Discrete Time Systems: Impulse responses and convolution; An - PowerPoint PPT Presentation

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Discrete Time Systems: Impulse responses and convolution; An introduction to the Z-transform Lecture 5 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Impulse response and system output Using impulse response, output can be calculated as: y[k] = h[k] * u[k]  Proof: Definition of impulse response Time-invariance Linearity Definition of convolution  Conclusion  The impulse of a system describes the input/output behavior completely. 2 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Impulse response and system output Visually: 3 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Z-transform  Discrete equivalent to the Laplace-transform  Converts time dependent descriptions of systems to the time- independent Z-domain.  Simplifies many calculations  Convolution theorem → convolution becomes multiplication  Linear difference equations become simple algebraic expressions  ... 4 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Z-transform  2 forms  Bilateral:  Requires knowledge of h for all values of k, including negative values  Can be used for non-causal systems  Unilateral:  Only requires knowledge of h for positive values of k  Can only be used for causal systems without loss of information 5 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Z-transform: properties 6 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Transfer function (DT) The transfer function of a discrete system is the Z-transform of the impulse response.  H(z) = Z {h[k]}  Recall: y[k] = h[k] * u[k]  Let U(z) = Z {u[k]} Y(z) = Z {y[k]}  Thus, applying the convolution theorem: Y(z) = H(z) . U(z)  BUT: Only applies when system starts from a null state (Reason: impulse response itself starts from a null state) 7 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Getting rid of convolutions (DT) 8 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics List of common Z-transform pairs 9 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics List of common Z-transform pairs 10 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Region of convergence  For the Z-transform to converge the following must hold:  We will look at convergence separately for positive and negative k, splitting the convergence criterion in 2: j ϴ  Using z = r e with R+ as small as possible and R- as large as possible we get: 11 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Region of convergence  The sums are finite if and Region of convergence:  R + < R - : Ring R - < R + : No ROC  Causal system, for negative k: cannot contain any poles of the system  ROC of a stable system always contains the unit circle R - R + Source: http://www.expertsmind.com/learning/z-transform-and-realization-of-digital-filters-assignment-help-7342873888.aspx 12 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Region of convergence 2 3  System 1: Causal: h[k] = { …, 0, 0, 0, 1, 1/3, (1/3) , (1/3) , … } 3 2 System 2: Anti- causal: h[k] = { …, 3 , 3 , 3, 1, 0, 0, 0, … } k  Analytical representation: h[k] = (1/3)  After Z-transform: H(z) = z / (z – 1/3)  2 systems with very different behaviors, but the same transfer function?  Answer: different ROC:  System 1: |z| > 1/3  System 2: |z| < 1/3 13 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Inverse Z-transform  Split the transfer function up in partial fractions  This is done by first factorizing the denominator  If all poles have multiplicity 1 then the following can be used:  The coefficients can be calculated by: 14 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Inverse Z-transform  If there are poles with multiplicity higher than 1 then the following approach is needed:  Where the highest coefficient for each pole can be calculated by: 15 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Inverse Z-transform Any remaining coefficients can be found by evaluating the equation:  for a number of values of z. 16 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Inverse Z-transform  Because of the linearity of the inverse Z-transform, each partial fraction can be transformed individually and the results can be added together afterwards.  The individual inverse Z-transforms can be found with the following: 17 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example  Transfer function:  Partial fraction decomposition:  Using the given formula’s:  This gives:  By evaluating the transfer function for z=1 we get: 18 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example The resulting transfer function is:  We can now find the inverse Z-transform for each individual fraction: 19 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Inverse Z-transform  Another technique for calculating the inverse Z-transform is direct division  The numerator of the transfer function is divided by the denominator via long division.  Example: -1  ⇒ Z {F(z)} = 1, 3, 12, 25, ... 20 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Solving difference equations with the Z-transform  A system is described by a difference equation of the following form:  After the Z-transform:  Rearranged: 21 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Solving difference equations with the Z-transform l We’ll apply the following transformation of the double summations: 22 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Solving difference equations with the Z-transform  The final simplified result is:  With this result it is easy to find the resulting output from a given input or vice-versa given a difference equation.  Right-hand fraction = output resulting from starting conditions: will vanish with time = transient behavior  Left-hand fraction = output resulting from input: will remain = steady state response  is the “transfer function” of the system 23 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Steady state behavior via Z-transform Starting from the previous result:  We wish to find the resulting output from the input:  To simplify derivation, we use: j(k α + θ ) u[k] = e  With Z-transform: 24 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Steady state behavior via Z-transform Filling in U(z) and splitting into partial fractions:  Calculating the coefficient c:  After the inverse Z-transform: 25 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Steady state behavior via Z-transform Because of linearity we can ignore the imaginary component, leading to the result:  In most applications (= stable system) we can ignore transient behavior as it will quickly die out  Using the transfer function steady state behavior can easily be determined by converting sinusoidal signals to phasors cos(k α + θ ) The input: |H(ej α )| cos(k α + θ + ∠ H(ej α )) Will produce steady state: 26 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example  We’ll have a look at the steady state response to the input for the system:  Evaluating the transfer function for the exponential with pulsation 3 gives:  The resulting output has been reduced to a third in amplitude and has undergone a small phase shift. 27 Systems and Control Theory