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ECE 562: Advanced Digital Communication
Lecture 17: Passband Wireless Communication
Introduction
Beginning with this lecture, we will study wireless communication. The focus of these lectures will be on point to point communication. Communication on a wireless channel is inherently different from that on a wireline channel. The main difference is that unlike wireline channel, wireless is a shared medium. The medium is considered as a federal resource and is federally regulated. The entire spectrum is split into many licensed and unlicensed bands. An example of the the point to point communication in the licensed band is the cellular phone communication, whereas wi-fi, cordless phones and blue tooth are some of the examples of communication in the unlicensed band. The transmission over a wireless channel is restricted to a range of frequencies
[
fc − W 2 , fc + W 2
]
around the central carrier frequency fc. Typically
fc ¿ W ; (1)
for example, W ≈ 1 MHz and fc ≈ 1 GHz for cellular communication. On the other hand, the wireline channel is quite a contrast: the carrier frequency fc = 0. Further more, the same wireless system (say cell phones) use different carrier frequencies in different cities. Clearly, it is not practical to tailor the communication strategies to the different carrier frequencies. It would be a lot better if we could design the system for a fixed carrier frequency and then have a simple mechanism to translate this design to suit the actual carrier frequency of operation. This is indeed possible, and the fixed carrier frequency might as well be zero. This has the added advantage that it lets us borrow our earlier understanding of communication strategies on the wireline channel. In other words, the plan is to design for “baseband” even though we are looking to communicate in passband. The focus of this lecture is on this conversion. Finally we also see how the effects of the wireless channel (which is in passband, after all) translate to the baseband: in other words, we derive a “baseband equivalent” of the passband wireless channel.
Baseband Representation of Passband Signals
Let’s begin with a baseband signal xb(t) (of double sided bandwidth W ) that we want to transmit over the wireless channel in a band centered around fc. From our discussion in the wireline channel, xb(t) would be the signal at the output of the DAC at the transmitter. We can “up convert” this signal by multiplying it by cos 2πfct:
x(t) =
2 cos 2πfct (2)
1
f
−fc f
|Xb(f )|
W − 2 W 2
|X(f )|
√^1 2
fc
Figure 1: Magnitude spectrum of the real baseband signal and its passband signal.
is a passband signal, i.e., it’s spectrum is centered around fc and −fc. Figure 1 shows this transformation diagrammatically. We scale the carrier by
2 as cos 2πfct has power 12. Thus, by scaling, we are keeping the power in xb(t) and x(t) same. To get back the baseband signal, we multiply x(t) again by
2 cos 2πfct and then pass the signal through a low pass filter with bandwidth W :
x(t)
2 cos 2πfct = 2 cos^2 (2πfct)xb(t) (3) = (1 + cos 4πfct)xb(t) (4)
The low pass filter will discard the signal xb(t) cos 4πfct as it is a passband signal (centered around 2fc). Figure 2 shows this transformation diagrammatically. One can see that if we multiply x(t) by
2 sin 2πfct instead of
2 cos 2πfct, we get xb(t) sin 4πfct which would be eliminated completely by the low pass filter. There will be a similar outcome had we modulated the baseband signal on
2 sin 2πfct and tried to recover it by using
2 cos 2πfct. This observation suggests the following engineering idea:
We should upconvert two baseband signals, one using the cosine and the other using the sine and add them to make up the passband transmit signal. The
The Passband AWGN Channel
As in the wireline channel, the wireless channel is also well represented by a linear system. The main difference is that this channel is time varying, as opposed to the time-invariant nature of the wireline channel. For now, let us suppose that the wireless channel is time invariant. This will allow us to focus on the passband nature of the signals and channels first. We will return to the time variation at a later point in the lectures. Denoting by h(t), the impulse response of the (time-invariant) wireless channel, the received passband signal is
y(t) = h(t) ∗ x(t) + w(t). (10)
Here w(t) is passband noise. We can use the complex baseband representation derived earlier (in the context of the transmit passband signal x(t)) for the passband received signal y(t) and the passband noise w(t) (denoted, naturally, by yb(t) and wb(t) respectively). As can be expected, the passband signal equation in Equation (10) turns into the following baseband signal equation: yb(t) = hb(t) ∗ xb(t) + wb(t). (11)
The key question is:
How is the “baseband equivalent” hb(t) related to the passband channel h(t)?
The answer to this question will let us work directly with the baseband channel in Equa- tion (11) which is almost the same as the one for the wireline channel (except that all signals are complex instead of real). To understand the relation between h(t) and hb(t), let us consider a few examples.
- We start with a very simple scenario:
h(t) = δ(t). (12)
Now y(t) = x(t) + w(t) (13) and hence yb(t) = xb(t) + wb(t). (14) We conclude for this case that
hb(t) = h(t) = δ(t). (15)
- Now we move to a slightly more complicated scenario:
h(t) = δ(t − t 0 ). (16)
Now
y(t) = x(t − t 0 ) = xIb (t − t 0 )
2 cos 2πfc(t − t 0 ) − xQb (t − t 0 )
2 sin 2πfc(t − t 0 ). (17)
The baseband signal yb(t) is composed of:
ybI (t) = low pass filter output of
y(t)
2 cos 2πfct
= low pass filter output of
(2 cos 2πfc(t − t 0 ) cos 2πfct)xIb (t − t 0 )
−(2 sin 2πfc(t − t 0 ) cos 2πfct)xQb (t − t 0 )
= low pass filter output of
(cos 2πfc(2t − t 0 ) + cos 2πfct 0 )xIb (t − t 0 )
−(sin 2πfc(2t − t 0 ) − sin 2πfct 0 )xQb (t − t 0 )
= xIb (t − t 0 ) cos 2πfct 0 + xQb (t − t 0 ) sin 2πfct 0 (21) = <
xb(t − t 0 )e−j^2 πfct^0
Similarly, we obtain yQb (t) as
yQb (t) = low pass filter output of
−yb(t)
2 sin 2πfct
xb(t − t 0 )e−j^2 πfct^0
Thus, yb(t) = xb(t − t 0 )e−j^2 πfct^0. (25) We conclude that
hb(t) = e−j^2 πfct^0 δ(t − t 0 ) (26) = e−j^2 πfct^0 h(t). (27)
We notice that the baseband signal also gets delayed by the same amount as the passband signal. The key change is in the phase of the baseband signal (remember, it is a complex signal) depends on both the carrier frequency and time shift.
We are now ready to generalize: a general wireless channel is typically a sum of attenuated and delayed copies of the transmit signal. Specifically:
- the attenuation is due to the energy lost at the transmit and receive antennas, as well as any absorption in the medium;
- the delay is due to the time taken to travel the distance between the transmitter and receiver;
- the multiple copies are due to the fact that there are many physical paths between transmitter and receiver that are of the same order of attenuation.
In mathematical notation,
h(t) =
i
aiδ(t − τi). (28)
The complex baseband equivalent of the channel is, as a natural generalization of the exam- ples we did earlier,
hb(t) =
i
ale−j^2 πfcτi^ δ(t − τi). (29)
