2. Discrete Fourier Transform.(/25) 1. N-th roots of unity are defined as solutions to the equation:...
2TT sinn (1) a) Let x1 [n] = πη Find the Discrete Time Fourier transform of this signal and plot it with all its critical values. (you can use only transform tables from the book). b) Now, define xzlv) = (**) GHS) Using transform properties, find the Discrete Time Fourier transform of x2[n] and plot it with all its critical values. In your calculations be sure to show your steps ! 2TT sinn sinn sinwon c) Let y[n] [( )...
Show that Discrete Fourier transform matrix W /Vn is a unitary matrix, where e2j/n and i = V-1, W = 1 1 1 1 1 (n-1) w-2(n-1) w-2 1 z- w~(n-1) -2(n-1) (n-1)(n-1) Show that Discrete Fourier transform matrix W /Vn is a unitary matrix, where e2j/n and i = V-1, W = 1 1 1 1 1 (n-1) w-2(n-1) w-2 1 z- w~(n-1) -2(n-1) (n-1)(n-1)
numerical analysis problem 4. Let s = (2,1, -4,3). Find the discrete Fourier transform F(s) of s. 5. Let w=i, s = (1, 2+2w, 3, 2-2w), t = (4,3w, 2w, -w). Find the pointwise multiplication ext.
8. Let n be a positive integer. The n-th cyclotomic polynomial Ф,a(z) E Z[2] is defined recursively in the following way: 1. Ф1(x)-x-1. 2. If n > 1, then Фп(x)- , (where in the product in the denomina- tor, d runs through all divisors of n less than n). . A. Calculate Ф2(x), Ф4(x) and Ф8(z): . B. n(x) is the minimal polynomial for the primitive n-th root of unity over Q. Let f(x) = "8-1 E Q[a] and ω...
The discrete Fourier transform of an mxn matrix X = (Xj.k) is an m x n matrix X = (ĉik) m,n ypj yqk pqSm Sn, p.q=1 where Šm = e270i/m The corresponding inverse Fourier transform is m.n Xj,k = (mn)-1 » počinje-k. p, Sm Sn . peq=1 Let X and Y be mxn matrices with the discrete Fourier transforms X and Y respectively. Define two dimensional circular convolution Z = X * Y to be min Za,b = XXj,kYa–j,b-k j,k=1...
2. Calculate the inverse Fourier transform of X(cfw) = {2 2j 0 <W <T -2j -n<w < 3. Given that x[n] has Fourier transform X(@j®), express the Fourier transforms of the following signals in terms of X(el“) using the discrete-time Fourier transform properties. (a) x1[n] = x[1 – n] + x[-1 - n] (b) x2 [n] = x*[-n] + x[n]
(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...
roblem 3: (15-7+8 points) Consider the left-sided discrete-time signal a(n)42+1). a) Find the discrete Fourier transform X(eju n-2 ). (b) Find the phase (o) of the discrete Fourier transform X
9.99 Walk-Through: Discrete Fourier Trans- forms. You've measured the following data points for a function f(x):f(0) = 2, /(2) = 3,f(4) =-6, f(6) = 0. (a) Use Equation 9.7.1 to calculate and f2 (b) Find /-, without using Equation 9.7.1. This should lake no more than 20 seconds (c) What are/2 and? Again, more than 20 seconds means you re doing it wrong. (d) What frequencies p are represented by the terms f 1, fo fi and /2? 1J0J1 The...
Determine the Discrete Time Fourier Transform (DTFT) of the following discrete-time signal. x[n]=n0.1" u(n) 1-0 1e112 0970.1e* 5) -0.12- e in 1-0.1e) C), ei (1+0.2e-in d) =-*+0.2e-10 e / +0.2012