代写Fundamentals of Digital Signal Processing Coursework Assignment 1代写Matlab编程

Fundamentals of Digital Signal Processing

Coursework Assignment 1

1. Consider the sequence

a) What is the Fourier transform. of x[n]?

b) Using MATLAB plot the Fourier transform. of sequences

for N = 1, N = 3, N = 5, N = 100, and N = 1000, and ω in the range ω ∈ (−π, π) with the step ∆ω = π/500.

2. The frequency response of an LTI system is

Determine the output of the system to x[n] = cos(5πn/2). Express your answer in as simple form. as you can.

3. What can you say about the periodicity of the output of an LTI system when its input is a periodic sequence of period N. Prove your answer.

4. When the input to an LTI system is

the output is

a) Find the transfer function of the system, i.e. the z-transform. of its impulse response.

b) Find the impulse response of the system.

c) Determine whether the system stable.

5. Consider the system in the figure below with Xc(jΩ) = 0 for |Ω| > 2000π, and the discrete-time system a squarer, i.e. y[n] = x[n]2. What is the largest value of T such that yc(t) = xc(t)2 ?