1. Determine the z-transforms of the following sequences: (a) x = [3, 0, 5, 6, 0,...
1. Determine the z-transforms of the following sequences:
(a) x = [3, 0, 5, 6, 0, 1]
(b) x = [1, 0, 0, 4]
2. Compute the transfer fuctions for the following impulse responses:
(a) h = [1, −5, 4, 0, 5]
(b) h = [1, −0.5, 0.25, −1.125]
3. If h(n) = 3^ -n for n ≥ 0, express H(z) as a ratio of polynomials.
4. Find the 10 roots of unity, that is, solve z^10 − 1 = 0.
5. What are the values of a1 and b1 for the sine function filter we created running at 8 kHz that will produce a 750 Hz sine wave?
Homework Answers
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1. Determine the z-transforms of the following sequences:
(a) x = [3, 0, 5, 6, 0,...
3.96 Determine the z-transforms of the following sequences and their respective ROCs: (a) x1[n] = -a"u[-n-1],...
3.96 Determine the z-transforms of the following sequences and their respective ROCs: (a) x1[n] = -a"u[-n-1], (b) x2[n] = "Min + 1], and (c) x3[n] = "u-n).
QUESTION 1 Characterise the following systems as being either causal on anticausal: yn)-ePyn-1)+u...
QUESTION 1 Characterise the following systems as being either causal on anticausal: yn)-ePyn-1)+u/n), where u/h) is the unit step and B is an arbitrary constant (B>0), Take y-1)-0. Answer with either causal or 'anticausal only QUESTION 2 For the following system: yn) -yn-1Va -x(n), for a 0.9, find y(10), assuming y(n) - o, for ns -1.Hint: find a closed form for yin) and use it to find the required output sample. (xin)-1 for n>-0) QUESTION 3 A filter has the...
Using the following two finite-length sequences: x = {0, 1, 7, 6, 1, 2, 0, 7, 1, 0, 3, 4}; h = {1, 1, -1}
Using the following two finite-length sequences: x = {0, 1, 7, 6, 1, 2, 0, 7, 1, 0, 3, 4}; h = {1, 1, -1}; a Obtain the linear convolution of the two sequences. b Obtain the circular convolution of the two sequences. c Obtain the linear convolution of the two sequences using the overlap-and-add method with a partition size of 4. d Obtain a factor of two interpolation of the sequence x with filter h using: (i) upsampling followed by filtering, (ii) the...
Given the following two sequences x (n)=[3 , 11,7 ,0 ,−1, 4 ,2 ],−3≤n≤ 3 ;...
Given the following two sequences x (n)=[3 , 11,7 ,0 ,−1, 4 ,2 ],−3≤n≤ 3 ; h (n )=[2,3 ,0 ,−5, 2,1 ] ,−1≤n≤ 4. (a) Use the deiniion to determine the convoluion y ( n )= x ( n )∗h (n ) (b) Compare your result in (a) with that obtained by MATLAB
Determine the z-transform of the following sequences and their ROCs: a) x(n) = (0.5)" for n>...
Determine the z-transform of the following sequences and their ROCs: a) x(n) = (0.5)" for n> 5, and zero for all other values of n; b) x(n)= (0.5)"[u(n) - u(n-7)]; c) x(n)=(-1)"a"u(n), 0 <a<1.
3) Compute the Z-Transforms of the following time series: (a) x(n)k2"u(n) (b) + 1) x(n) =...
3) Compute the Z-Transforms of the following time series: (a) x(n)k2"u(n) (b) + 1) x(n) = u(-n (c) x(n) -k2"u-1) (d) x(n) 0.5%1(n) + 3"11(-n) (e) x(n) = 4-nu(n) + 5-nu(n + 1) In the above, u(n) stand for the unit step signal in the discrete time domain. Also, if you can in each case determine the region of convergence of the Z-Transform you obtain.
Problem #1. Topics: Z Transform Find the Z transform of: x[n]=-(0.9 )n-2u-n+5] X(Z) Problem #2. T...
Problem #1. Topics: Z Transform Find the Z transform of: x[n]=-(0.9 )n-2u-n+5] X(Z) Problem #2. Topics: Filter Design, Effective Time Constant Design a causal 2nd order, normalized, stable Peak Filter centered at fo 1000Hz. Use only two conjugate poles and two zeros at the origin. The system is to be sampled at Fs- 8000Hz. The duration of the transient should be as close as possible to teft 7.5 ms. The transient is assumed to end when the largest pole elevated...
Problem 4: (a) and (b): Find the z-transform including the ROC for each of the following...
Problem 4: (a) and (b): Find the z-transform including the ROC for each of the following waveforms: [n] 3(금)"u[n] Xa | 지 = (c) Find the z-transfor by m of the impulse response hn] of an LTI system, when h[n] is given h[n] = 5(을)"u[n]. (d) and (e): Using z-transforms, find the responses (yan] and yb[n]) of the system described in part (c) to the inputs (an andn] described in parts (a) and (b
6.6a b 6.7c d e of the following irrational z transforms. (a) x(z) = ea", Izl...
6.6a b
6.7c d e
of the following irrational z transforms. (a) x(z) = ea", Izl > 0, o)X(z)log (1- az), zl < 1/lal. . 1 Show the following p roperties for the z transforms of even and odd 6.7 discrete-time functions. (a) If x[n] is even, that is, x[n-x(-n], then X(z) = X(z-1) (b) If x[n] is odd, that is, x[n] =-x|-n], then X(z)--X(2-1). (c) If x[n] is odd, then there is a zero in X(z) at z 1.
a) The transfer function of an ideal low-pass filter is and its impulse response is where...
a) The transfer function of an ideal low-pass filter is and its impulse response is where oc is the cut-off frequency i) Is hLP[n] a finite impulse response (FIR) filter or an infinite impulse response filter (IIR)? Explain your answer ii Is hLP[n] a causal or a non-causal filter? Explain your answer iii) If ae-0. IT, plot the magnitude responses for the following impulse responses b) i) Let the five impulse response samples of a causal FIR filter be given...
3.96 Determine the z-transforms of the following sequences and their respective ROCs: (a) x1[n] = -a"u[-n-1], (b) x2[n] = "Min + 1], and (c) x3[n] = "u-n).
QUESTION 1 Characterise the following systems as being either causal on anticausal: yn)-ePyn-1)+u/n), where u/h) is the unit step and B is an arbitrary constant (B>0), Take y-1)-0. Answer with either causal or 'anticausal only QUESTION 2 For the following system: yn) -yn-1Va -x(n), for a 0.9, find y(10), assuming y(n) - o, for ns -1.Hint: find a closed form for yin) and use it to find the required output sample. (xin)-1 for n>-0) QUESTION 3 A filter has the...
Determine the z-transform of the following sequences and their ROCs: a) x(n) = (0.5)" for n> 5, and zero for all other values of n; b) x(n)= (0.5)"[u(n) - u(n-7)]; c) x(n)=(-1)"a"u(n), 0 <a<1.
3) Compute the Z-Transforms of the following time series: (a) x(n)k2"u(n) (b) + 1) x(n) = u(-n (c) x(n) -k2"u-1) (d) x(n) 0.5%1(n) + 3"11(-n) (e) x(n) = 4-nu(n) + 5-nu(n + 1) In the above, u(n) stand for the unit step signal in the discrete time domain. Also, if you can in each case determine the region of convergence of the Z-Transform you obtain.
Problem #1. Topics: Z Transform Find the Z transform of: x[n]=-(0.9 )n-2u-n+5] X(Z) Problem #2. Topics: Filter Design, Effective Time Constant Design a causal 2nd order, normalized, stable Peak Filter centered at fo 1000Hz. Use only two conjugate poles and two zeros at the origin. The system is to be sampled at Fs- 8000Hz. The duration of the transient should be as close as possible to teft 7.5 ms. The transient is assumed to end when the largest pole elevated...
Problem 4: (a) and (b): Find the z-transform including the ROC for each of the following waveforms: [n] 3(금)"u[n] Xa | 지 = (c) Find the z-transfor by m of the impulse response hn] of an LTI system, when h[n] is given h[n] = 5(을)"u[n]. (d) and (e): Using z-transforms, find the responses (yan] and yb[n]) of the system described in part (c) to the inputs (an andn] described in parts (a) and (b
6.6a b
6.7c d e
of the following irrational z transforms. (a) x(z) = ea", Izl > 0, o)X(z)log (1- az), zl < 1/lal. . 1 Show the following p roperties for the z transforms of even and odd 6.7 discrete-time functions. (a) If x[n] is even, that is, x[n-x(-n], then X(z) = X(z-1) (b) If x[n] is odd, that is, x[n] =-x|-n], then X(z)--X(2-1). (c) If x[n] is odd, then there is a zero in X(z) at z 1.
a) The transfer function of an ideal low-pass filter is and its impulse response is where oc is the cut-off frequency i) Is hLP[n] a finite impulse response (FIR) filter or an infinite impulse response filter (IIR)? Explain your answer ii Is hLP[n] a causal or a non-causal filter? Explain your answer iii) If ae-0. IT, plot the magnitude responses for the following impulse responses b) i) Let the five impulse response samples of a causal FIR filter be given...
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