Digital Signal Processing Reference
In-Depth Information
Now consider what happens if we up-sample each signal rst, then add them. After
up-sampling them, we have:
[x
0
; 0;x
1
; 0;x
2
; 0;x
3
; 0]
and
[y
0
; 0;y
1
; 0;y
2
; 0;y
3
; 0]:
If we add them together now, we get:
[x
0
+ y
0
; 0;x
1
+ y
1
; 0;x
2
+ y
2
; 0;x
3
+ y
3
; 0]:
We see that this exactly matches the result from adding rst, then up-sampling.
Also, the above argument works for arbitrary lengths of x and y. Also note that we
can recursively apply this reasoning and conclude that the reconstructions shown in
Figures 9.13 and 9.14 are equivalent.
In Figure 9.15, we see the four signals that, when added together, reform the
impulse function. Figure 9.16 shows another four waves that reform the impulse
function when added. The dierence between these two is the wavelet used; in
Figure 9.15, we used the Haar wavelet, while we used the Daubechies-2 (db2) wavelet
in Figure 9.16. Both of these gures show how the DWT represents the input signal
as shifted and scaled versions of the wavelet and scaling functions. Since we use an
impulse function, the wavelet function appears in the \High wave" signals, while the
scaling function appears in the \Low wave" signals (or the \Low-Low-Low wave"
signals in the case of these two gures).
It may be dicult to see from the gure that these four signals do cancel each
other out. To make this easier, Table 9.1 shows the actual values corresponding
to the plots shown in Figure 9.15. The impulse signal used has 100 values, all of
which are zero except for a value of 1 at oset 50. Rather than give all values in
the table, we leave out the zeros and just show the values in the range 49 to 56.
As the reader can verify, all columns of this table add to zero except for the second
column. Belonging to the oset 50, the sum of this column is 1 just as the original
impulse signal has a 1 at oset 50.
Figure 9.17 shows the input (impulse function) versus the output. As expected,
the output signal exactly matches the input signal.
A more typical example is seen in Figure 9.18, a signal chosen because it clearly
shows the Haar wavelets in the rst three subplots. (The example arrayf9,2,
5,2, 7,6, 4,4gwas used here.) The top plot shows four repetitions of the
Haar wavelet, each of a dierent scale. The second subplot shows two repetitions,
but these are twice as long as the ones above it. In the third subplot, one Haar
