Digital Signal Processing Reference
In-Depth Information
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
0
10
20
30
0
10
20
30
(a) Sample, Cosine Correlator
(b) Sample, Test Sig
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
0
10
20
30
0
10
20
30
(c) Sample, Sine Correlator
(d) Sample, Recon Sig
Figure 4.6:
(a) and (c) Test Cosine and Sine correlators, respectively; (b) Test sinusoid of arbitrary phase;
(d) Test sinusoid reconstructed from the test waveforms at (a) and (c) and the correlation coefficients.
The following call
k = 0;C=sum(cos(2*pi*k*(0:1:7)/8).ˆ2)
can be used, letting
k
vary as 0:1:7 in successive calls, and noting the value of
C
for each value of
k
. The
function
cos
in the call should then be changed to
sin,
and the experiments performed again. You will
observe that for the cosine function (
cos
),
C
is
N
for the case of
k
= 0 or 4 (i.e.,
N/
2), but
C
is equal to
N/
2 for other values of
k
. In the case of the sine (
sin
) function,
C
is zero when
k
is0or
N/
2 since the
sine function is identically zero in those cases, and
C
is
N/
2 for other values of
k
. This information will
be prove to be relevant in the very next section of this chapter as well as in Volume II of the series, where
we study the complex DFT.
4.3.7 MULTIPLE FREQUENCY CORRELATION AND RECONSTRUCTION
For a test signal containing a single frequency, a pair of CZLs using cosine and sine waves of the same
frequency enables reconstruction of the original test signal using the correlation coefficients and the cosine
and sine test waves as basis functions.
This can be extended to reconstruct a complete sequence of length
N
containing any frequency
(integer- or noninteger- valued), between 0 and
N/
2. To achieve this, it is necessary, in general, to
do cosine-sine CZLs for all nonaliased integral frequencies possible within the sequence length. For a
