172
Capacity of wireless channels
transforming the channels back to the AWGN channel, or by using the type
of heuristic spherepacking arguments we have just seen.
5.2 Resources of the AWGN channel
The AWGN capacity formula (5.8) can be used to identify the roles of the
key resources of
power
and
bandwidth
.
5.2.1 Continuoustime AWGN channel
Consider a continuoustime AWGN channel with bandwidth
W
Hz, power
constraint
¯
P
watts, and additive white Gaussian noise with power spectral
density
N
0
/
2. Following the passband–baseband conversion and sampling at
rate 1
/W
(as described in Chapter 2), this can be represented by a discrete
time complex baseband channel:
y±m²
=
x±m²
+
w±m²³
(5.9)
where
w±m²
is
±²
´
0
³N
0
µ
and is i.i.d. over time. Note that since the noise is
independent in the I and Q components, each use of the complex channel can
be thought of as two independent uses of a real AWGN channel. The noise
variance and the power constraint per real symbol are
N
0
/
2 and
¯
P/´
2
Wµ
respectively. Hence, the capacity of the channel is
1
2
log
±
1
+
¯
P
N
0
W
²
bits per real dimension
³
(5.10)
or
log
±
1
+
¯
P
N
0
W
²
bits per complex dimension
¶
(5.11)
This is the capacity in bits per complex dimension or degree of freedom.
Since there are
W
complex samples per second, the capacity of the continuous
time AWGN channel is
C
awgn
´
¯
P³Wµ
=
W
log
±
1
+
¯
P
N
0
W
²
bits/s
¶
(5.12)
Note that
SNR
·
=
¯
P/´N
0
is the SNR per (complex) degree of freedom.
Hence, AWGN capacity can be rewritten as
C
awgn
=
log
´
1
+
SNR
µ
bits
/
s
/
Hz
¶
(5.13)
This formula measures the maximum achievable
spectral efficiency
through
the AWGN channel as a function of the SNR.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document173
5.2 Resources of the AWGN channel
5.2.2 Power and bandwidth
Let us ponder the significance of the capacity formula (5.12) to a communica
tion engineer. One way of using this formula is as a benchmark for evaluating
the performance of channel codes. For a system engineer, however, the main
significance of this formula is that it provides a highlevel way of thinking
about how the performance of a communication system depends on the basic
resources
available in the channel, without going into the details of specific
modulation and coding schemes used. It will also help identify the bottleneck
that limits performance.
The basic resources of the AWGN channel are the received power
¯
P
and
the bandwidth
W
. Let us first see how the capacity depends on the received
power. To this end, a key observation is that the function
f±
SNR
²³
=
log
±
1
+
SNR
²
(5.14)
is
concave
, i.e.,
f
±±
±x²
≤
0 for all
x
≥
0 (Figure 5.4). This means that increasing
the power
¯
P
suffers from a law of diminishing marginal returns: the higher
the SNR, the smaller the effect on capacity. In particular, let us look at the
low and the high SNR regimes. Observe that
log
2
±
1
+
x²
≈
x
log
2
e
when
x
≈
0
´
(5.15)
log
2
±
1
+
x²
≈
log
2
x
when
x
²
1
µ
(5.16)
Thus, when the SNR is low, the capacity increases linearly with the received
power
¯
P
: every 3 dB increase in (or, doubling) the power doubles the capacity.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 SNR, Diversity scheme, Rayleigh fading, AWGN

Click to edit the document details