Tasks to do:

• Properties of 2nd order systems, inverse Z-transform of higher-order systems.

• For discrete-time systems described by following difference equations draw block scheme, compute difference equation, transfer function, zeros and poles of transfer function, impulse response, and estimate frequency response.

  1. y[n] = 0.4455 x[n-1] + 1.2728 y[n-1] - 0.81 y[n-2]

     Result : Display for system defined above in MATLAB:
     - graphically in Z-plane zeros and poles of transfer function (compute also exact values numericaly),
     - frequency response of used filter,
     - impulse response of used filter.

  2. Only in MATLAB: "all=pole" filter with the pair of complex conjugate poles p_1 = 0.9 *exp ( +/- j * pi/4 )

  3. y[n] = x[n] + 2 x[n-1] + 1.5 y[n-1] - y[n-2]

  4. y[n] = x[n] + 0.81 x[n-2]

  5. Observe the change of the frequency response and zero-poles locations for the following systems:
     • y[n] = x[n] + 0.81 x[n-2]
     • y[n] = x[n] + 0.64 x[n-2]
     • y[n] = x[n] + 0.925 x[n-2]
     • y[n] = x[n] - 0.81 y[n-2]
     • y[n] = x[n] - 0.64 y[n-2]
     • y[n] = x[n] - 0.925 y[n-2]

  6. Perform the filtering of white noise using the system No. 1. The noise should have Gaussian distribution and the length of 10000 samples. Display:

     Result :

     • short-time power spectra of input and output signals computed from the short-time frame with the length of 256 samples (compute them directly using FFT and do not forget to weight the frame by Hamming window),
     • twosided smoothed estimation of PSD for input and output signals, the short-time frame should have the length of 256 samples,

  7. 

  8. y[n] = x[n] + x[n-1] + x[n-2] + ... + x[n-9] ;

  9. y[n] = x[n] - x[n-6]

  10. y[n] = x[n] - x[n-9]

• check also interactive tool fvtool