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ENGINEERING
Let $$ X(e^{jω}) $$ be the Fourier transform of a real signal x[n]. Show that x[n] can be written as $$ x[n]=∫_o^π(B(ω)cosω+C(ω)sinω)dω $$ by finding expressions for B(ω) and C(ω) in terms of $$ X(e^{jω}). $$
ENGINEERING
Consider a casual, nonrecursive (FIR) filter whose real-valued impulse response h[n] is zero for n≥N. (a) Assuming that N is odd, show that if h[n] is symmetric about (N-1)/2 (i.e., if h[(N-1)/2+n]=h[(N-1)/2-n]), then $$ H(e^{jω})=A(ω)e^{-j[(N-1)/2]ω} $$ , where A(ω) is a real-valued function of ω. We conclude that the filter has linear phase.
COMPUTER SCIENCE
Prove that o(g(n)) ∩ ω(g(n)) is the empty set.
COMPUTER SCIENCE
Show that RANDOMIZED-QUICKSORT’s expected running time is Ω(n lg n).
COMPUTER SCIENCE
Show that quicksort’s best-case running time is Ω(n lg n).
ENGINEERING
Find f(t) if: (a) F(ω) = 2sinπω[u(ω + 1) - u(ω - 1)] (b) F(ω) = 1/ω (sin 2ω - sin ω) + j/ω (cos 2ω - cos ω)
ENGINEERING
The Fourier transform of f(t) is given by F(ω) = 0, -∞ ≤ ω < -3; F(ω) = 4, -3 < ω < -2; F(ω) = 1, -2 < ω < 2; F(ω) = 4, 2 < ω < 3; F(ω) = 0, 3 < ω ≤ ∞. Find f(t).
ENGINEERING
Determine the inverse Fourier transforms of: (a) F(ω) = 4δ(ω + 3) + δ(ω) + 4δ(ω - 3) (b) G(ω) = 4u(ω + 2) - 4u(ω - 2) (c) H(ω) = 6 cos 2ω
COMPUTER SCIENCE
Show that the worst-case running time of HEAPSORT is Ω(n lg n).
COMPUTER SCIENCE
Argue that since sorting n elements takes Ω(n lg n) time in the worst case in the comparison model, any comparison-based algorithm for constructing a binary search tree from an arbitrary list of n elements takes Ω(n lg n) time in the worst case.
ENGINEERING
Suppose x[n] is a real-valued discrete-time signal whose Fourier transform $$ X(e^{jω}) $$ has the property that $$ X(e^{jω})=0 $$ for π/8≤ω≤π. We use x[n] to modulate a sinusoidal carrier c[n]=sin(5π/2)n to produce y[n]=x[a]c[n]. Determine the values of ω in the range 0≤ω≤π for which $$ Y(e^{jω}) $$ is guaranteed to be zero.
ENGINEERING
Determine the functions corresponding to the following Fourier transforms: $$ (a) F_1(ω) = e^{jω}/-jω + 1 (b) F_2(ω) = 2e^{|ω|} (c) F_3(ω) = 1/(1 + ω^2)^2 (d) F_4(ω) = δ(ω)/1 + j2ω $$
ENGINEERING
Prove that if F(ω) is the Fourier transform of f(t), 𝓕[ f(t) sinω₀t] = j/2 [F(ω + ω₀) - F(ω - ω₀)]
ENGINEERING
A series RCL circuit has R = 30Ω, $$ X_C = 50 Ω $$ and $$ X_L = 90 Ω. $$ The impedance of the circuit is: (a) 30 + j140 Ω (b) 30 + j40 Ω (c) 30 - j40 Ω (d) -30 - j40 Ω (e) -30 + j40 Ω
ENGINEERING
a) Show that F {df(t)/dt} = jωF(ω), where F(ω) = F {f(t)}. Hint: Use the defining integral and integrate by parts. b) What is the restriction on f(t) if the result given in (a) is valid? c) Show that $$ F {d^nf(t)/dt^n} = (jω)^nF(ω), $$ where F(ω) = F {f(t)}.
ENGINEERING
Determine whether each of the following statements is true or face. Justify your answers. In each statement, the Fourier transform of x[n] is denoted by $$ X(e^{jω}). $$ (a) if $$ X(e^{jω})=X(e^{j(ω-1)} $$ , then x[n]=0 for |n|>0. (b) If $$ X(e^{jω})=X(e^{j(ω-v)} $$ , then x[n]=0 for |n|>0, (c) if $$ X(e^{jω})=X(e^{j(ω-l2)} $$ , then x[n]=0 for |n|>0, (d) if $$ X(e^{jω})=X(e^{j2ω}) $$ , then x[n]=0 for |n|>0$$
ENGINEERING
Determine the inverse Fourier transforms of the following: $$ (a) F(ω) = e^{-j2ω}/1+jω $$ (b) H(ω) = 1/( jω + 4)² (c) G(ω) = 2u(ω + 1) - 2u(ω - 1)
COMPUTER SCIENCE
Show that the nth harmonic number is Ω(lg n) by splitting the summation.
DISCRETE MATH
Show that the average depth of a leaf in a binary tree with n vertices is Ω(log n).
COMPUTER SCIENCE
Show that when all elements are distinct, the best-case running time of HEAPSORT is Ω(n lg n).
ENGINEERING
Let $$ x_c(t) $$ be a continuous-time signal whose Fourier transform has the property that $$ X_c(jω)=0 $$ for |ω|≥2.000π. A discrete-time signal $$ x_d[n]=x_c(n(0.5x10^{-3})) $$ is obtained. For each of the following constraints on the Fourier transform $$ X_d(e^{jω}) $$ of $$ x_d[n] $$ , determine the corresponding constraint on $$ X_c(jω) $$ ; (a) $$ X_d(e^{jω}) $$ is real. (b) The maximum value of $$ X_d(e^{jω}) $$ over all ω is 1. (c) $$ X_d(e^{jω})=X_d(e^{j(ω-π)}). (d) $$ X_d(e^{jω})=X_d(e^{j(ω-π)}).$$
ENGINEERING
Find the inverse Fourier transforms of the following functions: (a) F(ω) = 100/jω(jω + 10) (b) G(ω) = 10jω/(-jω + 2)(jω + 3) (c) H(ω) = 60/-ω² + j40ω + 1300 (d) Y(ω) = δω/(jω + 1)(jω + 2)
ENGINEERING
An electric heater draws 10 A from a 120-V line. The resistance of the heater is: (a) 1200 Ω (b) 120 Ω (c) 12 Ω (d) 1.2 Ω
ENGINEERING
Obtain the inverse Fourier transform of the following signals. (a) G(ω) = 5/jω-2 (b) H(ω) = 12/ω²+4 (c) X(ω) = 10/(jω-1)(jω-2)
ENGINEERING
The Thevenin impedance of a network seen from the load terminals is 80 + j55 Ω. For maximum power transfer, the load impedance must be: (a) -80 + j55 Ω (b) -80 - j55 Ω (c) 80 - j55 Ω (d) 80 + j55 Ω