Ecet345 signals and systems homework #5

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                                                ECET345 Signals and Systems

                                                                                Homework #5

Name of Student   __________________________________________

 

 

1.       Using z-transform tables (page 776 of text or equivalent), find the z-transform of

 

(a) 

 

(b)  

 

(c)     and express it in positive powers of z. When computing the value of trigonometric functions, keep in mind that the arguments are always in radians and not in degrees.

 

(d)    and express it in positive powers of z

 

 

 

2.       Find the inverse z-transform, x(n), of the following functions by bringing them into a form such that you can look up the inverse z-transform from the tables. This will require some algebraic and /or trigonometric manipulation/calculation. You will also need a table of z-transforms (page 776 of text or equivalent). When computing the value of trigonometric functions, keep in mind that the arguments are always in radians and not in degrees.

 

(a)  

then use the z-transform tables. Having found the inverse z-transform, determine its numerical value at n = 2.

 

 

(b)   Using partial fraction expansion , find the inverse z-transform of

 

(c)    Using the tables, find the inverse z-transform of

 

 

(d)   Using the tables, find inverse z-transform of

 

3.       Find the first seven values (i.e., x(n) for n = 0 to 6) of the function given below.

Hint: Manually calculate the three parts separately for various values of n and add or subtract them point by point for various values of n. For example,  for n = 2 equals 2 * 2 * 1 (or 4);  for n = 5 equals 2 * u(2) or 2 * 1 = 2; and so on. Also keep in mind that u(n – k) is a unit step function delayed by k samples, and hence it will be zero for all values of (n – k), which are negative and 1 otherwise.

 

 

Write the numerical values here as a row vector.

 

 

4.       The simulation diagram of a discrete time system is shown below. Find the first six output (y(0) to y(6)) of the system when an input x(n) , as computed in problem 3, is applied to the discrete time system.

 

 

(a)    Determine the difference equation of the system.

(b)   Find the first six outputs (y(0) to y(5)) using the iteration method as explained in the lecture. Assume all initial conditions to be zero.

 

 

 

 

 

 

 

 

 

(c)    Given the transfer function shown below, convert it into a difference equation.

 

 

5.       The impulse response of a discrete time system is given by

h(n) = [ 1 -1 2 ].

To such a system we apply an input of the type

x(n) = [ 2 1 2 3 ].

Use MATLAB to convolve the two sequences and enter the answer below.

 

6.       A system is described by the transfer function below.

 

 

 

 Find the location of poles and zeros of the system and write them here.

 

 

 

 

 

Is this a stable system? Why or why not?

 

 

7.        The transfer function of a discrete time system is given by

 

 

The system has a sampling rate of 100 Hz. Calculate the frequency response of the system at 25 Hz input frequency as a complex number in both rectangular and polar form.