3. Digitization Flashcards
What is digitization?
On the one hand, the time axis is discretized, an operation which is called sampling. On the other hand, a discretization of the amplitude axis is required, which is referred to as quantization. The combined process of sampling and quantization is called digitization.
(p. 44)
What is sampling?
What is uniform sampling?
Sampling means that the amplitude of the analog signal x(t) is measured and recorded only at discrete time instances tk, k ∈ Z. Hence, the information carried by x(t) in between consecutive sampling instances tk and tk+1 is discarded and not further used.
Typically, the sampling is uniform, i.e. the sample instances are at equal distance tk+1 − tk = Ts from each other, whatever k. Upon sampling an analog signal of finite length, a finite number of signal samples x[k] result.
(p. 44)
What is the sampling time?
How does it relate to the sampling frequency?
Typically, the sampling is uniform, i.e. the sample instances are at equal distance tk+1 − tk = Ts from each other, whatever k. Parameter Ts is commonly referred to as the sampling time or sample period T.
The inverse of the sampling time is called the sampling frequency, sampling rate or sample rate:
fs = 1/Ts
(p. 44)
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What is quantization?
What is linear quantization?
To further scale down the amount of information in the signal, the sample amplitudes need to be quantized. To that purpose, the signal amplitude range [xmin, xmax] is divided into a number of quantization levels. The quantized signal xQ[k] is then commonly obtained by rounding off (or truncating) the signal amplitudes x[k] to the closest quantization level. Notice that this comes down to a discretization of the amplitude axis.
Most of the time the amplitude axis is divided into intervals of the same length, as in figure 3.2. This is referred to as linear quantization. In that case the distance between consecutive quantization levels is a constant, which is called the quantization step (size).
(p. 44)
Give a definition of quantization step.
Most of the time the amplitude axis is divided into intervals of the same length, as in figure 3.2. This is referred to as linear quantization. In that case the distance between consecutive quantization levels is a constant, which is called the quantization step (size). If the amplitude range [xmin, xmax] is divided into N equidistant quantization levels, the quantization step is (typically) given by:
Q = (xmax − xmin)/N
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What is the quantization error?
The difference between the quantized signal and the original signal amplitudes is called the quantization error :
eQ[k] = xQ[k] − x[k]
(p. 45)
What is aliasing?
When does it occur?
Why is it undesired?
Make sure you can give a graphical example as in figures 3.3 and 3.4.
The fact that several frequency components are projected onto the same digital waveform, and hence, provoke ambiguity, confusion and loss of information (i.e. prevent the reconstruction of x(t) from xfs[k]), is called aliasing.
The observation can be generalized to: zie formule 3.11
(p. 48 - 49)
What does the frequency spectrum of a discrete-time signal : fmax ≤ fs/2 look like? Draw it.
Make sure you can reproduce figure 3.5. You are not supposed to be able to replicate the values on the vertical axes. Observe that the frequency spectrum of an aperiodic discrete-time signal is continuous and periodic with period fs.
(p. 50)
What is the link between periodicity and discrete behavior/continuity between the two domaines?
Discrete behaviour in one domain brings about periodicity in the other domain and vice versa. Similarly, continuity in one domain leads to aperiodic behaviour in the other domain.
(p. 51)
What does the frequency spectrum of a discrete-time signal : fmax > fs/2 look like? Draw it.
Make sure you can reproduce figure 3.6. You are not supposed to be able to replicate the values on the vertical axes.
(p. 52)
Derive the Nyquist sampling theorem from figure 3.6
Observe that aliasing is inserted in figure 3.6 because the right tip of X(f) is at a higher frequency than the left tip of X(f − fs), in other words, fmax > fs − fmax. Note that this does not happen in figure 3.5 as fmax ≤ fs − fmax ⇔ 2fmax ≤ fs. It follows that the discrete-time signal is free of aliasing only if the sampling frequency fs is higher than twice the highest frequency fmax in the analog signal x(t), i.e. if:
fs ≥ 2fmax
This inequality represents the famous Nyquist sampling theorem. If fs < 2fmax, aliasing will most likely (but not always) occur.
(p. 51)
What does the Nyquist sampling theorem say?
Observe that aliasing is inserted in figure 3.6 because the right tip of X(f) is at a higher frequency than the left tip of X(f − fs), in other words, fmax > fs − fmax. Note that this does not happen in figure 3.5 as fmax ≤ fs − fmax ⇔ 2fmax ≤ fs. It follows that the discrete-time signal is free of aliasing only if the sampling frequency fs is higher than twice the highest frequency fmax in the analog signal x(t), i.e. if:
fs ≥ 2fmax
This inequality represents the famous Nyquist sampling theorem. If fs < 2fmax, aliasing will most likely (but not always) occur.
(p. 51)
What is the Nyquist frequency?
In practice, the analog signal is not perfectly bandlimited and therefore fmax is infinite. The sampling theorem needs to be relaxed to fs >= 2fmm with fmm the highest frequency in the signal that is of interest. Frequencies above fs/2 are considered parasitic. The frequency fs/2 is commonly called the Nyquist frequency.
(p. 52 - 53)
What is an analog-to-digital converter?
Which main building blocks are inside?
Why are they needed?
An analog-to-digital converter converts an analog signal into a digital signal.
The building blocks are an anti-aliasing filter followed by the sampling unit (see figure 3.7). In practice, an active or passive analog filter is used for the filtering circuit.
As aliasing is a severe type of nonlinear distortion that is to be avoided, it is mandatory to use an anti-aliasing filter in most applications.
(p. 53)
What is an anti-aliasing filter?
Why is it needed?
How do you set the cut-off frequency?
To avoid aliasing insertion the analog signal x(t) is first lowpass filtered to remove the signal components above fs/2. In this way, a lowpass filtered version xLPF(t) of x(t) is obtained that can be sampled at rate fs (almost) free of aliasing. Such an analog lowpass prefilter is called an anti-aliasing filter.
By removing parasitic frequencies above fs/2 > fmm, aliasing insertion is greatly avoided whilst preserving the signal content that is of interest to the user.
The cut-off frequency fc of the filter has to be kept slightly below the Nyquist frequency, by setting fc e.g. to α · fs/2 with α = 0.8 . . . 0.95. Unfortunately, in this way, also frequency components that are (just) below fs/2 and that fall in the transition band of the filter, are attenuated. To minimize this type of spectral distortion α should be close to 1.
Keep in mind that the anti-aliasing filter is an analog lowpass filter that is put in front of the sampling unit.
(p. 52 - 53)
