Digital Signal Processing Reference
In-Depth Information
When digitizing a signal, it is very important to capture most, if not all, of the
information present in the original analog signal without generating alias
frequencies. For this reason, a good deal of study has gone into the conditions
necessary to faithfully convert an analog signal into a digital form. We can see by
closely observing Figure 5.2(b) that if we are to eliminate the effect of aliasing,
the sampling frequency must be at least twice as high as the highest frequency in
the original signal. This relationship, as shown in (5.6), is known as the Nyquist
criteria, and is a well-known requirement in sampling theory:
f
> 2
⋅
f
(5.6)
s
h
In order to guarantee that this requirement is met at all times, it is normal
procedure to band-limit the input signal to one-half of the sampling frequency
once the sampling frequency has been set. This is a prudent measure, since
frequencies beyond those which are normally expected can occasionally occur in
all systems. The band-limiting process can be implemented by sending the analog
signal through an analog lowpass filter prior to sampling. (What an excellent use
of our analog filter theory!)
5.1.2 Quantization of Samples
Once the analog signal has been sampled, it is a discrete-time signal since values
of the signal exist only at particular moments of time, but the amplitude of the
signal is still continuous. Then, the next step in the analog-to-digital conversion
(ADC) is to quantize the continuous-amplitude signal to one of many discrete
values of amplitude. The number of possible values allowed for the amplitude is
determined by the size of the variable chosen to store the values. For example, if a
single byte of memory (8 bits) is chosen to store the information, then the
amplitude can take on one of 2
8
or 256 different values. If the original signal had a
range of amplitudes from +1 volt to −1 volt, then the difference between adjacent
amplitudes would be approximately 7.8 ⋅ 10
−3
volts. On the other hand, if two
bytes of memory (16 bits) were used to store each sample, there would be 2
16
or
65,536 different values to represent the signal. With this many values, the ±1 volt
signal would have adjacent amplitudes separated by only 3.05 ⋅ 10
−5
volts.
Obviously, the larger the variable used to store the sampled data, the more closely
we can approximate the analog signal with the digital representation. (In the
previous discussion it is assumed that uniform sampling was used where all levels
would be equally spaced. There are also techniques that use nonuniform spacing
to place more levels at lower levels and wider spacing for larger signals.)
However, the drawback of using larger and larger variables is twofold. First,
the storage requirements to store the digitized waveform are proportional to the
number of bits used to quantize the samples. For example, suppose we decide to
