DSP-Lec 06-Transfer functions and Filter Realization

Chapter p Transfer functions g Filter Realization and Digital Ha Hoang Kha, Ph.D.Click to edit Master subtitle style Ho Chi Minh City University of Technology @ Email: hhkha@hcmut.edu.vn ™ With the aid of z-transforms, we can describe the FIR and IIR filters in se several eral mathematically mathematicall equivalent eq i alent way a Ha H Kha Transfer functions and Digital Filter Realizations Content Transfer T f functions f ti ‰ Impulse response ‰ Difference equation ‰ Impulse response ‰ Frequency q y response p ‰ Block diagram of realization Digital filter realization ‰ Direct form ‰ Canonical form ‰ Cascade form Ha H Kha Transfer functions and Digital Filter Realizations Transfer functions ™ Given a transfer functions H(z) one can obtain: ((a)) the th impulse i l response h(n) h( ) (b) the difference equation satisfied the impulse response ( ) the (c) h I/O / difference diff equation i relating l i the h output y(n) ( ) to the h input i x(n) (d) the block diagram realization of the filter ( ) the sample-by-sample (e) p y p p processingg algorithm g (f) the pole/zero pattern (g) the frequency q y response p H(w) ( ) Ha H Kha Transfer functions and Digital Filter Realizations Impulse response ™ Taking the inverse z-transform of H(z) yields the impulse response h(n) Example: p consider the transfer function To obtain the impulse response, we use partial fraction expansion to write Assuming the filter is causal, we find Ha H Kha Transfer functions and Digital Filter Realizations Difference equation for impulse response ™ The standard approach is to eliminate the denominator polynomial of H(z) ( ) and then transfer back to the time domain Example: p consider the transfer function Multiplying both sides by denominator, we find Taking inverse z-transform z transform of both sides and using the linearity and delay properties, we obtain the difference equation for h(n): Ha H Kha Transfer functions and Digital Filter Realizations I/O difference equation ™ Write then eliminate the denominators and go back to the time domain Example: consider the transfer function We have which can write Taking the inverse z-transforms of both sides, we have Thus, the I/O difference equation is Ha H Kha Transfer functions and Digital Filter Realizations Block diagram ™ One the I/O difference equation is determined, one can mechanize it byy block diagram g Example: consider the transfer function We have the I/O difference equation The direct form realization is given by Ha H Kha Transfer functions and Digital Filter Realizations Sample processing algorithm ™ From the block diagram, we assign internal state variables to all the delays: We define v1((n)) to be the content of the x-delayy at time n: Similarly, y w1((n)) is the content of the y-delay y y at time n: Ha H Kha Transfer functions and Digital Filter Realizations Frequency response and pole/zero pattern ™ Given H(z) whose ROC contains unit circle, the frequency response H(w) can be obtained by replacing z=ejw Example: Using the identity we obtain b i an expression i ffor the h magnitude i d response ‰ Drawing peaks when passing near poles ‰ Drawing dips when passingg near zeros p Ha H Kha 10 Transfer functions and Digital Filter Realizations Example ™ Consider the system which has the I/O equation: a) Determine the transfer function b) Determine the casual impulse response c)) Determine the frequency q y response p and plot p the magnitude g response p of the filter d)) Plot the block diagram g of the system y and write the sample p processing algorithm Ha H Kha 11 Transfer functions and Digital Filter Realizations Digital filter realizations ™ Construction of block diagram of the filter is called a realization of the filter filter ™ Realization of a filter at a block diagram level is essentially a flow graph of the signals in the filter ™ It includes operations: delays, additions and multiplications of signals by a constant coefficients ™ The block diagram realization of a transfer function is not unique ™ Note that for implementation of filter we must concerns the accuracy of signal g values, accuracy of coefficients and accuracy of arithmetic operations We must analyze the effect of such imperfections on the performance of the filter Ha H Kha 12 Transfer functions and Digital Filter Realizations Direct form realization ™ Use the I/O difference equation ‰ The b-multipliers are feeding forward ‰ The a-multipliers are feeding backward Ha H Kha 13 Transfer functions and Digital Filter Realizations Example ™ Consider IIR filter with h(n)=0.5nu(n) a)) Draw D the th direct di t form f realization li ti off this thi digital di it l filter filt ? b) Given x=[2, 8, 4], find the first samples of the output by using the sample processing algorithm ? Ha H Kha 14 Transfer functions and Digital Filter Realizations Canonical form realization ™ Note that Y ( z) = H ( z) X ( z) = N ( z) X ( z) D( z ) ‰ The maximum number of common delays: K=max(L,M) Ha H Kha 15 Transfer functions and Digital Filter Realizations Cascade form ™ The cascade realization form of a general functions assumes that the transfer functions is the product of such second second-order order sections (SOS): ™ Each of SOS mayy be realized in direct or canonical form Ha H Kha 16 Transfer functions and Digital Filter Realizations Cascade form Ha H Kha 17 Transfer functions and Digital Filter Realizations Homework ™ Problems: 6.1, 6.2, 6.5, 6.16, 6.18, 6.19 ™ Problems: 7.1, 7.3, 7.5, 7.10 Ha H Kha 18 Transfer functions and Digital Filter Realizations ... diagram of realization Digital filter realization ‰ Direct form ‰ Canonical form ‰ Cascade form Ha H Kha Transfer functions and Digital Filter Realizations Transfer functions ™ Given a transfer functions. .. functions and Digital Filter Realizations Digital filter realizations ™ Construction of block diagram of the filter is called a realization of the filter filter ™ Realization of a filter at a block diagram... direct or canonical form Ha H Kha 16 Transfer functions and Digital Filter Realizations Cascade form Ha H Kha 17 Transfer functions and Digital Filter Realizations Homework ™ Problems: 6.1, 6.2,