Digital Signal Processing Reference
In-Depth Information
3.3 ALIASING
In 1928, Nyquist, working at the Bell Telephone Laboratories, discovered that in order to adequately
reconstruct a sinusoid, it was only necessary to obtain two samples of each cycle of the sinusoid. So if we
have a continuous-valued voltage representing a single frequency sinusoid, we need to obtain amplitude
samples of the signal twice per cycle. If we sample regularly at equal intervals, we can describe the sampling
operation as operating at a certain frequency, and obviously this frequency,
F
S
, will have to be at least
twice the frequency of the sinusoid we are sampling.
• If a sinusoid is sampled fewer than two times per cycle, a phenomenon called
Aliasing
will occur,
and the sampled signal cannot be properly reconstructed. When aliasing occurs, a signal's original,
pre-sampling frequency generally appears in the sampler output as a different apparent, or aliased,
frequency.
• In signals containing many frequencies, the sampling rate must be at least twice the highest fre-
quency in the signal-this ensures that each frequency in the signal will be sampled at least twice
per cycle.
The preceding statement leads to the question, “how do you know what the highest frequency is in
the signal you are quantizing?” The general answer is that the only way to know is to completely control
the situation by filtering the analog signal before you sample it; you would use an analog (continuous
domain) lowpass filter with a cutoff frequency at half the sampling frequency. This ensures that in fact
the sampling rate is more than twice the highest frequency in the signal.
Such an analog filter is called an
Anti-Aliasing Filter
, or sometimes, simply an
Aliasing Filter
.The
usual arrangement is shown in Fig. 3.1; the time domain signal, which might have unlimited or unknown
bandwidth, passes through an anti-aliasing filter in which all frequencies above one-half the sampling rate
are removed. From there, the actual sampling operation is performed: a switch is momentarily closed and
the instantaneous amplitude of the signal is stored or held on a capacitor while the unity-gain-buffered
output of the Sample-and-Hold is fed to an Analog-to-Digital Converter (ADC), an example of which
we'll cover in detail after completing our discussion of aliasing.
Figure 3.2 shows what happens when an 8 Hz sine wave is sampled at a rate of 9 Hz rather than
the minimum acceptable value of 16 Hz-the output sequence looks just like the output sequence that
would have been generated from sampling a 1 Hz-inverted-phase-sine wave at a rate of 9 Hz.
• An aliased frequency in a digital sequence has forever lost its original identity. There are, assuming
an unlimited bandwidth frequency being input to a sampler without an anti-aliasing filter, literally
an infinite number of frequencies which could be aliased into even a single frequency lying below
the Nyquist rate (half the sampling rate). In this situation, once a frequency becomes aliased, there
is no way to reverse the aliasing process and determine which of the infinite number of possible
source frequencies was in fact the original frequency.
• Samples are like snapshots or individual frames in a motion picture of action which is continuous
in nature. If we take 30 snapshots or frames per second of human actors, we are certain that we
have a good idea of everything that takes place in the scene simply because human beings cannot
move fast enough to do anything of significance in the time interval between any two adjacent
snapshots or frames (note that the human eye itself takes snapshots or samples of visually received
information and sends them one after another to the brain). Imagine then if we were to lower the
frame rate from one frame (or snapshot) every 30th of a second to, say, one frame every five seconds.
At this low rate, it is clear that the actors could engage in a huge range of activities and complete
