Torrieri D. Principles of Spread-Spectrum Communication Systems

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and (5-36) becomes

A communication system such as a mobile that receives a signal from an ele-

vated base station may be surrounded by many scattering objects. An isotropic

scattering model assumes that multipath components of comparable power are

reflected from many different scattering objects and hence arrive from many

different directions. For two-dimensional isotropic scattering, N is large, and

the lie in a plane and have values that are uniformly distributed over

Therefore, the summation in (5-37) can be approximated by an integral;

that is,

This integral has the same form as the integral representation of the

Bessel function of the first kind and order zero. Thus, the autocorrelation of

the channel response for two-dimensional isotropic scattering is

The normalized autocorrelation which is a real-valued function

of is plotted in Figure 5.2. It is observed that its magnitude is less than

0.3 when This observation leads to the definition of the coherence

time or correlation time of the channel as

where is the maximum Doppler shift or Doppler spread. The coherence time

is a measure of the time separation between signal samples sufficient for the

samples to be largely decorrelated. If the coherence time is much longer than

the duration of a channel symbol, then the fading is relatively constant over a

symbol and is called slow fading. Conversely, if the coherence time is on the

order of the duration of a channel symbol or less, then the fading is called fast

fading.

The power spectral density of a complex process is defined as the Fourier

transform of its autocorrelation. From (5-39) and tabulated Fourier transforms,

we obtain the Doppler power spectrum for two-dimensional isotropic scattering:

The normalized Doppler spectrum which is plotted in Figure 5.3

versus is bandlimited by the Doppler spread and tends to infinity as

approaches The Doppler spectrum is the superposition of contributions

5.2.

TIME-SELECTIVE FADING

239

Figure 5.2: Autocorrelation of r(t) for isotropic scattering.

Figure 5.3: Doppler spectrum for isotropic scattering.

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from multipath components, each of which experiences a different Doppler shift

upper bounded by

The received signal power spectrum may be calculated from (5-7), (5-11),

and (5-41). For an unmodulated carrier, and the received signal power

spectrum is

In general, when the scattering is not isotropic, the imaginary part of the auto-

correlation is nonzero, and the amplitude of the real part decreases much

more slowly and less smoothly with increasing than (5-39). Both the real

and imaginary parts often exhibit minor peaks for time shifts exceeding

Thus, the coherence time provides only a rough characterization of the channel

behavior.

Fading Rate and Fade Duration

The fading rate is the rate at which the envelope of a received fading signal

crosses below a specified level. Consider a time interval over which the fading

parameters are constant. For a level isotropic scattering, and Ricean

fading, it can be shown that the fading rate is [1]

where is the Rice factor and

For Rayleigh fading, and (5-43) becomes

Equations (5-43) and (5-45) indicate that the fading rate is proportional to the

Doppler spread Thus, slow fading occurs when the Doppler spread is small,

whereas fast fading occurs when the Doppler spread is large.

Let denote the average envelope fade duration, which is the amount

of time the envelope remains below the specified level The product

is the fraction of the time between fades during which a fade occurs. If the

time-varying envelope is assumed to be a stationary ergodic process, then this

fraction is equal to the probability that the envelope is below or equal

to the level Thus,

If the envelope has the Rice distribution, then integrating (5-28) and using

(5-43), (5-44), and (5-46), we obtain

5.2.

TIME-SELECTIVE FADING

241

Figure 5.4: Two antennas receiving plane wave that results in a signal copy at

each antenna.

where is defined by (D-15). For Rayleigh fading, (5-45), (5-46), and the

integration of (5-20) yields

For both Ricean and Rayleigh fading, the fade duration is inversely proportional

Spatial Diversity and Fading

To obtain spatial diversity in a fading environment, the antennas in an array

must be separated enough that there is little correlation between signal replicas

or copies at the antennas. To determine what separation is needed, consider the

reception of a signal at two antennas separated by a distance D, as illustrated

in Figure 5.4. If the signal arrives as an electromagnetic plane wave, then the

signal copy at antenna 1 relative to antenna 2 is delayed by where

is the arrival angle of the plane wave relative to a line perpendicular to the line

joining the two antennas. Let denote the phase of the complex envelope

of multipath component at antenna Consider a time interval small enough

that and each multipath component arrives

from a fixed angle. Thus, if multipath component of a narrowband signal

arrives as a plane wave at angle then the phase of the complex envelope

of the component copy at antenna 2 is related to the phase at antenna 1 by

where is the wavelength of the signal. If the multipath component

propagates over a distance much larger than the separation between the two

antennas, then it is reasonable to assume that the attenuation is identical

at the two antennas. If the range of the delay values exceeds then the

sensitivity of the phases to small delay variations makes it plausible that the

phases are well modeled as independent random

variables that are uniformly distributed over From (5-12), the complex

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envelope of the signal copy at antenna when the signal is a tone is

The cross-correlation between and is defined as

Substituting (5-50) into (5-51), using the independence of each and the

independence of and and the uniform distribution of each

and then substituting (5-49), we obtain

This equation for the cross-correlation as a function of spatial separation clearly

resembles (5-36) for the autocorrelation as a function of time delay. If all the

multipath components have approximately the same power so that

then

Applying the two-dimensional isotropic scattering model, a derivation sim-

ilar to that of (5-39) gives the real-valued cross-correlation

This model indicates that an antenna separation of ensures that

the normalized cross-correlation is less than 0.3. A plot of the

normalized cross-correlation is obtained from Figure 5.2 if the abscissa is inter-

preted as When the scattering is not isotropic or the number of scattering

objects producing multipath components is small, then the real and imaginary

parts of the cross-correlation decrease much more slowly with For ex-

ample, Figure 5.5 shows the real and imaginary parts of the normalized cross-

correlation when the are a nearly continuous band of angles between

and radians so that (5-53) can be approximated by an integral. Figure 5.6

depicts the real and imaginary parts of the normalized cross-correlation when

N = 9 and the are uniformly spaced throughout the first two quadrants:

In the example of Figure 5.5, an antenna

separation of at least is necessary to ensure approximate decorrelation of

the signal copies and obtain spatial diversity. In the example of Figure 5.6, not

even a separation of is adequate to ensure approximate decorrelation.

5.3. FREQUENCY-SELECTIVE FADING

243

Figure 5.5: Normalized cross-correlation for multipath components arriving

between and radians: (a) real part and (b) imaginary part.

5.3

Frequency-Selective Fading

Frequency-selective fading occurs because multipath components combine de-

structively at some frequencies, but constructively at others. The different path

delays cause dispersion of a received pulse in time and cause intersymbol in-

terference between successive symbols. When a multipath channel introduces

neither time variations nor Doppler shifts, (5-8) and (5-9) indicate that the

received complex envelope is

The number of multipath components includes only those components with

power that is a significant fraction, perhaps 0.05 or more, of the power of

the dominant component. The multipath delay spread is defined as the

maximum delay of a significant multipath component relative to the minimum

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Figure 5.6: Normalized cross-correlation for N = 9 multipath components ar-

riving from uniformly spaced angles in the first two quadrants: (a) real part,

and (b) imaginary part.

delay of a component; that is,

If the duration of a received symbol is much larger than then the multi-

path components are usually unresolvable,

and hence is proportional to Since all frequency components

of the received signal fade nearly simultaneously, this type of fading is called

frequency-nonselective or flat fading and occurs if In contrast, a

signal is said to experience frequency-selective fading if because then

the time variation or fading of the spectral components of may be different.

The large delay spread may cause intersymbol interference, which is accommo-

dated by equalization in the receiver. However, if the time delays are sufficiently

different among the multipath components that they are resolvable at the de-

modulator or matched-filter output, then the independently fading components

provide diversity that can be exploited by a rake receiver (Section 5.5).

5.3.

FREQUENCY-SELECTIVE FADING

245

It is conceptually useful to define the coherence bandwidth as

Let denote the bandwidth of In general, for practical

modulations, so flat fading occurs if Frequency-selective fading

requires

To illustrate frequency-selective fading, consider the reception of a tone at

frequency with two multipath components so that and in

(5-55). It then follows that the complex envelope has magnitude

where If another tone at frequency is received, then this equa-

tion is valid with substituted for Thus, the two complex envelopes can

differ considerably as and both range over a spectral band with bandwidth

equal to the coherence bandwidth. If then the difference between the

two complex-envelope magnitudes varies from 0 to

Channel Impulse Response

A generalized impulse response may be used to characterize the impact of the

transmission channel on the signal. The equivalent complex-valued baseband im-

pulse response of the channel is the response at time due to an impulse

applied seconds earlier. The complex envelope of the received signal is

the result of the convolution of the complex envelope of the transmitted

signal with the baseband impulse response:

In accordance with (5-8), the impulse response is usually modeled as a complex-

valued stochastic process:

For most practical applications, the wide-sense stationary, uncorrelated scat-

tering model is reasonably accurate. The impulse response is wide-sense sta-

tionary if the correlation between its value at and its value at depends

only on Thus, the autocorrelation of the impulse response is

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Uncorrrelated scattering implies that the gains and phase shifts associated with

two different delays are uncorrelated. Extending this notion, the wide-sense

stationary, uncorrelated scattering model assumes that the autocorrelation has

the form

where is the Dirac delta function. This equation implies that

is the result of integrating the autocorrelation over The multipath

intensity profile or delay power spectrum can be interpreted

as the channel output power due to an impulse applied seconds earlier. The

range of the delay over which the multipath intensity profile has a significant

magnitude is a measure of the multipath delay spread. The multipath intensity

profile has diffuse components if it is a piecewise continuous function and has

specular components if it includes delta functions at specific values of the delay.

A received signal from one source can often be decomposed into the sum of

signals reflected from several clusters of scatterers. Each cluster is the sum of

a number of multipath components with nearly the same delay. In this model,

the impulse response can be expressed as

where is the number of clusters and is the distinct delay associated

with the ith cluster. Each complex process has has the form of (5-23) and

an envelope with a Rayleigh, Rice, or Nakagami probability density function.

The Fourier transform of the impulse response gives the time-varying chan-

nel frequency response:

The autocorrelation of the frequency response for a wide-sense stationary chan-

nel is

For the wide-sense stationary, uncorrelated scattering model, the substitution

of (5-64), (5-61), and (5-62) into (5-65) yields

which is a function only of the difference If then the autocor-

relation of the frequency response is

5.4.

DIVERSITY FOR FADING CHANNELS

247

which is the Fourier transform of the delay power spectrum. From a funda-

mental characteristic of the Fourier transform, it follows that the coherence

bandwidth of the channel, which is a measure of the range of frequency shift

over which the autocorrelation has a significant value, is given by the recipro-

cal of the multipath delay spread. Thus (5-57) is confirmed for this channel

model.

The Doppler shift is the main limitation on the channel coherence time or

range of values of the difference for which is significant.

Thus, the Doppler power spectrum is defined as

The spectral extent of the Doppler power spectrum is on the order of the max-

imum Doppler shift. Thus, (5-40) is confirmed for this channel model.

5.4

Diversity for Fading Channels

Diversity combiners for fading channels are designed to combine independently

fading copies of the same signal in different branches. The combining is done in

such a way that the combiner output has a power level that varies much more

slowly than that of a single copy. Although useless in improving communications

over the additive-white-Gaussian-noise (AWGN) channel, diversity improves

communications over fading channels because the diversity gain is large enough

to overcome any noncoherent combining loss. Diversity may be provided by

signal redundancy that arises in a number of different ways. Time diversity

is provided by channel coding or by signal copies that differ in time delay.

Frequency diversity may be available when signal copies using different carrier

frequencies experience independent or weakly correlated fading. If each signal

copy is extracted from the output of a separate antenna in an antenna array,

then the diversity is called spatial diversity. Polarization diversity may be

obtained by using two cross-polarized antennas at the same site. Although this

configuration provides compactness, it is not as potentially effective as spatial

diversity because the received horizontal component of an electric field is usually

much weaker than the vertical component.

The three most common types of diversity combining are selective, maximal-

ratio, and equal-gain combining. The last two methods use linear combining

with variable weights for each signal copy. Since they usually must eventually

adjust their weights, maximal-ratio and equal-gain combiners can be viewed as

types of adaptive arrays. They differ from other adaptive antenna arrays in

that they are not designed to cancel interference signals.

Optimal Array

Consider a receiver array of L diversity branches, each of which processes a

different signal copy. Each branch input is translated to baseband, and then