Signal Processing

Signal Processing
29 October, 6- 8 pm December 14:6 -9 pm December 08: 5:30-8 :30 pm 27 April 2017, 12 : 00 – 3 PM 13 December 2017 12 14 April2018 10 18 2018 9 :00 AM-12 :00 11 4 9 4 8 9 9 5 10 7 5 8 12 10 11 55 4 10 8 10 5 9 4 8 10 10 4 7 8 10 PART I Problem 1. Signals x t   etu t  and x (t )  2 cos100t  are applied as the input signals to the following 1 2 system: Determine the energy and power of: x1 t  , x2 t  , y1 t  , y2 t  . Summarize your results in the form of the following table. Signal Energy Power x1 t  x2 t  y1 t  y2 t  Problem 2. Give concrete examples of systems that satisfy the following conditions: System A should be: 1. Linear . 2. Memory. 3. Causal. 4. Stable. 5. Time Varying. System B should be: 1. Non-Linear . 2. Memory. 3. Causal. 4. Stable. 5. Time Invariant. PART II Problem 1. Sketch the signal that is a result of of the following convolution Problem 2. We are given a LTI system with the impulse response h0 t  . We are told that when the input is x0 t  the output is y0 t  (which is depicted below), i.e., y0 t   x0 t  h0 t  . We are given the following set of new inputs to LTI systems with the corresponding new impulse responses. In each of these cases, determine whether or not we have enough information to determine the output yt  when the input is x t  and the system has impulse response ht  . If it is possible to determine yt   x t  ht  , provide an accurate sketch (and a formula) of yt  . x t  ht  yt   x t  ht  x t   2×0 t  ht   h0 t  x t   x0 t   x0 t  2 ht   h0 t  x t   x0 t  2 ht   h0 t  1 x t   x1 t  ht   h0 t  2 x t   x0 t  ht   h2 t  x t   x0 t  ht   h0 t  Note: x1 t  denotes the first derivative of x t  , h2 t  denotes the square of h t  and x t  is the absolute value of x0 t  . Use the basic properties of convolution. Problem 3. Consider the following a sample-and-hold circuit that is often used in a number of signal processing applications. The parameter   0 defines the value of the time shift applied in this circuit. Confirm that this is an LTI system. Find the impulse response function ht  of this system. Is this system BIBO stable ? Find the frequency response function H  . What is an impulse response function of a new system that consists of the above LTI system (represented by ht  ) followed by the squaring element ? xt Squaring Problem 4. Consider an LTI system described by the following differential equation d 2yt   dt 2 B dyt  dt 25yt   dx t  dt 23x t  . Compute the range of values of constant B so that the system impulse response ht  is Overdamped (non-oscillatory) . 2. Underdamped (oscillatory). 3. Unstable (blows up). If B  26 , compute the system impulse response ht  . PART III Problem 1. The two-sided spectra (only one side is shown) of a certain periodic signal x t  is shown below. Amplitude 4 3 9 12 Phase 24 30 What is the fundamental frequency 0 and fundamental period T0 of x t  ? Write a formula for x t  in terms of cos functions. Determine the signal power directly from the spectra. Problem 2. The input signal x t  of an LTI system with frequency response H (shown below) is the sawtooth periodic wave: 9 9 It is known (see Lecture Notes) that x t  has the following Fourier series representation. A  A    k1 x t   2   k cos k0t  2  . Find a formula (and sketch) for the output signal yt  . Assume A  1 and T0  1 . Calculate the average power of the input and output signals x t  and yt  . What percentage of the input signal power is available in the output signal ? PART IV Problem 1. Let x t  be a signal with the following Fourier transform (FT): 1 1 Formulate (and sketch) the FT rule “ yt  = ….has the FT Y  =.…” when yt  is given by 1. yt   2x t cos10t  ; 2. yt   x t ejt  x t ej3t ; 3. yt   2x 2t   1 x t / 2 ; 2 4. yt    x  e2t d ; 5. yt   x t    t  ,  t  – delta function.  Problem 2. The input voltage to an ideal bandpass filter is vt   120e24tu t  V. Find V   and sketch Find Vo   and sketch V  2 (energy spectrum) for the filter input voltage. V  2 (energy spectrum) for the filter output voltage. o What percentage of the input signal energy content is available at the output ?   Problem 3. One would like to get the spectra of the signal x t     cos5t , 0  t  2 0 else depicted below. The spectra X  of x t  is rather difficult to derive analytically and one has decided to use the numerical approximation based on a certain number of samples over the interval 0, 2 , i.e., the sampling interval is Ts  0.1 . 0, 2 . Let us use N  20 points over Evaluate the approximated spectra X!   of x t  and plot the amplitude spectra !X   . What is the proper frequency range where we obtain the unique approximation !X   ? Based on the shape of !X   draw conclusions on the frequency content of x t  , i.e, whether it is a low- pass, band-pass or high pass signal. Detect the important features of the spectra, i.e., X! 0 and X! max  – the maximum value of X!   . Find the approximated 3dB bandwidth of x t   Bonus Question PART V Problem 1. Derive the impulse response function ht  for the following ideal filters. Lowpass Filter Bandpass Filter Note. Express you result in terms of 0  1  2 and  2 c 2  1 . 2 Notch Filter Comment on the stability and causality properties of the above ideal filters. Hint. You may use the following FT pair combined with the duality property in order to obtain the inverse FT of the triangular shape in the frequency domain. Problem 2. Figure below shows a scheme to transmit two signals m1 t  and m2 t  simultaneously on the same channel (without causing spectral interference). Such a scheme, which transmits more than one signal, is known as signal multiplexing. In this case, we transmit multiple signals by sharing an available spectral band on the channel, and, hence, this is an example of the frequency-division multiplexing. The signal at point b is multiplexed signal, which now modulates a carrier of frequency 20,000 rad/sec. The modulated signal at point c is now transmitted over the channel. Sketch the spectra at points a, b, c. What must be the minimum bandwidth of the channel ? Design a receiver to recover signals m1 t  and m2 t  from modulated signal at c. 5000 5000 m t  5000 5000 2 cos10,000t  PART VI Problem 1. Proof the time differentiation property of the Laplace Transform: x t   X s  dx t   sX s  x 0  . dt Establish the time differentiation property using the two-sided Laplace Transform defined as follows: X s   x t est dt .  Compare the results in A and B. Proof the dual property to A, i.e., the differentiation in s – domain property x t   X s  tx t    dX s . ds Problem 2. Determine the Laplace Transform of the following signals: 1. x t   t  12 u t  1 . 2. 1 1 2 4 5 3. d e4tu t  .   dt Problem 3. Obtain the inverse Laplace transform of each of the following functions in the s – domain. 1. X1 s  s2  1 s2  3s  2 ; 2. X2 s  1 s  2 e2 s  1 s  2s  3 e3s ; 3. X s  2 s  1 s2  6s  13 . PART VII Problem 1. Consider the following pole diagram representing the transfer functions H1 s through H7 s of LTI lumped systems. The location of poles is denoted by . All poles are first-order (single) poles. Which of the pole-diagrams could correspond to LTI systems which are stable ? Which of the pole-diagrams could correspond to the impulse response ht  shown below ? Problem 2. The initial conditions in the following circuit are zero. Find the transfer function Hs  Give the numerical value of each pole and zero of Hs . Comment on the circuit stability. Io s Iin s Problem 3. Consider the following circuit, where the voltage sources v1 t  , v2 t  have been applied at t 0. Assume that no energy was stored in the circuits before t 0, i.e., i 0 0. Represent the circuit in the s -domain. Find the formula for I s – the inductor current in terms of V1 s and V2 s . Assume that v1 t   u t  2 and v2 t   u t  , where u t  is the unit step function. Find and plot i t  . Identify the steady-state response and transient response. Problem 4. Consider the following composite LTI system. It consists of two subsystems described by the transfer functions H s  and H s  . s 1 1 s  1 2 s  1 Determine the transfer function Hs  Y s X s of the whole system. Write the differential equation that relates the output signal yt  and input signal x t  . Determine the impulse response function ht  of the system. Determine whether the system is stable. 1/9