Digital Signal Processing Reference
In-Depth Information
Correlation of Signal (circles) & Test Cosine (stars) = 0
5
0
−5
0
5
10
15
20
25
30
(a) Test Correlator = Cosine, Frequency = 1; X−Axis = Sample
5
Correlation of Signal (circles) & Test Sine (stars) = −16
0
−5
0
5
10
15
20
25
30
(b) Test Correlator = Sine, Frequency = 1; X−Axis = Sample
Figure 4.7:
(a) Samples of test signal (circles, connected by solid line for visualization of waveform), test
cosine correlator (stars, connected with dashed line for visualization of waveform); (b) Samples of test
signal (circles, connected with solid line, as in (a)), test sine correlator (stars, connected with dashed line,
as in (a)).
Finally, Fig. 4.9 shows the reconstruction process, in which the coefficients are used to reconstruct
or synthesize the original signal. The script actually performs the reconstruction two ways and plots
both results together on the same axis: in the first way, one sample at a time of
x
[
n
]
is computed using
formula (4.13), and in the second method, all samples of
x
[
n
]
are computed at once for each basis cosine
and sine, and all weighted basis cosines and sines are summed to get the result, which is identical to
that obtained using the sample-by-sample method. The latter method, synthesis harmonic-by-harmonic,
gives a more intuitive view of the reconstruction process, and thus the upper plot of Fig. 4.9 shows
a 2-cycle cosine, and a 5-cycle cosine, each scaled by the amplitude of the corresponding correlation
coefficient and the middle plot shows the same for the sine component, in this case a 1-cycle sine. The
lower plot shows the original signal samples as circles, and the reconstructed samples are plotted as stars
(since the reconstruction is essentially perfect using both methods, the stars are plotted at the centers of
the corresponding circles).
As mentioned above, analysis formulas (4.11) and (4.12) and synthesis formula (4.13) form a
version of the Discrete Fourier Transform (DFT) known as the
Real DF T
since only real arithmetic is
used. The standard version of the DFT, which uses complex arithmetic, and which has far more utility
than the Real DFT, is discussed extensively in Volume II of the series (see Chapter 1 of this volume for
