Response of an LTI System using Fourier Transform

Introduction:
In this question, we are given an LTI (Linear Time-Invariant) system with an input signal x(n) and impulse response h(n). We are required to find the response of the system using Fourier Transform.

Step 1: Find the Fourier Transform of x(n) and h(n)
The Fourier Transform of a signal x(n) is given by X(f) = ∑[x(n)e^(-j2πfn)], where X(f) represents the spectrum of the signal x(n) at frequency f.

Similarly, the Fourier Transform of a signal h(n) is given by H(f) = ∑[h(n)e^(-j2πfn)], where H(f) represents the spectrum of the signal h(n) at frequency f.

Let's calculate the Fourier Transform of x(n) and h(n) using the given values.

For x(n) = (1, 2, 3, 1):
- x(0) = 1, x(1) = 2, x(2) = 3, x(3) = 1
- X(f) = x(0)e^(-j2πf0) + x(1)e^(-j2πf1) + x(2)e^(-j2πf2) + x(3)e^(-j2πf3)
= 1 + 2e^(-j2πf) + 3e^(-j4πf) + e^(-j6πf)

Similarly, for h(n) = (1, 2, 1, -1):
- h(0) = 1, h(1) = 2, h(2) = 1, h(3) = -1
- H(f) = h(0)e^(-j2πf0) + h(1)e^(-j2πf1) + h(2)e^(-j2πf2) + h(3)e^(-j2πf3)
= 1 + 2e^(-j2πf) + e^(-j4πf) - e^(-j6πf)

Step 2: Calculate the Frequency Response H(f)
The frequency response H(f) of an LTI system is obtained by multiplying the Fourier Transform of the input signal X(f) with the Fourier Transform of the impulse response H(f).

H(f) = X(f) * H(f)
= (1 + 2e^(-j2πf) + 3e^(-j4πf) + e^(-j6πf))(1 + 2e^(-j2πf) + e^(-j4πf) - e^(-j6πf))
= (1 + 4e^(-j2πf) + 4e^(-j4πf) + e^(-j6πf) + 2e^(-j4πf) + 4e^(-j2πf) + 2e^(-j4πf) + e^(-j6πf)
+ 3e^(-j4πf) + 6e^