Does Digital Signal Have Discrete or Continuous Amplitude

Continuous Signal

Communication Systems, Civilian

Simon Haykin , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.C.3 Quantizing

A continuous signal, such as voice, has a continuous range of amplitudes, and therefore its samples have a continuous amplitude range. In other words, within the finite amplitude range of the signal we find an infinite number of amplitude levels. It is not necessary in fact to transmit the exact amplitudes of the samples. Any human sense (the ear or the eye), as ultimate receiver, can detect only finite intensity differences. This means that the original continuous signal can be approximated by a signal constructed of discrete amplitudes selected on a minimum-error basis from an available set. The existence of a finite number of discrete amplitude levels is a basic condition of PCM. Clearly, if we assign the discrete amplitude levels with sufficiently close spacing, we can make the approximated signal practically indistinguishable from the original continuous signal.

The conversion of an analog (continuous) sample of the signal to a digital (discrete) form is called the quantizing process.

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Sampling Theory

Luis F. Chaparro , Aydin Akan , in Signals and Systems Using MATLAB (Third Edition), 2019

Abstract

Processing signals continuous in both time and amplitude with a computer requires one to sample, to quantize, and to code them to obtain digital signals—discrete in both time and amplitude. The uniform sampling Nyquist condition for band-limited signals indicates that the sampling period used depends on the maximum frequency present in the signal. Moreover, by using the correct sampling period, reconstruction of the original signal from the samples is possible by Shannon's sinc interpolation. Practical aspects of the sampling and reconstruction are discussed when considering analog-to-digital (A/D) and digital-to-analog (D/A) converters. Digital communications was initiated with the concept of pulse code modulation (PCM) for the transmission of binary signals. PCM is a practical implementation of sampling, quantization and coding of an analog message into a digital message. Efficient use of the radio spectrum has motivated the development of multiplexing techniques in time and in frequency. In this chapter, we highlight some of the communication techniques that relate to the sampling theory. MATLAB is used to illustrate concepts.

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Digital signal processing

A.C. Fischer-Cripps , in Newnes Interfacing Companion, 2002

3.5.1 Digital filters

If a continuous signal y(t) is sampled N times at equal time intervals Δt, then the resulting digitised signal includes the information of interest plus any noise that might have been present in the original signal.

The purpose of a digital filter is to take this set if data, perform mathematical operations on it, and produce another set of data possessing certain desirable properties (such as reduced noise).

Digital filters fall into two basic categories: Infinite Impulse Response (IIR) or Finite Impulse Response (FIR). These terms describe the time domain characteristics of the filter when presented with an impulse signal as an input.

There are two approaches to digital filtering. The data itself may be operated upon using a filter algorithm with the desired transfer function, or, the frequency spectrum of the data may be obtained using Fourier analysis, selected frequencies discarded, and then the filtered sequence recomputed from the modified spectrum. The second method is described in some detail in this chapter.

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The design of FIR filters

Bob Meddins , in Introduction to Digital Signal Processing, 2000

5.3 PHASE-LINEARITY AND FIR FILTERS

Imagine a continuous signal, let's say a rectangular pulse, being passed through a filter. Also imagine that the transfer function of the filter is such that the gain is 1 for all frequencies. Such filters exist and are called 'all-pass' filters. It would seem reasonable that the pulse will pass through an all-pass filter undistorted. However, this is unlikely to be the case. This is because we have not taken into account the phase response of the filter. When the signal passes through a filter, the different frequencies making up the rectangular pulse will usually undergo different phase changes – effectively, signals of different frequencies are delayed by different times. It is as though, as a result of passing through the filter, signals are 'unravelled' and then put back together in a different way. This reconstruction results in distortion of the emerging signal.

An example of a continuous, all-pass filter is one with the transfer function T(s) = (s − 4)/(s + 4). (Note that this transfer function has a zero on the right-hand side of the s-plane. This is fine – it is only poles which will cause instability if placed here.) This particular filter will have a gain of 1 for all frequencies – this should be fairly obvious from its p–z diagram. Interpretation of the signal response is made slightly easier if we imagine that we have an inverting amplifier in series with the filter. The frequency response, Fig. 5.1, confirms that the gain of the combination is 1. Figure 5.2 shows the response of the filter to a rectangular pulse – the signal has clearly been changed. Notice that the distortion occurs particularly at the leading and falling edges of the pulse. This would be expected, as the edges correspond to sudden changes in the signal magnitude and rapid changes consist of a broad band of signal frequencies. Figure 5.3 shows the output when a unit impulse passes through the filter. Theoretically, a unit impulse is composed of an infinite range of frequencies and so it should suffer major distortion – and it certainly does (notice how its width has spread to approximately 1 s). In a similar way, discrete all-pass filters will also cause distortion. Figure 5.4 shows what happens to a sampled rectangular pulse, consisting of three unit pulses, passed through such a filter.

Figure 5.1.

Figure 5.2.

Figure 5.3.

Figure 5.4.

It can be shown that only if a filter is such that the gradient of the plot of phase shift against frequency is constant will there be no distortion of the signal due to the phase response.

This is because the effective signal delay introduced by a filter is given by dϕ/dω, where ϕ is the phase change. It follows that we require that ϕ = kω, where k is a constant, if the delay is to be the same for all frequencies, as then dϕ/dω = k.

Clearly, the phase response of our all-pass filter is not linear (Fig. 5.1), and so the signals suffer distortion. If a filter has a phase response with a constant gradient, i.e. where the phase response is linear then, very sensibly, the filter is described as being a 'linear-phase filter'.

In the previous chapter we spent quite a lot of time converting continuous filters to their discrete IIR equivalents, and then comparing them very critically in terms of their magnitude responses. However, you might have noticed that not too much was made of any difference between their phase responses, even though the differences were sometimes very obvious. This is because the phase response of the original continuous filter itself was probably far from ideal and so it didn't matter too much if the phase response of the discrete filter differed from it.

While we're dealing with this subject, it should be mentioned that we can take an IIR filter with its poor, i.e. non-linear, phase response and place a suitable 'all-pass' discrete filter in series with it so as to linearize the combined phase responses. As well as having a gain of 1 for all signal frequencies, the compensating all-pass filter will have a phase response which is as close as possible to the inverse of that of the original filter.

So, to sum up, a case has been made for using FIR filters in preference to IIR filters, in certain circumstances. This is because FIR filters can be designed to have a linear phase response. As a result, such an FIR filter will not introduce any distortion into the output signal due to its phase characteristics. However, do not get the idea that all FIR filters are automatically linear-phase filters – far from it. If we want an FIR filter to have a linear phase response, and we usually do, then we need to design it to have this particular property.

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Power Spectral Density

Scott L. Miller , Donald Childers , in Probability and Random Processes, 2004

For a deterministic continuous signal, x(t), the Fourier transform is used to describe its spectral content. In this text, we write the Fourier transform as ¹

(10.1) $X(f)=F[x(t)]={\int }_{-\infty }^{\infty }x(t)e^{-j2\pi ft}dt,$

and the corresponding inverse transform is

(10.2) $x(t)=F[X(f)]={\int }_{-\infty }^{\infty }X(f)e^{j2\pi ft}df.$

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Chemometrics: a textbook

In Data Handling in Science and Technology, 2003

2.4 Digitization of analytical signals

With experimental data, continuous signals are sampled at finite intervals. As discussed in Sect. 1, the sampling interval between the data points determines the maximum observable frequency, f _max. The relationship between sample interval and f _max is given by the Nyquist sampling theorem, which states that f _max = 1/(2 Δ x). If Δ x is not small enough, the high frequency information present in the data is lost and may disturb the results at other frequencies by folding. This means that the sampling frequency has to be adjusted to the properties (time constant) of the signal. Applications of Nyquist's (also referred as Shannon's) sampling theorem to the Gaussian elution profile of a chromatogram, leads to the conclusion that approximately 8 samples per 6 σ-width are needed to preserve all the frequency information present in a Gaussian peak. If the peak elutes in 1 s, a sampling frequency of 8 Hz is required to retain all the information in the signal. Reconstruction of the originally measured signal from the digitized signal could be made via the frequency spectrum, but it is evident that this would require lengthy calculations. It is common practice to choose a smaller digitizing interval than the Nyquist interval, permitting the recovery of the signal through regression for the interpolation between the sampled values. Such a higher digitizing rate also enables correction for noise in the data.

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STROKE-BASED ILLUSTRATIONS

Thomas Strothotte , Stefan Schlechtweg , in Non-Photorealistic Computer Graphics, 2002

5.3.2 Approximating the Input as a Continuous Function

The reconstruction of a continuous signal f(x) from a set of uniformly spaced discrete samples (x_i,f_i ) may be performed by convoluting with a reconstruction kernel k(x):

(5.1) $f(x)=\underset{i}{\Sigma }{f}_{i}k(x-x_i)$

It is important that the function f(x) yields the original value f_i for x = x_i , that is, when it is evaluated exactly at the position of an input sample. Figure 5.14(a) illustrates the situation. Although many different convolution kernels are available to solve this problem from the signal processing point of view, experience has shown that a 4 × 4 cubic convolution kernel as in Equation (5.2) works well.

FIGURE 5.14. Reconstruction function to be evaluated at x. All pixels in the 4 × 4 neighborhood of x contribute to the value of f(x) at x(a) and the same neighborhood in the presence of discontinuity edges (b).

(5.2) $k(x)=\{\begin{array}{ll}1.5{\left|x\right|}^3-2.5{\left|x\right|}^2+1\hfill & 0\le \left|x\right|<1\hfill \\ -0.5{\left|x\right|}^2+2.5{\left|x\right|}^2-4\left|x\right|+2\hfill & 1\le \left|x\right|<2\hfill \\ 0\hfill & 2\le \left|x\right|\hfill \end{array}$

Figure 5.15 shows a plot of this kernel function. Notice that the values close to x = 1 contribute the most, whereas values between x = 1 and x = 2 contribute negatively to counterbalance this. The reconstruction of a value at point x is done using Equation (5.1) if no discontinuity edge crosses the area of the kernel (the 4 × 4 region). Otherwise, these discontinuities have to be taken into account, and a different kernel has to be chosen.

FIGURE 5.15. Kernel function k(x) versus x.

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Digital systems

Martin Plonus , in Electronics and Communications for Scientists and Engineers (Second Edition), 2020

9.3.3 Speech signal

Next we consider a continuous signal such as speech. Fig. 9.2a shows a typical speech signal plotted on a log scale as a function of time. The vertical log scale reflects the fact that the human ear perceives sound levels logarithmically. Ordinary speech has a dynamic range of approximately 30   dB, ² which means that the ratio of the loudest sound to the softest sound is 1000 to 1, as can be seen from Fig. 9.2a. Combining this with the fact that human hearing is such that it takes a doubling of power (3   dB) to give a noticeably louder sound, we can divide (quantize) the dynamic range into 3   dB segments, which gives us 10 intervals for the dynamic range, as shown in Fig. 9.2a. The 10 distinguishable states in the speech signal imply that the quantity of information at any moment of time t is

Fig. 9.2. (a) A typical speech signal showing a 30   dB dynamic range. Along the vertical scale the signal is divided into 10 information states (bit depth), (b) Screen of a picture tube showing 525 lines (called raster lines) which the electron beam, sweeping from side to side, displays. All lines are unmodulated, except for one as indicated.

(9.4) $I_o={log}_{2}10=3.32\text{bits}$

We need to clarify resolution along the vertical scale, which is the resolution of the analog sound's amplitude. Using 10 quantizing steps is sufficient to give recognizable digital speech. However, a much higher resolution is needed to create high-fidelity digital sound. Commercial audio systems use 8 and 16 bits. For example, the resolution obtainable with 16-bit sound is 2¹⁶  =   65,536 steps or levels. As bit depth (how many steps the amplitude can be divided into) affects sound clarity, greater bit depth allows more accurate mapping of the analog sound's amplitude.

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Frequency-domain representation of discrete-time signals

Edmund Lai PhD, BEng , in Practical Digital Signal Processing, 2003

4.6.3 Computation of real-valued FFTs

Most data sequences are sampled continuous signals and are therefore real-valued. The FFTs are therefore performed on real data instead of complex-valued data assumed earlier. Since the imaginary part will be zero, some computational savings can be achieved.

One way to achieve computational saving is by computing two real FFTs simultaneously with one complex FFT. Suppose x ₁(n) and x ₂(n) are two real data sequences of length N. We can form a complex sequence by using x ₁(n) as the real part and x ₂(n) as the imaginary part of the data.

Alternatively, the original sequences can be expressed in terms of x(n) by

$\begin{array}{l}{x}_1\left(n\right)=\frac{1}{2}\left[x\left(n\right)+x*\left(n\right)\right]\\ x_2\left(n\right)=\frac{1}{2j}\left[x\left(n\right)-x*\left(n\right)\right]\end{array}$

Since the DFT is a linear operation, the DFT of x(n) can be expressed as

where

$\begin{array}{l}{X}_1\left(k\right)=\frac{1}{2}\left\{DFT\left[x\left(n\right)\right]+DFT\left[x*\left(n\right)\right]\right\}\\ X_2\left(k\right)=\frac{1}{2j}\left\{DFT\left[x\left(n\right)\right]+DFT\left[x*\left(n\right)\right]\right\}\end{array}$

Since the DFT of x∗(n) is X∗(N−k),

(7) $\begin{array}{l}{X}_1\left(k\right)=\frac{1}{2}\left[X\left(k\right)+X*\left(N-k\right)\right]\\ X_2(k)=\frac{1}{2j}\left[X\left(k\right)-X*\left(N-k\right)\right]\end{array}$

In this way, we have computed the DFT of two real sequences with one FFT. Apart from the small amount of additional computation to obtain X ₁(k) and X ₂(k), the computational requirement is halved.

Another way of achieving computational savings on real sequences is to compute 2N-point DFT of real data with one N-point complex FFT. First, subdivide the original sequence g(n) into two sequences

with x ₁(n) the even and x ₂(n) the odd data elements of the sequence. Then form a complex sequence as before and perform the FFT on it. The DFT of x ₁(n) and x ₂(n) are given by equation 7. Notice that separating a sequence into odd and even sequences is basically what is done with the decimation-in-time FFT algorithm. Hence the 2N-point DFT can be assembled from the two N point DFTs using the formulas given by the decimation-in-time algorithm. That is,

$\begin{array}{l}G\left(k\right)=X_1\left(k\right)+W_{2\text{N}}^{\text{k}}{X}_2\left(k\right)k=0,1,\dots ,N-1\\ G\left(k+N\right)=X_1\left(k\right)+W_{2\text{N}}^k{X}_2\left(k\right)k=0,1,\dots ,N-1\end{array}$

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Data Hiding Using Steganographic Techniques

Nihad Ahmad Hassan , Rami Hijazi , in Data Hiding Techniques in Windows OS, 2017

Digital Signal

A digital signal is a type of continuous signal (discrete signal) consisting of just two states, on (1) or off (0). In computer systems any waveform that switches between two voltage levels representing the two states of a Boolean value (0 and 1) is called a digital signal (see Fig. 3.53). Computers store digital audio as a sequence of 0's and 1's [10].

Figure 3.53. Sample digital wave signal.

Pulse code modulation (PCM) digital schema was created in 1937 and is used to digitalize analog data. PCM has two main properties, sample rate and bit depth.

Sample rate measures how often per second the amplitude of a waveform is taken, while the bit depth measures the possible digital values.

PCM is considered the standard form of digital audio in computers and other storage devices like CD, DVD, and portable storage. PCM's main function is to sample a waveform and turn it into digital. The sample is stored in an uncompressed format using a lossless compression, thus consuming a lot of hard disk space.

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