Impulse Response Model of a Multipath Channel

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5.2 Impulse Response Model of a Multipath Channel

The small-scale variations of a mobile radio signal can be directly related to the impulse

response of the mobile radio channel. The impulse response is a wideband channel

characterization and contains all information necessary to simulate or analyze any type

of radio transmission through the channel. This stems from the fact that a mobile radio

channel may be modeled as a linear filter with a time varying impulse response, where

the time variation is due to receiver motion in space. The filtering nature of the channel

is caused by the summation of amplitudes and delays of the multiple arriving waves at

any instant of time. The impulse response is a useful characterization of the channel, since

it may be used to predict and compare the performance of many different mobile

communication systems and transmission bandwidths for a particular mobile channel condition.

To show that a mobile radio channel may be modeled as a linear filter with a time varying

impulse response, consider the case where time variation is due strictly to receiver motion in

space. This is shown in Figure 5.2.

In Figure 5.2, the receiver moves along the ground at some constant velocity v. For a fixed

position d, the channel between the transmitter and the receiver can be modeled as a linear time

invariant system. However, due to the different multipath waves which have propagation delays

which vary over different spatial locations of the receiver, the impulse response of the linear time

invariant channel should be a function of the position of the receiver. That is, the channel

impulse response can be expressed as h(d,t). Let x(t) represent the transmitted signal, then the

received signal y(d,t) at position d can be expressed as a convolution of with h(d,t)

For a causal system, h(d, t) = 0 for t < 0, thus Equation (5.3) reduces to

Since the receiver moves along the ground at a constant velocity v, the position of the

receiver can by expressed as

Figure 5.2 The mobile radio channel as a function of time and space.

Substituting (5.5) in (5.4), we obtain

Since v is a constant, y(vt, t) is just a function of t. Therefore, Equation (5.6) can be expressed

as

From Equation (5.7), it is clear that the mobile radio channel can be modeled as a linear time

varying channel, where the channel changes with time and distance.

Since v may be assumed constant over a short time (or distance) interval, we may let

x(t) represent the transmitted bandpass waveform, y(t) the received waveform, and   the impulse

response of the time varying multipath radio channel. The impulse response   completely

characterizes the channel and is a function of both t and . The variable t represents

the time variations due to motion, whereas   represents the channel multipath delay for a fixed

value of . One may think of   as being a vernier adjustment of time. The received signal

y(t) can be expressed as a convolution of the transmitted signal

x(t) with the channel impulse response (see Figure 5.3a)

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Figure 5.3 (a) Bandpass channel impulse response model; (b) baseband equivalent channel

impulse response model.

 

If the multipath channel is assumed to be a bandlimited bandpass channel, which is reasonable,

then  may be equivalently described by a complex baseband impulse response

 with the input and output being the complex envelope representations of the transmitted

and received signals, respectively (see Figure 5.3b). That is,

where c(t)  and r(t) are the complex envelopes of x(t) and y(t), defined as

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Since the received signal in a multipath channel consists of a series of attenuated, timedelayed,

phase shifted replicas of the transmitted signal, the baseband impulse response of a

multipath channel can be expressed as

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phase shift due to free space propagation of the ith multipath component, plus any additional

phase shifts which are encountered in the channel. In general, the phase term is simply represented

by a single variable which lumps together all the mechanisms for phase shifts of a

single multipath component within the ith excess delay bin. Note that some excess delay bins

may have no multipath at some time t and delay since  may be zero. In Equation

(5.12), N is the total possible number of multipath components (bins), and δ(•)  is the unit

impulse function which determines the specific multipath bins that have components at time t

and excess delays  Figure 5.4 illustrates an example of different snapshots of , where

t varies into the page, and the time delay bins are quantized to widths of   Modern wireless

communication systems have recently used spatial filtering to increase capacity and coverage,

and often it is useful to modify Equation (5.12) to include the effects of angle of arrival of each

multipath component [Lib99], [Ert98], [Mol01].

It is important to note that depending on the choice of   and the physical channel delay

properties, there may be two or more multipath signals that arrive within an excess delay bin that

are unresolvable and that vectorially combine to yield the instantaneous amplitude and phase of a

single modeled multipath component. Such situations cause the multipath amplitude within an

excess delay bin to fade over the local area. However, when only a single multipath component

arrives within an excess delay bin, the amplitude over the local area for that particular time delay

will generally not fade significantly.

Figure 5.4 An example of the time varying discrete-time impulse response model for a multipath

radio channel. Discrete models are useful in simulation where modulation data must be convolved

with the channel impulse response [Tra02].

If the channel impulse response is assumed to be time invariant, or is at least wide sense

stationary over a small-scale time or distance interval, then the channel impulse response may be

simplified as

The assumption of time invariance over a local area is valid when the time delay resolution of

the channel impulse response model accurately and uniquely resolves every multipath component

over the local area.

When measuring or predicting  a probing pulse p(t) which approximates a delta

function is used at the transmitter. That is,

is used to sound the channel to determine 

 

For small-scale channel modeling, the power delay profile of the channel is found by taking

the spatial  average of over a local area. By making several local area measurements

 of in different locations, it is possible to build an ensemble of power delay

profiles, each one representing a possible small-scale multipath channel state [Rap91a].

Based on work by Cox [Cox72], [Cox75], if p(t) has a time duration much smaller than

the impulse response of the multipath channel, p(t) does not need to be deconvolved from the

received signal r(t) in order to determine relative multipath signal strengths. The received

power delay profile in a local area is given by

where the bar represents the average over the local area and many snapshots of  are typically

averaged over a local (small-scale) area to provide a single time-invariant multipath power

delay profile . The gain k in Equation (5.15) relates the transmitted power in the probing pulse

p(t) to the total power received in a multipath delay profile.

5.2.1 Relationship Between Bandwidth and Received Power

In actual wireless communication systems, the impulse response of a multipath channel is measured

in the field using channel sounding techniques. We now consider two extreme channel

sounding cases as a means of demonstrating how the small-scale fading behaves quite differently

for two signals with different bandwidths in the identical multipath channel.

Consider a pulsed, transmitted RF signal of the form

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Note that if all the multipath components are resolved by the probe p(t), then  for

all j ≠ i , and

For a wideband probing signal p(t), T_bb is smaller than the delays between multipath components

in the channel, and Equation (5.18) shows that the total received power is simply related

to the sum of the powers in the individual multipath components, and is scaled by the ratio of the

probing pulse’s width and amplitude, and the maximum observed excess delay of the channel.

Assuming that the received power from the multipath components forms a random process

where each component has a random amplitude and phase at any time , the average small-scale

received power for the wideband probe is found from Equation (5.17) as

In Equation (5.19), Ea, θ[•] denotes the ensemble average over all possible values of

a_i and θ_i in a local area, and the overbar denotes sample average over a local measurement area which is

generally measured using multipath measurement equipment. The striking result of Equations

(5.18) and (5.19) is that if a transmitted signal is able to resolve the multipaths, then the average

small-scale received power is simply the sum of the average powers received in each multipath

component. In practice, the amplitudes of individual multipath components do not fluctuate

widely in a local area. Thus, the received power of a wideband signal such as p(t) does not fluctuate

significantly when a receiver is moved about a local area [Rap89].

Now, instead of a pulse, consider a CW signal which is transmitted into the exact same

channel, and let the complex envelope be given by c(t) = 2 . Then, the instantaneous complex

envelope of the received signal is given by the phasor sum

and the instantaneous power is given by

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As the receiver is moved over a local area, the channel induces changes on r(t), and the

received signal strength will vary at a rate governed by the fluctuations of

a_i and θ _i . As mentioned earlier, a_i varies little over local areas, but θ_i will vary greatly due to

changes inpropagation distance over space, resulting in large fluctuations of r(t) as the

receiver is moved over small distances (on the order of a wavelength). That is, since r(t ) is

the phasor sum of the

individual multipath components, the instantaneous phases of the multipath components cause

the large fluctuations which typifies small-scale fading for CW signals. The average received

power over a local area is then given by

 

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and the overbar denotes time average for CW measurements made by a mobile receiver over the

local measurement area [Rap89]. Note that when

= 0 and/or r_ij = 0, then the average power for a CW signal is equivalent to the average received

power for a wideband signal in asmall-scale region. This is seen by comparing Equation (5.19)

and Equation (5.24).

This can occur when either the multipath phases are identically and independently distributed

(i.i.d uniform) over  or when the path amplitudes are uncorrelated. The i.i.d uniform distribution

of θ is a valid assumption since multipath components traverse differential path lengths that measure

 hundreds of wavelengths and are likely to arrive with random phases. If for some reason it is

believed that the phases are not independent, the average wideband power and average CW power

 will still be equal if the paths have uncorrelated amplitudes. However, if the phases of the paths are

dependent upon each other, then the amplitudes are likely to be correlated, since the same mechanism

 which affects the path phases is likely to also affect the amplitudes. This situation is highly unlikely

at transmission frequencies used in wireless mobile systems.

Thus it is seen that the received local ensemble average power of wideband and narrowband

signals are equivalent. When the transmitted signal has a bandwidth much greater than the bandwidth

of the channel, then the multipath structure is completely resolved by the received signal at

any time, and the received power varies very little since the individual multipath amplitudes do not

change rapidly over a local area. However, if the transmitted signal has a very narrow bandwidth

(e.g., the baseband signal has a duration greater than the excess delay of the channel), then multipath

is not resolved by the received signal, and large signal fluctuations (fading) occur at the

receiver due to the phase shifts of the many unresolved multipath components.

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Figure 5.5 Measured wideband and narrowband received signals over a 5λ (0.375 m) measurement

track inside a building. Carrier frequency is 4 GHz. Wideband power is computed using

Equation (5.19), which can be thought of as the area under the power delay profile. The axis into

the page is distance (wavelengths) instead of time.

 

Figure 5.5 illustrates actual indoor radio channel measurements made simultaneously with

a wideband probing pulse having T_bb = 10 ns, and a CW transmitter. The carrier frequency was

4 GHz. It can be seen that the CW signal undergoes rapid fades, whereas the wideband measurements

change little over the 5λ measurement track. However, the local average received powers

of both signals were measured to be virtually identical [Haw91].

Example 5.2

Assume a discrete channel impulse response is used to model urban RF

radio channels with excess delays as large as 100 μs and microcellular

channels with excess delays no larger than 4 μs. If the number of multipath

bins is fixed at 64, find (a)  and (b) the maximum RF bandwidth

which the two models can accurately represent. Repeat the exercise for

an indoor channel model with excess delays as large as 500 ns. As

described in section 5.7.6, SIRCIM and SMRCIM are statistical channel

models based on Equation (5.12) that use parameters in this example.

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Example 5.3

Assume a mobile traveling at a velocity of 10 m/s receives two multipath

components at a carrier frequency of 1000 MHz. The first component is

assumed to arrive at τ = 0 with an initial phase of 0° and a power of

–70 dBm, and the second component which is 3 dB weaker than the first

component is assumed to arrive at   μs, also with an initial phase of

0°. If the mobile moves directly toward the direction of arrival of the first

component and directly away from the direction of arrival of the second

component, compute the narrowband instantaneous power at time intervals

of 0.1 s from 0 s to 0.5 s. Compute the average narrowband power

received over this observation interval. Compare average narrowband and

wideband received powers over the interval, assuming the amplitudes of

the two multipath components do not fade over the local area.

Solution

Given v = 10 m/s, time intervals of 0.1 s correspond to spatial intervals of

1 m. The carrier frequency is given to be 1000 MHz, hence the wavelength

of the signal is

The narrowband instantaneous power can be computed using Equation (5.21).

Note –70 dBm = 100 pW. At time t = 0, the phases of both multipath components

are 0°, hence the narrowband instantaneous power is equal to

Now, as the mobile moves, the phase of the two multipath components

changes in opposite directions.

Since the mobile moves toward the direction of arrival of the first component,

and away from the direction of arrival of the second component, θ₁ is positive,

and θ₂ is negative.

Therefore, at t = 0.1 s, θ₁ = 120°, and θ ₂ = –120°, and the instantaneous

power is equal to

Similarly, at t = 0.2 s, θ₁ = 240°, and θ ₂ = –240°, and the instantaneous

power is equal to

Similarly, at t = 0.3 s, θ₁ = 360° = 0°, and θ ₂ = –360° = 0°, and the instantaneous

power is equal to

It follows that at t = 0.4 s, |r (t )|² = 79.3pW, and at t = 0.5 s, |r (t )| ² = 79.3 pW.

The average narrowband received power is equal to

Using Equation (5.19), the wideband power is given by

As can be seen, the narrowband and wideband received power are virtually

identical when averaged over 0.5 s (or 5 m). While the CW signal fades over

the observation interval, the wideband signal power remains constant over

the same spatial interval.

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