ECE 5670 : Digital Communications
Lecture 19: The Discrete Time Complex Baseband Wireless Channel
1
3/31/2011
Instructor: Salman Avestimehr
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
x
b
[
m
].
The real and imaginary parts of
x
b
[
m
] pass through the D/A converter to give baseband signal
x
b
(
t
). Real and imaginary
parts of
x
b
(
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
) =
summationdisplay
i
a
i
x
(
t
−
τ
i
) +
w
(
t
)
,
(1)
1
Based on lecture notes of Professor Pramod Viswanath at UIUC.
1
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x(t)
Information
Packet
Coding
Coded
Packet
Modulation
sequence of
voltage levels
D/A
D/A
x
I
b
[
m
]
x
Q
b
[
m
]
x
I
b
(
t
)
x
Q
b
(
t
)
√
2 cos 2
πf
c
t
√
2 sin 2
πf
c
t
Figure 1: Diagrammatic representation of transmitter.
where
a
i
is the attenuation of the
i
th
path and
τ
i
is the delay it experiences.
w
(
t
) denotes
the additive noise.
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 Spring '11
 SCAGLIONE
 Signal Processing, Bandwidth, Baseband, Passband, wireless channel

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