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6.3 System-Theoretic Description of Wireless Channels
As we have seen in the previous section, a wireless channel can be described by an impulse response;
it thus can be interpreted as a linear filter. If the BS, MS, and IOs are all static, then the channel is time invariant, with an impulse response h(τ ). In that case, the well-known theory of Linear Time Invariant (LTI) systems [Oppenheim and Schafer 2009] is applicable. In general, however, wireless channels are time variant, with an impulse response h(t, τ ) that changes with time; we have to distinguish between the absolute time t and the delayτ. Thus, the theory of theLinear Time Variant (LTV) system must be used. This is not just a trivial extension of the LTI theory, but gives rise to considerable theoretical challenges and causes the breakdown of many intuitive concepts. Fortunately, most wireless channels can be classified asslowlytime-variant systems, also known asquasi-static. In that case, many of the concepts of LTI systems can be retained with only minor modifications.
6.3.1 Characterization of Deterministic Linear Time Variant Systems
As the impulse response of a time-variant system,h(t, τ ), depends on two variables,τ andt, we can perform Fourier transformations with respect to either (or both) of them. This results in four different, but equivalent, representations. In this section, we investigate these representations, their advantages, and drawbacks.
From a system-theoretic point of view, it is most straightforward to write the relationship between the system input (transmit signal)x(t) and the system output (received signal)y(t) as:
y(t )= ∞
−∞x(τ )K(t, τ ) dτ (6.7)
whereK(t, τ )is thekernel of the integral equation, which can be related to the impulse response.
For LTI systems, the well-known relationshipK(t, τ )=h(t−τ )holds. Generally, we define the time-variant impulse response as:
h(t, τ )=K(t, t−τ ) (6.8)
so that
y(t )= ∞
−∞
x(t−τ )h(t, τ ) dτ (6.9)
An intuitive interpretation is possible if the impulse response changes only slowly with time – more exactly, the duration of the impulse response (and the signal) should be much shorter than the time over which the channel changes significantly. Then we can consider the behavior of the system at one timet like that of an LTI system. The variablet can thus be viewed as “absolute” time that
3In wireless communications, it has become common to denote systems whose bandwidth is larger than 20% of the carrier frequency asUltra Wide Bandwidth (UWB) systems (see Section 6.6). This definition is similar to the RF definition of wideband.
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parameterizes the impulse response, i.e., tells us which (out of a large ensemble) impulse response h(τ )is currently valid. Such a system is also calledquasi-static.
Fourier transforming the impulse response with respect to the variableτresults in thetime-variant transfer function H(t, f):
H (t, f )= ∞
−∞h(t, τ )exp(−j2πf τ ) dτ (6.10) The input–output relationship is given by:
y(t )= ∞
−∞X(f )H (t, f )exp(j2πf t ) df (6.11) The interpretation is straightforward for the case of the quasi-static system – the spectrum of the input signal is multiplied by the spectrum of the “currently valid” transfer function, to give the spectrum of the output signal. If, however, the channel is quickly time varying, then Eq. (6.11) is a purely mathematical relationship. The spectrum of the output signal is given by a double integral
Y (f )˜ = ∞
−∞
∞
−∞X(f )H (t, f )exp(j2πf t )exp(−j2πf t ) df dt˜ (6.12) which doesnotreduce toY (f )=H (f )X(f )[Matz and Hlawatsch 1998].
A Fourier transformation of the impulse response with respect to t results in a different representation – namely, the Doppler-variant impulse response, better known asspreading function s(ν, τ ):
s(ν, τ )= ∞
−∞h(t, τ )exp(−j2π νt ) dt (6.13) This function describes the spreading of the input signal in the delay and Doppler domains.
Finally, the functions(ν, τ )can be transformed with respect to the variableτ, resulting in the Doppler-variant transfer function B(ν, f ):
B(ν, f )= ∞
−∞s(ν, τ )exp(−j2πf τ ) dτ (6.14) A summary of the interrelations between the system functions is given in Figure 6.5. Figure 6.6 shows an example of a measured impulse response; Figure 6.7 shows the spreading function com- puted from it.
h(t, τ)
H(t, f)
B(ν, f)
s(ν, τ)
−1 −1
−1 −1
Figure 6.5 Interrelation between deterministic system functions.
6.3.2 Stochastic System Functions
We now return to the stochastic description of wireless channels. Interpreting them as time- variant stochastic systems, a complete description requires the multidimensional pdf of the impulse
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Figure 6.6 Squared magnitude of the impulse response |h(t, τ )|2 measured in hilly terrain near Darmstadt, Germany. Measurement duration 140 s; center frequency 900 MHz.τdenotes the excess delay.
Reproduced with permission from Liebenow and Kuhlmann [1993]©U. Liebenow.
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Figure 6.7 Spreading function computed from the data of Figure 6.6.
Reproduced with permission from U. Liebenow.
response – i.e., the joint pdf of the complex amplitudes at all possible values of delay and time.
However, this is usually much too complicated in practice. Instead, we restrict our attention to a second-order description – namely, theAutoCorrelation Function(ACF).
Let us first repeat some facts about the ACFs of one-dimensional stochastic processes (i.e., processes that depend on a single parametert). The ACF of a stochastic processyis defined as:
Ryy(t, t)=E{y∗(t )y(t)} (6.15)
Wideband and Directional Channel Characterization 109
where the expectation is taken over theensemble of possible realizations(for the definition of this ensemble see Appendix 6.A at www.wiley.com/go/molisch). The ACF describes the relationship between the second-order moments of the amplitude pdf of the signaly at different times. If the pdf is zero-mean Gaussian, then the second-order description contains all the required information.
If the pdf is non-zero-mean Gaussian, the mean
y(t )=E{y(t )} (6.16)
together with the auto covariance function
˜
Ryy(t, t)=E{[y(t )−y(t )]∗[y(t)−y(t]} (6.17) constitutes a complete description. If the channel is non-Gaussian, then the first- and second-order statistics are not a complete description of the channel. In the following, we mainly concentrate on zero-mean Gaussian channels.
Let us now revert to the problem of giving a stochastic description of the channel. Inserting the input–output relationship into Eq. (6.15), we obtain the following expression for the ACF of the received signal:
Ryy(t, t)=E ∞
−∞
x∗(t−τ )h∗(t, τ ) dτ ∞
−∞
x(t−τ)h(t, τ) dτ
(6.18) The system is linear, so that expectation can be interchanged with integration. Furthermore, the transmit signal can be interpreted as a stochastic process that is independent of the channel, so that expectations over the transmit signal and over the channel can be performed independently. Thus, the ACF of the received signal is given by:
Ryy(t, t)= ∞
−∞
∞
−∞E{x∗(t−τ )x(t−τ)}E{h∗(t, τ )h(t, τ)}dτ dτ
= ∞
−∞
∞
−∞Rxx(t−τ, t−τ)Rh(t, t, τ, τ) dτ dτ (6.19) i.e., a combination of the ACF of the transmit signal and the ACF of the channel:
Rh(t, t, τ, τ)=E{h∗(t, τ )h(t, τ)} (6.20) Note that the ACF of the channel depends on four variables since the underlying stochastic process is two dimensional.
We observe a formal similarity of the channel ACF to the impulse response of a determinis- tic channel: we can form stochastic system functions by Fourier transformations. In contrast to the deterministic case, we now have to perform a double Fourier transformation, with respect to the pair of variables t, t and/or τ, τ. From that, we obtain in an elementary way the relation- ships between the different formulations of the ACFs of input and output – e.g.,Rs(ν, ν, τ, τ)= E{s∗(ν, τ )s(ν, τ)} = Rh(t, t, τ, τ)ãexp(+j2π νt )exp(−j2π νt)dt dt.