Fig. 9(a) shows the real part of


Fig. 8. Multipath group model for a Rician fading channel. (a) Relative powers in each group. (b) ratio corresponding to the best fit Rician model for each group. (c) Correlation coefficient for the complex amplitude variations of each group with respect to those of the first group. (d) Number of multipath components in each group.

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the 19-kHz spectral line. It can be seen to have a near-zero mean and a sinusoidal nature concomitant with the presence of a strong line of sight signal. The oscillations are a result of the motion of the receiver. Fig. 9(b) shows that there were ten runs, which, after reference to tables in [12], indicates that a WSS model is reasonable. Fig. 9(c) shows the real part of the same spectral line after synchronization. It is clear that the oscillations resulting from the vehicle motion have been removed, and there is a relatively large nonzero mean value, consistent with the Rician model. The results in Fig. 9(d) show that in this case also there were ten runs, allowing the conclusion that the synchronization has not altered the WSS properties exhibited by the sample record. As for the Rayleigh case, the foregoing demonstration that an assumption of Gaussian WSS properties is reasonable, also indicates that an assumption of ergodicity is reasonable and time series analysis can be applied. Fig. 10 shows FCFs resulting from , repreapplication of (7)–(9). The presynchronization FCE, sented by the circles, exhibits some asymmetry. However, such minor deviations from the perfect case can be expected in results from the analysis of measurements with a limited number of samples. Thus, it can be concluded that the Fourier transform in (7) is valid for practical purposes. In some results the author has previously obtained from the analysis of indoor measurements, asymmetries are very large, and clearly not the result of errors related to the number of samples.

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Fig. 9. Intermediate results from the test for WSS characteristics on a Rician fading channel. (a) Real part of the 19-kHz line before synchronization. (b) Subinterval variances estimated from the data represented in subplot (b). (c) Real part of the 19-kHz line after synchronization. (d) Subinterval variances estimated from the data represented in subplot (c).

Fig. 10.

FCF estimates for a Rician fading channel.

The dashed curve is a plot of , the unadjusted FCP before receiver synchronization. Comparison of this with the solid , after syncurve with no markers, which shows the FCP, chronization and shape adjustment, exemplifies the importance of the shape adjustment. The “shoulders” at about 3 MHz from midband result from the division in (9). On comparison of with it was found that in the midband region, the normalized curve corresponding to the latter was very slightly and greater than the normalized plot of the magnitude of the two curves crossed at approximately 2 MHz on either side of midband. This slight difference, which could have resulted from

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Fig. 11.


Midband section of a 3-D FCE for a Rician fading channel.

noise, or even some uncalibrated antenna input impedance imperfections, is the cause of the aberrations (shoulders) in the adjusted FCP, which are not typical of correlation functions. The FCE (circles), however, is free from these artifacts. Since no adjustments were required in the computation of this, it is considered likely to be the more accurate of the two estimators. The post-synchronization FCE is shown by the crosses. Differences between the two FCEs are considered negligible; again leading to the conclusion that synchronization has little effect on the estimation of FCEs from measured data. The dots, comprising the curve labeled “FKE,” in Fig. 10 represent the postsynchronization FCE that results when the long-term mean along the axis of each spectral line is removed before cross correlation. This represents decorrelation in the random components of the received signal as a function of their spacing from midband, but ignores the contribution of the deterministically evolving specular component. It indicates there would be performance degradation as a result of ISI at symbol rates far lower than those that could actually be transmitted on the channel without ISI problems. V. FCFS IN CASES WHERE AN ASSUMPTION OF UNCORRELATED SCATTERING CANNOT BE MADE When the channel of interest cannot be demonstrated to be an , given by (8) is asymmetrical. This US channel, the FCE, means that frequency correlation depends on the actual frequen, and the channel’s cies under consideration, rather than just FCF must be expressed in terms of the actual frequencies, and , making it 3-D. In addition, for predicting digital link performance, channel soundings must be done at the actual center frequency intended for use in the final system. However, it is considered probable that, on a statistical basis, results under ergodic conditions average out in time for center frequencies spaced by small percentage bandwidths, say 10%–20%. The 3-D FCE, over a bandwidth of 400 kHz from midband for the Rician channel discussed in Section IV-B, is plotted in Fig. 11. Two things can be observed. The first is the nonuniformities in correlation as a function of the reference frequency that make the 3-D representation necessary. For two diagonal cuts

near the center of the plot, results would be different. The differences in this example would be slight, and probably result from estimation errors due to limited data, as pointed out earlier, rather than truly correlated scattering. Sometimes, however, significant changes with reference frequency can be observed. The second observation is that correlation drops very slowly as a function of frequency. In fact, it never decreased below 90% over the whole measurement bandwidth for this example. This makes it impossible to define a correlation bandwidth in the usual way. It is clear, however, that on a frequency flat fading channel, the volume defined by this FCE would be a cube having unity height. Thus, the nonuniformities in this function for a channel with correlated scattering, as well as the degradation in frequency correlation, whether there is US or not, are well represented by the decrease in volume, with respect to that of a cube with unit height. In [16], it is recommended that the bandwidth over which this volume should be determined is that which would be occupied by transmitted signals. It is additionally shown that a scatter plot of FCE volumes defined in this way against analytically derived estimates of irreducible probability of error for DQPSK transmission on Rayleigh fading channels demonstrates a good correlation between the plotted parameters. Equations for polynomial fits to the running means of such plots for different transmission rates are also given. On Rician channels, error performance is better correlated with than with FCF volumes. The latter, in such cases are always close to unity. VI. SUMMARY AND CONCLUSION This paper reviewed the Fourier transform relationship between a fading channel’s APDP and its FCF. Emphasis was placed on validity requirements, including a demonstration of ergodicity, US, and that the channel’s transfer function has a constant mean in . The need for time averages rather than just snapshot results was also highlighted. Finally, it was pointed out that, although the channel’s transfer function must exhibit time variations, there is no requirement for either Gaussian or zero-mean distributions of the complex components of its baseband equivalent. For application in the analysis of propagation measurements, a method for ensuring the assumption of ergodicity is reasonable, and one for demonstrating that the channel exhibits US were outlined. The latter also provides an alternative to the Fourier transform for estimating the channel’s FCF from a time series of experimentally derived IREs. This method is concluded herein to be a better approach than the Fourier transform method, as it avoids some of the validity requirements listed in the foregoing paragraph. Additionally, it can be applied to estimate 3-D FCEs, needed when US cannot be demonstrated. Finally, a new measure of frequency correlation, which is the volume defined by the envelope of 3-D FCEs normalized to the intended transmission bandwidth, is proposed as a replacement for channel characterizations in terms of correlation bandwidth, regardless of whether there is US or not. This avoids any ambiguities in the values (e.g., 1/ , or 0.5) of correlation coefficients that should be taken to represent correlation band edges.


ACKNOWLEDGMENT The author would like to thank the anonymous IEEE referees and N. Adnani at The Communications Research Centre for their good constructive criticisms and for recommending changes to the original text to improve clarity. REFERENCES [1] R. J. C. Bultitude. (2000, Oct.) Estimating frequency correlation functions on fading radio channnels. Online Symposium for Electrical Engineers, Tech. [Online]. Available: www.techonline.com [2] P. A. Bello and B. D. Nelin, “The effect of frequency-selective fading on the binary error probabilities of incoherent and differentially-coherent matched filter receivers,” IRE Trans. Commun. Syst., vol. COM-11, pp. 170–186, June 1963. [3] D. C. Cox, “Delay Doppler characteristics of multipath propagation at 910 MHz in a suburban mobile radio environment,” IEEE Trans. Antennas Propagat., vol. AP-20, pp. 625–635, Dec. 1972. [4] M. J. Gans, “A power-spectral theory of propagation in the mobile-radio environment,” IEEE Trans. Veh. Technol., vol. VT-21, pp. 27–38, Feb. 1972. [5] B. H. Fleury, “New bounds for the variation of mean-square-continuous wide-sense-stationary processes,” IEEE Trans. Inform. Theory, vol. 41, pp. 849–852, May 1995. [6] R. J. C. Bultitude, “A study of coherence bandwidth measurements for frequency selective radio channels,” in Proc., IEEE Vehicular Technology Conf., Toronto, May 1983, pp. 269–278. [7] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1983, ch. 7. [8] R. J. C. Bultitude, “Measured characteristics of 800/900 MHz fading radio channels with high angle propagation through moderately dense foliage,” IEEE J. Select. Areas Commun., vol. SAC-5, pp. 116–127, Feb. 1987. [9] R. F. Linfield, R. W. Hubbard, and L. E. Pratt, “Transmission channel characterization by impulse response measurements,” U.S. Dep. of Commerce, OT Rep. 76–96, Aug. 1976. [10] D. C. Cox, “Time- and frequency-domain characterizations of multipath propagation at 910 MHz in a suburban mobile-radio environment,” Radio Sci., vol. 7, pp. 1069–1077, Dec. 1972. [11] T. Hagfors, “Some properties of radio waves reflected from the moon and their relation to the lunar surface,” J. Geophys. Res., vol. 66, pp. 777–781, Mar. 1961. [12] J. S. Bendat and A. G. Piersol, Measurement and Analysis of Random Data. New York: Wiley, 1966, pp. 219–223. [13] R. H. Clarke, “A statistical theory of mobile radio reception,” Bell Syst.Tech. J., pp. 957–1000, July–Aug. 1968. [14] P. Beckmann, Probability in Communication Engineering. New York: Harcourt, Brace and World, 1967, pp. 103–105. [15] M. Slack, “The probability distributions of sinusoidal oscillations combined in random phase,” J. Inst. Elect. Eng., pt. III, vol. 93, pp. 76–81, 1946.

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[16] R. J. C. Bultitude and A. W. Leslie, “Propagation measurement based probability of error predictions for digital land mobile radio,” IEEE Trans. Veh. Technol., vol. 46, pp. 717–729, Aug. 1997. [17] W. J. Dixon and F. J. Massey, Introduction to Statistical Analysis, 3rd ed. New York: McGraw-Hill, 1969, pp. 345–347. [18] E. S. Sousa, V. M. Jovanovic, and C. Daigneault, “Delay spread measurements for the digital cellular channel in Toronto,” IEEE Trans. Veh. Technol., vol. 43, pp. 837–847, Nov. 1994. [19] G. Xu, R. H. Roy, and T. Kailath, “Detection of number of sources via exploitation of centro-symmetry property,” IEEE Trans. Signal Processing, vol. 42, pp. 102–112, Jan. 1994. [20] T. J. Shan, M. Wax, and T. Kailath, “On spatial smoothing for direction-of-arrival estimation of coherent signals,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 806–811, Aug. 1985. [21] G. V. Serebryakov, “Direction of arrival estimation of correlated sources by adaptive beamforming,” IEEE Trans. Signal Processing, vol. 43, pp. 2782–2787, Nov. 1995. [22] L. J. Greenstein, D. G. Michelson, and V. Erceg, “Moment method estimation of the Ricean -factor,” IEEE Commun. Lett., vol. 3, pp. 175–176, June 1999. [23] W. T. Webb and L. Hanzo, Modern Quadrature Amplitude Modulation. London, U.K.: IEE Press, 1994, p. 60.

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Robert J. C. Bultitude (S’74–M’78–SM’00) received the B.Sc.Eng. degree in electrical engineering from the University of New Brunswick, Fredericton, NB, Canada, in 1975, and the M.Eng. and Ph.D. degrees in electronics engineering from Carleton University, Ottawa, ON, Canada, in 1979 and 1987, respectively. He was with Hoyles Niblock Associates in Vancouver, Canada, in 1975, where he worked for two years as a Telecommunications Systems Engineer. In 1980, he worked on radar head end systems with Leigh Instruments, Ottawa, then joined the Communications Research Centre (CRC), Ottawa, in 1981. From 1981 to 1989, he was a Telecommunications Research Engineer working on Radio Propagation Research for applications in mobile radio systems. From 1989 to 1996, he was a manager of CRC’s Land Mobile and Indoor Radio Propagation Group. During 1998–1999, he was Research Leader, Land Mobile and Point-Multipoint Radio Propagation Research at CRC and a Visiting Scientist at Eindhoven University of Technology, Eindhoven, The Netherlands, collaborating on radio channel modeling. Since 2000, he has been a Senior Research Scientist at CRC, working on mobile radio propagation and channel modeling. He is an Adjunct Professor in the Department of Systems and Computer Engineering at Carleton University, Dr. Bultitude is a member of Sigma Xi.