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A system that possesses two basic properties namely linearity and timeinvariant is known as **linear time-invariant system** or **LTI system**.

There are two major reasons behind the use of the LTI systems −

The mathematical analysis becomes easier.

Many physical processes through not absolutely LTI systems can be approximated with the properties of linearity and time-invariance.

The LTI systems are always considered with respect to the impulse response. That means the input is the impulse signal and the output is the impulse response.

Consider a continuous-time LTI system as shown in the block diagram of Figure-1.

Here, the input to the system is an impulse signal, i.e.,

𝑥(𝑡) = 𝛿(𝑡)

And the impulse response of the system is

𝑦(𝑡) = ℎ(𝑡) = 𝑇[𝛿(𝑡)]

According to the shifting property of signals, any signal can be expressed as a combination of weighted and shifted impulse signal, i.e.,

$$\mathrm{x(t)=\int_{-\infty }^{\infty }x(\tau )\delta \left ( t-\tau \right )d\tau} $$

Then, the impulse response is,

$$\mathrm{y(t)=T\left [x(t) \right ]=\int_{-\infty }^{\infty }x(\tau )T\left [ \delta \left ( t-\tau \right ) \right ]d\tau}$$

$$\mathrm{\Rightarrow y(t)=\int_{-\infty }^{\infty }x(\tau )h\left ( t-\tau \right )d\tau\: \:\: ...(1)} $$

The expression in equation (1) is known as **convolution integral**.

The symbolic representation of the convolution integral is given by,

𝑦(𝑡) = 𝑥(𝑡) * ℎ(𝑡) … (2)

The discrete time LTI system is shown in Figure-2.

Here, the input to the system is an impulse signal, i.e.,

𝑥(𝑛) = 𝛿(𝑛)

And the discrete-time impulse response of the system is,

𝑦(𝑛) = ℎ(𝑛) = 𝑇[𝛿(𝑛)]

According to the shifting property of signals, any signal can be expressed as a combination of weighted and shifted impulse signal, i.e.,

$$\mathrm{x(n)=\sum_{k=-\infty }^{\infty }x(k)\, \delta (n-k)} $$

Therefore, the impulse response of the system is,

$$\mathrm{y(n)=T\left [ \delta (n) \right ]=\sum_{k=-\infty }^{\infty }x(k)\, T\left [ \delta (n-k) \right ]} $$

$$\mathrm{\Rightarrow y(n)=\sum_{k=-\infty }^{\infty }x(k)\,h(n-k)\: \: \cdot \cdot \cdot (3)} $$

The expression of Eqn. (3) is known as convolution sum. The convolution sum can be represented symbolically as,

𝑦(𝑛) = 𝑥(𝑛) * ℎ(𝑛) … (4)

Also,

$$\mathrm{ y(n)=\sum_{k=-\infty }^{\infty }x(k)\,h(n-k)=\sum_{k=-\infty }^{\infty }x(n-k)\,h(k)\: \: \cdot \cdot \cdot (5)} $$

- Related Questions & Answers
- Signals and Systems – Properties of Linear Time-Invariant (LTI) Systems
- Signals and Systems: Time Variant and Time-Invariant Systems
- Signals and Systems – Response of Linear Time Invariant (LTI) System
- Signals and Systems – Transfer Function of Linear Time Invariant (LTI) System
- Signals and Systems: Linear and Non-Linear Systems
- Signals and Systems – Filter Characteristics of Linear Systems
- Signals and Systems – Time Scaling of Signals
- Signals and Systems: Classification of Systems
- Signals and Systems – Time Convolution Theorem
- Signals and Systems – What is a Linear System?
- Signals and Systems: Invertible and Non-Invertible Systems
- Signals and Systems: Multiplication of Signals
- Signals and Systems: Even and Odd Signals
- Signals and Systems: Periodic and Aperiodic Signals
- Signals and Systems: Energy and Power Signals

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