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- Fourier Series
- Fourier Series
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- Laplace Transform
- Laplace Transforms
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- Z Transform
- Z-Transforms (ZT)
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- What is Inverse Z Transform?
- Inverse Z-Transform by Convolution Method
- Transform Analysis of LTI Systems using Z-Transform
- Convolution Property of Z Transform
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- System Realization
- Cascade Form Realization of Continuous-Time Systems
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- Discrete Fourier Transform
- Discrete-Time Fourier Transform
- Properties of Discrete Time Fourier Transform
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- Inverse Discrete-Time Fourier Transform
- Time Convolution and Frequency Convolution Properties of Discrete-Time Fourier Transform
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- Miscellaneous Concepts
- What is Mean Square Error?
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- Region of Convergence
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- What is the Frequency Response of Discrete Time Systems?
- Basic Elements to Construct the Block Diagram of Continuous Time Systems
- BIBO Stability Criterion
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- Distortion Less Transmission
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- Rayleigh’s Energy Theorem
Properties of Linear Time-Invariant (LTI) Systems
Linear Time Invariant System
A system for which the principle of superposition and the principle of homogeneity are valid and the input/output characteristics do not with time is called the linear time invariant (LTI) system.
Properties of LTI System
A continuous-time LTI system can be represented in terms of its unit impulse response. It takes the form of convolution integral. Hence, the properties followed by the continuous time convolution are also followed by the LTI system. The impulse response of an LTI system is very important because it can completely determine the characteristics of an LTI system.
In this article, we will highlight some of the important properties of an LTI system (or continuous-time convolution).
Commutative Property of LTI System
The convolution in continuous-time is a commutative operation, i.e.,
$$\mathrm{x(t)\:\cdot\:h(t)\:=\:h(t)\:\cdot\:x(t)\:=\:\int_{-\infty }^{\infty}x(\tau)\:h(t\:-\:\tau )d\tau\:=\:\int_{-\infty }^{\infty}h(\tau)\:x(t\:-\:\tau )d\tau}$$
Therefore, according to the commutative property of an LTI system, the output of the LTI system with input x(t) and unit impulse response h(t) is same as the output of the LTI system with input h(t) and impulse response x(t).
Distributive Property of LTI System
The convolution in continuous-time distributes over addition, i.e.,
$$\mathrm{x(t)\:\cdot\:[h_{1}(t) \:+\:h_{2}(t)]\:=\:x(t)\:\cdot\:h_{1}(t)\:+\:x(t)\:\cdot\:h_{2}(t)}$$
The distributive property of the LTI system has a useful interpretation in terms of system interconnection. Hence, according to this, the two LTI systems with impulse responses $h_{1}(t)$ and $h_{2}(t)$ connected in parallel can be replaced by a single system with impulse response [$h_{1}(t)+h_{2}(t)]$. Also, the distributive property of continuous-time convolution can be used to break a complicated convolution into several simpler convolutions.
Associative Property of LTI System
The convolution in continuous-time is associative, i.e.,
$$\mathrm{x(t)\:\cdot\:[h_{1}(t)\:\cdot\:h_{2}(t)] \:=\:[x(t)\:\cdot\:h_{1}(t)]\:\cdot\:h_{2}(t)}$$
Therefore, according to the associative property the signals can be convolved in any order.
Causality Property of LTI System
A causal system is non-anticipatory and does not produce an output before an input is applied. Therefore, the output of a causal system depends only on the present and past values of input but not on the future inputs.
Hence, for a causal LTI system, we get,
$$\mathrm{h(t)\:=\:0;\:for\:t \:\lt\:0}$$
Therefore,
- The output of a causal LTI system for a non-causal input is given by,
$$\mathrm{y(t)\:=\:\int_{0}^{\infty }h(\tau )\:x(t\:-\:\tau)d\tau\:=\:\int_{-\infty}^{t}x(\tau)\:h(t\:-\:\tau)d\tau}$$
- The output of a causal LTI system for a causal input is given by,
$$\mathrm{y(t) \:=\: \int_{0}^{t}h(\tau )\:x(t\:-\:\tau)d\tau\:=\:\int_{0}^{t}x(\tau )\: h(t\:-\:\tau)d\tau}$$
Stability of LTI System
If for a given system every bounded input produces a bounded output, then the system is stable. The stability of an LTI system can be determined from its impulse response. For a continuous-time LTI system to be stable, its impulse response h(t) must be absolutely integrable, i.e.,
$$\mathrm{\int_{-\infty }^{\infty}\left | h(\tau )\right |d\tau\:\lt\:\infty}$$
Invertibility of LTI System
A continuous LTI system with impulse response is called invertible, if an inverse system with impulse response ${h}'(t)$ which when connected in series with the original system produces an output equal to the input of the first system, i.e.,
$$\mathrm{h(t)\:\cdot\:{h}'(t)\:=\:\delta(t)}$$
LTI System with and without Memory
An LTI system is called static or memoryless system if its output at any time depends only upon the value of the input at that time. Hence, a continuous-time LTI system is said to be memoryless system if,
$$\mathrm{h(t) \:=\: 0;\:for\:t\:=\:0}$$
Such a memoryless LTI system is represented as,
$$\mathrm{y(t)\:=\:k\:x(t)}$$
The system has some memory if,
$$\mathrm{h(t)\:=\:0;\:for\:t\:=\:0}$$
The memory system is also known as dynamic system.
Unit Step Response of LTI System
When the unit step input u(t) is applied to an LTI system, then the corresponding output is called the unit step response s(t) of the LTI system.
The unit step response of an LTI system can be obtained by convolving the unit step input u(t) with the impulse response h(t) of the system, i.e.
$$\mathrm{s(t) \:=\: u(t)\:\cdot\:h(t) \:=\: h(t)\:\cdot\:u(t)}$$
$$\mathrm{\Rightarrow \: s(t) \:=\: \int_{-\infty }^{t}h(\tau)d\tau}$$