Digital Signal Processing Reference
In-Depth Information
In Fig. 4.2, plots (a) and (b), the sequence (a single cycle of a sine wave) has been correlated with
itself; resulting in a large positive value, 16.
Plots (c) and (d) show that when the second sine wave is shifted by 180 degrees, the correlation
value, as expected, becomes a large negative value, indicating that the two waveforms are alike, but in the
opposite or inverted sense.
Plots (e) and (f ) demonstrate an important concept: the correlation (at the zeroth lag) of a sine
and cosine of the same integral-valued frequency is zero. This property is called orthogonality, which
we'll consider extensively below.
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) = 16
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) = −16
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) = 0
Figure 4.2:
(a) First sequence, a sine wave; (b) Second sequence, identical to the first, yielding a large
positive correlation; (c) First sequence; (d) Second sequence, the negative of the first, showing a large
negative correlation; (e) First sequence; (f ) Second sequence, a cosine wave, yielding a correlation value
of 0.
4.3.1 CZL EQUAL-FREQUENCY SINE/COSINE ORTHOGONALITY
The correlative relationship of equal-frequency sines and cosines noted in the examples above may be
stated mathematically as
0
N
−
1
if
k
=[
0
,N/
2
]
sin
(
2
πkn/N)
sin
(
2
πkn/N)
=
(4.2)
N/
2
if
otherwise
n
=
0
