This chapter is from the book
This chapter is from the book
This chapter is from the book
2.2 SAMPLING LOW-PASS SIGNALS
Consider sampling a continuous real signal whose spectrum is shown in Figure 2-4(a). Notice that the spectrum is symmetrical about zero Hz, and the spectral amplitude is zero above +B Hz and below –B Hz, i.e., the signal is band-limited. (From a practical standpoint, the term band-limited signal merely implies that any signal energy outside the range of ±B Hz is below the sensitivity of our system.) Given that the signal is sampled at a rate of fs samples/s, we can see the spectral replication effects of sampling in Figure 2-4(b), showing the original spectrum in addition to an infinite number of replications, whose period of replication is fs Hz. (Although we stated in Section 1.1 that frequency-domain representations of discrete-time sequences are themselves discrete, the replicated spectra in Figure 2-4(b) are shown as continuous lines, instead of discrete dots, merely to keep the figure from looking too cluttered. We'll cover the full implications of discrete frequency spectra in Chapter 3.)
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Figure 2-4 Spectral replications: (a) original continuous signal spectrum; (b) spectral replications of the sampled signal when fs/2
> B; (c) frequency overlap and aliasing when the sampling rate is too low because fs/2 < B.
Let's step back a moment and understand Figure 2-4 for all it's worth. Figure 2-4(a) is the spectrum of a continuous signal, a signal that can only exist in one of two forms. Either it's a continuous signal that can be sampled, through A/D conversion, or it is merely an abstract concept such as a mathematical expression for a signal. It cannot be represented in a digital machine in its current band-limited form. Once the signal is represented by a sequence of discrete sample values, its spectrum takes the replicated form of Figure 2-4(b).
The replicated spectra are not just figments of the mathematics; they exist and have a profound effect on subsequent digital signal processing.† The replications may appear harmless, and it's natural to ask, "Why care about spectral replications? We're only interested in the frequency band within ±fs/2." Well, if we perform a frequency translation operation or induce a change in sampling rate through decimation or interpolation, the spectral replications will shift up or down right in the middle of the frequency range of interest ±fs/2 and could cause problems[1]. Let's see how we can control the locations of those spectral replications.
In practical A/D conversion schemes, fs is always greater than 2B to separate spectral replications at the folding frequencies of ±fs/2. This very important relationship of fs ≥ 2B is known as the Nyquist criterion. To illustrate why the term folding frequency is used, let's lower our sampling frequency to fs = 1.5B Hz. The spectral result of this undersampling is illustrated in Figure 2-4(c). The spectral replications are now overlapping the original baseband spectrum centered about zero Hz. Limiting our attention to the band ±fs/2 Hz, we see two very interesting effects. First, the lower edge and upper edge of the spectral replications centered at +fs and –fs now lie in our band of interest. This situation is equivalent to the original spectrum folding to the left at +fs/2 and folding to the right at –fs/2. Portions of the spectral replications now combine with the original spectrum, and the result is aliasing errors. The discrete sampled values associated with the spectrum of Figure 2-4(c) no longer truly represent the original input signal. The spectral information in the bands of –B to –B/2 and B/2 to B Hz has been corrupted. We show the amplitude of the aliased regions in Figure 2-4(c) as dashed lines because we don't really know what the amplitudes will be if aliasing occurs.
The second effect illustrated by Figure 2-4(c) is that the entire spectral content of the original continuous signal is now residing in the band of interest between –fs/2 and +fs/2. This key property was true in Figure 2-4(b) and will always be true, regardless of the original signal or the sample rate. This effect is particularly important when we're digitizing (A/D converting) continuous signals. It warns us that any signal energy located above +B Hz and below –B Hz in the original continuous spectrum of Figure 2-4(a) will always end up in the band of interest after sampling, regardless of the sample rate. For this reason, continuous (analog) low-pass filters are necessary in practice.
We illustrate this notion by showing a continuous signal of bandwidth B accompanied by noise energy in Figure 2-5(a). Sampling this composite continuous signal at a rate that's greater than 2B prevents replications of the signal of interest from overlapping each other, but all of the noise energy still ends up in the range between –fs/2 and +fs/2 of our discrete spectrum shown in Figure 2-5(b). This problem is solved in practice by using an analog low-pass anti-aliasing filter prior to A/D conversion to attenuate any unwanted signal energy above +B and below –B Hz as shown in Figure 2-6. An example low-pass filter response shape is shown as the dotted line superimposed on the original continuous signal spectrum in Figure 2-6. Notice how the output spectrum of the low-pass filter has been band-limited, and spectral aliasing is avoided at the output of the A/D converter.
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Figure 2-5 Spectral replications: (a) original continuous signal plus noise spectrum; (b) discrete spectrum with noise contaminating
the signal of interest.
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Figure 2-6 Low-pass analog filtering prior to sampling at a rate of fs Hz.
This completes the discussion of simple low-pass sampling. Now let's go on to a more advanced sampling topic that's proven so useful in practice.