Homework week3 1. Given a sequence x(n) for , where x(0) = 1, x(1) = 1, x(2) = -1, and x(3) = -1 . Compute its DFT X(k) and obtain its inverse. Use hand computation and verify with MATLAB. 2. Given a...

I have this homework that I need help with, some of the questions need to done using Matlab, and the rest need to handwritten. Please write the answer neatly.The name of the Course isCEE430 Digital Signal Processing


Homework week3 1. Given a sequence x(n) for , where x(0) = 1, x(1) = 1, x(2) = -1, and x(3) = -1 . Compute its DFT X(k) and obtain its inverse. Use hand computation and verify with MATLAB. 2. Given a sequence x(n) for , where x(0) = 0.2, x(1) = 0.2, x(2) = 0.2, and x(3) = 0 . Compute its DFT X(k) and obtain its inverse. Use hand computation and verify with MATLAB. 3. Given a sequence x(n) for , where x(0) = 4, x(1) = 3, x(2) = 2, x(3) = 1 with two additional zero- padded date points x(4) = 0, and x(5) = 0 . Compute its DFT X(k) and obtain its inverse. Use hand computation and verify with MATLAB. 4. Use MATLAB – Consider three sinusoidal signals with the following amplitudes and phases: ?1(?) = 5 ???( 2? ⋅ 500?) ?2(?) = 5 ???( 2? ⋅ 1200? + 0.25?) ?3(?) = 5 ???( 2? ⋅ 1800? + 0.5?) a. Create a MATLAB program to sample each sinusoid and generate a sum of three sinusoids, that is, ?(?) = ?1(?) + ?2(?) + ?3(?), using a sampling rate of 10,000 Hz. Plot x(n) over the range of 0 to 4 milisecond. b. Use MATLAB function fft() to compute DFT coefficients, then plot and examine the spectrum of the original signal. 5. Given a sequence x(n) for 30  n , where x(0) = 4, x(1) = 3, x(2) = 2, and x(3) = 1 . Compute its DFT X(k) using the decimation in frequency FFT method. Use decimation in frequency method to evaluate the inverse DFT 6. Find the Z-transform of the following sequences a. ?(?) = 4(0.8)? ???( 0.1??)?(?) b. ?(?) = 4?−3? ???( 0.1??)?(?) 7. Given two sequences )2(2)(5)(1 −−= nnnx  and )3(3)(2 −= nnx  a. Determine the Z transform of the convolution of the two sequences using the convolution property of Z transform that says : ?(?) = ?1(?) ⋅ ?2(?) b. Determine the convolution by the inverse Z transform: ?(?) = ?−1(?1(?) ⋅ ?2(?)) 8. Find the inverse z-transform of the following functions: a. ?(?) = −5? (?−1) + 10? (?−1)2 + 2? (?−0.8)2 b. ?(?) = 2? (?−0.1) (?−0.6) 9. Given the following difference equation with input-output relationship with zero initial conditions, ?(?) − 0.7?(? − 1) + 0.1?(? − 2) = ?(?) + ?(? − 1), use the Z-transform in order to: a. Find the impulse response sequence y(n) due to the impulse sequence )(n . b. Find the output response of the system when the unit step function u(n) is applied.