A data sequence x[n] = {1, 2, 3, 4, 5} passes through a linear time-invariant system with, impulse response h[n] = {5, 4, 3, 2, 1}, The output of the filter is

Concept:

• For linearity and time invariance output must be weighted superposition of time-shifted impulses.
• This weighted superposition is termed as convolution sum for discrete-time systems and convolution integral for continuous-time. And it is determined by the symbol (∗ )
• If two systems are cascaded then the resultant signal is convolution in the time domain and multiplication in the frequency domain, below diagrams, shows that.

Symbolically resultant is represented as:

y(t) = x(t) ∗ h(t)

symbolically this is represented as:

y[n] = x[n] ∗ h[n]

calculation:

Given input signal is

X[n] = {1, 2, 3, 4, 5}

LTI system is

h[n] = {5, 4, 3, 2, 1}

The output response will be convolution of the input and system. This can be done in two ways.

1.Traditional method

2.Sum by column

Traditional method

y[n] = {5, 14, 26, 40, 55, 40, 26, 14, 5}

Sum by column

Extra concept:

UNIT IMPULSE RESPONSE

The impulse response is defined as the output of an LTI system due to a unit impulse signal applied at time t = 0 or n = 0

As x(n) → y(n), δ (n) → h(n)

As x(t) → y(t), δ (t) → h(t)