Digital Signal Processing Reference
In-Depth Information
Source signal (16 samples))
x
1
x
2
x
3
x
4
x
5
x
6
x
7
x
8
x
9
x
10
x
11
x
12
Xn
10
14
10
12
14
8
14
12
10
8
10
12
Sliding mean value from two adjacent samples: y
n
= (x
n
+ x
n-1
)/2 This is equivalent to a 2-tap FIR-
low
pass with pulse response h
TP
(n) = h
1
x
n
+ h
2
x
n-1
Yn
10
12
12
11
13
11
11
13
11
9
9
11
Difference between two adjacent samples: z
n
= (x
n
- x
n-1
)/2. This is equivalent to a 2-tap FIR-
high
pass with pulse response h
HP
(n) = h
1
x
n
- h
2
x
n-1
Zn
0
2
-2
1
1
-3
3
-1
-1
-1
1
1
Now LP und HP samples must be devided in half (downsampling). To this end the even values of y
n
und z
n
are simply omitted and the rest is than
arranged one after the othe. As a result the sequence w has exactly the same number of samples as the source signal !
y
2
y
4
y
6
y
8
y
10
y
12
z
2
z
4
z
6
z
8
z
10
z
12
w
10
12
13
11
11
9
0
-2
1
3
-1
1
Lowpass samples after filtering and downsampling
Highpass samples after filtering and downsampling
The source signal can be completely reconstructed from the sequence
w
as:
y
2n
+ z
2n
= (x
2n
+ x
2n - 1
)/2
+
(x
2n
- x
2n-1
)/2 = x
2n
(!)
y
2n
- z
2n
= (x
2n
+ x
2n-1
)/2
-
(x
2n
- x
2n - 1
)/2 = x
2n-1
(!)
This process of lowpass and highpass filtering with subsequent downsampling can be repeated with the (reduced) LP data. Overall this is equivalent to
dyadic subband coding or the Discrete Wavelet Transformation. The coding process was not examined here.
Illustration 244:
Complete reconstruction of a signal with a 2-tap FIR filter
The signal is represented here by 12 samples or concrete figures x
n
. The simplest low pass imaginable is
selected: a sliding averager which always forms the average value from two adjacent values:
y
n
= (x
n
+ x
n-1
)/2 . Note that this lowpass is really “inert” because the readings of yn change less abruptly
than those of xn! From the point of view of the signal the pulse response of this lowpass or scaling function
is accordingly h(t) = 0.5 x
n
+ 0.5 x
n-1
.
Now the concomitant bandpass which represents the rapid changes is still missing. These are formed by
the difference z
n
between two adjacent values of x
n
, hence z
n
= (x
n
- x
n-1
)/2 . By downsampling the two sets
of data y
n
and z
n
are halved and re-arranged to form a new set of data which also contains 12 samples.
The original signal can be completely reconstructed from these 12 samples in w!
The fact that in the last 10 years thousands of scientific articles have been written on the
problem of optimal scaling functions and wavelets shows how difficult it is to find the
optimal “basic pattern” - for instance in medicine to recognise a certain coronary valve
defect by means of the ECG signal pattern.
A very simple example: houses are usually rectangular. Hence, one can regard the
rectangular brick as a possible basic pattern. Almost every house can be built from this.
Is this too simple an analogy for signal processing? Wrong! The first wavelet - called the
HAAR function after its discoverer - has precisely a rectangle as its basic pattern. In this
connection examine the example in Illustration 244. Here a 2-tap FIR filter is used, a filter
the pulse response h(t) or h(n) of which comprises only two values. This is shown in
Illustration 245 in a concrete experiment. The scaling function h(n)tp is a “rectangle”, the
wavelet is h(n)
BP
