Digital Signal Processing 1 Lab course Winter Term Assignment 4

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Digital Signal Processing 1

Lab course Winter Term

Assignment 4

1. Continuous-timetoDiscrete-timetransformation

▪ Define the following transfer function in MATLAB: ▪ MATLABcommand:tf

▪ Hint: Requires the MATLAB Control System Toolbox G(s) = 1

  • ▪  Plot the impulse response, step response, bode diagram and pol-zero map of the given system using the following MATLAB commands:

    ▪ impulse ▪ step
    ▪ bode
    ▪ pzmap

  • ▪  Determine a corresponding discrete-time transfer function with a sample time T=0.2s with an equal impulse, respectively step response. Use the impulse invariant and zero order hold transformation methods by using the following MATLAB command:

    ▪ c2d(…,…,’impulse’)

    ▪ c2d(…,…,’zoh’)

  • ▪  Which effect would have the multiplication of z with the resulting transfer function?
  • ▪  Compare the transformed discrete system with sample time T=0.2s and T=0.15s.
  • ▪  Determine a discrete system with the given parameters using the bilinear transformation

method. Use the following MATLAB command:

▪ c2d(…,…,’tustin’)

2. Symboliccomputation:Bilineartransform

▪ Transform the following transfer function by hand: G(s) = 1

▪ Check the result with the MATLAB using the following commands: ▪ syms

▪ simplify
▪ pretty
▪ Hint: Requires the MATLAB Symbolic Toolbox

Applied Image & Signal Processing Informationstechnik & System-Management Salzburg University of Applied Sciences

s2 +2s+4

s2 + 2s + 4

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Digital Signal Processing 1

Lab course Winter Term

Home Assignment: Cont. Butterworth for discrete applications

▪ MATLAB uses the function butter (signal processing toolbox) to create discrete butterworth coefficients (b and a).

Internally it uses the bilinear transform method to create the discrete coefficients from the analog equivalent. In the following we want to verify with our own implementation that the bilinear transformation can be used in this scenario.

Hint: The function butter does some processing to make the coefficients more stable, but those optimizations can be neglected in the following. You will get minor differences in the pole/zero map.

  • ▪  Furthermore, the function butter can also be used to create analog butterworth coefficients. See MATLAB manual (F1) for further help.
  • ▪  Create the following
    ▪ discrete butterworth coefficients with Fs=100Hz and ▪ analog butterworth coefficients using
    ▪ common parameters

    ▪ Order: 5

    ▪ Cut-off frequency: 0.1 normalized Nyquist frequency

  • ▪  Transform the analog butterworth transfer function by using the bilinear transformation.
  • ▪  Overlay the pole zero maps of both the discrete (MATLAB reference) and the c2d

converted (analog matched version) transfer functions.

hold on;
pzmap(Hz_mat,’r-‘); % MATLAB reference pzmap(Hz_c2d,’b-‘); % c2d matched version hold off;

  • ▪  Show that the overlayed zeros and poles match very closely.
    ▪ Hint: Really read the help for butter function thoroughly. Especially for the

    creation of analog coefficients.

  • ▪  Required upload

    ▪ demobutter.m

Applied Image & Signal Processing Informationstechnik & System-Management Salzburg University of Applied Sciences

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