Digital Signal Processing Reference
In-Depth Information
1
1
0
0
−1
−1
0
10
20
30
0
10
20
30
(a) Sample
(b) Sample
Correlation Value of Sequences at (a) & (b) = 0
1
1
0
0
−1
−1
0
10
20
30
0
10
20
30
(c) Sample
(d) Sample
Correlation Value of Sequences at (c) & (d) = 0
1
1
0
0
−1
−1
0
10
20
30
0
10
20
30
(e) Sample
(f) Sample
Correlation Value of Sequences at (e) & (f) = −1.1657
Figure 4.5:
(a) One cycle of a sine wave; (b) Two cycles of a sine wave, with correlation at the zeroth lag
(CZL) value of zero; (c) Three cycles of a sine wave; (d) Five cycles of a sine wave; (e) Four cycles of a
sine wave; (f ) Five point eight (5.8) cycles of a sine wave.
Suppose in one case that
A
[
n
]
= sin
[
2
πn/N
+
θ
]
and
θ
= 0. The correlation of
A
[
n
]
with a test
sine wave
TS
will yield a large positive correlation value. If, however,
θ
=
π/
2 radians
(90 degrees), then the same correlation would yield a value of zero due to orthogonality.
In another case, suppose
A
[
n
]
= sin
[
2
πn/N
]
yields zero irrespective of
the value of
θ
. Unfortunately, we don't know if that means the unknown signal is simply identically zero
(at least at samples where the test sine wave is nonzero), or if the unknown signal happens to be a sinusoid
that is 90 degrees out of phase with the test sine wave
TS
[
n
]
= 0. The correlation of
A
[
n
]
with
TS
[
n
]
[
n
]
.
In yet a third case, if
A
[
n
]
= sin
[
4
πn/N
+
θ
]
, any correlation with
TS
[
n
]
will also yield zero for
any value of
θ
.
The solution to the problem, then, is to perform two correlations for each test frequency with
A
, one with a test sine and one with a test cosine. In this manner, for each frequency, if the sinusoid of
unknown phase is present, but 90 degrees out of phase with one test signal, it is perfectly in phase with
the other test signal, thus avoiding the ambiguous correlation value of zero.
A
[
n
]
[
n
]
, then, must be separately
correlated with sin
[
2
πn/N
]
and cos
[
2
πn/N
]
to adequately detect the presence of sin
[
2
πn/N
+
θ
]
, and
A
[
n
]
must also be separately correlated with sin
[
4
πn/N
]
and cos
[
4
πn/N
]
to detect the presence of
sin
is presumed to be
equal to zero, assuming that it could only assume the values given in the statement of the problem.
[
4
πn/N
+
θ
]
. If all four correlations (CZLs) are equal to zero, then the signal
A
[
n
]
