Digital Signal Processing Reference
In-Depth Information
1
30
0.5
20
0
10
−0.5
0
−1
0
10
20
30
0
1
2
3
4
5
(a) Sample
(b) Freq of 2nd Cosine
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
0
10
20
30
0
1
2
3
4
5
(c) Sample
(d) Freq of 2nd Sine
Figure 4.3:
(a) First cosine,
k
1 = 0; (b) CZL between first and second cosines, the second cosine frequency
k
2 varying between 0 and 5 cycles per 32 samples; (c) First sine,
k
1 = 0; (d) CZL between first and second
sines, the second sine frequency
k
2 varying between 0 and 5 cycles per 32 samples.
4.3.3 ORTHOGONALITY OF COMPLEX EXPONENTIALS
The sum of the product (i.e., the CZL) of two complex exponentials each having an integral number of
cycles
k
1
,
k
2
, respectively, over the sequence length
N
, where
k
1
and
k
2
are modulo-
N
,is
0
N
−
1
if
k
1
=
k
2
e
j
2
πk
1
n/N
e
−
j
2
πk
2
n/N
=
N
if
otherwise
n
=
0
4.3.4 SUM OF SAMPLES OF SINGLE COMPLEX EXPONENTIAL
An interesting and useful observation is that if one of the complex exponentials has its frequency as zero,
it is identically equal to 1.0, and the sum reduces to
N
N
−
1
if
k
1
=
...
−
N,
0
,N,
2
N, ...
e
±
j
2
πk
1
n/N
=
0
if
otherwise
n
=
0
For the sum of the samples of a single complex exponential, use the following:
functions=LVSumCE(k,N)
%s=LVSumCE(2,32)
n = 0:1:N-1; s = sum(exp(j*2*pi*n*k/N))
