When Does An Fir Filter Reach Steady State?

Steady State Response

From the Basis Up!

Luis Chaparro , in Signals and Systems Using MATLAB (2d Edition), 2015

0.4.4 The Phasor Connection

The cardinal belongings of a circuit made up of constant—valued resistors, capacitors, and inductors is that its steady-state response to a sinusoid is too a sinusoid of the same frequency. The effect of the circuit upon the input sinusoid is on its magnitude and phase and it depends on the frequency of the input sinusoid. This is due to the linear and time-invariant nature of the circuit. As we will see in Chapters 3, 4, v, ten and eleven Chapter three Chapter 4 Affiliate 5 Chapter 10 Chapter eleven , this beliefs can be generalized to more complex continuous-time also as discrete-time systems.

To illustrate the connection of phasors with dynamic systems consider the RC circuit ( $R=1\mathrm{\Omega }$ and C  =   i F) in Figure 0.7. If the input to the circuit is a sinusoidal voltage source $v_i(t)=A\cos ({\mathrm{\Omega }}_{0}t)$ and the voltage beyond the capacitor v _c (t) is the output of interest, the circuit can be easily represented by the starting time-order ordinary differential equation

${\mathrm{five}}_i(t)=\frac{{\mathit{dv}}_c(t)}{\mathit{dt}}+{\mathrm{five}}_c(t)$

Presume that the steady-country response of this excursion (i.due east., v _c (t) every bit t  →   ∞) is likewise a sinusoid

$v_c(t)=C\cos ({\mathrm{\Omega }}_{0}t+\psi )$

of the same frequency as the input, but with amplitude C and phase ψ to exist determined. Since this response must satisfy the ordinary differential equation, we have

$\begin{array}{cc}\hfill v_i(t)=& \frac{{\mathit{dv}}_c(t)}{\mathit{dt}}+v_c(t)\hfill \\ \hfill A\cos ({\mathrm{\Omega }}_{0}t)=& -C{\mathrm{\Omega }}_0\sin ({\mathrm{\Omega }}_{0}t+\psi )+C\cos ({\mathrm{\Omega }}_{0}t+\psi )\hfill \\ \hfill =& C{\mathrm{\Omega }}_0\cos ({\mathrm{\Omega }}_{0}t+\psi +\pi /2)+C\cos ({\mathrm{\Omega }}_{0}t+\psi )\hfill \\ \hfill =& C\sqrt{1+{\mathrm{\Omega }}_0^2}\cos ({\mathrm{\Omega }}_{0}t+\psi +{\tan }^{-\mathrm{ane}}{\mathrm{\Omega }}_0)\hfill \end{array}$

Comparing the two sides of the above equation gives

$C=\frac{A}{\sqrt{1+{\mathrm{\Omega }}_0^{\mathrm{two}}}}\text{,}\psi =-{\tan }^{-1}({\mathrm{\Omega }}_0)$

for a steady-country response

$5_c(t)=\frac{A}{\sqrt{1+{\mathrm{\Omega }}_0^{\mathrm{two}}}}\cos ({\mathrm{\Omega }}_{0}t-{\tan }^{-\mathrm{one}}({\mathrm{\Omega }}_0))$

Comparing the steady-land response 5 _c (t) with the input sinusoid v _i (t), we see that they both take the same frequency ${\mathrm{\Omega }}_0$ , but the amplitude and phase of the input are inverse by the circuit depending on the frequency ${\mathrm{\Omega }}_0$ . Since at each frequency the excursion responds differently, obtaining the frequency response of the circuit is useful non simply in analysis but in design of circuits.

The in a higher place sinusoidal steady-state response can also be obtained using phasors. Expressing the steady-state response of the circuit as

$5_c(t)=\mathcal{R}\mathrm{due\; east}\left[5_c{e}^{j{\mathrm{\Omega }}_{0}t}\right]$

where V _c   = Ce ^jψ is the respective phasor for v _c (t) nosotros find that

$\frac{{\mathit{dv}}_c(t)}{\mathit{dt}}=\frac{d\mathcal{R}e[V_c{e}^{j{\mathrm{\Omega }}_{0}t}]}{\mathit{dt}}=\mathcal{R}e\left[V_c\frac{{\mathit{de}}^{j{\mathrm{\Omega }}_{0}t}}{\mathit{dt}}\right]=\mathcal{R}\mathrm{eastward}\left[j{\mathrm{\Omega }}_0{V}_c{\mathrm{eastward}}^{j{\mathrm{\Omega }}_{0}t}\right]$

Replacing 5 _c (t), dv _c (t)/dt, obtained to a higher place, and

$v_i(t)=\mathcal{R}e\left[{\mathrm{Five}}_i{e}^{j{\mathrm{\Omega }}_{0}t}\right]\mathrm{where}{5}_i={\mathit{Ae}}^{j0}$

in the ordinary differential equation nosotros obtain:

$\mathcal{R}\mathrm{east}\left[V_c(\mathrm{i}+j{\mathrm{\Omega }}_0){\mathrm{eastward}}^{j{\mathrm{\Omega }}_{0}t}\right]=\mathcal{R}\mathrm{due\; east}\left[{\mathit{Ae}}^{j{\mathrm{\Omega }}_{0}t}\right]$

then that

$V_c=\frac{A}{\mathrm{i}+j{\mathrm{\Omega }}_0}=\frac{A}{\sqrt{1+{\mathrm{\Omega }}_0^2}}{\mathrm{due\; east}}^{-j{\tan }^{-1}({\mathrm{\Omega }}_0)}={\mathit{Ce}}^{j\psi }$

and the sinusoidal steady-state response is

$5_c(t)=\mathcal{R}e\left[{\mathrm{Five}}_c{e}^{j{\mathrm{\Omega }}_{0}t}\right]=\frac{A}{\sqrt{1+{\mathrm{\Omega }}_0^2}}\cos ({\mathrm{\Omega }}_{0}t-{\tan }^{-\mathrm{one}}({\mathrm{\Omega }}_0))$

which coincides with the response obtained above. The ratio of the output phasor 5 _c to the input phasor Five _i

$\frac{5_c}{V_i}=\frac{1}{1+j{\mathrm{\Omega }}_0}$

gives the response of the circuit at frequency ${\mathrm{\Omega }}_0$ . If the frequency of the input is a generic $\mathrm{\Omega }$ , irresolute ${\mathrm{\Omega }}_0$ above for $\mathrm{\Omega }$ gives the frequency response for all possible frequencies.

The concepts of linearity and fourth dimension-invariance will be used in both continuous every bit well as discrete-fourth dimension systems, along with the Fourier representation of signals in terms of sinusoids or complex exponentials, to simplify the analysis and to allow the blueprint of systems. Thus, transform methods such equally Laplace and the Z-transform will be used to solve differential and departure equations in an algebraic setup. Fourier representations will provide the frequency perspective. This is a full general approach for both continuous and discrete-time signals and systems. The introduction of the concept of transfer function will provide tools for the analysis as well as the pattern of linear time-invariant systems. The pattern of analog and discrete filters is the most important application of these concepts. We will look into this topic in Chapters 5, 7, and 12 Chapter 5 Affiliate vii Chapter 12 .

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Discrete-Fourth dimension Signals and Systems

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

8.xx Steady Land of IIR Systems—MATLAB

Suppose an IIR organization is represented by a difference equation

$\begin{array}{l}\hfill y\left[\mathrm{northward}\right]=ay\left[n-\mathrm{one}\right]+x\left[n\right]\end{array}$

where 10[n] is the input and y[north] is the output.

(a)

If the input x[north] = u[n] and information technology is known that the steady-state response is y[northward] = two, what would be a for that to be possible (in steady state ten[n] = one and y[n] = y[northward − 1] = ii since n → ∞).

(b)

Writing the organization input as x[n] = u[n] = δ[n] + δ[n − ane] + δ[n − 2] + ⋯ then according to the linearity and fourth dimension invariance, the output should exist

$\begin{array}{l}\hfill y\left[n\right]=h\left[n\right]+h\left[\mathrm{north}-1\right]+h\left[n-\mathrm{two}\right]+\cdots \end{array}$

Use the value for a found in a higher place, that the initial condition is cypher (i.due east., y[−i] = 0) and that the input is x[n] = u[n], to notice the values of the impulse response h[due north] for n ≥ 0 using the above equation. The system is causal.

(c)

Utilise the MATLAB function filter to compute the impulse response h[n] and compare it with the i obtained above.

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Physical Oceanography, Oceanic Adjustment

Ping Chang , in Encyclopedia of Concrete Scientific discipline and Applied science (Tertiary Edition), 2003

II.B Extratropical Planetary Waves

The Sverdrup flow represents a steady-state residuum between local air current stress curl and oceanic response. To accomplish this steady-land response, the sea must undergo an adjustment from an initial unbalanced state to a final balanced state. The adjustment relies on Rossby waves, whose dynamics obey the conservation of potential vorticity, i.e., the homogenous role of (4). Assuming a moving ridge solution of the course ψ   =   ψ₀ cos(kx  + ly  −   ωt), we obtain for the homogenous role of (4) the following wave dispersion relation:

(6) $\omega =-\beta \text{yard}/\left({\text{1000}}^{\mathrm{two}}+{\text{fifty}}^{\mathrm{two}}+{\text{f}}_0^2/{\text{c}}^2\right),$

where ω is the moving ridge frequency and chiliad and l are the zonal and meridional wave numbers, respectively. Notation that the moving ridge frequency ω and zonal wave number g are of opposite sign, which means that the Rossby wave always has a due west phase propagation. In other words, wave crests always movement from east to due west regardless of wavelength. This is not the case for moving ridge free energy propagation, which is given by the moving ridge group velocity. The zonal and meridional components of the wave group velocity are obtained by differentiating wave frequency ω with respect to moving ridge numbers 1000 and 50, respectively:

(7) $\begin{array}{c}\hfill {\omega }_{\text{m}}=\beta \left({\text{thousand}}^2-{\text{l}}^2-{\text{f}}_0^{\mathrm{two}}/{\text{c}}^{\mathrm{two}}\right)/{\left({\text{chiliad}}^{\mathrm{two}}+{\text{l}}^2+{\text{f}}_0^2/{\text{c}}^2\right)}^2\\ \hfill {\omega }_{\text{l}}=2\text{k}\text{fifty}\beta /{\left({\text{thou}}^{\mathrm{two}}+{\text{l}}^2+{\text{f}}_0^2/{\text{c}}^2\right)}^2.\end{array}$

For Rossby waves whose zonal wavelength is much greater than the Rossby radius of deformation, i.e., k  ≪ f ₀/c, the zonal grouping velocity is negative and independent of zonal moving ridge number k. This ways that in the long-moving ridge limit, wave energy propagates westward at the same rate for all the waves and thus the long Rossby waves are able to keep their initial shape during propagation. These waves are called nondispersive waves. In contrast, for short Rossby waves, g  ⪢ f ₀/c, the zonal group velocity is positive and decreases like g ⁻² as k increases. These short Rossby waves move energy eastward at a much slower rate than the long waves and are highly dispersive.

Figure 1 illustrates the dispersion relation for a typical Rossby wave. Note that the highest possible wave frequency, ω_max  =   βc/2f ₀, occurs at zonal wave number one thousand  = f ₀/c (assuming l  =   0) with zero group velocity. Since the Coriolis parameter f ₀ increases poleward, the maximum frequency of a Rossby moving ridge in high latitudes is lower than in low latitudes. The zonal grouping velocity ω ₁₀₀₀ at the long-wave limit k  =   0, given by ω _k   =   βc ²/f ₀ ² (assuming l  =   0) also decreases with an increase in latitude. This means that the west energy transport by the waves in the high-latitude oceans is much slower than that at low latitudes. On the other hand, the group velocity for the short waves is independent of breadth.

FIGURE one. A dispersion diagram for Rossby waves with a zero meridional wave number fifty  =   0. The zonal wave number is nondimensionalized in terms of the radius of deformation c/f ₀. The moving ridge frequency is nondimensionalized in terms of β c/f ₀. The slope of the dispersion curve gives the zonal group velocity, which changes from negative values (westward propagation) for k  < f ₀/c to positive values (east propagation) for k  > f ₀/c. The zonal group velocity vanishes at thou  = f ₀/c, where the moving ridge frequency is maximum.

In a closed sea basin, wave reflection takes place as a wave reaches a boundary. At the western boundary an incoming long Rossby wave converts its free energy into a brusk Rossby wave of the same frequency only with much shorter zonal wavelength and slower east energy propagation. This means that western boundaries of the oceans are regions where wave energy accumulates. This wave energy tin exist transferred into mean period free energy through frictional furnishings. This is one of reasons for the existence of intense western boundary currents in the oceans. At northern and southern boundaries incoming and reflected Rossby waves are of the same frequency and wavelength, and consequently are propagating at a aforementioned speed merely in the opposite direction. Superposition of these waves forms continuing waves or modes. Therefore, in the presence of northern and southern boundaries, a fix of discrete continuing Rossby waves, instead of a continuum of waves, emerges. Mathematically, these meridional modes are solutions of the following eigenvalue problem:

${\phi }_{\text{y}\text{y}}-(\beta \text{k}/\omega +{\text{k}}^2+{\text{f}}_0^{\mathrm{ii}}/{\text{c}}^2)\phi =0$

with boundary conditions

$\phi =0\text{at}\text{y}=0\mathrm{and}\text{y}=\text{Fifty}.$

The resultant eigenvalues are given past

${\omega }_{\text{n}}=-\beta \text{k}/\left[{\text{k}}^{\mathrm{ii}}+{\left(\text{northward}\pi /\text{L}\right)}^{\mathrm{ii}}+{\text{f}}_0^2/{\text{c}}^2\right],\text{n}=0,\mathrm{one},2,\mathrm{\dots },$

and the corresponding eigenfunctions are given by

${\phi }_{\text{n}}=\text{s}\text{i}\text{n}(\text{n}\pi \text{y}/\text{Fifty}),\text{due north}=0,1,2\mathrm{\dots }.$

These eigenfunctions form a consummate set of orthonormal vector ground. The excitation of these waves depends on the meridional structure of the forcing. Since the meridional scale of high-club modes is smaller than that of low-order modes, a wide-calibration forcing excites more often than not depression-order modes. A narrow forcing, on the other hand, excites a broad spectrum of modes.

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System parameters

West. Bolton , in Control Systems, 2002

4.i Introduction

When a organization is subject to, say, a unit pace input it may give an output which somewhen settles down to some steady state response. The response that it gives before settling down to this steady state is chosen its transient response. This affiliate is almost the parameters used to specify the transient response of systems and whether the transients lead to unstable systems.

For example, if nosotros accept a leap organization (Figure 4.1) and of a sudden utilise a load to information technology, it has a transient response which results in it taking some time to reach its steady state value and too information technology is likely to overshoot the steady state value before it finally settles down to the steady country value. What factors tin we change with the spring organisation in order to get it to respond more quickly to an input and also to minimise the overshooting? These are questions that are often posed for control systems. As some other illustration, consider a control system used with an automated machine to position a workpiece before some machining operation, we need to know how fast the system will reply to an input indicate and position the item in the required position and will the system exist similar the spring system when a load is practical to it and overshooting of the required position occur. Overshooting is undesirable in such a situation and then, if information technology occurred in such a control organisation, we would demand to consider what steps can be taken to eliminate information technology. Parameters are used every bit a mode of specifying how fast a arrangement volition answer to an input and how quickly it will settle down to its steady country value.

Figure iv.1. A spring arrangement with an output to a step input which takes fourth dimension to reach the steady state value and shows overshooting

With the above bound system, the outcome of applying a load is that, afterward some oscillations with ever decreasing aamplitude, the transients die away and the system settles down to a stead country value. The organisation, is said to be stable. If, however, the oscillations had continued with ever increasing amplitude, then no steady country value would have been reached and the organisation would be unstable. This affiliate takes a brief wait at the conditions for stability of systems.

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Spectral envelope and source signature assay

Mikio Tohyama , in Acoustic Signals and Hearing, 2020

five.one.ii Frequency characteristics of steady-country vibrations of string

Presume an impulse-like external forcefulness or a sinusoidal function of a frequency as the external force. The frequency characteristics for the steady-state response of a vibrating string can be written equally a closed form such that

(5.4) $H(x_s,\mathrm{ten},k)=\mathrm{i}\frac{\sin k{x}_{\mathrm{due\; south}}\cdot \sin k(L-x)}{\sin k\mathrm{50}},$

where the decaying constant is negligibly small, and ${\mathrm{ten}}_s$ and x $(x_s\le \mathrm{ten}\le L)$ are locations for the source and observation points on the string of the length $L(\mathrm{m})$ [1] [2] [3] [4] [5].

Equally described in Section 2.v, the numerator of Eq. (5.4) yields the troughs or the zeros in the complex frequency plane. On the other hand, the denominator makes the poles corresponding to the eigenfrequencies. The poles are independent of the locations of the source and observation points; however, this is non the instance for the zeros. The zeros are interlaced with the poles when the location of the ascertainment bespeak is very close to the source: the zeros move to the right (to the higher frequencies) and the number of zeros in the frequency interval of interest decreases as the observation position goes farther from the source location [4] [5].

Suppose an external source function $f(t)$ and its Fourier transform $F(\omega )$ . The response $y(t)$ and its Fourier transform $Y(\omega )$ are given by

(5.5) $y(t)=f⁎h(t)\mathrm{or}Y(\omega )=F(\omega )\cdot H(\omega )$

post-obit Eq. (v.1), where $h(t)$ denotes the impulse response divers past the changed Fourier transform of $H(\omega )$ . Every bit Eq. (5.4) shows, the frequency characteristic $H(\omega )$ is a serial of pulse-like resonance responses including the zeros, under the status that the decaying constant is negligibly pocket-sized. In dissimilarity, the external function tin be a brief bespeak in the time domain but with a wide frequency range in the frequency domain. Taking the production of $H(\omega )$ and $F(\omega )$ , $H(\omega )$ works for a sampling office of the spectral function of the external forcefulness $F(\omega )$ . The spectral part of the external force yields the spectral envelope of the response. Fig. 5.2 displays an image of power spectral response that is the product of the spectral function of the external force and frequency characteristics given by Eq. (v.iv). The vibration of a string of a instrument such every bit a piano is excited by a hammer that strikes the string. The overall spectral characteristics of string vibration are mostly adamant by the spectral envelope (or the outer envelope) made by the hit hammer. The inner envelope due to the zeros depends on the locations of observation and source positions. Fine structures of the vibration represented by line spectral components renders sensation of pitch of the sound in principle.

Figure 5.2. Ability spectral response of a multi-resonance arrangement (e.one thousand., cord vibrations, whose frequency characteristics are given by Eq. (v.4)) to a broad frequency ring external source (east.grand., an impulsive source).

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Introduction to the Design of Discrete Filters

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

12.ane Introduction

Filtering is an important awarding of linear time-invariant (LTI) systems. According to the eigenfunction property of discrete-time LTI systems, the steady-land response of a discrete-fourth dimension LTI arrangement to a sinusoidal input is also a sinusoid of the same frequency as that of the input, only with magnitude and phase afflicted past the response of the system at the frequency of the input. Since periodic also every bit aperiodic signals have Fourier representations consisting of sinusoids of different frequencies, these betoken components tin can exist modified past accordingly choosing the frequency response of a LTI system, or filter. Filtering tin can thus be seen equally a way to alter the frequency content of an input bespeak.

The advisable filter is specified using the spectral characterization of the input and the desired spectral characteristics of the output of the filter. Once the specifications of the filter are gear up, the trouble becomes one of approximation, either by a ratio of polynomials or by a polynomial (if possible). After establishing that the filter resulting from the approximation satisfies the given specifications, it is then necessary to check its stability (if not guaranteed past the pattern method)—in the instance of the filter being a rational approximation—and if stable, nosotros need to figure out what would be the best possible fashion to implement the filter in hardware or in software. If not stable, we need either to echo the approximation or to stabilize the filter before its implementation.

In the continuous-time domain, filters are obtained by means of rational approximation. In the discrete-fourth dimension domain, there are 2 possible types of filters: ane that is the result of a rational approximation—these filters are chosen recursive or Space Impulse Response (IIR) filters. The other type is the non-recursive or Finite Impulse Response (FIR) filter, which results from a polynomial approximation. Equally we volition see, the discrete filter specifications can be in the frequency or in the fourth dimension domain. For recursive or IIR filters, the specifications are typically given in the form of magnitude and phase specifications, while the specifications for non-recursive or FIR filters tin can be in the time domain equally a desired impulse response. The discrete filter design trouble then consists in: Given the specifications of a filter we look for a polynomial or rational (ratio of polynomials) approximation to the specifications. The resulting filter should exist realizable, which too causality and stability requires that the filter coefficients be existent-valued.

There are different ways to attain a rational approximation for detached IIR filters: past transformation of analog filters, or by optimization methods that include stability as a constraint. We volition encounter that the classical analog design methods (Butterworth, Chebyshev, elliptic, etc.) tin be used to blueprint discrete filters by ways of the bilinear transformation that maps the analog s-plane into the Z-plane. Given that the FIR filters are unique to the discrete domain, the approximation procedures for FIR filters are unique to that domain.

The divergence between discrete and digital filters is in quantization and coding. For a discrete filter we assume that the input and the coefficients of the filter are represented with infinite precision, i.eastward., using an infinite number of quantization levels, and thus no coding is performed. The coefficients of a digital filter are binary, and the input is quantized and coded. Quantization thus affects the operation of a digital filter, while information technology has no consequence in discrete filters.

Considering continuous-to-discrete (CD) and discrete-to-continuous (DC) ideal converters simply as samplers and reconstruction filters, respectively, theoretically it is possible to implement the filtering of band-limited analog signals using discrete filters (Fig. 12.ane). In such an application, an additional specification for the filter design is the sampling menstruation. In this process it is crucial that the sampling flow in the CD and DC converters be synchronized. In practice, filtering of analog signals is washed using analog-to-digital (A/D) and digital-to-analog (D/A) converters together with digital filters.

Effigy 12.1. Discrete filtering of analog signals using ideal continuous-to-discrete (CD), or a sampler, and discrete-to-continuous (DC) converter, or a reconstruction filter.

Two-dimensional filtering finds an important application in epitome processing. Edge detection, de-blurring, de-noising, and compression of images are possible using two-dimensional linear shift-invariant filters. The overall construction of images is characterized by low-frequency components, while edges and texture are typically high-frequency components. Thus edge detection tin be implemented using high-frequency filters capable of detecting sharp changes, or edges, in the epitome. Likewise, denoising consists in smoothing or preserving the low-frequency components of the prototype. In many situations, not-linear filters perform better than linear filters. Decomposing a bespeak into components with specific frequency ranges is useful in image compression. Bank of filters are capable of separating an image into components in frequency sub-bands that when added requite the whole spectrum of frequencies; each of the components can be represented peradventure more efficiently. MATLAB provides a comprehensive gear up of tools for image processing and the pattern of 2-dimensional filters.

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Fourier Assay in Communications and Filtering

Luis F. Chaparro , Aydin Akan , in Signals and Systems Using MATLAB (3rd Edition), 2019

seven.4 What Accept We Accomplished? What Is Adjacent?

In this chapter we have illustrated the application of the Fourier analysis to communications and filtering. Different from the awarding of the Laplace transform in control problems where transients also as steady-country responses are of involvement, in communications and filtering there is more interest in steady-land response and frequency characterization which are more appropriately treated with the Fourier transform. Different types of modulation systems are illustrated in the communication examples. Finally, this affiliate provides an introduction to the design of analog filters. In all the examples, the application of MATLAB was illustrated.

Although the material in this chapter does non accept sufficient depth, reserved for texts in communications and filtering, information technology serves to connect the theory of continuous-fourth dimension signals and systems with applications. In the next part of the book, nosotros will consider how to process signals using computers and how to use the resulting theory over again in control, communication and signal processing problems.

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Application to Control and Communications

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

Publisher Summary

This chapter introduces issues in classical control and communications and links them with the Laplace and Fourier analyses. The Laplace transform is very appropriate for control problems, where transients too as steady-land responses are of interest. In communications and filtering there is more interest in steady-state responses and frequency characterizations, which are more appropriately, treated using the Fourier transform. Control and communication systems consist of interconnection of several subsystems such as pour, parallel, and feedback. Pour and parallel issue from backdrop of the convolution integral, while the feedback connexion relates the output of the overall arrangement to its input. The control examples show the importance of the transfer function and transient and steady-state computations. Cake diagrams assistance to visualize the interconnection of the unlike systems. Different types of modulation systems are illustrated in the advice examples. The affiliate also discusses analog filter design. Filtering is a very important application of linear time invariant (LTI) systems in communications, control, and digital point processing. This In all the examples, the application of MATLAB was illustrated.

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Introduction

Mikio Tohyama , in Acoustic Signals and Hearing, 2020

1.2.3 Transient and steady-state responses to sinusoidal input

The response expressed past the convolution shows the output signal every bit a function of time after the input point is fed into the linear system. The time dependence of the response can be understood as the transient and steady-state responses. Suppose that the impulse response $h(t)$ is given by

(1.55) $h(t)={\mathrm{east}}^{-\delta t}$

for $t\ge 0$ assuming $\delta >0$ , which shows an exponential decay curve. The response to a sinusoidal input

(i.56) $x(t)={\mathrm{east}}^{\mathrm{i}\omega t}$

for $t\ge 0$ becomes

(ane.57) $y(t)={\int }_0^{t}h(\tau )x(t-\tau )d\tau =\frac{\mathrm{i}}{\delta +\mathrm{i}\omega }(1-{\mathrm{e}}^{-(\delta +\mathrm{i}\omega )t}){\mathrm{e}}^{\mathrm{i}\omega t}\to H(\omega ){\mathrm{due\; east}}^{\mathrm{i}\omega t},(t\to \infty )$

where

(i.58) $H(\omega )=\frac{1}{\delta +\mathrm{i}\omega }.$

The result of the convolution shows that the response to a sinusoidal input is still the sinusoidal part of time with the identical frequency after the fourth dimension goes long (or at the steady country), but with a different magnitude and phase expressed as $H(\omega )$ . The response that the output bespeak reaches equally time passes long is called the steady-country response. Interestingly, $H(\omega )$ , which represents the magnitude and phase at the steady-state response, can be derived:

(i.59) ${\int }_0^{\infty }h(t){\mathrm{east}}^{-\mathrm{i}\omega t}dt=H(\omega ).$

The integral formulation given by Eq. (ane.59) is called the Fourier transform of the impulse response, which gives the magnitude and phase of the response of a linear organisation to a sinusoidal input at the steady country. The formulation by the convolution says the frequency of the input indicate does not modify, even in a reverberant space at the steady state.

The exponentially decaying impulse response approximately represents the reverberation in a room. The decaying constant δ gives the speed of the disuse of the impulse response. The reverberation becomes longer (or shorter) as the decomposable constant is smaller (or larger). The transient response before reaching the steady-state response becomes quicker (or slower) equally the reverberation is shorter (or longer).

In a larger hall the reverberation can be rich, but sometimes musical instruments may not be accurately localized by hearing. This may be true in a listening room every bit well. Conditions of the recording or reproduction could exist basically specified such that

(1.60) $D_{T_D}=\frac{{\int }_0^{T_D}{h}^2(t)dt}{{\int }_0^{\infty }{h}^2(t)dt},$

where $T_D\cong \mathrm{30}(\mathrm{ms})$ , [8] [ix] [10] where $h(t)$ denotes the impulse response in the recording or reproducing space. The ratio subjectively corresponds to the free energy ratio of the directly to the delayed sounds in the superposition constructing the impulse response.

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Excursion Fundamentals

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

The Underdamped Example

Nosotros volition now go on to decrease losses (i.due east., continue decrease damping) by increasing resistance essentially to, say R   =   56 Ω, only keeping the storage elements L, C and the resonant frequency ${\omega }_0=\sqrt{7}=2.65$ the same. This decreases the damping coefficient to α = 0.v. Using Eqs. (1.63b) and (i.65), we obtain for

(one.72) ${\mathrm{southward}}_{\begin{array}{c}1\\ 2\end{array}}=-\alpha \pm \mathrm{j}\sqrt{{{\omega }_0}^2-{\alpha }^2=}-\alpha \pm \mathrm{j}{\omega }_d=-0.5\pm \mathrm{j}\mathrm{two.sixty}$

Since, ${\alpha }^2\ll {{\omega }_0}^2\left(0.25<<7\right)$ , the exponential damping or attenuation is much smaller than for overdamping and critical damping.

We at present proceed to the solution for this example; using Eq. (1.66), we can write

(ane.73) $\mathrm{v}\left(\mathrm{t}\right)={\mathrm{due\; east}}^{-\mathit{\alpha t}}\left[{\mathrm{A}}_{\mathrm{three}}cos{\omega }_{d}t+{\mathrm{A}}_{\mathrm{iv}}sin{\omega }_{d}t\right]$

Again nosotros need to determine the constants from the given initial weather, five(0)   =   0 and i(0)   =   3. Using v(0)   =   0 in the to a higher place equation results in A₃  =   0, which when practical in the above equation gives us

(1.74) $\mathrm{5}\left(\mathrm{t}\right)=e^{-\mathit{\alpha t}}{\mathrm{A}}_{4}sin{\omega }_{d}t=e^{-0.5t}{\mathrm{A}}_{4}sin2.6t$

To determine the remaining coefficient, we use our 2d initial condition, i(0)   =   3. To do this, nosotros first differentiate Eq. (1.73)

$\mathrm{dv}/\mathrm{dt}={\mathrm{due\; east}}^{-\mathit{\alpha t}}{\mathrm{A}}_4\left({\omega }_{d}cos{\omega }_{d}t-\alpha sin{\omega }_{d}t\right)$

Evaluating the in a higher place expression at t   =   0 gives dv(0)/dt   =   A_iv ω _d . Since nosotros know that for a capacitor i  =   C   dv/dt, and likewise that all of the inductor current initially must flow through the capacitor (meet discussion following Eq. 1.67), we can say that dv(0)/dt   =   i(0)/C   = 3/(1/56)   =   168. Equating both expressions for dv(0)/dt, we tin can now solve for the remaining constant and obtain A_four  =   168/ω _d   =   168/2.6   =   64.6. Our concluding expression is

(i.75) $\mathrm{v}\left(\mathrm{t}\right)=64.6e^{-\mathrm{0.five}t}sin\mathrm{2.vi}t$

Nosotros plot the underdamped response, given past the above equation, in Fig. ane.29 . We discover that voltage five starts at zero, increases with time merely then begins to decrease to zip; zero is the steady state response just as in the case of overdamped and critically damped. Notwithstanding, earlier v reaches steady country, it outset overshoots negatively and so positively, in other words it looks like a sinusoidal oscillation that is exponentially damped and could oscillate like that for a long time if resistance R is sufficiently large ²⁹ . We can now easily visualize that voltage 5 could be a pure sinusoid if we had no exponential attenuation, that is, α   =   0. This is possible when R   =   ∞, then losses vⁱⁱ/R vanish in the arrangement, (or excursion in our case) and oscillations would continue unabated at the resonant frequency f₀  =   w₀/2π. Such a pure sinusoid suggests perpetual motion. However, in our case it only means that after shocking the circuit with an initial inductor electric current, there is no mechanism anymore to dissipate the initial energy. Such an un-attenuated sinusoid is a useful waveform that commercial signal generators provide.

To generate a pure sinusoidal indicate, allow u.s. imagine that an electronically generated "negative resistance R" is connected in parallel to the RLC circuit shown in Fig. i.28. Since the resistance of two parallel-connected resistors is given past R₁R_ii/(R₁  +   R₂), nosotros obtain that −   RR/(−   R   +   R)   =   ∞. The infinite resistance reduces the RLC to an LC circuit which equally expected has no damping (α   =   0). As suggested higher up, such a excursion when excited will therefore oscillate forever at the resonant frequency w₀, while periodically exchanging electric energy 1/2Cv^two in the capacitor with magnetic energy 1/2Liⁱⁱ in the inductor. A commercial indicate generator that provides sinusoidal signals, the resnant frequency f₀ of which a user can change, is such a device by changing Fifty or C. Active circuits in the betoken generator readily synthesize the negative resistance alluded to here.

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https://www.sciencedirect.com/scientific discipline/article/pii/B9780128170083000012

When Does An Fir Filter Reach Steady State?,

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