Discrete Time SystemsExamTimo Heikkinen

The best result is 6+11+4 = 21 points.

1What do the following terms and acronyms mean? Give a short description of each term or acronym! Concentrate on the essential, and what is the difference between this and other closely related terms or acronyms. If you can write just to the point, one or two sentences will suffice, but if you are not quite sure, give a longer explanation.

1.1IIR (2 p)

  • A filter structure. Most important properties:
  • Infinite Impulse Response (½ p)
  • impulse response is infinite in length (1 p)
  • infinite impulse response (½ p, ½ p reduced)
  • in practice, impulse response is finite in length because of limited calculation accuracy (½ p)
  • feedback (½ p)
  • must be checked for stability (½ p)
  • value of output samples are calculated using current and previous input samples and previous output samples (½ p)
  • Impulse Response (2 p)
  • system’s response to unit impulse function (1 p)
  • notation (1p):

1.3 (2 p)

  • delay operator (1 p)
  • is the Z-transform of adsamples long delay (1 p)
  • Z-transform of x(n-d) (1 p)

2Each subtask in this problem is based on the previous subtask's result. If, however, you are not able to carry out a subtask, but you think you could do better in the following one, use a dummy result (a result you have just made up yourself) as initial data for the following subtask.
A system is defined by the following difference equation:

2.1Draw Block Diagram of the defined system! (2 p)

2.2Define its Transfer Function H(z)! (2 p)

2.3Calculate the zeros and poles of the system. Plot them in complex domain with Unit Circle! (2 p)

  • System’s zeros are the roots of the divider polynomial of the transfer function. To calculate the zeros, the divider polynomial is set to zero and z is solved:


  • System’spoles are the roots of the divident polynomial of the transfer function. To calculate the poles, the divident polynomial is set to zero and z is solved:


  • Unit circle is plotted in complex domain. Zeros (o) and poles (x) are also drawn in.

2.4Calculate system's gain at angular frequencies 0, and at the Transfer Function's zero's angular frequency!(3 p)

  • The system’s gain can be calculated by taking an absolute value of the transfer function after complex variable z has been replaced by .

  • The zero lies on angular frequency of

2.5Sketch system's Amplitude Response using the results of subitem 2.3 and 2.4! (2 p)

  • Using the zero/pole-plot of 2.3 it can be concluded that the gain is quite close to unity on frequencies 0 ja as both zero and pole are at almost the same distance from these frequencies.Atthere is a deep notch as system’s zero is very close to unit circle on this frequency. As the pole is very close to the zero the notch becomes very steep.
  • Using the gain values of 2.4 the amplitude response plot can be fixed to three frequency points.

3A signal is sampled, and the following series of four samples is chosen to be analyzed:

3.1Calculate DFT with k = 1! (2 p)

  • k=1, N=4and all fourx(n)values are substituted into DFT formula:

3.2What does the result of 3.1 mean, if a sampling frequency of fs = 44.1 kHz is assumed to have been used? (2 p)

  • DFT is used to analyze frequency contents of a signal.The result means that there is 0.5590 units of signal in 26.57 degreesphase, on a frequency that corresponds to k ‘s value of 1.
  • The frequency that corresponds to k ‘s value of 1 is: