EEL5701: Exam 1

1. (25 points) You are given an unknown LTI system. You observe that when the input is , the output with .
   1. (10 points) What is h[n]? Show all of your work.

   2. (5 points) What is ?

   3. (5 points) Sketch the magnitude and phase of .

   4. (5 points) If the unknown system is a nonlinear time invariant system, suggest a nonlinear system that would produce when given the same input .

    

2. (25 points) An analog signal has the Fourier Transforms sketched above. This signal is sampled to produce the sequence .
   1. (10 points) Sketch the Fourier transform of the sequence when sampling is performed at twice the highest frequency present in the signal (i.e. ).

   2. (15 points) What is the lowest sampling frequency that can be used without incurring any aliasing distortion, i.e., so that can be recovered from x[n]?

3. (25 points)

   In the above diagram, an attempt was made to reduce the sampling rate by a factor of 2 to reduce the amount of processing. Before reconstruction of the final analog signal ( ), an upsampling is performed. Unfortunately, the usual ideal low-pass filters that are required in such systems were completely neglected. Answer the following questions, assuming that is a 60Hz sine wave and T is chosen to achieve a 100Hz sampling rate.

   1. (5 points) Write an expression for . Explain.

   2. (5 points) Write an expression for . Explain.

   3. (5 points) Discuss where aliasing occurs in the diagram.

   4. (10 points) Write an expression for . Explain.

4. (25 points) Short Answer
   1. (5 points)

      Under what conditions can an analog system be replaced (mathematically speaking) by a discrete time system?

   2. (5 points) We know that aliasing occurs if we sample at less than the Nyquist rate. What happens (if anything) when we sample at exactly the Nyquist rate.

   3. (5 points) A student thinks that we can simplify the definition of a linear system to those systems that satisfy

      

      Is this valid? Why or why not?

   4. (5 points)

      A student builds a device that is to be connected to a telephone to disguise her voice. By changing T', the frequencies in the input signal can be increased or decreased. Discuss the major problems with the real-time operation of such a device.

   5. (5 points) Consider the system in the figure above with

      

      and the discrete-time system is an exponential function, i.e., . What is the largest value of T such that .