Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
6.2 The Causes of Delay Dispersion
Why does a channel exhibit delay dispersion – or, equivalently, why are there variations of the channel over a given frequency range? The most simple picture arises again from the two-path model, as introduced in the beginning of Chapter 5. The transmit signal gets to the receiver (RX) via two different propagation paths with different runtimes:
τ1=d1/c0 and τ2=d2/c0 (6.1)
We assume now that runtimes do not change with time (this occurs when neither transmitter (TX), RX, nor Interacting Objects (IOs) move). Consequently, the channel is linear and time invariant, and has an impulse response:
h(τ )=a1δ(τ−τ1)+a2δ(τ−τ2) (6.2) where again the complex amplitudea= |a|exp(j ϕ). Clearly, such a channel exhibits delay disper- sion; the support (duration) of the impulse response has a finite extent, namelyτ2−τ1.
A Fourier transformation of the impulse response gives the transfer functionH (j ω):
H (f )= ∞
−∞h(τ )exp[−j2πf τ]dτ=a1exp[−j2πf τ1]+a2exp[−j2πf τ2] (6.3) The magnitude of the transfer function is
|H (f )| =
|a1|2+ |a2|2+2|a1||a2|cos(2πfãτ−ϕ)
withτ =τ2−τ1andϕ=ϕ2−ϕ1 (6.4) Figure 6.1 shows the transfer function for a typical case. We observe first that the transfer function depends on the frequency, so that we havefrequency-selective fading. We also see that there are dips (notches) in the transfer function at the so-callednotch frequencies. In the two-path model, the notch frequencies are those frequencies where the phase difference of the two arriving waves becomes 180◦. The frequency difference between two adjacent notch frequencies is
fNotch= 1
τ (6.5)
Wideband and Directional Channel Characterization 103
Frequency (MHz)
|H(f)|/max|H(f)|| (dB)
−40
−30
−20
−10 0
900.0 900.2 900.4 900.6 900.8 901.0
Figure 6.1 Normalized transfer function for|a1| =1.0,|a2| =0.95,ϕ=0,τ1=4μs,τ2=6μs at the 900- MHz carrier frequency.
The destructive interference between the two waves is stronger the more similar the amplitudes of the two waves are.
Channels with fading dips distort not only the amplitude but also the phase of the signal. This can be best seen by considering the group delay, which is defined as the derivative of the phase of the channel transfer functionφH=arg(H (f )):
τGr= −1dφH
2π df (6.6)
As can be seen in Figure 6.2, group delay can become very large in fading dips. As we will see later, this group delay can be related to ISI.
900.0 900.2 900.4 900.6 900.8 901.0
Frequency (MHz)
Group delay (às)
−40
−30
−20
−10 0 10
Figure 6.2 Group delay as a function of frequency (same parameters as in Figure 6.1).
6.2.2 The General Case
After the simple two-path model, we now progress to the more general case where IOs can be at any place in the plane. Again, the scenario is static, so that neither TX, RX, nor IOs move. We now draw the ellipses that are defined by their focal points – TX and RX – and the eccentricity determining the runtime.1 All rays that undergo a single interaction with an object on a specific ellipse arrive at the RX at the same time. Signals that interact with objects on different ellipses arrive at different times. Thus, the channels aredelay-dispersiveif the IOs in the environment are not all located on a single ellipse.
It is immediately obvious that in a realistic environment, IOs never lie exactly on a single ellipse.
The next question is thus: How strict must this “single ellipse” condition be fulfilled so that the channel is still “effectively” nondispersive? The answer depends on the system bandwidth. An RX with bandwidthWcannot distinguish between echoes arriving atτandτ+τ, ifτ1/W(for many qualitative considerations, it is sufficient to consider the above condition withτ=1/W).
Thus echoes that are reflected in the donut-shaped region corresponding to runtimes betweenτ and (τ+τ )arrive at “effectively” the same time (see Figure 6.3).
IOs on ellipses IOs in annular rings
"Impulse response"
Delay in excess of direct path 0 Δτ 2Δτ3Δτ4Δτ
Figure 6.3 Scatterers located on the same ellipses lead to the same delays.
A time-discrete approximation to the impulse response of a wideband channel can thus be obtained by dividing the impulse response into bins of width τ and then computing the sum of echoes within each bin. If enough nondominant IOs are in each donut-shaped region, then the MPCs falling into each delay bin fulfill the central limit theorem. In that case, the amplitude of each bin can be described statistically, and the probability density function (pdf) of this amplitude is Rayleigh or Rician. Thus, all the equations of Chapter 5 are still valid; but now they apply for the field strengthwithinone delay bin. We furthermore define the minimum delay as the runtime of the direct path between the Base Station (BS) and the Mobile Station (MS)d/c0and we define the maximum delay as the runtime from the BS to the MS via the farthest “significant” IO – i.e., the farthest IO that gives a measurable contribution to the impulse response.2Themaximum excess delay τmaxis then defined as the difference between minimum and maximum delay.
1These ellipses are thus quite similar to the Fresnel ellipses described in Chapter 4. The difference is that in Chapter 4 we were interested in excess runtimes that introduce a phase shift ofiãπ, while here we are interested in delays that are typically much larger.
2We see from this definition that the maximum delay is a quantity that is extremely difficult to measure, and depends on the measurement system.
Wideband and Directional Channel Characterization 105
The above considerations also lead us to a mathematical formulation fornarrowbandandwide- band from a time domain point of view: a system is narrowband if the inverse of the system bandwidth 1/W is much larger than the maximum excess delay τmax. In that case, all echoes fall into a single delay bin, and the amplitude of this delay bin isα(t ). A system is wideband in all other cases. In a wideband system, theshape and duration of the arriving signal is different from the shape of the transmitted signal; in a narrowband system, they stay the same.
If the impulse response has a finite extent in the delay domain, it follows from the theory of Fourier transforms (FTs) that the transfer function F{h(τ )} =H (f )is frequency dependent.
Delay dispersion is thus equivalent tofrequency selectivity. A frequency-selective channel cannot be described by a simple attenuation coefficient, but rather the details of the transfer function must be modeled. Note that any real channel is frequency selective if analyzed over a large enough bandwidth; in practice, the question is whether this is true over the bandwidth of the considered system. This is equivalent to comparing the maximum excess delay of the channel impulse response with the inverse system bandwidth. Figure 6.4 sketches these relationships, demonstrating the variations of wideband systems in the delay and frequency domain.
We stress that the definition of a wideband wireless system is fundamentally different from the definition of “wideband” in the usage of Radio Frequency (RF) engineers. The RF definition
Wideband system
Narrowband system
|Hc(f)| |Hs(f)| |H(f)|
1/Δtmax f f f
X =
=
|hc(t)| |hs(t)| |h(t)|
|Hc(f)| |Hs(f)| |H(f)|
|hc(t)| |hs(t)| |h(t)|
Δtmax t t t
X =
=
f f f
t t t
1/Δtmax
Δtmax
Figure 6.4 Narrowband and wideband systems. HC(f ), channel transfer function; hC(τ ), channel impulse response.
Reproduced with permission from Molisch [2000]©Prentice Hall.
of wideband implies that the system bandwidth becomes comparable with carrier frequency.3 In wireless communications, on the other hand, we compare the properties of the channel with the properties of the system. It is thus possible that the same system is wideband in one channel, but narrowband for another.