Lecture Digital signal processing: Chapter 6 - Nguyen Thanh Tuan

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Lecture Digital signal processing - Chapter 6 introduce transfer function and digital filter realization. In this chapter, you will learn to: Transfer functions (Impulse response, difference equation, impulse response,...), digital filter realization (Direct form, canonical form, cascade form).

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Nội dung Text: Lecture Digital signal processing: Chapter 6 - Nguyen Thanh Tuan

Chapter 6 Transfer function and Digital Filter Realization Nguyen Thanh Tuan, Click M.Eng. to edit Master subtitle style Department of Telecommunications (113B3) Ho Chi Minh City University of Technology Email: nttbk97@yahoo.com
 With the aid of z-transforms, we can describe the FIR and IIR filters in several mathematically equivalent way Digital Signal Processing 2 Transfer function and Digital Filter Realization
Content 1. Transfer functions  Impulse response  Difference equation  Impulse response  Frequency response  Block diagram of realization 2. Digital filter realization  Direct form  Canonical form  Cascade form Digital Signal Processing 3 Transfer function and Digital Filter Realization
1. Transfer functions  Given a transfer functions H(z) one can obtain: (a) the impulse response h(n) (b) the difference equation satisfied the impulse response (c) the I/O difference equation relating the output y(n) to the input x(n). (d) the block diagram realization of the filter (e) the sample-by-sample processing algorithm (f) the pole/zero pattern (g) the frequency response H(w) Digital Signal Processing 4 Transfer function and Digital Filter Realization
Impulse response  Taking the inverse z-transform of H(z) yields the impulse response h(n) Example: consider the transfer function To obtain the impulse response, we use partial fraction expansion to write Assuming the filter is causal, we find Digital Signal Processing 5 Transfer function and Digital Filter Realization
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: consider the transfer function Multiplying both sides by denominator, we find Taking inverse z-transform of both sides and using the linearity and delay properties, we obtain the difference equation for h(n): Digital Signal Processing 6 Transfer function and Digital Filter Realization
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 Digital Signal Processing 7 Transfer function and Digital Filter Realization
Block diagram  One the I/O difference equation is determined, one can mechanize it by block diagram Example: consider the transfer function We have the I/O difference equation The direct form realization is given by Digital Signal Processing 8 Transfer function and Digital Filter Realization
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-delay at time n: Similarly, w1(n) is the content of the y-delay at time n: Digital Signal Processing 9 Transfer function and Digital Filter Realization
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 an expression for the magnitude response  Drawing peaks when passing near poles  Drawing dips when passing near zeros Digital Signal Processing 10 Transfer function and Digital Filter Realization
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 response and plot the magnitude response of the filter. d) Plot the block diagram of the system and write the sample processing algorithm Digital Signal Processing 11 Transfer function and Digital Filter Realization
2. Digital filter realizations  Construction of block diagram of the filter is called a realization of the 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 values, accuracy of coefficients and accuracy of arithmetic operations. We must analyze the effect of such imperfections on the performance of the filter. Digital Signal Processing 12 Transfer function and Digital Filter Realization
Direct form realization  Use the I/O difference equation  The b-multipliers are feeding forward  The a-multipliers are feeding backward Digital Signal Processing 13 Transfer function and Digital Filter Realization
Example  Consider IIR filter with h(n)=0.5nu(n) a) Draw the direct form realization of this digital filter ? b) Given x=[2, 8, 4], find the first 6 samples of the output by using the sample processing algorithm ? Digital Signal Processing 14 Transfer function and Digital Filter Realization
Canonical form realization 1 1  Note that Y ( z)  H ( z) X ( z)  N ( z) X ( z)  N ( z) X ( z) D( z ) D( z )  The maximum number of common delays: K=max(L,M) Digital Signal Processing 15 Transfer function and Digital Filter Realization
Cascade form  The cascade realization form of a general functions assumes that the transfer functions is the product of such second-order sections (SOS):  Each of SOS may be realized in direct or canonical form. Digital Signal Processing 16 Transfer function and Digital Filter Realization
Cascade form Digital Signal Processing 17 Transfer function and Digital Filter Realization
Review Digital Signal Processing 18 Transfer function and Digital Filter Realization
Homework 1 Digital Signal Processing 19 Transfer function and Digital Filter Realization
Homework 2 Digital Signal Processing 20 Transfer function and Digital Filter Realization