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# Signals and Systems â€“ Response of Linear Time Invariant (LTI) System

## 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 change with time is called the *linear time-invariant (LTI) system.*

## Impulse Response of LTI System

When the impulse signal is applied to a linear system, then the response of the system is called ** the impulse response**. The impulse response of the system is
very important for understanding the behaviour of the system.

Therefore, if

$$\mathrm{\mathit{\mathrm{Input}, x\left(t\right)=\delta\left(t\right)}}$$

Then,

$$\mathrm{\mathit{\mathrm{Output}, y\left(t\right)=h\left(t\right)}}$$

As the Laplace transform and Fourier transform of the impulse function is given by,

$$\mathrm{\mathit{L\left [\delta\left(t\right) \right ]\mathrm{=}\mathrm{1}\:\:\mathrm{and} \:\:F\left [\delta\left(t\right) \right ]\mathrm{=}\mathrm{1}}}$$

Hence, once the transfer function $\mathit{H\left(\omega\right)}$ of an LTI system is known in frequency domain, then the impulse response of the system can be determined by taking the inverse Fourier transform of $\mathit{H\left(\omega\right)}$, i.e.,

$$\mathrm{\mathit{h\left(t\right)=F^{-\mathrm{1}}\left [ H\left(\omega\right) \right ]}}$$

Similarly, once the transfer function $\mathit{ H\left(s\right)}$ of an LTI system is known in s-domain, then the impulse response of the system can be determined by taking the inverse Laplace transform of $\mathit{ H\left(s\right)}$, i.e.,

$$\mathrm{\mathit{h\left(t\right)=L^{-\mathrm{1}}\left [ H\left(s\right) \right ]}}$$

Once the impulse response $\mathit{ h\left(t\right)}$ of the linear system is known, then the response of the linear system $\mathit{ y\left(t\right)}$ for any given input signal $\mathit{ x\left(t\right)}$ can be obtained by convolving the input with the impulse response of the system, i.e.,

$$\mathrm{\mathit{ y\left(t\right)=h\left(t\right)*x\left(t\right)=x\left(t\right)*h\left(t\right)}}$$

## Step Response of LTI System

The convolution integral can be used to obtain the step response of a continuous-time LTI system. If the unit step signal $\mathit{u\left(t\right)}$ is an input signal to a system having impulse response $\mathit{h\left(t\right)}$, then the step response of the system is given by,

$$\mathrm{\mathit{s\left(t\right)\mathrm{=}h\left(t\right)*u\left(t\right)}}$$

If the given system is non-causal, then the step response is given by,

$$\mathrm{\mathit{s\left(t\right)\mathrm{=}\int_{-\infty }^{t}h\left(\tau\right)u\left(t-\tau\right)d\tau =\int_{-\infty }^{t}h\left(\tau\right)d\tau }}$$

And, when the given system is causal, then the step response is,

$$\mathrm{\mathit{s\left(t\right)=\int_{\mathrm{0}}^{t}h\left(\tau\right)d\tau} }$$

Hence, the step response of an LTI system is the running integral of impulse response of the system.

Now, when causal and non-causal signals are applied to causal and non-causal systems, we get the following outputs −

When a non-causal signal is applied to a non-causal system, then,

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

When a causal signal is applied to a non-causal system, then,

$$\mathrm{\mathit{y\left(t\right)\mathrm{=}\int_{-\infty }^{t }h\left(\tau\right)x\left(t-\tau\right)d\tau \mathrm{=}\int_{\mathrm{0} }^{\infty }x\left(\tau\right)h\left(t-\tau\right)d\tau }}$$

When a non-causal signal is applied to a causal system, then,

$$\mathrm{\mathit{y\left(t\right)\mathrm{=}\int_{\mathrm{0} }^{\infty }h\left(\tau\right)x\left(t-\tau\right)d\tau \mathrm{=}\int_{-\infty }^{t}x\left(\tau\right)h\left(t-\tau\right)d\tau }}$$

When a causal signal is applied to a causal system, then,

$$\mathrm{\mathit{y\left(t\right)\mathrm{=}\int_{\mathrm{0} }^{t }h\left(\tau\right)x\left(t-\tau\right)d\tau \mathrm{=}\int_{\mathrm{0} }^{t}x\left(\tau\right)h\left(t-\tau\right)d\tau }}$$

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