A data sequence x[n] = {1, 2, 3, 4, 5} passes through a linear ti
A. {6, 6, 6, 6}
B. {5, 8, 9, 5}
C. {5, 14, 26, 40, 55, 40, 26, 14, 5}
D. {1, 4, 10, 20, 35, 44, 46, 40, 25}
Please scroll down to see the correct answer and solution guide.
Right Answer is: C
SOLUTION
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, sows 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)