Is a Movie Discrete or Continuous Signal

Continuous Signal Processing

Steven W. Smith , in Digital Signal Processing: A Practical Guide for Engineers and Scientists, 2003

Continuous signal processing is a parallel field to DSP, and most of the techniques are nearly identical. For example, both DSP and continuous signal processing are based on linearity, decomposition, convolution and Fourier analysis. Since continuous signals cannot be directly represented in digital computers, don't expect to find computer programs in this chapter. Continuous signal processing is based on mathematics; signals are represented as equations, and systems change one equation into another. Just as the digital computer is the primary tool used in DSP, calculus is the primary tool used in continuous signal processing. These techniques have been used for centuries, long before computers were developed.

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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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Multicarrier transmission in a frequency-selective channel

S.K. Wilson , O.A. Dobre , in Academic Press Library in Mobile and Wireless Communications, 2016

9.3.1.1 The discrete Fourier transform

With continuous signals, convolution in the time domain is equivalent to multiplication in the frequency domain. However, with discrete signals, multiplication in the frequency domain is equivalent to cyclic convolution in the time domain. That is, the cyclic convolution between two discrete and finite sequences of length N, h _n , and x _n , is defined as follows:

$\begin{array}{l}\hfill h_n\otimes x_n=\sum _{k=0}^{N-1}{h}_{\text{mod}(n-k,N)}{x}_k.\end{array}$

For example, given N = 4, (h _n ⊗x _n ) _n=2 = h ₂ x ₀ + h ₃ x ₁ + h ₀ x ₂ + h ₁ x ₃.

The DFT of a sequence of length N is

$\begin{array}{l}\hfill X_n=\sum _{k=0}^{N-1}{x}_k{\text{e}}^{\frac{-j2\pi kn}{N}},\end{array}$

and the inverse DFT (IDFT) of a sequence of length N is

$\begin{array}{l}\hfill x_k=\frac{1}{N}\sum _{n=0}^{N-1}{X}_n{\text{e}}^{\frac{j2\pi kn}{N}}.\end{array}$

The DFT of x _n ⊗ h _n is X _n × H _n . The cyclic aspect of the convolution is the key point here. It must be considered to ensure orthogonality between subcarriers in OFDM.

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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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BEARING DIAGNOSTICS

C.J. Li , K. McKee , in Encyclopedia of Vibration, 2001

Wavelet transform

For a continuous signal x(t), the wavelet transform (WT) is defined as:

[7] $W_x\left(a,b\right)=\int g^{\ast \left(a,b\right)}\left(t\right)x\left(t\right)\text{d}t$

where ^* denotes the complex conjugate, and g(t) represents the mother wavelet, e.g.:

[8] $\begin{array}{l}g\left(t\right)=\exp \left(-\sigma t\right)\sin \left({\omega }_{0}t\right)\text{for}t=0\hfill \\ \text{and}:g\left(t\right)=-g\left(-t\right)\text{for}t<0\hfill \\ g^{\left(a,b\right)}\left(t\right)=\frac{1}{\surd a}g\left(\frac{t-b}{a}\right)\hfill \end{array}$

where a is the dilation parameter which defines a baby wavelet for a given value, and b is the shifting parameter.

For a given a, carrying out WT over a range of b is like passing the signal through a filter whose impulse response is defined by the baby wavelet. Therefore, one may consider WT as a bank of band-pass filters defined by a number of a's. The salient characteristic of the WT is that the width of the passing band of the filters is frequency-dependent. Therefore, the WT can provide a high-frequency resolution at the low-frequency end while maintaining good time localization at the high-frequency end. This is advantageous for processing transient bearing ringings.

When applied to a bearing signal as a preprocessing tool, the passing band of one or more of the filters could overlap with some of the resonances that are being excited by the roller defect impacts. This results in an enhanced signal-to-noise ratio. For example, Figure 4 shows a bearing vibration measured from a roller-damaged bearing. (Note that it is already high-pass-filtered.) The periodic ringings are not obvious. Figure 5 is the result of WT with one of the baby wavelets. The periodic ringings can be seen more readily and therefore easily identified by, say, an envelope analysis.

Figure 4. Bearing vibration with inner-race defect.

Figure 5. Wavelet transform of the vibration in Figure 4.

By breaking up a broad-band bearing signal into a number of narrow-band subsignals and then scanning them for evidence of bearing defect, WT avoids the risk of selecting a wrong band that does not include any resonance and then missing the defect. The price is that one has to repeat the same bearing diagnostic algorithm on more than one subsignal.

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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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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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Signals, Systems, and Spectral Analysis

Ali Grami , in Introduction to Digital Communications, 2016

3.6.1 Orthogonal Functions

Suppose there are two continuous signals x ₁(t) and x ₂(t), and the signal x ₁(t) can be linearly approximated over a time interval [t ₁, t ₂] by x ₂(t) whose energy is E ₂. In other words, we can have $x_1\left(t\right)\cong c{x}_2\left(t\right)$ , where c is a constant. It can be shown that the optimum value of c that minimizes the energy of the error signal $e\left(t\right)=x_1\left(t\right)-c{x}_2\left(t\right)$ is as follows:

(3.32a) $c=\frac{1}{E_2}{\int }_{t_1}^{t_2}{x}_1\left(t\right)x_2^{*}\left(t\right)dt$

where the asterisk * denotes the complex conjugate operation. Note that in (3.32a), both the shapes of the two signals and the time interval of interest play important roles.

Equation (3.32a) in turn implies x ₁(t) contains a component cx ₂(t), and cx ₂(t) is thus the projection of x ₁(t) on x ₂(t). If the component of a signal x ₁(t) of the form x ₂(t) is zero (i.e., we have $c=0)$ , meaning there is zero contribution from one signal to the other, then x ₁(t) and x ₂(t) are said to be orthogonal over an interval [t ₁, t ₂]. Mathematically, x ₁(t) and x ₂(t) are orthogonal signals over a time interval [ $t_0,t_0+T_0$ ], of duration T ₀, if and only if we have the following:

(3.32b) $c={\int }_{t_0}^{t_0+T_0}{x}_1\left(t\right)x_2^{*}\left(t\right)dt=0$

As a vector can be represented as a sum of orthogonal (perpendicular) vectors, which form the coordinate system of a vector space, a signal can be represented as a sum of orthogonal signals, which form the coordinate system of a signal space. A signal can be linearly approximated by a set of N mutually orthogonal signals over a certain interval. An orthogonal set of time functions is said to be a complete set if the approximation can be made into equality. The set of orthogonal functions, which are all linearly independent, is called a set of basis functions . A vector can be represented as a sum of its vector components in various ways, depending on the choice of a coordinate system. Similarly, a signal can be represented as a sum of its signal components in various ways, depending on the choice of a set of basis functions. It can be shown that for a periodic signal, with period T ₀, the sinusoids or complex exponentials of all harmonics of the fundamental frequency (i.e., n/T ₀, where n is a positive integer) are an orthogonal complete set over the period, and form a set of basis functions.

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Describing signals in terms of frequency

Grant E Hearn , Andrew V Metcalfe , in Spectral Analysis in Engineering, 1995

4.2.3 Leakage

Let us suppose that a continuous signal is sampled at equal intervals and the total number of data is N. A finite Fourier series will fit harmonics at frequencies 2πk/N (radians/ sampling interval) for k = 1,…, N/2. If the signal is itself harmonic with frequency equal to 2πm/N, for some integer m, then its Fourier line spectrum will consist of a single spike at 2πm/N. If the signal is harmonic at some frequency between 2πm/N and 2π(m + 1)IN its line spectrum is not confined to these two frequencies but 'leaks' out to others. The three harmonic signals below illustrate this phenomenon

(a)

sin(πt/2)

(b)

sin(3πt/4)

(c)

sin(5πt/8)

sampled at times t = 0, 1, 2, 3,…, 7.

The corresponding Fourier line spectra are:

	(a)	(b)	(c)
(Mean)2	0.0000	0.0000	0.0070
1st harmonic [π/4 rad/samp int]	0.0000	0.0000	0.0225
2nd harmonic [π/2 rad/samp int]	0.5000	0.0000	0.1821
3rd harmonic [3π/4 rad/samp int]	0.0000	0.5000	0.2543
4th harmonic [π rad/samp int]	0.0000	0.0000	0.0350

They are shown in Fig. 4.3.

Fig. 4.3. Line spectra of harmonic signals; (a) and (b) no leakage; (c) leakage

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