Biomedical Engineering Reference
In-Depth Information
T
represents the sampling interval or the time between adjacent samples. In real applica-
tions, finite data sequences are generally used in digital signal processing. Therefore, the
range of a data points is
is the total number of discrete samples.
The sampling frequency, f
s
, or the sampling rate, is equal to the inverse of the sampling
period, 1/
k
¼
0, 1,
...
N
-1, where
N
, and is measured in units of Hertz (s
1
).
The following are digital sequences that are of particular importance:
T
The unit-sample of impulse sequence:
d
(k)
¼
1ifk
¼
0
0ifk
6¼
0
The unit-step sequence:
u(k)
¼
1ifk
>
0
0ifk
<
0
The exponential sequence:
a
k
u(k)
a
k
¼
if k
>
0
0ifk
<
0
The sampling rate used to discretize a continuous signal is critical for the generation of
an accurate digital approximation. If the sampling rate is too low, distortions will occur
in the digital signal. Nyquist's theorem states that the minimum sampling rate used,
f
s
, should be at least twice the maximum frequency of the original signal in order to
preserve all of the information of the analog signal. The Nyquist rate is calculated as
f
nyquist
¼
2
f
max
ð
11
:
2
Þ
where f
max
is the highest frequency present in the analog signal. The Nyquist theorem
therefore states that f
s
must be greater than or equal to 2
f
max
in order to fully represent
the analog signal by a digital sequence. Practically, sampling is usually done at five to ten
times the highest frequency, f
Max
.
The second step in the A/D conversion process involves signal quantization. Quanti-
zation is the process by which the continuous amplitudes of the discrete signal are digi-
tized by a computer. In theory, the amplitudes of a continuous signal can be any of an
infinite number of possibilities. This makes it impossible to store all the values, given
the limited memory in computer chips. Quantization overcomes this by reducing the
number of available amplitudes to a finite number of possibilities that the computer
can handle.
Since digitized samples are usually stored and analyzed as binary numbers on com-
puters, every sample generated by the sampling process must be quantized. During quan-
tization, the series of samples from the discretized sequence are transformed into binary
numbers. The resolution of the A/D converter determines the number of bits that are
available for storage. Typically, most A/D converters approximate the discrete samples
with 8, 12, or 16 bits. If the number of bits is not sufficiently large, significant errors may
be incurred in the digital approximation.
A/D converters are characterized by the number of bits that they use to generate the
numbers of the digital approximation. A quantizer with
bits is capable of representing
a total of 2
N
possible amplitude values. Therefore, the resolution of an A/D converter
N
