Consider the DT LTI system defined by the mpulse response h[n] = ?[n] The input to...
So sorry for the long question, I am able to do a) and b) but not sure about the rest 2. Consider the DT LTI system defined by the impulse response h[n]-i[n]-?[n-1]. The input to this system is the signal rn: (a) Sketch hn and n (b) Determine the output of the system, y[n], using convolution. Sketch y[n (c) Determine the DTFTs H(ei) and X(e). Make fully-labeled sketches of the magn tudes of these DTFTs. (d) Recall that the discrete...
Problem 5 (score 30%) You are given a final sequence of N samples of a time signal sampled withk samples/sec. a) Define the Discrete Fourier-Transform (DFT) and inverse DFT of the sequence. Are there any restrictions on the samples? b) Define the z-transform of the sequence c) d) Sketch a block-diagram of a recursive, digital filter and relate the filter-coefficients to the z-transform of the sequence.
Question 2: Consider the signal h[n] given by 11 n=0 h[n] = { -1 n=4 10 otherwise a) Calculate the z-transform H(z). Find its poles and zeros. b) Let H[k] be the 512-point DFT of h[n]. Show that H[0] = H[128] = H (256) = H[384] = 0 by substituting k = 0, 128, 256, 384 in the DFT formula 511 H[k] => b[m]e-jkan n=0 c) Now, show H[0] = H[128] = H (256] = H[384] = 0 directly using...
(a) Consider a discrete-time signal v[n] satisfying vn0 except if n is a multiple of some fixed integer N. i.e oln] -0, otherwise where m is an integer. Denote its discrete-time Fourier transform by V(eJ"). Define y[nl-v[Nn] Express Y(e) as a function of V(e). Hint : If confused, start with N-2 (b) Consider the discrete-time signal r[n] with discrete-time Fourier transform X(e). Now, let z[n] be formed by inserting two zeroes between any two samples of x[n]. Give a formula...
(b) Perform convolution to obtain the discreet if input x[n] = [1 3 2 1] and impulse response, h[n] signal output of y[n], [1 -4 2]. [3 marks] (c) An analogue signal is sampled every 50ms for a duration of 1000 seconds. i) Calculate how much data (samples) are collected. [2 marks] If a Discrete Fourier Transform (DFT) is performed what is the maximum frequency information that can be obtained? [2 marks] Calculate the minimum frequency (the frequency resolution) a...
[20 total pts) Consider the sequences x1n] = { 1, 2,-2, 1}, and x2 [n] = { 1, 2,-2, 1, 0, 0, 0, 0). The sequence x2In] is known as a zero-padded version of x,[n]. When answering the questions below, please use Table 1, provided on Page 3. a. [3 pts] Compute X1(eo), the Discrete-Time Fourier Transform (DTFT) of x1 [n], and evaluate it for the following values of normalized frequency: a-63 笎哮,쯤뀨 write the values in the table b....
Consider the signal x(n-õn-4] + 2õn-5] + õn_6]. (a) Find X(el the discrete-time Fourier transform of xin]. Write expressions for the magnitude and phase of X(elu), and sketch these functions (b) Find all values of N for which the N-point DFT is a set of real numbers (c) Can you find a three-point causal signal x1n i.e., x1In] 0 for n <0 and n > 2) for which the three-point DFT of x (n] is: xn[nl (ie, xiin] O for...
Question 4 (a) Find the DFT of the series x[n)-(0.2,1,1,0.2), and sketch the magnitude of the resulting spectral components [10 marks] (b) For a discrete impulse response, h[n], that is symmetric about the origin, the spectral coefficients of the signal, H(k), can be obtained by use of the DFT He- H(k)- H-(N-1)/2 Conversely, if the spectral coefficients, H(k), are known (and are even and symmetrical about k-0), the original signal, h[n], can be reconstituted using the inverse DFT 1 (N-D/2...
It has been shown n that the discrete Pourier transform(DFT) of a time-varying process discrete h(tk) for (k0,1,2,.. ,N - 1). is given by carry out the Cooley-Tukey formulation of FFT by following the steps below. (a) Write the expressions for DFT H, in terms of h(ta) and the inverse DFT h(tk) in terms of Hn for N =8. (b) Define W e2x/N and rewrite (a) using W. (c) Express (b) in matrix form. (d) Express n and k in...
DSP 4. (12 points) (a) (4 points) Let x[n] = {1,2, 1, 2} and h[n] = {1,-1,1, -1} be two length-4 sequences defined for 0 <n<3. Determine the circular convolution of length-4 y[n] = x[n] 4 hin). (b) (6 points) Find the 4-point discrete Fourier transform (DFT) X[k], H[k], and Y[k]. (c) (2 points) Find the 4-point inverse DFT (IDFT) of Z[k] = {X[k]H[k].