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H nunx. 24 c JFessler,May27,04,1310(studentversion) 212 Classication of discretetime signals The energy of a discretetime signal is dened as Ex 4= X1 n=1 jxnj2 The average power of a signal is dened as Px 4= lim N!1 1 2N 1 XN n= N jxnj2 If E is nite (E < 1) then xn is called an energy signal and P = 0 If E is innite, then P can be either nite or innite. 6003 Homework #3 Solutions / Fall 11 3 3 Z transforms DeterminetheZtransform(includingtheregionofconvergence)foreachofthefollowing signals a x. Answer to h(n) = u(n), x(n) = u(n) Find the output y(n) for each system using the convolution sum equation Are the systems stable.
6 l y(n) You are told that h1(n)=u(n)2u(n1)u(n2) You observe that the step response of the total system is s(n)=2r(n)3r(n1)r(n2) where r(n)=nu(n) Find and sketch h2(n) 1218 The impulse response of a discretetime LTI system is given by h(n)= ⇢ 1ifn is a positive prime number 0 otherwise (a) Is the system causal?. Consider the following systems 1 h(n) u(n), x(n) u(n) 2 h(n) (2)" u(n), x(n) (n) 3, h(n)(3)" u(n), x(n) = u(n) 4, h(n) = (3)" u(n), x(n) = (2)" u(n) 5 hn) (3. If we convolve some input x with h, and then convolve the result with g x # h # g this is equivalent to convolving x with the convolution of h with g x # ( h # g ) By the convolution rule for the ztransform, the ztransform of ( h # g ) will be the product H(z)G(z).
Eq1) The utility of this frequency domain function is rooted in the Poisson summation formula Let X (f) be the Fourier transform of any function, x (t) , whose samples at some interval T (seconds) are equal (or proportional) to the x n sequence, ie T ⋅ x (nT) = x n Then the periodic function represented by the Fourier series is a periodic summation of X (f) in terms of frequency f. Consider system H with impulse response h(n)=(1/2) n u(n) Input x(n)=e jπn/2 Let’s compute output y(n) Will show 𝐻𝐻𝜔𝜔= 1 1− 1 2 𝑒𝑒 −𝑗𝑗𝜔𝜔 Input corresponds to ω=π/2 Response is 𝐻𝐻𝜔𝜔= 𝜋𝜋 2 = 1 1− 1 2 𝑒𝑒 −𝑗𝑗𝜋𝜋/2 = 1 1− 1 2 (−𝑗𝑗) = 1 1 𝑗𝑗 2 Output y(n)=𝑒𝑒. Let X 1, j, 1, j , H 0, 1, 1, 1 be the DFT's of two sequences x and h respectively Using the properties of the DFT (do not compute the sequences) determine the DFT's of the following Solution a) x n 1 4 Recall DFT x n m N w km X k.
H n u n x n u n n n = = b) 6S For each of the following pairs of waveforms, use the convolution integral to find response y(t) of the LTI system with impulse response h(t) and x(t) Sketch your results a) ( ) ( ) ( ) ( ) h t e u t x t e u t t t. Hn = un, xn = (I)"un Evaluate the system output yn and sketch P45 (a) Using convolution, determine and sketch the responses of a linear, timeinvar iant system with impulse response h(t) = et 2 u(t) to each of the two inputs x 1(t), x 2(t) shown in Figures P451 and P452 Use yi(t) to denote the response.
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