Download The Discrete Time Complex Baseband Wireless Channel - Lecture Notes | ECE 562 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity!
ECE 562: Advanced Digital Communication
Lecture 18: The Discrete Time Complex Baseband Wireless
Channel
Introduction
In the previous lecture we saw that even though the wireless communication is done via passband signals, most of the processing at the transmitter and the receiver happens on the (complex) baseband equivalent signal of the real passband signal. We saw how the baseband to passband conversion is done at the transmitter. We also studied simple examples of the wireless channel and related it to the equivalent channel in the baseband. The focus of this lecture is to develop a robust model for the wireless channel. We want the model to capture the essence of the wireless medium and yet be generic enough to be applicable in all kinds of surroundings.
A Simple model
Figure 1 shows the processing at the transmitter. We modulate two data streams to generate the sequence of complex baseband voltage points xb[m]. The real and imaginary parts of xb[m] pass through the D/A converter to give baseband signal xb(t). Real and imaginary parts of xb(t) then modulates cos and sin parts of the carrier to generate the passband signal x(t). The passband signal x(t) is transmitted in the air and the signal y(t) received. Given all the details of the reflectors and absorbers in the surroundings, one can possibly use Maxwell’s equations to determine the propagation of the electromagnetic signals and get y(t) as an exact function of x(t). However, such a detailed model is neither required nor is desired. The transmitter and receiver antennas are typically separated by several wavelengths apart and far field approximations of the signal propagation are good enough. Secondly, we do not want the model to be very specific to certain surrounding. We want the model to be applicable to most of the surroundings and still be meaningful. We can model the electromagnetic signal as rays. As the rays travel in the air, they get attenuated. There is a nonzero propagation delay that each ray experiences. Further, the rays gets reflected by different reflectors before reaching the receiver. Thus, the signal arrives at the receiver via multiple paths, each of which sees different delay and attenuation. There is also an additive noise present at the receiver. Hence, we can have a simple model for the received signal y(t) as
y(t) =
i
aix(t − τi) + w(t), (1)
where ai is the attenuation of the ith^ path and τi is the delay it experiences. w(t) denotes the additive noise.
x(t)
Information Packet
Coding
Coded Packet
Modulation
sequence of voltage levels
D/A
D/A
xIb [m]
xQb [m]
xIb (t)
xQb (t)
√2 cos 2πf ct
√2 sin 2πf ct
Figure 1: Diagrammatic representation of transmitter.
The delay τi is directly related to the distance traveled by the path i. If di is the distance traveled by the path i, then the delay is
τi =
di c
where c is the speed of light in air. The typical distances traveled by the direct and reflected paths in the wireless scenario ranges from of the order of 10 m (in case of Wi-Fi) to 1000 m (in case of cellular phones). As c = 10^8 m/s, this implies that the delay values can range from 33 ns to 3.3 μs. The delay τ depends on the path length and is same for all the frequencies in the signal. Another variable in Equation 1 is the attenuation ai. In free space the attenuation is inversely proportional to the distance traveled by the path i, i.e., ai ∝ (^) d^1 i. In the terrestrial communication, the attenuation depends on the richness of the environment with respect to the scatterers. Depending on the environment, it can vary from ai ∝ (^) d^12 i to ai ∝ e−di^. Scatterers can have different absorption coefficients for the different frequencies and the attenuation can depend on the frequency. However, we are communicating in a narrow band (in KHz) around a high frequency carrier (in GHz). Thus, the variation within the band of interest are insignificant. However, the most important aspect of the wireless communication is that the transmit- ter, the receiver and the surrounding are not stationary during the communication. Hence the number of path arriving at the receiver and the distance they travel (and hence the delay and the attenuation they experience) change with time. All these parameters are then functions of time. Hence, Equation 1 should be modified to incorporate this factor.
y(t) =
i
ai(t)x(t − τi(t)) + w(t). (3)
At the receiver y(t) is down-converted to the baseband signal yb(t). Its real and imaginary parts are then sampled at the sampling rate W samples per second. Figure 2 depicts these operations.
Recall that xb(t) is obtained from xb[n] by passing it through the pulse shaping filter. As- suming the ideal pulse shaping filter sinc( (^) Tt ), xb(t) is
xb(t) =
n
x[n]sinc
t − nT T
Substituting in Equation 9, we get
yb[m] =
n
i
xb[n]sinc
m − n −
τi T
aie−j^2 πfcτi^ + wb[m] (11)
n
xb[n]
i
aie−j^2 πfcτi^ sinc
m − n −
τi T
Substituting ` := m − n, we get
yb[m] =
`
xb[m − `]
i
aie−j^2 πfcτi^ sinc
` −
τi T
L∑− 1
`=
hxb[m −] + wb[m], (14)
(15)
where the tap coefficient h` is defined as
h` def =
i
aie−j^2 πfcτi^ sinc
` −
τi T
We recall that these are exactly the same calculations as for obtaining the tap coefficients for the wireline channel in Lecture 9. From Lecture 9, we recall that if Tp is the pulse width and Td is the total delay spread, then the number of taps L are
L = b
Tp + Td T
c (17)
where the delay spread Td is the difference between the delays between the shortest and the longest path.
Td def = max i 6 =j |τi − τj |. (18)
Note that Equation 14 also has the complex noise sample wb[m]. It is the sampled baseband noise wb(t). We model the discrete noises wb[m], m ≥ 1, as i.i.d. complex Gaussian random variables. Further we model both the real and imaginary parts of the complex noise as i.i.d. (real) Gaussian random variables with mean 0 and variance N 20. Equation 14 is exactly the same as that of wireline channel equation. However, the wireline channel and wireless channel are not the same. We recall that the delays and attenuations of the paths are time varying and hence the tap coefficients are also time varying. We also note unlike in the wireline channel, both hl and xb[m] are complex numbers.
The fact that the tap coefficients hl and even the number of taps L are time varying is a distinguishing feature of the wireless channel. In wireline channel the taps do not change and hence they can be learned. But now we cannot learn them once and use that knowledge for rest of the communication. Further, the variations in the tap coefficients can be huge. It seems intuitive that the shortest paths will add up in the first tap and since these paths are not attenuated much, h 0 should always be a good tap. It turns out that this intuition is misleading. To see this, let’s consider Equation 16. Note that the paths whose delays are separated by at most T seconds. For the tap h 0 , τi ≤ T and we can approximate sinc
−τ Ti
But note that the phase term e−j^2 πfcτi^ can vary a lot. The paths that have a phase lag π will have
fc(τ 1 − τ 2 ) =
τ 1 − τ 2 =
2 fc
With fc = 1 GHz, τ 1 − τ 2 = 0. 5 ns. This corresponds to the difference in their path lengths to be of 15 cm. Thus, there could be many paths adding constructively and destructively and we could have a low operating SNR even when the transmitter and receiver are right next to each other. We can now see that the key primary difference between the wire line and the wireless channels is in the magnitudes of the channel filter coefficients: in a wireline channel they are usually in a prespecified (standardized) range. In the wireless channel, however:
- the channel coefficients can have a widely varying magnitude. Since there are multiple paths from the transmitter to the receiver, the overall channel at the receiver can still have a very small magnitude even though each of the individual paths are very strong.
- the channel coefficients change in time as well. If the change is slow enough, relative to the sampling rate, then the overhead in learning them dynamically at the receiver is not much.
Even if the wireless channel can be tracked dynamically by the receiver, the communication engineer does not have any idea a priori what value to expect. This is important to know since the resource decisions (power and bandwidth) and certain traffic characteristics (data rate and latency) are fixed a priori by the communication engineer. Thus there is a need to develop statistical knowledge of the channel filter coefficients which the communication engineer can then use to make the judicial resource allocations to fit the desired performance metrics. This is the focus of the rest of this lecture.
Statistical Modeling of The Wireless Channel
At the outset, we get a feel for what statistical model to use by studying the reasons why the wireless channel varies with time. The principal reason is mobility, but it helps to separate the role as seen by different components that make up the overall discrete time baseband channel model.
the distance between the transmitter and receiver and on some gross topographical properties (such as indoors vs outdoors).
- Rician Fading Model: Sometimes one of the paths may be strongly dominant over the rest of the paths that aggregate to form a single channel filter coefficient. The dominant path could be a line of sight path between the transmitter and receiver while the weaker paths correspond to the ones that bounce off the objects in the immediate neighborhood. Now we can statistically model the channel filter coefficient as a Rayleigh fading with a non-zero mean. The stronger the dominant path relative to the aggregation of the weaker paths, the larger the ratio of the mean to the standard deviation of the Rayleigh fading model.
Looking Ahead
Starting next lecture we turn to using the statistical knowledge of the channel to communi- cate reliably over the wireless channel.
