IEEE TRANSACTIONS ON COMMUNICATION TECHNOLOGY VOL. 13, NO. 3 SEPTEMBER, 1965

Some Techniques for the Instantaneous Real-Time Measurement of Multipath and Doppler Spread

Abstract-The design optimization of digital communication sys- tems using selectively fading transmission media such as HF and troposcatter requires knowledge of the selective fading characteris- tics of the media. A minimal characterization is provided by the gross channel parameters of multipath spread and Doppler spread. This paper presents several techniques which will allow the simul- taneous measurement of multipath and Doppler spread in real time with a minimum of equipment complexity. The techniques are first divided into two classes depending upon whether measurements are made on in-phase and quadrature components of received carriers or on envelopes of received carriers. A further subdivision is made according to two different techniques of multipath spread measure- ment, one of which, called the SSB method, uses two transmitted carriers while the other, called the FM method, uses a frequency- modulated carrier. The envelope measurements are the simplest and should provide considerable help in any large scale testing of the selective fading propert:.es of the HF and troposcatter media.

D I. INTRODUCTION

URING RECENT YEARS there has been an in- creasing interest in the transmission of digital in-

formation over HF and troposcatter links. The perform- ances of such digital comnlunication systems are much more sensitive to multipath and Doppler spreading effects than the more common voice communication systems. It has been shown [5]-[SI, [ l o ] , [ l l ] that relatively small amounts of frequency selective and time selective (fast) fading can cause noticeable system degradation and can result in the existence of an “irreducible” error probability, i.e., an error probability that cannot be reduced by in- creasing transmitter power. It follows that the design Optimization of digital communications systems, using selectively fading channels (such as HF and troposcatter), requires some knowledge of the statistics of the fading parameters of these media. Although some measurements on fading channels have been performed, they have not re- sulted in the statistical data necessary for the prediction of digital communication system performance. The present paper proposes methods of obtaining the minimum essential data on the fading properties of the HF and troposcatter channels so as to fill the gap in available knowledge and thus aid in the design of digital data systems.

If the statistics of the channel were time-invariant and if equipment complexity and cost were unimportant, one could give a clear-cut answer as to what should be meas- ured to characterize the fading channel, namely, all order probability density functions for the transfer function

Manuscript received May 19, 1965; revised June 4, 1965. Pre- sented as Paper CP65-524 at the 1965 IEEE Communications Convention, Bolllder, Colo.

was formerly wi th ADCOM, Inc., Cambridge, Mass. The author is with SIGNATRON, Inc., Lexington, Mass. He

(or impulse response) of the channel. With regard to the channel transfer function, this involves the transmission of an arbitrary number of carriers at different frequencies and the measurement of the joint probability density function of the amplitudes and phases of the received fading carriers. Such information would be sufficient to characterize statistically the received process correspond- ing to any digital signal.

Unfortunately, the channel is nonstationary and equip- ment cost and complexity are important. Thus, a realistic measurement program must take into consideration two problems : the nonstationary behavior of the channel and the complexity and cost of the associated equipment. In prac- tice, the latter problem is handled by measuring the bare minimum number of essential “gross” parameters that are needed to characterize those aspects of the channel that are important with respect to the communication problem at hand. The former problem is handled by measuring the gross channel parameters over some set time interval, say TI , over which the statistics are (hopefully) nearly sta- tionary and then determining probability distribution func- tions of the T1-measured values throughout the day, month, year, etc.

The heavy early emphasis on received signal power meas- urements was a direct reflection of the fact that the essential unknown parameter in early communication systems was received SNR. While received SNR will always be im- portant, other parameters related to the selective fading of the channel have become important. In particular it is clear that the multipath spread and Doppler spread in addition to SNR are basic parameters needed to ascertain the performance of digital data modems. Since both param- eters are important in an optimized modem, one should measure both individual and joint probability distribu- tions of multipath and Doppler spread.

Strictly speaking, even for a channel with Gaussian fad- ing statistics, the error probability caused by time and frequency selective fading depends upon correlation func- tions of the channel transfer function. Thus, if the neces- sary means are available, such correlation functions should be measured [3]. However, when such measurements can- not be made, the simpler measurements of Doppler spread and multipath spread would provide invaluable informa- tion ’or digital system design [4].

In later subsections we shall present two different ap- proaches to the “instantaneous” real-time measurement of Doppler and multipath spread. One approach, called the complex envelope approach, involves the in-phase and quadrature components of received carriers, while the

255

286 IEEE TRANSACTIONS ON COMMUNICATION TECHNOLOGY SEPTEMBER

other approach, called the envelope approach, deals only with the envelopes of received carriers. The former pro- cedure is more difficult, but, in principle, more accurate. The latter approach is quite sinlple and inexpensive to implement hut involves the assunlption that a received carrier is a narrow-band Gaussian process. This assump- tion appears to be sufficiently accurate for the HF and troposcatter channels to render this latter approach of considerable help in any large-scale testing of the HF and troposcatter media.

The determination of the Doppler spread parameter re- quires the transnlission of a single carrier, and only one measurement technique is presented for each approach. In the case of multipath spread measurement, two meas- urement techniques are given for each approach. One technique, called the single sideband (SSB) technique, involves the transnlission of two carriers. The other tech- nique, called the FRil technique, involves the transmission of a sinusoidally frequency-modulated carrier. It appears that the SSB technique is particularly adapted to HF measurenlent since the carriers may be unnlodulated sub- carrier tones, allowing data transnlission to take place during testing. On the other hand, the FRil technique is probably better adapted to troposcatter measurement due to the almost universal existence of FRiI equipment on troposcatter links.

11. DEFINITION OF MULTIPATH SPREAD AND DOPPLER SPREAD

The transnlission of a sinusoid through a fading radio channel results in the reception of a narrow-band random process. Let z(t) denote the received narrow-band wave- form resulting from the transnlission of a sinusoid f C/S

away from a “carrier” frequency of f o c/s. Then s(t) may be expressed as

z(t) = Re( T ( f , t)ej2”uo+’)t} (1)

where T u , t ) , the complex envelope of the received narrow- band process, is the time variant transfer function of the medium when measured with respect to the carrier fre- quency fo.

The correlation properties of this transfer function are defined by the correlation function

R/,t(T, 0) = T*(f, OT(S 0, t + TI (‘2)

where ( ) * denotes the complex conjugate and the overline indicates an ensemble average.

Bello [l] has derived the quasi-WSSUS channel to ac- count for the fading characteristics observed on many radio channels. A WSSUS (wide-sense stationary uncorrelated scattering) channel is one that, for analytical purposes, may be represented as a continuum of uncorrelated scat- ters fluctuating with wide-sense stationary statistics. A necessary and sufficient condition for a channel to be WSSUS is that the channel corre.lation function Rf, 1 ( ~ , 0) be independent off, t , i.e., independent of the location of the time interval and the frequency band within which the

measurement of the correlation function was made. Thus

Rf. t ( T , 0) = R(T, 0) (3)

for the WSSUS channel. The quasi-WSSUS or QWSSUS channel is one in which

Ef, t ( ~ , h) changes sufficikntly “slowly” with changes in f and t. In the context of the present discussion we define the QWSSUS channel as one for which R f v l ( T , 0) changes very little for values of f , t within a time-frequency rec- tangle whose time and frequency dimensions are equal to the time interval and bandwidth, respectively, required to measure the Doppler spread and multipath spread param- eters.

The utility of the QWSSUS channel concept is that, when applicable, i t allows one to represent the chanuel as a WSSUS channel, which is easy to deal with analytically.

The measurement problem is complicated by the fact that physical channels do not exhibit statistical regularity over indefinitely long intervals of time. Thus, there is an inherent conflict between the desire to have a long meas- urement time in order to reduce measurement errors due to finite sample size, and the desire to have a short enough nleasurenlent time so that the desired statistical param- eters have changed negligibly due to the instability of the statistical structure of the channel.

Most radio channels exhibit a “fast” fading (e.g., a few cycles per second) superimposed upon a “slow” fading (e.g., hourly or diurnal variations). In order to measure the channel parameters by time averages, the measurement time T , must satisfy the inequality

where e,,, is the bandwidth of the slow fluctuations. On the other hand, to keep the measurement errors due to finite sample size small, one nlust have

1 T , >> -

Bf where B, is the bandwidth of the fast fluctuations. It follows that the short-time correlation functions can be measured only if the bandwidth of the fast fluctuations is a few orders of magnitude greater than the slow fluctua- tions. Fortunately, it appears that many radio channels exhibit this property. The following discussion assumes that (4) and (5) are satisfied, and that over the measure- ment interval we can represent the channel by a hypo- thetical WSSUS channel.

When (3) is satisfied, the transfer function of the channel TU, t ) becomes a wide-sense stationary process in both the time and frequency variables. The time correlation function p ( ~ ) given by

p ( ~ ) = R(T, 0) = T*U, t ) T ( f , t + 7) (6)

is the autocorrelation function of the (complex) sinsuoidal response of the channel. The frequency correlation func- tion g(0) given by

1965 BELLO: MULTIP~A.TH‘I~R’DIDOPP,LER.SPRE;ID 287

q(f22) = N O , Q) = T*(f, + f2, 0 (7)

is the cross correlation function between the complex en- velopes of two received carriers spaced f2 c/s apart.

The Doppler spread parameter is a measure of the dis- persion in Doppler shifts suffered by a process in passing through a channel. This is entirely equivalent to the spec- tral width of a received carrier or to the “width” of the power spectrum of the received carrier. Thus, i t is appro- priate to deal with the Fourier transform of p ( ~ ) , given by

~ ( v ) = ,f p(T)e-j21rvrcLT (8)

where P(v) is the power density spectrum of T(f, 2) con- sidered as a time function. Note that P(v) is the power spectrum of the complex envelope of the sinsuoidal re- sponse. The power spectrum of the actual real sinusoidal response is given by

H(v) = ‘/~[P(v - j n ) + l’( - v - f o ) ] . (9)

i\Jany measures of “width” are possible but we use the rms width to define Doppler spread. Thus, we define

as the Doppler spread parameter, where

- s vP(v)dv J P(v)dv v =

is the “centroid” of the power spectrum or the “mean” Doppler shift. The quantity D/2 is analogous to the stand- ard deviation of a probability density function

The standard deviation is a useful measure of width if the corresponding density function has tails that drop suffi- ciently rapidly. I n a physical channel the .Doppler shifts are caused by the motion of physical entities, and it ap- pears reasonable to assume that the corresponding range ‘of velocities is sufficiently limited to make the rrns Doppler spread a meaningful channel parameter.

The multipath spread parameter is a measure of the ,dispersion in path delays suffered by a process in passing through the channel. This is entirely equivalent to the ‘“width” of the impulse response of the channel. Now it may be shown ( [9] and Section IVD of [ 1 1) that the Fourier transform of the frequency correlation function q(f2) yields a density distribution of path delays Q(E) given by

:Since q(f2) is the correlation function of T u , t ) with t fixed, & ( E ) is actually the power density spectrum of T ( f , t ) considercd as a function of the frequency variable f.

To give a feeling for the physical meaning of & ( E ) , we note that &([ )dt is the amount of power received due to transmission delays in the range ( E , + c@), where the total power received is given by 1 T ( j , t ) [ 2. Thus we note from (7) and (13) that

d o ) = IT(f, 0 1 2 = .f Q(tM. (14)

We shall also use the rms measure of width to define multipath spread. Thus we define

as the nwltipath spread parameter where

is the centroid of the delay power density spectrum or the ‘(mean” path delay. It is felt that the tails of Q([) drop sufficiently rapidly to make J4 a practically useful measure of multipath spread.

111. THE COMPLEX ENVELOPE APPROACH

I n this section we shall present nleasurenlent schemes for multipath and Doppler spread which make use of in- phase and quadrature components of received carriers. Section III-A will deal with the Doppler spread rneasure- ment, and Section III-B will deal with the multipath spread measurement.

A. Doppler Spread Measurement In the definition of Doppler spread in (lo), we deal with

moments of the power spectrum of T ( f , t ) , where f is con- sidered fixed, i.e., we deal with moments of P(v) , the power spectrum of the complex envelope of a received carrier transmitted at the frequency fo + f. Since the statistical properties of T(f, t ) are assumed independent off,we may deal with

xr(t) T ( f , t ) ; f fixed (17)

the complex envelope of a received carrier which was transmitted at a frequency of fo + f C/S.

The various moments of P(v) needed to determine D, the Doppler spread parameter, can be evaluated in terms of the elements of the moment matrix of the pair of random variables [ ~ ~ ( t ) , (cZ/dt)z,(t) ] :

where we have used the dot to denote differentation with respect to t. To determine the relationship between 1110-

ments of P ( v ) and the foregoing averages, we note first that the cross-power density spectrum between the mth and nth derivatives of xr( t ) is just (-$TV)~ (~~Tv)”P(v).

285 IEEE TRANSACTIONS ON COMMUNICATION TECHNOLOGY SEPTEMBER

Thus,

clmx,*(t) dnxf(t) _______ - dt" dt"

- ( - j 2 7 r ) " ( j 2 ~ ) ~ J V ~ + ~ P ( V ) ~ U (18)

and

m= J P(u)dv ~ I$,( = (2a)2J v2P(v)dv (20) __

~ f * S f = j 2 ~ J vP(v)dv. (21)

Using (19), (20), and (21) in (lo), the following expres- sion for D results:

g ( t ) m = jaaJ.fp,(f)df = 0 (2'7)

where P g ( f ) is the power spectrum of g(t) and the integral in (27) vanisher because PJf) is an even function. Prom (27) we deduce that

X f 2 f = y,jf = 0 __ -_

(28)

and that __

Z,*i, = j(x$ - Uk). (29) - -

Thus, use of (24), (25) , and (29) i n (22) yields the follow- ing expression for D in terms of averages involving the in- phase and quadrature components of a received carrier:

In order to determine D experimentally, the various averages must be interpreted as time averages as shown in (30) by triangular brackets. The in-phase and quadrature components zf, yf can be determined by multiplying the received carrier by both a local carrier and a 90" shifted local carrier at the same frequency as the received carrier (or as near to the same frequency as possible) and then ex- tracting the low-frequency components. Strictly speaking, D is independent of mean Doppler shift, and thus precise knowledge of the received carrier frequency is not neces-

sary. However, as the local carrier frequency departs from the received carrier frequency the extracted z(t) and y(t) increase in bandwidth, necessitating larger bandwidth fdters and passing more noise. Thus, from the point of view of maximizing signal-to-noise ratio, it is desirable to keep the local carrier frequency as near as possible t80 the received carrier frequency.

The operations indicated in (30) are readily performed with the aid of standard analog computer components to produce an "instantaneous" real-time measurement of D.

B. Multipath Spread Measwentent

From a strictly mathematical point of view the multi- path spread measurement problem is entirely analogous to the Doppler spread measurement problem. Thus, while the latter case involves the stationary time process T(f, t) with f considered fixed, the former case involves the sta- tionary f process T(f, t ) with t fixed. The Doppler spread parameter has an identical expression in terms of the mo- ments of the power spectrum of T u , t), f fixed, as the multi- path spread parameter has in terms of the moments of the power spectrum of T ( f , t ) , t fixed. Thus, if we write

Z,(f) = T ( f , 2) ; t fixed, (31)

then by direct analogy with (22) me may express the multi- path spread parameter in the form

in which the dot, denotes differentiation with respect to f , not t.

Unfortunately it does not appear possible to obtain the frequency functions Z,df) and Z,(;f) in any simple fashion. This difficulty can be resolved if me realize that only cer- tain averages involving these functions are required, rather than the functions themselves. Because of the assumed mide-sense stationary properties of T(f, t) in both f and t , it is possible to convert the ensemble averages of the fre- quency functions to ensemble average of time function and thence to time averages of t h e functions (assuming t,he ergodic hypothesis, of course). Thus we note

_ _ _ _ _ lZ,(f)]2 = lT( f , t)12; t fixed

1965 BELLO: MULTIPATH AND DOPPLER SPREAD 359

The function dz,(t)/df is a time function to be inter- preted as the rate of change in the complex amplitude of the received carrier with change in carrier frequency. Using (33) to (3.5) we may rewrite our expression for ill in terms of time averages as follows:

The two different techniques of multipath spread meas- urement to be described below differ in their method of obtaining the time function dz,/df. Both methods are approximate but appear capable of yielding accurate re- sults.

1) The SSB Technique: An approximate determination of dz,/dj may be achieved by transmitting two carriers separated by a frequency small compared to the correlation bandwidth. Thus, suppose two carriers are transmitted separated by F c/s. Ignoring temporarily the problems of oscillator instability, the complex envelopes of the two received carriers may be represented as ~ , + ~ ( t ) and x f ( t ) . By forming the normalized difference

(37)

when F is snlall compared to the correlation bandwidth, we can obtain a close approxinmtion to bz,(t)/df.

By analogy with (30), then, we can immediately es- press M in terms of averages over in-phase and quadrature components as follows :

where

(39)

The components dx,, dy, can be determined experiment- ally by multiplying the two received carriers by correspond- ing local carriers at the same frequency (and with appro- priate 90" phase shifts) to obtain the in-phase and quad- rature components corresponding to ~ ~ + ~ ( t ) and z,(t).

Then taking appropriate differences as indicated in (39) (40), clx, and dy, can be determined and M evaluated.

For the foregoing measurements to be meaningful, care- ful attention must be given to the method of generation of the two carrier frequencies. Instabilities in the two carrier frequencies may be divided into conmlon and differential frequency instabilities. The common instabilities are pres- ent during use of the medium as a communication system and may be regarded as part of t.he channel. Thus critical requirements center around the stability of the difference frequency between the two carriers (both at Branslnitter and receiver). This difference frequency can be made as stable as state-of-the-art by SSB modulating a frequency standard output to produce one carrier, the other one being the SSB ca,rrier. A study of the effects of oscillator in- stabilities has indicated that crystal contro'led frequency standards exist with sufficient stability to allow t'he use of the foregoing measurement technique.

2) The F M Technique: Another technique for the approximate deternlination of dz,/df involves the use of a frequency-modulated carrier. It will be assumed that the conditions are satisfied to use the quasi-stationary ap- proximation to the channel response to an Fl\4 signal. Thus, if the frequency modulation is given by Q ( t ) , where 27r+(t) is the corresponding phase modulation, the complex envelope of the channel response ~ ( t ) is given by

w(t> = e j 2 T ( ~ ( f ) + ' t ) / 1 ( ~ ( t ) + f , t ) . (41)

At the receiver it is assumed that the received mave- form is multiplied by both a replica and a 90" shifted replica of the transmitted signal and the resulting low- frequency components extracted. These operations result in the extraction of the real and imaginary parts of T ( Q + f , t ) . Consequently, we can assunle that the receiver ex- tracts

, ~ 4 ( t ) zs T ( Q ( t ) + f , t ) . (42)

Now assuming that the peak deviation, Qnlaxt is much smaller than the correlation bandwidth,

or

Since Q ( t ) is a sinusoid whose frequency is chosen much higher than the fading rate, it follows that the second term is a narrow-band process centered on the modulation fre- quency whose complex amplitude, apart from a complex constant, is equal to the desired function dz,(t)/d.f. Al- though the magnitude of the complex constant is readily seen to be given by the peak frequency deviation Qmax, the phase of the complex constant is unknown. However, it is quickly seen by examination of (36) that M is unaffected if dz,/dj is multiplied by an arbitrary phase factor ej'.

I n view of the foregoing, it is clear that, from the real

290 IEEE TRANSACTIONS ON COMMUNICATION TECHNOLOGY SEPTEMBER

and imaginary parts of eo,(t) which have been extracted, one may obtain the real and imaginary parts of both zf ( t ) and bz,( t ) /df . Thus, by analogy wit,h (30), one may com- pute

34 =

V ((sf2) + (YrZ))Z

(45)

Although it may not appear at first glance that the PA4 technique is as vulnerable to oscillator instability as the SSB technique, closer study shows that similar stabilities are required in both cases.

IV. THE ENVELOPE APPROACH

Section I11 presented techniques for the “instantaneous” measurement of Doppler and multipath spread, which re- quire the extraction of in-phase and quadrature compo- nents of receiver carriers. In this section we shall present simpler techniques for the extraction of Doppler and multi- path spread, which require the extraction of only envelopes of received carriers. I n addition, and perhaps most im- portant, the technique can be used with nonlinear ampli- fiers such as the commonly used and readily available (approximately) logarithmic amplifiers, which do not re- quire large dynamic ranges at their outputs. For the tech- niques proposed in this section to be strictly correct, it is necessary to assume that the transmission of a carrier results in the reception of narrow band Gaussian noise. However, slight departures from Gaussianness should not affect the measured parameter significantly.

In Section IV-A following, we present the relationship between the rnls bandwidth of the output of a nonlinear (no-memory) envelope detector [equivalent to a cascade of nonlinear (no-memory) amplifier with linear envelope detector] and the rnls bandwidth of an input narrowband Gaussian process. Then in Sections IV-B and IV-C we develop techniques for the measurement of the Doppler and nmltipath spread parameters in terms of received envelopes.

A . The vms Bandwidth of Nonlinearly Envelope Detected Nuwow-Bund Gaussian Noise

It has been demonstrated by Bello [2] that the ratio of the rms bandwidth of the output of a nonlinear envelope detector to the rms bandwidth of a narrow-band input Gaussian process is a constant depending only upon the nonlinear device and not upon the shape of the input spec- trum. Thus, it is shown that this constant, say a, is given by

where K ( s ) is a function directly related to the nonlinear characteristics of the envelope detector. Specifically, if I ( t ) denotes the envelope of the input process and O(t) the envelope of the output process, then

0 = K [ P ] . (47)

With this definition for K ( .), a square law envelope detec- tor corresponds to K ( z ) = x, while a linear envelope de- tector corresponds to K ( s ) = di. It is readily shown by direct calculation that for a general vth law device, i.e.,

tbe constant a is given sinlply by

a = dv/2. (49)

Thus, if a linear envelope detector and linear amplifiers are used, the output envelope has an rrns bandwidth 1/ a times the rms bandwidth of the input narrow-band process. In practice, a nonlinear amplifier would probably be used. It is a relatively sinlple matter to measure the character- istics of the amplifier-envelope-detector cascade and then evaluate the proportionality constant a by numerical methods. In the discussion that follows it will be assumed that a is known.

B. Doppler Spread 2liIeusurenaent From the results of the previous section it is clear that

we may measure the rnls bandwidth of a Gaussian channel by transmitting a carrier, measuring the rnls bandwidth of the envelope detected received waveform, .and then divid- ing this value by the a of the amplifier-envelope-detector combination. Thus, defining e f ( t ) as the detected envelope and P, (v ) as ihs power spectrum, we have the following expression for the Doppler spread parameter

(50)

Note that the term involving F is nlissing because P,(v) is of necessity an even function, er(t) being real.

If the nonlinear characteristic is defined by K ( .) as in the previous section, then

er(t) = K [ , T ( f , t);”]; f fixed. (51)

We may determine an expression for D in terms of operations upon e f ( t ) by noting that

[8,(t)]Z = (27r)y- v 2 P e ( v ) d v . (53) Thus

where we have converted to time averages. Note that this. method of computing D is considerably simpler than its counterpart involving in-phase and quadrature compo- nents.

19G5 BELLO: MULTIPATH AND DOPPLER SPREAD 291

Fig. 1. SSB technique for the simultaneous measurement of Doppler spread and multipath spread using envelope measurements alone.

spread measurement discussed in Section IV-B exists also for multipnth spread measurement.

Thus we write immediately by analogy with (54) that

(55)

where

E,O = K[/TCf, t ) lz] l ; t fixed (56)

(57)

The ensemble averages of the frequency functions in (55) may be converted into time averages of time functions as was done for the analogous situation in Section 111-B. Thus,

- Et2Cf) = (e,"t)> (58)

which, assuming a judicious choice of F as in Section 111-B, will yield a close approximation to de,/bf. The correspond- ing approximate expression for multipath spread is then

Figure 1 presents a block diagram which combines the SSB method of M measurement with the D measurement scheme to produce a simple device for the simultaneous measurement of multipath spread and Doppler spread.

2) T h e F M Technique: In the FRiI technique we trans- mit a frequency-modulated carrier with &(t) as the fre- quency modulation. Using the quasi-stationary approxima- tion again as in Section 111-B, 2 ) we may represent the received envelope approximately as

Assuming a small enough peak deviation

-,

Here also we have an SSB and an F M technique de- Thus, the detected envelope consists of a low-frequency pending upon how the function de,(t)/bf is determined. term e,(t) and a component at the modulation frequency

292 lEEE TRANSACTIONS ON COMMUNICATION TECHNOI.OGY

C Squore A W o p c I Mul l lpa th

r S p r e o d

& & - M no+,.

t P

Sq“0W Arerope

Fig. 2. FA4 techniyrle for the simultaneous measurement of Doppler spread and nlulti path spread using envelope measurements alone.

anlplitude modulated by [de,(t) ]/bf. To extract Q(be,/df) it is necessary only to filter out the conlponent at the mod- ulation frequency and envelope detect. Appropriate aver- ages then yield the required numbers to determine M as given by (60). Figure 2 shows a block diagram that com- bines the FRiI method of M measurement with the D meas- urement scheme to produce a simple device for the simul- taneous measurement of multipath and Doppler spread.

V. CONCLUSION In this paper several techniques have been derived for

the measurement of the multipath and Doppler spread parameters of fading radio channels, It is fell; that these techniques will be of considerable .help i n the collection of the statistical information necessary for the evaluation of digital communication system performance over such channels.

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