From LNTwww

<Linear and time-invariant systems

Jump to:navigation,to look for

- Linear and time-invariant systems
- Fundamentals of systems theory
- Description of the system in the time domain

- previous page
- Next page

- previous page
- Next page

## Contents

- 1 impulse response
- 2 Some Fourier transform laws
- 3 causal systems
- 4 Calculation of the output signal
- answer in 5 steps
- 6 chapter exercises

## impulsive reaction

in the area"First Fourier Integral"The book "Signal Representation" explains that for any deterministic signal $x(t)$ you can give the spectral function $X(f)$ by means of a Fourier transform. Often $X(f)$ is abbreviated as "ghost".

However, all information about the spectral function is already contained in the time domain representation, although not always immediately recognizable. The same facts apply to linear time-invariant systems.

$\text{Definition:}$ The most important quantity that describes a linear time-invariant system in the time domain is the inverse Fourier transform $H(f)$, which is denoted $\text{impulse response}$:

- $$h(t) = \int_{-\infty}^{+\infty}H(f) \cdot {\rm e}^{\hspace{0,05cm}{\rm j}2\pi ft} \ hspace{0,15cm} {\rm d}f.$$

In this context, the following must be noted:

- The frequency response $H(f)$ and the impulse response $h(t)$ are equivalent descriptive quantities that contain exactly the same information about the LTI system.
- If a Dirac-shaped input $x(t) = δ(t)$ is used, then $X(f) = 1$ and $Y(f) = H(f)$ or $y(t) = h ( t)$ are correct.
- The term "impulse response" reflects this statement: $h(t)$ is the response of the system to a (Dirac delta) function as input.
- The above definition implies that each impulse response must have the unit $\text{Hz = 1/s}$.

Quadratic impulse response and related quantity spectrum

$\text{Example 1:}$ The impulse response $h(t)$ of the so-called "rectangle-in-time" filter is constant in the $T$ time interval and zero outside this time interval.

- The associated amplitude response is the magnitude of the frequency response

- $$\vert H(f)\vert = \vert {\rm si}(\pi fT)\vert \hspace{0,5cm}\tekst{med}\hspace{0,5cm}{\rm si}( x) =\sin(x)/x={\rm sinc}(x/\pi).$$

- The area over $h(t)$ is equal to $H(f = 0) = 1$. This is caused by:

In the interval $0 < t < T$, the impulse response must be constant and equal to $1/T$. - The phase response is given by

- $$b(f) = \left\{ \begin{array}{l} \hspace{0,25cm}\pi/T \\ - \pi/T \\ \end{array} \right.\quad \ quad \begin{array}{*{20}c} \text{for} \\ \text{for} \\ \end{array}\begin{array}{*{20}c}{\left \vert \ hspace {0,05cm} f\hspace{0,05cm} \right \vert > 0,} \\{\vert \hspace{0,05cm} f \hspace{0,05cm} \vert < 0.} \\ \end{tablica} $$

- With symmetric $h(t)$ around $t = 0$ (ie, not causal) ⇒ $b(f)=0$.

## Some Fourier transform laws

The$\text{set of Fourier transforms}$has already been explained in detail in the book "Signal Representation".

Below is a brief summary where $H(f)$ describes the frequency response of the LTI system and its inverse Fourier transform $h(t)$ is the impulse response. Laws and regulations are generally applied more often$\text{Exercise}$to this chapter "System description in the time domain".

Here, too, we refer to a (German) didactic film"Laws for the Fourier Transform"⇒ "Truths of the Fourier Transform".

The Fourier transform is used in the following equations. The filled circle indicates the spectral range, the white the time interval.

- »
**multiplication**« at constant factor:

- $$k \cdot H(f)\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,k \cdot h(t).$$
- $k \lt 1$ is attenuation, while $k \gt 1$ is amplification.

- »
**equality theorem**«:

- $$H({f}/{k})\bullet\!\!-\!\!\!\!\!\!\!-\!\!\circ\,|k| \cdot h(k\cdot t).$$
- This means: Compression $(k < 1)$ of the frequency response leads to a wider and lower impulse response.
- The expansion $(k > 1)$ $H(f)$ makes $h(t)$ narrower and taller.

- »
**offset set**« in the frequency and time domain:

- $$H(f - f_0) \bullet\!\!-\!\!\!\!-\!\!\!-\!\!\circ\, h( t )\cdot {\rm e}^{ \hspace{0,05cm}{\rm j}2\pi f_0 t},$$
- $$H(f) \cdot {\rm e}^{-{\rm j}2\pi ft_0}\bullet\!\!-\!\!\!-\!\!\!\!-\!\ !\cyrk\, h(t-t_0).$$
- Shift $t_0$ ("runtime") thus leads to a multiplication with a complex exponential function in the frequency domain.
- This does not change the amplitude response $|H(f)|$.

- »
**differential set**« in the frequency and time domain:

- $$\frac{1}{{{\rm j}2\pi}} \cdot \frac{{{\rm d}H( f )}}{{{\rm d}f}} \bullet\! \!-\!\!\!-\!\!\!-\!\!\circ\,- t \cdot h( t ),$$
- $${\rm j}\cdot 2\pi f \cdot H( f ){}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \frac{{{\rm d}h(t)}}{{\rm d}t}.$$
- The differential element in the LTI system leads to a multiplication by ${\rm j}\cdot 2πf$ in the frequency domain

that is, among other things, to a phase shift of $90^{\circ}$.

- The differential element in the LTI system leads to a multiplication by ${\rm j}\cdot 2πf$ in the frequency domain

## causal systems

$\text{Definition:}$ An LTI system is called $\text{causal}$ if the impulse response $h(t)$ - the inverse Fourier transform of the frequency response $H(f)$ - fulfills the following condition :

- $$h(t) = 0 \hspace{0,25cm}{\rm dla}\hspace{0,25cm} t < 0,$$

If this condition is not met, the system will »**uncaused**' (or 'causeless').

$\text{Note:}$ Every viable system is causal.

$\rm A$ the non-causal system and $\rm B$ the causal system

$\text{Example 2:}$ The diagram illustrates the differences between the $\rm A$ noncausal system and the $\rm B$ causal system.

- In the $\rm A$ system, the effect $($w $t =\hspace{0.05cm} –T)$ starts earlier than the cause $($Dirac delta function at $t = 0)$ , which is obviously not possible in practice.

- Almost any noncausal system can be converted to a realizable causal system using $\tau$ at runtime.

- For eksempel, for $\tau = T$: $h_{\rm B}(t) = h_{\rm A}(t - T).$

All statements so far apply to both causal and non-causal systems.

However, some specific properties can be used to describe causal systems, as explained in the book's third main chapter "Description of Causally Realized Systems" in the book$\text{this book}$.

In this first and next second main chapter, we will mainly deal with non-causal systems, since their mathematical description is usually simpler.

- In this example, the frequency response $H_{\rm A}(f)$ is real,
- while for $H_{\rm B}(f)$ the extra term ${\rm e}^{–{\rm j2π}f\hspace{0.05cm}T}$ must be included.

## Calculation of the output signal

Consider the following problem: The input $x(t)$ and the frequency response $H(f)$ are known. Specify the output signal $y(t)$.

To determine LTI system performance figures

If the solution is to be determined in the frequency domain, the spectrum X(f)$ must first be determined from the given input signal $x(t)$ using$\text{Fourier Transform}$and multiplied by the frequency response $H(f)$. till$\text{inverse Fourier transform}$product, the signal $y(t)$ is obtained.

Here is an overview of the entire solution process:

- $${\rm 1.\,\, Schritt\hspace{-0,1cm} :}\hspace{0,5cm} X(f)\bullet\!\!-\!\!\!\!-\!\ !\ !-\!\!\circ\, x( t )\hspace{1,55cm}{\rm input\:widmo},$$
- $${\rm 2nd\,\, step\hspace{-0.1cm}:}\hspace{0.5cm}Y(f)= X(f) \cdot H(f) \hspace{0.82cm } {\rm output\:spektrum},$$
- $${\rm 3.\,\, Schritt\hspace{-0,1cm}:}\hspace{0,5cm} y(t)\circ\!\!-\!\!\!\!-\!\ !\ !-\!\!\bullet\, Y(f )\hspace{1,55cm}{\rm Output\:Signal}.$$

The same result is obtained when calculated in the time domain by first deriving the impulse response $h(t)$ from the frequency response $H(f)$ using the "inverse Fourier transform" and then applying the convolution operation:

- $$y(t) = x (t) * h (t) = \int_{ - \infty }^{ + \infty } {x ( \tau )} \cdot h ( {t - \tau } ) \hspace {0,1cm}{\rm d}\tau.$$

- The results are identical for both approaches.
- You should specifically choose a solution method that requires less computational effort.

$\text{Example 3:}$ At the input of the rectangular impulse response filter $h(t)$ with a width of $T$ (see$\text{Example 1}$) a rectangular pulse $x(t)$ of duration $2T$ was applied.

Trapezoid output because $x(t)$ and $h(t)$ are rectangular

In this case, it is more practical to calculate directly in the time domain:

- Combining two rectangles $x(t)$ and $h(t)$ with different widths leads to the trapezoid output $y(t)$.

- The low-pass property of the filter can be recognized by the final edge steepness $y(t)$.

- The pulse height of $(3\text{ V)}$ is preserved in this example thanks to

- $$H(f = 0) = 1/T · T = 1,$$

## trin respons

$\text{Definition:}$ In practice, one of the commonly used input functions $x(t)$ is to measure $H(f)$ »**drinking function**«

- $${\rm \gamma}(t) = \left\{ \begin{array}{l} \hspace{0,25cm}0 \\ 0,5 \\ \hspace{0,25cm} 1 \\ \ end{array} \right.\quad \quad\begin{array}{*{20}c} \text{for} \\ \text{for}\\ \text{for} \\ \end{array}\ start{array} {*{20}c}{\vert \hprzestrzeń{0,05cm} t\hprzestrzeń{0,05cm} \vert < 0,} \\ {\vert \hprzestrzeń{0,05cm}t\hprzestrzeń {0,05cm } \vert = 0,} \\ {\vert \hspace{0,05cm} t \hspace{0,05cm} \vert > 0.} \\ \end{array}$$

„**trin respons**« $\sigma(t)$ is the response of the system when the step function $\gamma(t)$ is applied to the input:

- $$x(t) = {\rm \gamma}(t)\hspace{0,5cm}\højre pil \hspace{0,5cm}y(t)={\rm \sigma}(t).$ $

The calculation in the frequency domain would be a bit complicated here, as the following equation would have to be used:

- $${\rm \sigma}(t)\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, X(f ) \cdot H(f) =\venstre({1}/{2}\cdot \delta(f) + \frac{1}{{\rm j}\cdot 2\pi f} \right) \cdot H(f).$$

Whereas a calculation in the time domain leads directly to the result:

- $${\rm \sigma}(t) = \int_{ - \infty }^{ t } {h ( \tau )} \hspace{0,1cm}{\rm d}\tau.$$

For causal systems $h(\tau) = 0$ for $\tau \lt 0$, so the lower limit of integration in the above equation can be set to $\tau = 0$.

$\text{Proof:}$ The above result is also instructive for the following reason:

- The step function $\gamma(t)$ is related to the Dirac delta function $\delta(t)$ as follows:

- $${\rm \gamma}(t) = \int_{ - \infty }^{ t } {\delta ( \tau )} \hspace{0,1cm}{\rm d}\tau.$$

- Since we have assumed that linearity and integration are linear operations, the corresponding relationship also applies to the output signal:

- $${\rm \sigma}(t) = \int_{ - \infty }^{ t } {h ( \tau )} \hspace{0,1cm}{\rm d}\tau.$$

what should be displayed.

Calculation of the step response of a square wave impulse response

$\text{Example 4:}$ The graphic illustrates the situation for the rectangular impulse response $h(\tau)$.

- The abscissa has been renamed $\tau$.
- The step function $\gamma(\tau)$ is drawn in blue.
- $\gamma(t - \tau)$ is obtained by reflection and displacement ⇒ purple dashed curve.
- The red shaded region therefore indicates the step response $\sigma(\tau)$ at time $\tau = t$.

## chapter exercises

Exercise 1.3: Measured step response

Exercise 1.3Z: Exponentially decaying impulse response

Lab 1.4: Second-order low-pass filter

Problem 1.4Z: Everything rectangular

Downloaded from ""

## FAQs

### What is time domain description? ›

What is Time Domain Analysis? A time domain analysis is **an analysis of physical signals, mathematical functions, or time series of economic or environmental data, in reference to time**.

**What is time domain description and frequency domain description? ›**

Time-domain data consists of one or more input variables u(t) and one or more output variables y(t), sampled as a function of time. Frequency-domain data consists of either transformed input and output time-domain signals or system frequency response sampled as a function of the independent variable frequency.

**What is an example of a time domain? ›**

Time domain refers to variation of amplitude of signal with time. For example consider a typical Electro cardiogram (ECG). **If the doctor maps the heartbeat with time say the recording is done for 20 minutes, we call it a time domain signal.**

**What important information does the time domain signal give? ›**

Time domain signal processing analyzes the input signal depending on the waveforms observed over a period of time. Time domain techniques **emphasize the amplitude variation in a specific time period**. This has many applications in speech processing and heavy vehicle classification.

**What is time domain solver? ›**

A time domain solver **calculates the development of fields through time at discrete locations and at discrete time samples**. It calculates the transmission of energy between various ports or other excitation sources and/or open space of the investigated structure.

**What are the list of time domain features? ›**

extracted time-domain features are; **Mean, Median, Mode, Standard Deviation, Variance, Covariance, Zero Cross Rate, Minimum, Maximum, Root Mean Square, and Distance**.

**What is frequency domain description? ›**

The Frequency Domain refers to **the analytic space in which mathematical functions or signals are conveyed in terms of frequency, rather than time**. For example, where a time-domain graph may display changes over time, a frequency-domain graph displays how much of the signal is present among each given frequency band.

**Which of the following is an example of a domain? ›**

An example of a domain name is **usps.com**. This is made up of a second-level domain ("usps") and top-level domain (".com).

**What are time domain models? ›**

The state-space approach (also referred to as the modern or time-domain approach) is **a unified method for modeling, analyzing and designing a wide range of systems**. We can use the state-space approach both linear and nonlinear systems. Also it can handle the systems with nonzero initial conditions.

**What is the difference between time domain? ›**

Time domain is the domain for analysis of mathematical functions or signals with respect to time. Frequency domain is the domain for analysis of mathematical functions or signals with respect to frequency. The time domain systems tend to use photon counting detectors which are slow but highly sensitive.

### What is time frequency domain? ›

Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies.

**What is the difference between time and frequency domain? ›**

**A time-domain graph shows how a signal changes with time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies**.

**What is the advantage of time domain? ›**

The advantage of the time-domain analysis is **its capability to model all system nonlinearities, including mass, damping, and stiffness terms, and time-varying load as input**. However, the computation can be time-consuming.

**How is a time domain system analyzed? ›**

The Time Domain Analyzes of the system is to be done on basis of time. The analysis is only be applied when nature of input plus mathematical model of the control system is known. Expressing the main input signals is not an easy task and cannot be determined by simple equations.

**What is time domain reflection method? ›**

The TDR technique **measures the velocity of propagation of a high-frequency signal down waveguides in the soil**. The velocity is related to the dielectric constant of the soil, which is then related to the water content.

**What are the time domain characteristics of control system? ›**

Basically the response of any control system can be analyzed in frequency as well as time domain. So, the analysis of a system that involves **defining input, output and other variables of the system as a function of time** is known as time-domain analysis. It is also known as the time response of the control system.

**What is frequency response description? ›**

A frequency response describes **the steady-state response of a system to sinusoidal inputs of varying frequencies** and lets control engineers analyze and design control systems in the frequency domain.

**What is spatial domain and frequency domain? ›**

Spatial domain — **enhancement of the image space that divides an image into uniform pixels according to the spatial coordinates with a particular resolution**. The spatial domain methods perform operations on pixels directly. Frequency domain — enhancement obtained by applying the Fourier Transform to the spatial domain.

**How do you read frequency domain? ›**

Frequency domain graphs

**Frequency is plotted along the x-axis and amplitude is plotted along the y-axis**. FFTs often look like a series of mountain peaks. The horizontal location of peaks indications which frequencies are strongly present in the sound. The valleys show which frequencies are absent.

**How do you represent signal in time and frequency domain? ›**

Time domain representation – **In frequency domain, a signal is represented by its frequency spectrum**. To obtain frequency spectrum of a signal, Fourier series and Fourier transformation are used. Fourier series is used to get frequency spectrum of time-domain signal, when the signal is periodic function of time.

### Why do we use frequency domain? ›

The frequency domain representation of a signal **allows you to observe several characteristics of the signal that are either not easy to see, or not visible at all when you look at the signal in the time domain**. For instance, frequency-domain analysis becomes useful when you are looking for cyclic behavior of a signal.

**How do you convert time into frequency? ›**

Frequency is expressed in Hz (Frequency = cycles/seconds). To calculate the time interval of a known frequency, simply **divide 1 by the frequency** (e.g. a frequency of 100 Hz has a time interval of 1/(100 Hz) = 0.01 seconds; 500 Hz = 1/(500Hz) = 0.002 seconds, etc.)

**What are the 3 domains and examples? ›**

The three domains are **the Archaea, the Bacteria, and the Eukarya**. Prokaryotic organisms belong either to the domain Archaea or the domain Bacteria; organisms with eukaryotic cells belong to the domain Eukarya.

**What are the three types of domains? ›**

The three domains of learning are **cognitive, affective, and psychomotor**. There are a variety of methods in professional development events to engage the different learning domains.

**What is time domain image? ›**

Fourier Transform of a Time-Domain Image: The output time-domain image is **the Fourier transform of the diffracted spectrum from the hologram**. If the hologram corresponds to the space-domain image itself, then the resulting time-domain image is related to the Fourier transform of the space image.

**What is the difference between time domain and frequency domain solver? ›**

**Time domain solver uses FDTD method to solve the maxwell's equations.** **Frequency domain solvers uses FEM (finite element method) to solve the maxwell's equation**. The two solvers uses two different methods to solve the same problem.

**What are the values of time domain specifications? ›**

Time-domain specifications ( TDS ) include the lower and/or upper bounds of the quantities of the time response such as the **first peak time, maximum peak time, rise time, maximum overshoot, maximum undershoot, setting time, and steady-state error**.

**What is the difference between time domain and frequency domain filtering? ›**

In the time domain, the filtering operation involves a convolution between the input and the impulse response of the finite impulse response (FIR) filter. In the frequency domain, the filtering operation involves the multiplication of the Fourier transform of the input and the Fourier transform of the impulse response.

**What is time domain and Laplace domain? ›**

The function f(t), which is a function of time, is transformed to a function F(s). The function F(s) is a function of the Laplace variable, "s." We call this a Laplace domain function. So **the Laplace Transform takes a time domain function, f(t), and converts it into a Laplace domain function, F(s)**.

**What is the difference in time domain? ›**

Time domain is the domain for analysis of mathematical functions or signals with respect to time. Frequency domain is the domain for analysis of mathematical functions or signals with respect to frequency. The time domain systems tend to use photon counting detectors which are slow but highly sensitive.

### How many types of time domain analysis are there? ›

**em, Undamped System & Critically Damped System**.

**What is the advantage of frequency domain over time domain? ›**

The frequency domain representation of a signal **allows you to observe several characteristics of the signal that are either not easy to see, or not visible at all when you look at the signal in the time domain**. For instance, frequency-domain analysis becomes useful when you are looking for cyclic behavior of a signal.

**What is time domain reflection? ›**

The time domain reflectometry (TDR) method is the most established and widely used measuring method for the determination of: the total length of a cable. the location of low resistive cable faults. the location of cable interruptions. the location of joints along the cable.