1 Analog Signals Are a Continuous B Discrete C Discontinuous D All of the Above

Analogue Signal

Communications and networking

Stuart Ferguson , Rodney Hebels , in Computers for Librarians (Third Edition), 2003

Analogue

Analogue signals are transmitted using a continuously changing quantity as a reference. Data transmission consists of sending signals as continuous waves. An example of an analogue signal is the transmission of radio waves, television waves or sound waves. Analogue signals are of particular interest to computer systems since the majority of the world's telephone systems transmit analogue signals and it is these telephone systems that are often used as the medium for data communications. The telephone system, which is designed primarily to transmit the human voice, translates sound waves into analogue signals for transmission, which are reconverted into sound waves at the receiving handset. An analogue signal may look something like the following:

Figure 7.12. Analogue signals

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Signal Processing, Analog

W.K. Jenkins , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

I Introduction

Analog signals are processed by specially designed devices, circuits, or systems to extract parametric information or to alter the characteristics of the input signal in some prescribed way. Spectral shaping is probably the most common form of analog signal processing and is typically done with either passive (containing resistors, capacitors, and inductors) or active [containing resistors, capacitors, and operational amplifiers (op amps)] analog filters. The active filter is probably the most common analog signal processor in use today. During the past few years, switched-capacitor filters have become popular as high-precision replacements for certain types of active filters that are to be realized in monolithic form as part of a more extensive integrated system. Switched-capacitor circuits, which contain capacitors, transistor switches, and operational amplifiers, have been successfully used for analog pre- and postfilters required at the interface between analog and digital systems. This subject is treated in more detail in Section VI. Both active filters and switched-capacitor filters are closely related historically to the analog computer, from which the term analog first originated.

Analog signals can be processed by many different types of devices, some of which are surface acoustic wave (SAW) filters, charge-coupled devices (CCDs), high-frequency distributed-parameter circuits, or optical circuits, to name only a few. Since it will not be possible to discuss all the different classes of analog processing techniques and devices, and because many of these are treated elsewhere, attention will be devoted to reviewing important mathematical tools used in analog signal analysis, discussing the connection between modern analog signal processing and its origins in the analog computer, and examining several particularly important historical topics in analog signal processing. These special topics include switched-capacitor filters, electronically tunable analog filters, nonlinear modeling by means of the analog computer, and image reconstruction for synthetic aperture radar by optical processing techniques.

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Digital Data Acquisition

JOHN DEMPSTER , in The Laboratory Computer, 2001

3.3 THE A/D CONVERTER

Analogue signals are digitised using a device known as an analogue-to-digital (A/D) converter. This is essentially a computer-controlled voltmeter which accepts an analogue signal as input and produces a computer-readable binary number as output. The A/D converter (ADC) is one of the most important components of the data acquisition system, its performance determining the accuracy and rate at which digitised samples can be acquired. ADCs have three key specifications:

•

Input voltage range

•

Resolution

•

Conversion time

The input voltage range specifies the range of analogue voltages which the ADC is capable of digitising. This voltage range may be bipolar, encompassing both negative and positive voltages, or unipolar (positive only). For general laboratory work, a bipolar range provides a more flexible option, since most signals have the potential to swing both positive and negative. Most ADCs are designed to handle bipolar voltages in the range ±5 V or ±10 V.

The precision with which an analogue voltage is digitised depends upon the number of bits in the binary output word generated by the ADC. This is known as the ADC resolution. The greater the number of bits the greater the number of integer quantisation levels, and hence the finer the division of the input voltage range. An ADC with an 8-bit resolution, for instance, represents the analogue signal as a binary number with 8 bits, providing 256 quantisation levels (0–255). A signal spanning nearly the full voltage range is thus measured with a precision of around 0.4%. The number of quantisation levels, n _q1, is related to the resolution by:

[3.3] $n_{\text{ql}}=2^n\text{bits}$

where n _bits is the number of bits in the binary word. For an ADC with a given input voltage range and resolution, the smallest voltage difference, V _min, that can be measured is given by

[3.4] $V_{\min }=\frac{V_{+}-V_{-}}{n_{\text{q}\text{l}}}$

where V ₊ and V _- are the positive and negative limits of the voltage range. Thus the smallest voltage difference that the 8-bit ADC, with a ± 5 V range, can measure is 39 mV.

ADCs are available with resolutions varying from 8 to 24 bits. Most of those in common use within the laboratory have at least a 12-bit resolution, yielding 4096 quantisation levels. This – a precision of around 0.025% – is usually sufficient for most purposes. ADCs with higher resolutions, such as 20 and even 24 bit, are available, and are used in applications where very high precision is required, the digitisation of signals from an HPLC (high-performance liquid chromatograph), for example.

The A/D conversion process is not instantaneous; a certain amount of time is required to measure the analogue voltage and to generate the binary output value. This conversion time places a limit on the rate at which an analogue signal can be sampled. The 12-bit ADCs, typically found in the laboratory, have conversion times in the range 1–10 µs, and are thus capable of sampling at rates of 100 kHz to 1 MHz. Generally speaking, the higher the precision of the ADC, the longer it takes to perform a conversion, thus a 16-bit ADC will tend to have a longer conversion time than an 8-bit one. ADCs intended for the digitisation of video signals can have conversion times of 10 ns, but may be restricted to 8-bit resolution to achieve this speed. Conversely, the 24-bit ADCs designed for high-precision work may require 20 ms per conversion.

3.3.1 A/D conversion methods

A/D conversion can be implemented in a number of different ways, depending upon whether precision, conversion speed or cost is most important. The three most common designs are:

•

Successive approximation

•

Parallel or 'flash' conversion

•

Dual slope integration

Their performance features are compared in Table 3.1.

Table 3.1. Performance range and application of different ADC designs

ADC	Resolution (bits)	Conversion Time	Applications
Successive approximation	8–16	0.5–2 µs	General purpose
Flash	7–8	10–100 ns	High speed/video
Dual slope integration	18–24	0.1–2 s	High precision

Most ADCs use the successive approximation method, which provides a compromise in terms of conversion speed and precision, and is relatively inexpensive to implement. The basic elements of a 0–5 V, 8-bit, successive approximation ADC are shown in Fig. 3.3. A unipolar ADC is illustrated for simplicity but the principles apply equally to bipolar devices. The conversion process starts by storing a snapshot of the analogue input voltage, V _in, using a sample-and-hold circuit. A reference voltage, V _ref, is digitally generated from the value of an 8-bit binary data word, by summing a set of eight fixed voltage levels (2.5, 1.25, 0.625, 0.312, 0.156, 0.078, 0.039, 0.020 V). Each voltage level is associated with a particular bit of the binary word, the bit setting determining whether that voltage is added to the reference voltage. Reference voltages between 0.02 V and 4.98 V, in steps of 0.02 V, can thus be produced by setting the binary number between 1 and 255.

Figure 3.3. ADC designs. (a) Successive approximation ADC. (b) Convergence of reference voltage, V _ref, to analogue input voltage, V _in, with each successive comparison during an A/D conversion. (c) Dual slope integration ADC. (d) Integrator charge/discharge cycle during an A/D conversion.

V _ref and V _in are fed into a comparator circuit which allows the ADC to determine whether V _ref exceeds V _in. The value of V _ref is adjusted to match V _in by successively comparing the effect of setting each bit in the binary word to one. All bits are initially set to zero. Starting with the highest bit (7) which generates the greatest voltage, each bit is set to one. If this causes V _ref to exceed V _in, the bit is set back to zero, otherwise it is retained. The process is then repeated with the next lower bit, until all 8 bits have been tested. This is, in effect, a binary search procedure which forces V _ref to converge towards V _in with each successive step, as shown in Fig. 3.3(b). On completion of the procedure, V _ref = V _in, to within the accuracy of the ADC, and the value of the binary number representing that voltage can be read out by the host computer. Accuracy depends upon the reference voltage levels for each bit in the binary word being exact binary multiples of each other. Any inaccuracy in these levels produces discontinuities in the ADC voltage response, which can result in some of the binary quantisation levels being missed out at particular input voltages. This phenomenon is known as missing codes and a guarantee of 'no missing codes' is something one should look for in the specification of a high-quality ADC.

The conversion speed of the successive approximation design is constrained by the number of voltage comparisons that have to be made, each of which requires a short period of time to allow V _ref to settle down after a bit is changed. Since one comparison is required for each bit, this is one reason why high-resolution ADCs have longer conversion times. A typical 12-bit successive approximation ADC tends to have a conversion time in the region of 0.5–2 µs, supporting sampling rates up to 2 MHz. The parallel or flash ADC avoids these constraints by providing a comparator for each quantisation level and making all voltage comparisons simultaneously, in parallel. This approach allows conversion times in the region of 10 ns and sampling rates as high as 100 MHz. Flash ADCs are typically used in frame grabbers for digitising video signals, or in high-speed digital oscilloscopes. The National Instruments NI 5112 Digital Oscilloscope card, for instance, uses a flash ADC which supports 100 MHz, 8-bit A/D sampling. The high speed of the flash design, however, is bought at the expense of resolution. The need to have one analogue comparator for each quantisation level means that the circuit becomes excessively complex and costly with increasing resolution. Flash ADCs thus typically have a resolution of only 8 bits (256 comparators).

At the other extreme, the integrating ADC design provides increased resolution at the expense of conversion speed. Rather than comparing the analogue input voltage against a series of fixed reference voltage levels, like the successive approximation or flash designs, integrating ADCs measure the time taken to charge or discharge a capacitor. The basic elements of the integrating ADC are shown in Fig. 3.3(c,d). The capacitor, C, is charged by connecting the integrator to the analogue input, V _in, via resistor R, for a fixed period of time, resulting in a voltage at the integrator output proportional to the analogue input voltage. The integrator is then switched to a fixed reference voltage, V _ref, so that the capacitor begins to discharge at a constant rate. A high-speed clock is started which increments the binary output word at regular intervals until the integrator output is zero again, producing a binary number proportional to the time taken for the capacitor to discharge. Since the time taken to discharge the capacitor is proportional to the voltage applied to the capacitor during the charging phase, the resulting binary number is also proportional to the analogue input voltage. This dual slope integration design, as it is known, provides a relatively inexpensive means of achieving a high degree of precision, with ADC resolutions commonly in the order of 18–24 bits.

The time taken to charge and discharge the integrating capacitor can be significant, resulting in conversion times of hundreds of milliseconds, or even seconds, compared to the microseconds achievable using successive approximation designs. However, a useful feature of the integrating design is its ability to automatically reject 50/60 Hz AC mains interference. Both successive approximation and flash ADCs measure the instantaneous analogue voltage, store it with a sample-and-hold circuit, and then perform the conversion. The integrating ADC, on the other hand, averages the analogue voltage over whole charging phase of its conversion cycle. If this is chosen to be equivalent to a single cycle of the mains frequency (20/16.7 ms), the mains interference will be cancelled out.

Integrating ADCs are used in applications where precision is more important than conversion speed. They are widely used in digital multimeters, both computer-based and hand held, where 5.5 decimal digit precision (0.003%) is not uncommon. There are also a number of laboratory applications where high precision can be valuable, HPLC (high-performance liquid chromatography) being one example. Chromatograms consist of a series of peaks whose height can vary by orders of magnitude. A high-resolution ADC permits the accurate measurement of both very large and very small peaks within the same record. The Data Translation DT2802 laboratory interface, for instance, is specially designed for this purpose, utilising a 24-bit integrating ADC which provides a one part in 16 777 216 precision. More details of the operation of these and other types of ADC can be found in Horowitz & Hill (1989) or Carr (1991).

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The basics

Bob Meddins , in Introduction to Digital Signal Processing, 2000

1.5 RECAP

•

Analogue signal processing systems have a variety of disadvantages, such as components needing to be changed in order to change the processor function, inaccuracies due to component ageing and temperature changes, processors built in the same way not performing identically.

•

Digital processing systems do not suffer from the problems above.

•

Digital signal processing systems sample the input signal and convert the samples to equivalent digital values. These values are processed and the resulting digital outputs converted back to analogue voltages. This series of discrete voltages is then smoothed to produce the processed analogue output.

•

The analogue input signal must be sampled at a frequency which is at least twice as high as its highest frequency component, otherwise 'aliasing' will take place.

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Signal Sampling and Quantization

Lizhe Tan , Jean Jiang , in Digital Signal Processing (Third Edition), 2019

2.4 Summary

1.

Analog signal is sampled at a fixed time interval so the ADC will convert the sampled voltage level to the digital value; this is called the sampling process.

2.

The fixed time interval between two samples is the sampling period, and the reciprocal the sampling period is the sampling rate. The half of sampling rate is the folding frequency (Nyquist limit).

3.

The sampling theorem condition that the sampling rate be larger than twice of the highest frequency of the analog signal to be sampled, must be met in order to have the analog signal be recovered.

4.

The sampled spectrum is explained using the following well-known formula

$X_s\left(f\right)=\cdots +\frac{1}{T}X\left(f+f_s\right)+\frac{1}{T}X\left(f\right)+\frac{1}{T}X\left(f-f_s\right)+\cdots \text{.}$

That is, the sampled signal spectrum is a scaled and shifted version of its analog signal spectrum and its replicas centered at the frequencies that are multiples of the sampling rate.

5.

The analog anti-aliasing lowpass filter is used before ADC to remove frequency components higher than the folding frequency to avoid aliasing.

6.

The reconstruction (analog lowpass) filter is adopted after DAC to remove the spectral images that exist in the sampled-and-hold signal and obtain the smoothed analog signal. The sample-and-hold DAC effect may distort the baseband spectrum, but it also reduces image spectrum.

7.

Quantization means the ADC unit converts the analog signal amplitude with infinite precision to digital data with finite precision (a finite number of codes).

8.

When the DAC unit converts a digital code to a voltage level, quantization error occurs. The quantization error is bounded by half of the quantization step size (ADC resolution), which is a ratio of the full range of the signal over the number of the quantization levels (number of the codes).

9.

The performance of the quantizer in terms of the signal-to-quantization noise ratio (SNR), in dB, is related to the number of bits in ADC. Increasing 1 bit used in each ADC code will improve 6-dB SNR due to quantization.

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Introduction to Digital Signal Processing

Robert Oshana , in DSP Software Development Techniques for Embedded and Real-Time Systems, 2006

Conclusion

Though analog signals can also be processed using analog hardware (that is, electrical circuits containing active and passive elements), there are several advantages to digital signal processing:

•

Analog hardware is usually limited to linear operations; digital hardware can implement nonlinear operations.

•

Digital hardware is programmable, which allows for easy modification of the signal processing procedure in both real-time and non real-time modes of operation.

•

Digital hardware is less sensitive than analog hardware to variations such as temperature, and so forth.

These advantages lead to lower cost, which is the main reason for the ongoing shift from analog to digital processing in wireless telephones, consumer electronics, industrial controllers and numerous other applications.

The discipline of signal processing, whether analog or digital, consists of a large number of specific techniques. These can be roughly categorized into two families:

•

Signal-analysis/feature-extraction techniques, which are used to extract useful information from a signal. Examples include speech recognition, location and identification of targets from radar signals, detection and characterization of changes in meteorological or seismographic data.

•

Signal filtering/shaping techniques, which are used to improve the quality of a signal. Sometimes this is done as an initial step before analysis or feature extraction. Examples of these techniques include the removal of noise and interference using filtering algorithms, separating a signal into simpler components, and other time-domain and frequency-domain averaging.

A complete signal processing system usually consists of many components and incorporates multiple signal processing techniques.

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Volume 1

M. Ortmanns , ... Y. Manoli , in Comprehensive Microsystems, 2008

1.16.2.4 Digital Postprocessing for Electronic Interfaces

After analog signal processing, an analog-to-digital (A/D) conversion provides a digital output that is more appropriate for transmitting the data over longer distances. Further, a bus interface can be implemented, simplifying the interconnection of the sensor or a number of sensors to a microcontroller or a network. Although analog calibration is possible, it might be easier from the system point of view to do the calibration in the digital domain. Calibration is, apart from packaging, one of the less obvious cost-determining factors for a sensor. It has to be considered thus as a part of the system design and has to be dealt with early in the design cycle. Calibration is necessary to correct any offset, gain, linearity, or other errors such as temperature dependence of the sensor. The process requires subjecting the sensor to a number of well-known conditions and extracting the necessary parameters. These parameters can be used to adjust the sensor signal or the readout electronics to eliminate or reduce the nonidealities.

Doing the correction in the digital domain makes it possible to use polynomial linearization functions or a table-look-up. Applications with extensive signal processing on the digital side or complicated control sequences would require a large chip area, if done in a standard cell application-specific integrated circuit (ASIC) manner. To avoid this and the long design time that such implementations would require, a configurable microcontroller that can be easily adapted to the application represents the most economic solution. Not only the size of the nonchip RAM and ROM is variable but the configuration and the kind of peripheral units are flexible as well (Lerch et al. 1995).

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Sampling Theory

Luis F. Chaparro , in Signals and Systems using MATLAB, 2011

7.2.4 Signal Reconstruction from Sinc Interpolation

The analog signal reconstruction from the samples can be shown to be an interpolation using sinc signals. First, the ideal low-pass filter $H_{lp}\left(s\right)$ in Equation (7.14) has as impulse response

(7.15) $\begin{array}{ll}\hfill h_{lp}\left(t\right)=\frac{T_s}{2\pi }\underset{-{\mathrm{\Omega }}_s∕2}{\overset{{\mathrm{\Omega }}_s∕2}{\int }}{e}^{j\mathrm{\Omega }t}d\mathrm{\Omega }=\frac{\sin \left(\pi t∕T_s\right)}{\pi t∕T_s}& \\ \hfill & & \hfill \end{array}$

which is a sinc function that has an infinite time support and decays symmetrically with respect to the origin t = 0. The reconstructed signal x _r (t) is the convolution of the sampled signal x _s (t) and $h_{lp}\left(t\right)$ , which is found to be

$\begin{array}{llll}\hfill x_r\left(t\right)& =\left[x_s\ast h_{lp}\right]\left(t\right)=\underset{-\mathfrak{\infty }}{\overset{\mathfrak{\infty }}{\int }}{x}_s\left(\tau \right)h_{lp}\left(t-\tau \right)d\tau & \hfill & \end{array}$

(7.16) $\begin{array}{llll}\hfill & =\underset{-\mathfrak{\infty }}{\overset{\mathfrak{\infty }}{\int }}\left[\sum _{n}x(n{T}_s)\delta (\tau -n{T}_s)\right]h_{lp}(t-\tau )d\tau & \hfill & \\ \hfill & =\sum _{n}x(n{T}_s)\frac{\sin (\pi (t-n{T}_s)∕T_s)}{\pi (t-n{T}_s)∕T_s}\end{array}$

after replacing x _s (τ) and applying the sifting property of the delta function. The recovered signal is thus an interpolation in terms of time-shifted sinc signals with amplitudes the samples {x(nT _s )}. In fact, if we let t = kT _s , we can see that

$\begin{array}{lll}\hfill x_r(k{T}_s)=\sum _{n}x(n{T}_s)\frac{\sin (\pi (k-n))}{\pi (k-n)}=x(k{T}_s)& & \hfill \end{array}$

since

$\frac{\sin (\pi (k-n))}{\pi (k-n)}=\{\begin{array}{ll}1\hfill & k-n=0\text{or}k=n\hfill \\ 0\hfill & k\ne n\hfill \end{array}$

This is because the above sinc function by L'Hˆopital's rule is shown to be unity when k = n, and it is 0 when k ≠ n since the sine is zero at multiples of π. Thus, the values at t = kT _s are recovered exactly, and the rest are interpolated by a sum of sinc signals.

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Continuous-Time Signals

Luis F. Chaparro , in Signals and Systems using MATLAB, 2011

1.3.3 Periodic and Aperiodic Signals

A useful characterization of signals is whether they are periodic or aperiodic (nonperiodic).

An analog signal x(t) is periodic if

■

it is defined for all possible values of t, −∞ < t < ∞, and

■

there is a positive real value T ₀, the period of x(t), such that

(1.9) $\begin{array}{l}\hfill x\left(t+k{T}_0\right)=x\left(t\right)\end{array}$

for any integer k.

The period of x(t) is the smallest possible value of T ₀ > 0 that makes the periodicity possible. Thus, although NT ₀ for an integer N > 1 is also a period of x(t) it should not be considered the period.

Remarks

■

The infinite support and the unique characteristic of the period make periodic signals nonexistent in practical applications. Despite this, periodic signals are of great significance in the Fourier representation of signals and in their processing, as we will see later. The representation of aperiodic signals is obtained from that of periodic signals, and the response of systems to periodic sinusoids is fundamental in the theory of linear systems.

■

Although seemingly redundant, the first part of the definition of a periodic signal indicates that it is not possible to have a nonzero periodic signal with a finite support (i.e., the analog signal is zero outside an interval t ∈ [t ₁, t ₂]). This first part of the definition is needed for the second part to make sense.

■

It is exasperating to find the period of a constant signal x(t) = A; visually x(t) is periodic but its period is not clear. Any positive value could be considered the period, but none will be taken. The reason is that x(t) = A = A cos(0t) or of zero frequency, and as such its period is not determined since we would have to divide by zero—not permitted. Thus, a constant signal is a periodic signal of nondefinable period!

Example 1.8

Consider the analog sinusoid

$x\left(t\right)=A\cos \left({\mathrm{\Omega }}_{0}t+\theta \right)-\mathfrak{\infty }<t<\mathfrak{\infty }$

Determine the period of this signal, and indicate for what frequency Ω₀ the period of x(t) is not clearly defined.

Solution

The analog frequency is Ω₀ = 2π/T ₀ so T ₀ = 2π/Ω₀ is the period. Whenever T ₀ > 0 (or Ω₀ > 0) these sinusoidals are periodic. For instance, consider

$x(t)=2\cos (2t-\pi /2)-\infty <t<\infty$

Its period is found by noticing that this signal has an analog frequency Ω₀ = 2 = 2π f ₀(rad/sec), or a hertz frequency of f ₀ = 1/π = 1/T ₀, so that T ₀ = π is the period in seconds. That this is the period can be seen for an integer N,

$\begin{array}{ll}x(t+N{T}_0)\hfill & =2\cos (2(t+N{T}_0)-\pi /2)=2\cos (2t+2\pi N-\pi /2)\hfill \\ \hfill & =2\cos (2t-\pi /2)=x(t)\hfill \end{array}$

since adding 2π N(a multiple of 2π) to the angle of the cosine gives the original angle. If Ω₀ = 0—that is, dc frequency—the period cannot be defined because of the division by zero when finding T ₀ = 2π/Ω₀.

Example 1.9

Consider a periodic signal x(t) of period T ₀. Determine whether the following signals are periodic, and if so, find their corresponding periods:

(a)

y(t) = A + x(t).

(b)

z(t) = x(t) + v(t) where v(t) is periodic of period T ₁ = NT ₀, where N is a positive integer.

(c)

w(t) = x(t) + u(t) where u(t) is periodic of period T ₁, not necessarily a multiple of T ₀. Determine under what conditions w(t) could be periodic.

Solution

(a)

Adding a constant to a periodic signal does not change the periodicity, so y(t) is periodic of period T ₀—that is, for an integer k, y(t + kT ₀) = A + x(t + kT ₀) = A + x(t) since x(t) is periodic of period T ₀.

(b)

The period T ₁ = NT ₀ of v(t) is also a period of x(t), and so z(t) is periodic of period T ₁ since for any integer k,

$z\left(t+k{T}_1\right)=x\left(t+k{T}_1\right)+v\left(t+k{T}_1\right)=x\left(t+kN{T}_0\right)+v\left(t\right)=x\left(t\right)+v\left(t\right)$

given that v(t + kT ₁) = v(t), and that kN is an integer so that x(t + kNT ₀) = x(t). The periodicity can be visualized by considering that in one period of v(t) we can place N periods of x(t).

(c)

The condition for w(t) to be periodic is that the ratio of the periods of x(t) and of u(t) be

$\begin{array}{l}\hfill \frac{T_1}{T_0}=\frac{N}{M}\end{array}$

where N and M are positive integers not divisible by each other so that MT ₁ = NT ₀ becomes the period of w(t). That is,

$w\left(t+M{T}_1\right)=x\left(t+M{T}_1\right)+u\left(t+M{T}_1\right)=x\left(t+N{T}_0\right)+u\left(t+M{T}_1\right)=x\left(t\right)+u\left(t\right)$

Example 1.10

Let x(t) = e ^j2t and y(t) = e ^{jπ t} , and consider their sum z(t) = x(t) + y(t), and their product w(t) = x(t)y(t). Determine if z(t) and w(t) are periodic, and if so, find their periods. Is p(t) = (1 + x(t))(1 + y(t)) periodic?

Solution

According to Euler's identity,

$\begin{array}{ll}\hfill x\left(t\right)& =\cos \left(2t\right)+j\sin \left(2t\right)\\ \hfill y\left(t\right)& =\cos \left(\pi t\right)+j\sin \left(\pi t\right)\end{array}$

indicating x(t) is periodic of period T ₀ = π (the frequency of x(t) is Ω₀ = 2 = 2π/T ₀) and y(t) is periodic of period T ₁ = 2 (the frequency of y(t) is Ω₁ = π = 2π/T ₁).

For z(t) to be periodic requires that T ₁/T ₀ be a rational number, which is not the case as T ₁/T ₀ = 2/π. So z(t) is not periodic.

The product is w(t) = x(t)y(t) = e ^j(2+π)t = cos(Ω₂ t) + j sin(Ω₂ t) where Ω₂ = 2 + π = 2π/T ₂ so that T ₂ = 2π/(2 + π), so w(t) is periodic of period T ₂.

The terms 1 + x(t) and 1 + y(t) are periodic of period T ₀ = π and T ₁ = 2, and from the case of the product above, one would hope this product be periodic. But since p(t) = 1 + x(t) + y(t) + x(t)y(t) and x(t) + y(t) is not periodic, then p(t) is not periodic.

■

Analog sinusoids of frequency Ω₀ > 0 are periodic of period T ₀ = 2π/Ω₀. If Ω₀ = 0, the period is not well defined.

■

The sum of two periodic signals x(t) and y(t), of periods T ₁ and T ₂, is periodic if the ratio of the periods T ₁/T ₂ is a rational number N/M, with N and M being nondivisible. The period of the sum is MT ₁ = NT ₂.

■

The product of two sinusoids is periodic. The product of two periodic signals is not necessarily periodic.

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Introduction to Digital Signal Processing

Lizhe Tan , Jean Jiang , in Digital Signal Processing (Third Edition), 2019

1.5 Summary

1.

An analog signal is continuous in both time and amplitude. Analog signals in the real world include current, voltage, temperature, pressure, light intensity, and so on. The digital signal contains the digital values converted from the analog signal at the specified time instants.

2.

Analog-to-digital signal conversion requires an ADC unit (hardware) and a lowpass filter attached ahead of the ADC unit to block the high-frequency components that ADC cannot handle.

3.

The digital signal can be manipulated using arithmetic. The manipulations may include digital filtering, calculation of signal frequency content, and so on.

4.

The digital signal can be converted back to an analog signal by sending the digital values to DAC to produce the corresponding voltage levels and applying a smooth filter (reconstruction filter) to the DAC voltage steps.

5.

DSP finds many applications in areas such as digital speech and audio, digital and cellular telephones, automobile controls, vibration signal analysis, communications, biomedical imaging, image/video processing, and multimedia.

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https://www.sciencedirect.com/science/article/pii/B9780128150719000014

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Source: https://www.sciencedirect.com/topics/computer-science/analogue-signal

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