Digital Signal Processing Reference
In-Depth Information
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
x
Figure 21.6
. Convexity of the
Q
(
x
) function for
x ≥
0
.
0.25
0.2
concave
0.15
0.1
0.042
convex
0
0.5
1
1.5
2
2.5
3
x (noise/signal)
1/3
Figure 21.7
. The probability of error for binary modulation as a function of the
noise-to-signal ratio
x.
The plot is convex for
x ≤
1
/
3 and concave for
x ≥
1
/
3
.
21.2.4 Convex and concave functions of two variables
We now consider examples of two-variable functions.
1. First consider
g
1
(
x
0
,x
1
)=
x
0
x
1
.Wehave
∂
2
g
1
∂x
0
=
∂
2
g
1
∂x
1
∂
2
g
1
∂x
0
∂x
1
∂
2
g
1
∂x
1
∂x
0
= 0
and
=
=1
,
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