Digital Signal Processing Reference
In-Depth Information
Figure 3.32 shows the matrix columns after being weighted by the appropriate samples, and at
column zero, the sum of all the weighted columns is shown, which forms the reconstruction.
Σ
x[n −4] x[n −3] x[n −2] x[n −1]
x[n]
x[n+1]
x[n+2]
x[n+3]
x[n+4]
x[n+5]
0
200
400
600
800
1000
1200
1400
1600
1800
0
2
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10
Sinc Matrix Column
Figure 3.32:
Sinc matrix columns, each weighted by the sample shown at the top of the column, and
the sum of all columns (i.e., the interpolated reconstruction), at the left with the symbol
at the
column head. The sample values corresponding to samples x[n-4] to x[n+5] are plotted as circles on the
reconstruction, with x[n-4] being topmost. The dashed horizontal lines intersect the 10 matrix columns
at the values that are summed to equal the corresponding interpolated value marked with a circle; the
particular rows summed correspond to the sample values x[n-4] up to x[n+5]. Any row may be summed
(as a single operation) to obtain an interpolated value between the original samples. Note that the smallest
reconstruction errors occur in the middle of the reconstruction. Note also that all signal amplitudes have
been scaled by 0.5 to avoid overlap with adjacent plots, but are not so scaled in the computation.
Figure 3.33 shows more detail of the reconstruction: the simulated continuous-domain signal (solid
line), 10 samples extracted therefrom by decimating the original signal by a factor of 100 (stem plot),
and the reconstruction (dashed line) of the original signal according to the matrix convolution method
described above.
The results of the linear convolution method are shown in Fig. 3.34. Note again that the best
approximation to the original signal is found in the middle of the reconstruction.
